Quale Mechanics, Page 1

Unpublished.1998–2013 by Ellis D. Cooper.1942–

1 Table of Contents

1998 Wholes and Parts in Quale Mechanics, Page 1

Although employed as a software engineer at Varian Semicon- ductor Equipment Associates, my real passion was – and still is – mathematical philosophy. When I learned of this conference on “Wholes and Parts” and that it would include talks by philosophers and also a former teacher of mine, F. William Lawvere, I felt doubly motivated to attend. 2001 Notes for “The Bent Note”, Page 21

Alden Long, The best software engineer I ever knew, with whom I collaborated at Varian Semiconductor Equipment Associates, is also a Fundamentalist Christian. We had a lively and deep conversation and correspondence. His need for Absolute Truth contrasted with my scientiﬁc viewpoint. One day he had provided to me on my desk the gift of a book, “Jesus Among Other Gods,” by Ravi Zacharias, a so-called Christian apologist. I studied that work, and was rather taken aback for its neglect to mention Jews and Judaism. Thus began for me a new and com- manding interest in religion and Moral Philosophy. Thank you, Alden. I decided to take one year, 2001, to write a screenplay. I think the notes for it are more entertaining than the completed work itself. 2003 Abstract for “Conference on Virtuality”, Page 39

A core element of my interest in mathematical philosophy is the theory of virtuality. I wrote a forty page dialog between several characters representing diﬀerent viewpoints on the topic. Mercifully, it is omitted here. 2003 Fictitious Negative Review of “Conference on Virtuality”, Page 40

Then I wrote a ﬁctitious and rather vicious negative review of the ﬁctional conference. 2005 Four Essences: Judaism, Christianity, Islam, Humanism, Page 41

My interest in religion led me to attempt to summarize the essential characteristics of the Abrahamic faiths in contrast to Humanism.

2 2006 The Great American Debate, Proposal for a New Television Program, Page 41

I wrote a memo to Simon Cowell – never delivered – to propose a new show, based on central questions about religion in the U.S.

2007 The Great American Debate, letter to the Editor, Page 51

The debate idea resurfaced in the form of a letter to the Editor. 2007 Science on a Napkin: Virtual Drawing is to Drawing as the Telephone is to Speaking, Page 53

A third major interest of mine has to do with using modern technology in the service of pedagogy and collaboration.

2007 “Being good for God’s sake,” letter to the Editor, Page 65

Jeﬀ Jacoby is a right-wing Jewish columnist for The Boston Globe who almost always infuriates me.

2009 Dear Islam and Christianity, Page 67

This is a letter from a Jew to Christians and Muslims. Be- hind the scene, I was inﬂuenced by works such as “Christian Antisemitism, A History of Hate,” by William Nicholls, “Con- stantine’s Sword,” by James Carroll, and Bernard Lewis’ “What Went Wrong, The Clash Between Islam and Modernity in the Middle East.”

2007-2009 Me, And Benoit, Page 71

My wife, Carolyn, was employed as an “Operations Assistant” by IBM Research in Cambridge, Massachusetts, where, as it turned out Benoit Mandelbrot was a researcher Emeritus. She introduced us, and before long he had hired me as what he called a “helper.” I maintained a journal of that relationship. It is revealing of both his character and mine. Maybe, more mine. 2009 Mathematics & The World-Wide Financial Crisis, Page 137

3 In that time-frame of 2008 I was witness to the renewed interest in Benoit Mandelbrot’s work in mathematical ﬁnance, which in some indirect, theoretical ways explains what had happened. 2009 Dear Henry, a letter on Moral Philosophy, Page 149

My son, Henry, was a college student with a major in Moral Philosophy, for some reason. 2012 Qualitative Smooth Shape in the Plane, Page 155

In the 1970’s I had invented the idea of “point-humans” who land, as it were, by parachute in the midst of letters with the intention of determining by looking in every direction what letter they are in. This was supposed to be an approach to an algorithm for optical character recognition. Over thirty years later I realized that there must be an algebraic correspondence, between the intricate structure entwining partial views of shapes, and the overall qualitative shape itself. 2012 On Salman Khan’s “Flipped Classroom”, letter to Time magazine, Page 201

Announcement of my idea for a world-wide virtual classroom. 2012 Virtuality, Reality and Phenomenological Spirituality, Page 203

Ever alert to the slightest possibility of being funded to pursue my passionate interests in mathematics, philosophy, and religion, I composed an application to the John Templeton Foun- dation. 2012 The U.S. Electoral College For Liberal Arts Mathematics Students, Page 211

My career veered over the years back towards education. In particular, I had an opportunity to teach a course in mathematics for liberal arts students exactly during the re-election season of Barack Obama. 2012 Memo To The Gods, Memo To The Humans, Page 223

I have always tended towards the bird’s-eye view, The Big Pic- ture story, not to mention grandiosity.

4 2013 A Category of Natural Numbers, Page 227

In developmental mathematics courses I had enjoyed explaining conversions between units of measurement by drawing arrows from the smaller unit to the larger unit, with a label specifying the number that must be used to divide the quantity at the tail end to calculate the quantity at the right end. A natural setting for this idea is a certain category whose objects are numbers and whose morphisms are additions and multiplications. 2013 Ground and Foundation For Mobile Diagram-Oriented Functional Pro- gramming, Page 233

My passion for using modern technology in pedagogy and collaboration requires a mathematical underpinning.

2013 The Human Condition, Page 261

At the age of 71 I decided to stop trying to do big things, and stick to writing small, self-contained works.

2013 The Design and Journal of Cognocity.org, Page 275

Then I committed myself to a business model using the world wide web to immensely improve how education works throughout the world. The idea is based on my experience in the developmental mathematics classroom. Namely, it is just great to see students helping each other, sometimes in Spanish. The plan is to turn the whole world into a virtual classroom of students working together. I own the domain name, cognocity.org, and am building the web-site for a “knowledge auction market” using Apache, PHP, HTML, and MySQL. This is an experiment.

5 Unpublished.1998-2013

Wholes and Parts in Quale Mechanics

by Ellis D. Cooper, Ph.D.

A talk at Maretsch Castle, Bolzano - Bozen, Italy

Istituto Mitteleuropeo di Cultura - Mitteleuropaisches Kulturinstitut

June 17, 1998

Wholes and Parts in Quale Mechanics

Ellis D. Cooper, Ph.D., Thursday, June 04, 1998

ABSTRACT

In all of nature that is amenable to scientific study, how shall I understand consciousness? Understanding is defined in terms of mental models, which are potentials for telling stories about how things work. Mental models include procedures for symbol manipulation (Turing computation), physical operations (objective force-varying procedure), and mental operations (subjective attention-directing procedure). A computational model of objective procedure is ordinarily called "virtual reality". An objective model of subjectivity is "virtual subjectivity", that is, "robotics". I propose a mathematical mental model of nature as a whole. As complex numbers are fundamental in mathematical physics, so links are to nature. As a complex number has magnitude and phase, a link is from one part of nature to another or the same part, with a certain quale. Assuming that links form a kind of mathematical structure called a topos provides for the long chains of inference and calculation which are essential for a scientific method. Spacetime is a part of nature, and some parts of nature have correlates in spacetime. In particular, a feel is the neural correlate of a quale. The principles and laws of quale mechanics are to nature as the principles and laws of quantum mechanics are to physics. An example of a law of quale mechanics is that the probability of remembering is proportional to the probability of attention.

Diagrams of links drawn with tagged arrows between dots can represent vast tracts of mathematics and science. Even though, I argue, there is more to nature than physics and mathematics, there is much that can be modeled. In particular, a simple extension of the standard idea of finite state machine, called timing machine, may be represented by links. I have used timing machinery to model biological neurons, thermodynamic theorems, and cybernetic machines.

An explanation of purpose in terms of a mental model of the situation, called a now, is advanced. Nows are the subjective parts of nature. The concept of situation must be explained, ultimately, in terms of the history of the whole physical universe, and an indication is provided of how that might be done using elementary concepts from a theory of quantum cosmology.

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INTRODUCTION

quale (KWALEE) noun, plural qualia (KWALEEA) (Latin: ‘quantis’ of what kind) A property (as redness) considered apart from things having the property, that is, a universal; a property as it is experienced as distinct from any source it might have in a physical object.

quantum (KWANTUM) noun, plural quanta (KWANTA) (Latin: ‘quantus’ how much) One of the very small increments or parcels into which many forms of energy are subdivided.

In all of nature that is amenable to scientific study, how shall I understand consciousness? I phrase the question in that way to imply, first, that nature is one whole; second, that not all of nature is necessarily scientifically comprehensible; and third, that consciousness is part of nature. It follows that there may be part of consciousness that cannot be understood by science. I hope there is a part of consciousness that can be. My approach takes apart the question, by defining key terms: "understanding", "science", and "nature". (Defined words are displayed in italic boldface.) With terms defined, the leading question is rephrased. Then I wonder why mathematics is so under-represented among people thoughtful about consciousness, and offer a mathematical mental model of nature. In preparation for defining the concepts of "now", "purpose" and "free will" it is essential to define what is meant by the "situation", and in the penultimate section I discuss how this might be done in terms of one particular alternative theory of quantum cosmology. Finally, I introduce four principles and laws of quale mechanics, which is my name for the mathematical mental model of nature that emphasizes neural correlates of the qualia of links between parts of nature.

UNDERSTANDING

Understanding comes in "molecules" I call mental models. This term "mental model" is at the start a primitive term, but one whose meaning should emerge with increasing precision in due course (1, p. 151)(2). A mental model is a "potential" to tell a story about how something works. The concept "mental model" is a second-order universal: its instances are particular mental models, and the instances of particular mental models are somehow recorded in neural machinery of individual organisms.

The universe is made of stories,

not of atoms. (3)

Atoms are widely studied by smashing them with relatively simple things like photons or electrons, and observing minutely the resulting debris. We may study mental models, similarly, by directing natural language at them. "Mom, how does television work?" There are parts of

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nature called human beings. These beings are linked by natural language signals. Some signals are queries and some are stories. A story is often a description of the trajectories of a timing machine (see the Appendix). Every organism is born with some mental models. For most creatures the mental models at birth remain the mental models for life. For human beings with natural language, throughout life mental models are acquired, transformed, and possibly obliterated. A mental model is a mathematical mental model if its stories are expressed in mathematics. Mathematics is an extension of natural language characterized by the use of declared symbols ("Let x be …") manipulated according to rules for inference and calculation (4). Mathematics is a "metaphor erector set". It can, has, and should be used for building "designer metaphors", quantum mechanics being a perfect example. That is to say, in efforts to achieve an understanding of some natural phenomenon, one is not limited to the metaphors whose vehicles pertain to daily life: one may freely invent extraordinary new ones. Furthermore, (intricate chains of) inference and calculation allow derivation of consequences from assumptions, and prediction or retrodiction of natural phenomena. In summary, understanding is encapsulated in mental models that are indirectly discerned through stories people tell about how things work. (The stories they tell, in a wide variety of instances, may be formalized in terms of timing machinery.) With mathematical mental models we reach into the future, the past, and imagination. Formalization, the slinging about of symbols by following declared rules, freezes for study a play of ideas. Rigor cleans the window through which intuition shines.

SCIENCE

Natural science, as a whole, has for its parts the specialized sciences, such as physics. This is not a disjoint union of parts, it is more like a combining of compatibly linked parts. Mr. Gell-Mann (5) perspicaciously notes, regarding the opposed urges towards reductionism and holism, that each scientific discipline in principle can be reduced to "a more fundamental discipline plus accidental, genuinely accidental, aleatory circumstances that permit the higher level to exist." (6) In particular, all sciences in principle can be reduced to physics plus their specific laws of nature resulting from the amplification of purely random events in the history of the universe. For example, the laws of chemistry are derivable from the laws of quantum mechanics, plus the accidents of the history of the universe that allowed atoms and molecules to exist. "Biology emerges from physics and chemistry", he says, "plus the unpredictable accidents of evolution of life here on earth." And, he persists, "Human consciousness, I am sure, is no different, and emerges in the same way from the laws of biology plus certain accidents of primate evolution."

A better understanding of "science" is achieved by enlarging on the idea of mental model. Not only has a mental model the potential for the activity of telling stories about how things work. It also includes procedures for other kinds of activity. "Telling a story" is the production of physical signals by operating muscles according to a procedure. An enlargement on the concept of mental model is to include its potential for physical activity of other muscles according to procedures (1, p.156). In other words, a mental model includes the potential for the production of time-varying force fields in space. Thus, one may produce speech signals, type on a keyboard, or wind a coil of copper wire. All such activities are trajectories of timing machines that include time-varying force fields among the signals that are emitted when active states time out. An objective procedure is a part of a mental model that may be instantiated in a trajectory of a forceful timing machine (7). I need to go even further with the potential of a mental model. A

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mental model also includes procedures for re-direction of attention. For example, counting the marks along a ruler to measure distance necessitates directing attention from one mark to the next, and at each moment recording the incremented counter value. I consider counting, the sine qua non of measurement, to be a "subjective procedure". In general, at any given time, attention distinguishes parts of awareness, bringing these distinguished parts into consciousness (8). Extreme control of attention is achieved by exercising "the attention muscle". Meditators are expert at attention gymnastics (9). A subjective procedure is a part of a mental model that may be instantiated in a trajectory of an attentive timing machine. Science is an activity of conscious human beings with mental models whose parts include objective and subjective procedures. The purpose of natural science is to create long-lived mental models of nature. The survival of a scientific mental model depends on whether the stories it potentiates by inference and calculation are consistent with the outcomes of its objective and subjective procedures. The scientific method is to forfeit non-scientific mental models.

The laws of nature are statements about natural phenomena characterized by regularity, necessity, and counterfactuality (10). Laws of nature constrain the construction of mental models of particular natural phenomena. (In nearly all cases, these mental models are tentative and approximate.) A science of consciousness should explain the phenomena of mental models. Conversely, to have a science of consciousness we must have a consensual mental model of consciousness. It is this unique relationship between scientific mental models and consciousness that forces mental models to be both the primitive starting point and ultimately I hope, the deeply understood ending point of the science of consciousness.

In my view at least, understanding consciousness is part of understanding nature as a whole. To kick things off it would help a lot to have a mathematical mental model of nature. We have plenty of mathematics for models of "generation," in other words, for constructing sets given generators for the sets. There is less mathematics of "emergence" (11), for example the registration of parts in a whole, or the evolution of new beings. "Mathematics will need to be overhauled." (12, p. 354) Nevertheless, there are intimations of new mathematics that does not assume generators or "ur-elements" for the fundamental parts. Inter-subjectivity is what is left after a process of communication fails; evolved beings are what is left after reproduction fails to occur; consciousness is what is left after everything else is forgotten.

NATURE

I argue that there is more to nature than physics. Physics is the science of amplitudes in the sense of Mr. Feynman (13). That is to say, an amplitude is a complex number whose squared magnitude is a probability that a specified arrangement of matter in space at one time takes on some other specified arrangement in space at some later time. Mr. Feynman's "path integral formula" is an infinite-dimensional summation of amplitudes determined by all possible ways that the initial arrangement could become the final arrangement. Using his formula, for example, it is possible to derive Mr. Schrodinger's formula for the evolution of the wave function of a system modeled by a choice of energy function. What none of these formulas tell us, and cannot tell us, is which way the dice will roll, so to speak. But once the condition of the system is determined one way or another, these formulas are excellent at prediction and retrodiction. Laws of nature resulting from "unpredictable accidents" are not derivable from the laws of physics.

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Therefore, there is more to understanding nature than physics. Whatever mathematical construct I choose for a mathematical mental model of nature, it is unlikely to be the complex number. There is also more to nature than mathematics, as I shall argue below.

I do have an idea for a mathematical mental model of nature, however. Here is the start of the story. Just as in physics deductions and calculations ultimately refer to numbers, in my model of nature, which is called NATURE, all deductions and calculations ultimately refer to parts, links, and qualia. NATURE has parts, every whole is a part of NATURE, its parts have links one to another, and every link has a quale. Just as in physics the fundamental symbolic entity is the complex number, so the fundamental symbolic entity in NATURE is the link. As a complex number has magnitude and phase, so a link is from one part to another part, with a quale. Every link and every quale is also a part of NATURE, so in my model of nature there are no "ultimate, unstructured, point-like parts". This means nature is modeled as infinitely deep, ultimately ineffable. A part with no links cannot possibly participate in any inferences or calculations in NATURE, nor can any aggregate of such insular parts.

If nature is the territory, then NATURE is the map. Draw the parts as dots, draw a link between parts as an arrow between dots (call the starting dot the bow and the ending dot the target of the arrow), and on each arrow draw a short tab labeled with the name of a quale (Fig. 1).

For example, put your attention on a clock and memorize the time when you get to the period at the end of this sentence, now. Keep reading, and relate that time to the time when you get to the period at the end of this sentence, now. The drawing in NATURE has two dots, called, say, now1 and now2. There is an arrow from now2 to now1, labeled with the quale of "pastness" (Fig. 2).

The quale of an arrow is separate from the name of the arrow itself: two different readers have different arrows, but the same quale. NATURE is a mathematical structure whose links are also parts of nature. This means there may be links between links, drawn as arrows between arrows, and links between links between links, drawn as arrows between arrows between arrows, and so on. Do you remember the arrow from now2 to now1 with the quale of "pastness"? If so, the model in NATURE is an arrow from a dot representing the part of nature, now, to that arrow you drew back then, with the quale of "remembrance" (Fig. 3).

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The arrow or arrows you have drawn are parts of nature with the peculiar property that they are models of parts of nature that are outside of space and time. For example, although now2 and now1 are in spacetime, the link between them is outside of space and time. The thought of that quale is in spacetime, it is the neural correlate of the quale. The neural correlate of a quale is called a feel (Fig. 4).

The system of principles and laws of qualia, whatever it turns out to be, shall be called quale mechanics.

REPHRASING THE QUESTION

With key terms explained, the opening question may be rephrased. How, in the mathematical model NATURE, shall I construct quale mechanics? There is an analogy between understanding, say, electricity, and understanding consciousness.

Understanding "electricity" depends on understanding numerous physical principles and laws:

a. Superposition, b. Conservation, c. Quantization, d. Charge Mobility, e. Force, f. Potential, g. Coulomb's Inverse Square Law, h. The Poisson and Laplace Equations, i. Capacitance, j. Energy,

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k. The Ampere-Biot-Savart Law, l. Vector Potential, m. Faraday's Law of Induction, n. Relative Motion, o. Charge Moving in Matter (14).

That is not all, for these principles and laws constitute only the classical understanding of electricity. The modern understanding depends on quantum electrodynamics and quantum statistical thermodynamics. Our understanding of electricity is quite mature. Nothing remotely like that can be said about my understanding of consciousness, but there is a bit of progress.

MATHEMATICS

In a survey of six recent volumes of The Journal of Consciousness Studies, there are authors from various academic departments, roughly as follows:

o 2 literature and religion; o 9 philosophy; o 15 ecology, sociology, psychology, psychiatry; o 5 neuro-psychology, neuro-biology, bio-physics, inter-disciplinary; o 5 cognitive science, computer science; o 4 physics, quantum chemistry; o 1 mathematics.

Putting aside my own bias in presenting the data, and the editorial bias of the editors of the journal, the fact is that mathematics is severely under-represented in efforts to understand consciousness.

My question is, why should mathematics be way out on the tail of this distribution? No doubt some weighted combination of the following answers:

a. The population in general suffers from "mathematics anxiety", on top of anti-science attitudes; b. A great deal of work is required to learn and do mathematics; c. The philosophical position that understanding consciousness is intrinsically not "reducible" to intricate chains of inference and calculation; d. Conviction that someday mathematics might be applicable to understanding consciousness, but the mathematics appropriate to this task does not exist yet, so it is way too early to try; e. There is no evidence yet that mathematics fruitfully can be applied to understanding consciousness, no one has had a good idea for a theorem; f. Mathematicians tend to be Platonists, hence have an in-built bias against alternative theoretical options.

Regarding (a)-(b), plenty of people enjoy great music without being composers or performers, and appreciate fine art without being artists. Llikewise, if people could be trained to "hear" and

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"see" high-quality mathematics, they would enjoy that as well. As for (c)-(e), I am here to give it a try. The reply to (f) is that pure mathematicians need to reconsider the foundations of mathematics beyond the conventional trilogy of "formalism, intuitionism, and Platonism".

The notational convention of drawing arrows to represent links between mathematical objects was turned boldly into a mathematical industry called category, functor, and natural transformation theory by Mr. Eilenberg and Mr. Mac Lane in the early 1940's (15)(16)(17)(18). Once, I was bold enough to tell Mr. Eilenberg I intended to work on mathematical metaphysics. Instantly, he harumphed that it would be better to work on metaphysical mathematics. Mr. Lawvere and colleagues invented a new kind of category, the topos, in the 1970's, thus revealing links between geometry and logic (19). One marvelous outcome of that work, over and above new questions, new proofs and better understanding of previously known theorems, is a bigger picture of mathematics. Previously, it was generally considered a truism that in principle all of mathematics may be formalized within axiomatic set theory. Mr. Lawvere and colleagues have proven that set theory (the theory of parts called "elements") is just one among many kinds of topos. Every topos has an "internal language" in terms of which vast parts of mathematics may be expressed, including, for example, the construction of the complex numbers and hence the amplitudes invented by Mr. Feynman to explain physics. I assume that links in NATURE form a kind of topos. Thus, mathematics is part of NATURE, and in particular all the apparatus for deduction and calculation in a topos is available for reasoning about parts of NATURE, just as the apparatus for deduction and calculation in the field of complex numbers is available for reasoning about physics.

There is another sense in which NATURE is shot through with mathematics. Some links between parts of NATURE have a quale of "membership", and axioms may be stated in the internal language which guide the quale of "membership" to satisfy the axioms of set theory. So a part of NATURE may have elements, namely, the parts linked with the quale of "membership" to it. It follows that in general, a part of NATURE may have elements, and it may have links with other qualia as well (Fig. 5).

This means that almost all parts of NATURE are more than the totality of their elements. (Those rare parts having only elements are absolutely pure mathematical parts of NATURE.) Thus, NATURE is more than mathematics.

SITUATION

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Spacetime is part of NATURE. Physics provides mental models so comprehensive that phenomena from the quantal to the cosmological are encompassed. Nevertheless, as the foregoing indicates, there is much in nature that is beyond physics, e.g., laws of biology emerging from unpredictable physical accidents. If my mental model of nature were drawn as some closed region in the plane, then one dot within that region would represent spacetime (Fig. 6).

There are links from non-physical parts of nature to spacetime, such as links with the quale of neural correlate (cf. Fig. 4), and there are links between non-physical parts of nature that completely bypass spacetime, such as laws that are relationships between second-order universals. Within spacetime we can know much, such as the behavior of objects. An object may be modeled mathematically by the set of trajectories of its points. I define a strand to be the map of time to space that traces the motion of a single point of an object (Fig. 7).

(Conventional literature calls this a "world-line".) Hence, a physical object is a bundle of strands (Fig. 8).

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As a plant grows, for example, from a tiny seed, molecules originating perhaps from very far away are "drawn into" the composition of the plant. The strands of the most elementary particles of those molecules all become strands of the plant. More and more strands join the plant until it is full grown. As the plant sheds dead leaves, and projects seeds elsewhere, and eventually dies and withers away, the strands of its most elemental parts separate from one another and merge into unlimited distant parts of the universe, most likely never to be together again. A physical object is, therefore, in general, a temporary bundling together of strands. Take care to consider an object to be a historical object, it is a map of time to subsets of space. The gathering together of the strands of an object may be gradual, such as in the growth of an organism, or relatively violent as in the manufacture of a computer, for example. The time-varying force fields of objective procedures for assembling a computer involve bringing together in a very brief time hundreds of thousands of parts. This implosion of strands makes for a characteristic distinction between what is grown and what is made, i.e., between the natural and the artificial (20). In any case, however, the position at a given time of all the points of all the objects in the universe, is by definition, the classical situation. (In the conventional literature, what I call "classical situation" is called "configuration", and the collection of "strands" is called "motion" (21).)

To simplify describing my mental model of Mr. Gell-Mann and colleagues’ theory of history that, they hope, is "sufficient for every prediction in science" (22), consider an object with some mass but exactly one strand. An electron is an example. Restricting consideration to one electron among many is a form of "coarse-graining", because it means that the model ignores what is going on with the omitted electrons. Another form of coarse-graining is to ignore what is going on with the selected electron at all but a specified set of times. The third form of coarse-graining is to consider only ranges for the values of the coordinates of the electron. In greater detail, define a set of sets of strands as follows. Choose a sequence of times. Associate with each of those times a partition into intervals of each space coordinate. (All of this construction can be carried out in the topos, NATURE.) There are infinitely many ways to choose, at each of those times, one of the intervals along each of the space coordinates. Imagine a box of space at each of

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those times, where the edges of the box are the particular chosen intervals. The set of all strands passing through those boxes at those times is called a coarse-grained history (Fig. 9).

Hence, there are infinitely many coarse-grained histories, called the alternative histories, for the given choice of times and partitions. "The important theoretical construct for giving the rule that determines whether probabilities may be assigned to a given set of alternative histories, and what these probabilities are, is the decoherence functional", which is a formula that, given two alternative histories, computes an amplitude in the sense of Mr. Feynman. The construction of a set of alternative histories starts with the choice of a set of times, and ends with a choice of three partitions of the real numbers for each of those times. The set of all subsets of a given set is partially ordered by set inclusion, and the set of all partitions of a set is partially ordered by the partition refinement relation. Therefore, the set of all sets of alternative histories is a partially ordered set (22, Fig. 2). Mr. Gell-Mann directs our attention to the possibility that for a given point in this partially ordered set, it may be the case that the value of the decoherence functional on any two distinct alternative histories at that point, is (or is nearly) zero. In that case, he says the point - a set of coarse grained alternative histories - is decoherent. "This is a generalization of the condition for the absence of interference in the two-slit experiment". He then argues plausibly that "to the extent that a set of alternative histories decoheres, probabilities can be assigned to its individual members." Consequently, "decoherence ... generalizes and replaces the notion of "measurement" [as used formerly in the orthodox interpretation of quantum mechanics]. Decoherence is a more precise, more objective, more observer-independent idea."

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Indeed, one of his main points is that the whole universe is a self-contained quantum mechanical system and that there is no need to postulate an "observer" somehow outside the universe to "collapse the wave function" by making measurements on the universe (23). My point is that a correct concept of situation taking quantum mechanics into account must be defined in terms of the whole universe. Discussing a situation means discussing the probability of the situation. The variables occurring in any scientific experiment depend on the entire history of the universe. We do not live in a context-free nature, nor even in a context dependent nature. We live in a history- dependent nature (24).

PURPOSE

A purpose is a mental model of a preferred future (objective or subjective) situation (1, p. 102). To "imagine" means to have a mental model of a situation other than the actual situation. A now is a mental model of the actual situation. All one has is now (25)(26). I can summon a feeling of anxiety just by thinking about how trapped I am in this time cocoon, my now. Every now is a part of NATURE and the set of all nows is partitioned into my nows and your nows and everyone else's nows. Nobody has anybody else's nows. Within any one's set of nows there are links with the quale of pastness, so that, normally, one's nows are totally ordered by pastness. Experience is a retentive succession of nows (27)(Fig. 10).

While you are reading I attempt to guide your attention with each of these successive words. This requires that your now retain enough of the preceding words, or better, the integration of the preceding words, into the mental model you must be re-forming continuously to track my story (1, p. 188).

Even if you are drawn away from reading this, you may "tell the story" potentiated by your mental model of my mental model. The manifestations of purpose are trajectories of objective and subjective procedures that may bring about the preferred future situation. The discrepancy between the actual unfolding situation and the imaginary preferred situation is "the error signal which drives the process forward" (1, p. 101). Those mental models that fulfill purposes are more likely to be retained and transmitted, e.g., via stories, or instruction manuals, between human beings (1, pp. 101-5). Imagining alternative future situations provides the quale of "choice", and so the quale of "free will". Attention is governed by purpose. Normally, there is in

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one's life exactly one "top-level" timing machine, of which the sub-machines represent one's purpose. Already at birth one "parses" awareness: attention actively selects parts of awareness to enter consciousness. Attention is a time-varying predicate on the parts of awareness (Fig. 11).

Whereat the predicate is "true", that part of awareness is a part of consciousness. This is called "the moving spotlight of consciousness". Natural language amplifies the capacity for parsing awareness. There is a probability distribution on the values of the attention field. The probability calculation factors through the principles and laws of quale mechanics, just as the probability of a physical event factors through the principles and laws of quantum mechanics. Some laws of quale mechanics are as follows:

a. The probability of remembering is proportional to the probability of attention(1, p. 189). b. Novelty, and especially difficulty, gets attention (28). c. Attention guided by purpose selects mental models.

d. A thought is an object in spacetime, and any thought gets attention.

CONCLUSION

Moving on to a mathematical science of consciousness requires breaking out of spacetime to a new mathematics of parts and wholes. Subsequent to the Hayden Colloquium on parts and wholes in 1963 (29), there have been major advances in physics and mathematics. Quale mechanics is a mathematical mental model of nature astride the coupled steeds of the quantum cosmology of Mr. Gell-Mann, and the element-free mathematics of Mr. Lawvere.

Appendix. TIMING MACHINERY

Timing machinery is my mathematical model of parallel activity (30). The parts of a timing machine are called states. A state may be linked, in one way only, to itself or to some other state with a quale of "timeout-in-N-seconds": this is called the timeout link of the state (Fig. 12).

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It may or may not be linked with a quale of "signal-X" to itself or some other state, called a signal link (Fig. 13).

This means that when its time runs out, the state emits the signal, becomes inactive, and the timer of its next state is started. The state receiving the signal may have links to other states with the quale of "recognition-K", called trigger links (Fig. 14).

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If matching-K recognizes the signal received by a state, then the receiving state is de-activated, and simultaneously the state to which it is linked by that recognition becomes the active state, which means its timer is started. On this simple foundation I have constructed many examples of timing machines, including approximations to differential equations, cybernetic machines, spike train filters, thermodynamics, and models of biological neurons (31). These examples are all purely computational, they have no links to physics except for the explicit appearance of physical time on the timeout links. The start of my climb towards using timing machinery for more than computation was triggered by Mr. Edelman’s attempt to explain what an algorithm is. He gives an "algorithm" for boiling an egg (32).

BOILING AN EGG

1. Put water in pot. 2. Turn on heater. 3. If water is boiling, go to step 4. Otherwise, go to step 3. 4. Put egg in pot. 5. Set timer to 3 minutes. 6. If timer has reached 3 minutes, go to step 7. Otherwise, go to step 6. 7. Turn off heater.

My point is that boiling an egg is not a computation with symbols, it is a "computation with nature". That is, in the sense that the primitive operations such as filling the pot with water, placing the pot on the stove, placing the egg in the water in the pot, turning the knob to start the heater, etc., are all time-varying force fields, not mathematical operations. I believe there must be a complete distinction between operations on abstract symbols, and operations on physical objects in spacetime. That is to say, it is one thing to send signals to states with one or more links having the quale of recognition, and quite another to apply forces to objects. But Mr. Edelman is still sort of correct, and my way of saying that he is correct is that both computational procedure and objective procedure may be guided by rules expressed in timing machinery. The mountain I am climbing has its foothills in computation, and it has trails leading to physical operations. The summit is subjective procedure (Fig. 15).

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The laws of thought are subjective procedures, and such procedures are distinguished by directed attention.

COUNTING A SET OF PARTS OF AWARENESS

1. Direct attention to the set of parts. 2. Remember the set of parts. 3. Direct attention to one of the parts. 4. Link the selected part to the natural number 1. 5. Remember the natural number last linked. 6. If there is an un-linked part in the set of parts, then direct attention to one of the un-linked parts, and go to step 7. Otherwise, go to step 8. 7. Link the selected part to the successor of the last linked natural number, remember the latest linked natural number, and go to step 6. 8. Direct attention to the last linked natural number.

Every scientific measurement requires counting a set of parts of awareness, whether it be counting marks on a ruler, or bright spots in a photograph. A mental model of a scientific experiment will include inference and calculation, physical manipulation, and directed attention.

NOTES AND REFERENCES

1. Newton, N., Foundations of Understanding, John Benjamins Publishing Company, Philadelphia, 1996.

2. Johnson-Laird, P. N., Mental Models, Harvard University Press, Cambridge, 1983.

3. Rukeyser, M., "The Speed of Darkness", in A Muriel Rukeyser Reader, Edited By Jan Heller Levi, W. W. Norton & Company, New York, 1994.

4. Implicit in this statement is my conviction that all meanings in mathematics are rooted in natural language. No one learns or creates any mathematics without first being a member of a natural language community.

5. In The Feynman Lectures on Physics there is a special lecture "intended for entertainment" on The Principle of Least Action, which is ultimately the inspiration for Mr. Feynman's theory of amplitudes. He relates that his high school physics teacher, "Mr. Bader", first introduced him to The Principle. He did not say, "Bader", or "Abram Bader", or "Mr. Abram Bader". My interpretation is that on days when students know how to respect the teachers, they show their respect by referring to teachers by last name prefixed by the appropriate designation, "Mr." or "Mrs.", etc. Likewise, I wish to acknowledge my respect for the authors from whom I try to learn by the same sign of respect used by Mr. Feynman.

6. Gell-Mann, M., "Hotline Nobel Direct", http://www.smau.it/nobel95/direct/gellq.htm

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7. Strong, J., Procedures in Experimental Physics, Lindsay Publications, Inc., Bradley, 1986. 8. "[P]rimary consciousness equates with wakefulness, and this level of consciousness directs attention to whatever is interesting or important at the time, so that the appropriate cortical region may have primacy, and the relevant higher-level conscious activity occupies center stage of the theatre of the mind." Petty, P. G., "Consciousness: A Neurological Perspective," Journal of Consciousness Studies, Volume 5, Number 1, 1998. 9. Tart, C. T., Waking Up, Shambala, Boston, 1987. 10. Armstrong, D. M., What is a Law of Nature?, Cambridge University Press, Cambridge, 1983. 11. "A wondrous vine emerges when Jack plants the seed for his beanstalk, and it unfolds into a world of giants and magic harps. When we were children, Jack’s miraculous beanstalk wasn’t so far removed from the everyday miracles of fall colors and germinating seeds. Now that we’re grown, seeds still fascinate us. Somehow these small capsules enclose specifications that produce structures as complicated and distinctive as a giant redwood, the common day’s-eye (daisy), and a beanstalk. They are the very embodiment of emergence - much coming from little." "It is tempting to take the inability to anticipate – surprise – as a critical aspect of emergence. It is true that surprise, occasioned by the antics of a rule-based system, is often a useful psychological guide, directing attention to emergent phenomena. However, I do not look upon surprise as an essential element in staking out the territory." Holland, J. H., Emergence, from Chaos to Order, Addison-Wesley Publishing Company, Reading, 1998, pp. 1, 5.

12. "Ontology is the projection of registration on the world. Representation is the projection of registration onto the subject or vehicle." Smith, B. C., On The Origin of Objects, The MIT Press, Cambridge, 1996, p. 349.

13. Feynman, R. P., Hibbs, A. R., Quantum Mechanics and Path Integrals, McGraw-Hill Book Company, New York, 1965.

14. Besancon, R. M., Editor, The Encyclopedia of Physics, Van Nostrand Reinhold, 1990.

15. Eilenberg, S., Mac Lane, S., "General Theory of Natural Equivalences," Transactions of the American Mathematical Society, Volume 58, Number 2, 1945.

16. Mac Lane, S., Mathematics Form and Function, Springer-Verlag, New York, 1986.

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17. Lawvere, F. W., Schanuel, S. H., Conceptual Mathematics, A First Introduction to Categories, Cambridge University Press, 1997.

18.

19. Mac Lane, S, Moerdijk, l., Sheaves in Geometry and Logic, A First Introduction to Topos Theory, Springer-Verlag, New York, 1992. 20. In some religious mental models, all things are the work of God, that is, all things are "made". Certainly, however, there is a distinction even in such religions, between what is made by God, and what is the work of man. In scientific mental models, the distinction between what naturally grows and what is man made is even more obvious. 21. Marsden, J. E., Hughes, T. J. R., Mathematical Foundations of Elasticity, Dover Publications, Inc., New York, 1983. 22. Gell-Mann, M., Hartle, J. B., "Quantum Mechanics in the Light of Quantum Cosmology", in Complexity, Entropy, and the Physics of Information, SFI Studies In the Sciences of Complexity, vol. VIII, Ed. W. H. Zurek, Addison-Wesley Publishing Company, Reading, 1990. 23. There is more than one effort to do quantum mechanics without collapse of the wave function. (See, for example, the accessible review, Goldstein, S. "Quantum Theory Without Observers", Parts I and II, Physics Today, March and April, 1998.) There is also a clique of physicists who care not at all to eliminate collapse and observers from quantum mechanics, in fact they elevate it to a central ontological position in their theories of consciousness. Naturally, that sort of reduction of consciousness to physics is alien to my point of view. 24. A timing machine, for example, is context-free if the response to a signal is the same regardless of what state happens to be active. More generally, it is context-dependent if the response to a signal depends on the active state. Most generally, it is history- dependent if the response to a signal depends on the prior sequence of active states. 25. "Only in the ‘now’ is anything directly given in experience. Past and future are also given in the now, as present memories and present anticipations." Hut, P., Shepard, R. N., "Turning the Hard Problem Upside Down & Sideways,’ Journal of Consciousness Studies, Volume 3, Number 4, 1996.

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26. "If only you could make now last forever," Frank said on one of those nights while they lay on their back watching a huge half-moon roar up out of the dark shoulders of the mountain. Frank was eleven and not by nature a philosopher. They had all lain still, thinking about this for a while. Somewhere, a long way off, a coyote called. "I guess that’s all forever is," his father replied. "Just one long trail of nows. And I guess all you can do is try and live one now at a time without getting too worked up about the last now or the next now." It seemed to Tom as good a recipe for life as he’d yet heard." Evans, N., The Horse Whisperer, A Dell Book, New York, 1998, p. 123. 27. Miller, I., "Husserl's Account of Our Temporal Awareness," in Dreyfus, H. L., Editor, Husserl Intentionality and Cognitive Science, The MIT Press, Cambridge, 1982.

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VISUAL + EXAGGERATE + TRAJECTORY The Bent Note By Ellis D. Cooper January 18, 2001

Buenos Aires, Summer, 1977 A tomboy Jewish little girl and her friends are playing in the backyard of her house on a side street of Avenida Indepencia, in southern Buenos Aires. They are building with discarded planks of wood, empty refrigerator cartons, and pieces of rope. The main project is to dig a tunnel. The kid at the bottom shouts up to her, “Hey, Achas, check this out! There’s a tunnel here!” She tilts her head of black hair, and sneers, “Right, that’s what we’re making.” He shouts, “No, you don’t understand, its a real tunnel!”

Buenos Aires is undergirded by a network of mysterious tunnels revealed by accident in 1912 during excavations for public construction. There are literary premonitions of the tunnels in writings of the past century, including implications that the tunnels served for esoteric practices. Achas rushes into her father’s study, a room that self-evidently is that of an extremely well-organized intellectual. “Dad, look what I made!” She shows him some of those ﬂat plastic refrigerator door magnets cut into triangles and squares. She slides them around on the refrigerator door, and triumphantly declares, “See, its a proof of Pythagoras’ Theorem!” He looks into her blue eyes, “Did you invent that yourself, Achas?” “No, I saw it on the tunnel wall. But I ﬁgured people would understand it better if they could do it with their hands.”

There is musical background now surging with a Spanish-Arabic sound tinged with a plaintive quality, something like the Muslim call to prayer.

Achas is the daughter of a blind Sephardic Jewish esoteric poet (a la Borges). He is a revered high school teacher, and had emigrated from Buchenwald, where he witnessed the murders of his gypsy and Jewish friends and relatives. The SS members who he witnessed torturing and murdering also fled to Argentina, and they learn that he belongs to a cabal which seeks them to bring them to trial for the sake of justice. His group almost finds them, but they find and murder him. While studying mathematics with Nicholas in Chile, she hears of this murder, and is inconsolable. She decides to leave South America to study mathematics in New York. She travels to New York on a beat up Harley-Davidson which she carefully maintains herself. She studies at the University of Mexico on the way to New York, and publishes a paper on category theory which is acclaimed. She travels alone. When she arrives in New York we see a slim, tawny black haired woman with penetrating blue eyes and an especially sensitive mouth, whose face can freeze - a stoneface to hide feelings while calculating moves. She is The One for Echod. She is killed by Arabic terrorists from Iraq in an explosion at 14

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street and 7 avenue during the July 23 Spanish street fair.

Achas Misrahi, Sephardic Jewess [Patai TJM 380] ”achas” is Hebrew femi- nine “one”.

That night, after Achas shows her father the proof of Pythagoras’ Theorem as found on the wall of a tunnel beneath their backyard, as he puts her to bed, she asks, “Daddy, tell me again where we come from. And you said it has to do with the tunnels.” He tells her the following.

The Jewish mind starts with the belief in one God only, and belief that there is a special relationship between God and the Jews. The Jews have a duty to God, and a duty to all other human beings. From 500 to 700 Christian Era, Talmudic scholars labored in Baby- lonia, and Babylonia became Arabized, and today is called Iraq. The Arabs believe that the Qoran is the word of Allah, dictated in beautiful Arabic to Muhammed, his Prophet, by the angel Gabriel. Muhammed died in 632 Christian Era. By 700 the Arabs had con- quered the entire Middle East, and eventually Southern Spain. Jews lived in all parts of this huge Arabized territory. For over 2000 years, from 1300 Before Christian Era until 900 Christian Era, the Jewish intellects concerned themselves with reﬁning the meaning of Judaism, but while living with Arabs in Spain they turned to science. Under the House of Islam, non-idolators (Jews, Christians,...) were allowed to live. Idolators were murdered. The Jews assimilated to Arab culture. They loved Spain, and produced mathematical and scholarly treatises in Arabic. In fact, Arabic was their everyday colloquial language, and Hebrew was reserved for poetry. Politi- cal forces pushed the Jews in Arabized southern Spain towards the more Christian northern Spain, leading to a new language formed from Castilian Spanish and Hebrew, the language of the Sephardic Jews called Ladino. When Achas ﬁrst meets Nicholas at the University of Chile, Santiago, she is more waif than woman, but just about to pop. Nicholas is courtly and gallant, much older, and truly wants nothing more than to teach her category theory. However, he is also not blind, and we see a visual representation of his arousal.

Rapid zoom to corner of an eye to see slightly slower or faster the welling up of a tear.

Rapid zoom to the lower or upper lip of a mouth to see twitch of uncertainty.

Qawwali “utterance” - Nusrat Fateh Ali Khan, The Sabri Brothers Cease- lessly repeated sound of the name of Allah.

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We are “treated” to both the fundamentalist argument, leading by tiny degrees up to but not quite including the murder of abortionists, and to the terrorist argument from the Qoran, leading carefully to full retaliation by murder in the name of Muhammed. Movie music: Jazz, Klezmer, Qawwali - all emphasizing bent notes. The death of Achas Mizrahi is marked by a Klezmer broken wailing note. All the academics always ask right away, “Where did you study and with whom did you get your degree?” Echod Eloim college colleagues who deny him tenure are (a) a brilliant fundamentalist who is a genius but only at one thing; (b) an extremely smart but seriously insecure mathematician who uses sugar-coated condescension to praise what he wishes he could do; (c) a black and brilliant mathematician who, however, gives poor names to his inventions, and lacks the courage to learn new languages; (d) an Irish mathematician who is always telling mathematics jokes, but also is always the ﬁrst to laugh, and often the last. The whole movie is a vicious, relentless attack on fundamentalism. The Iraqi terrorists are fundamentalists. The handsome, scowling terrorist leaves the room because a hostage baby is crying. There is a shot, the crying stops. Time stops. A woman screams, there is another shot. The screaming stops. The terrorist returns, with a smile, says, “That’s better,” and continues talking. 1977 Achas a child in Buenos Aires 1997 Achas arrives in New York 1998 Achas meets Echod

”echod” is Hebrew masculine “one” Achas and Echod = one plus one Both dead at the end of the movie, so one plus one is zero.

When Achas ﬁrst sees Echod at the jazz club in New York, and is told who he is - a superbly brilliant mathematician but with esoteric interests - and she notes the shape and sensitivity of his mouth, we get a visual indication of her arousal.

After Echod’s anti-tenure humiliation and Achas’ surprising and completely senseless death due to Arabic terrorists, he tells Muhammed (his black friend and colleague) that he wants to meet these terrorists. Through increasingly dangerous Muslim contacts, he faces The Real Bad Guys in a cell in Brooklyn near a mosque. We hear the Muslim call to prayer in the background. There is no common ground, and they kidnap him to Iraq. They torture him but not before he has an emotional epiphany. He realizes that what matters is truth, not proof, but that rigor cleans the window through which intuition shines. A ray of desert sunlight breaks into his cell, there is the sound of a pop, another pop, and a third unevenly spaced pop. Extraordinarily fast moving Suﬁ ninjas rescue him and bring him to The Master. These are an invisible group of Arabic Anti-Fundamentalist Suﬁ Militants who are experimenting with eye-movement electronics to accelerate development of Higher Consciousness. They relax him, and he attends a musical performance of heart-lifting qawwali music. His theory

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converges with their experiments, and he returns to the West with full Knowl- edge. Muhammed is a highly skilled electronic engineer and mathematician who unites coherently both the belief in God and his scientific interests. His family, all black people, attended church on Sunday (Baptist). He was not sheltered, the family was not introverted, his parents spoke their minds. He grew up in urban Philadeplphia, and was exposed to white people early. His interests in science and family background allowed not to develop a “chip on his shoulder” about being black in a mostly white society. Whenever we meet a new major character,we get a tour of their library. Achas, Echod, Muhammed, Achas father Milton, all have “Principles of Psychology” by William James. Milton Mizrahi, the father of Achas, is dying of cancer. So, you see a neat pile of technical reference works on cancer of the brain, and also “How We Die”, by Nuland. Milton has a precise habit of measuring his coffee with a graduated flask.

When Milton Mizrahi moves about, he does it often with a ﬂourish, a grand gesture, even just stretching his arm to pick up his cup of coﬀee.

Achas: “Mr. Dad, why do you do that?” Milton: “What? The extra move?” Achas: “Yeh.” Milton: “It’s the icing on the cake of my life.” Is there a “baruch atau adenoid” for fucking? Achas is the daughter of a white father and a black mother. She is remembered as Echod’s equal in formulating Consciousness Engineering, but is a tragic figure. How the Consciousness Engineering works to re-structure the terrorists’ mental model within the Qoran: “electronically” detect the spaces between his thoughts (corresponding to the silences between the notes in music), and stimulate him to re-direct to other parts of his love of God. This is related to the special effect in the movie, shared by Achas, Echod, the Sufi Ninjas, The Master: there is a 10 frame freeze of all action, while they continue to move. That is why they look so fast. Fast magnetic resonance imaging is replaced by injecting “fluid” of self- destruc- ting molecular pico-technology radio-transmitting sensors which measure local physical parameters on the micron scale and coordinate position. Thus, neural correlates of consciousness of two people having a conversation are made objective. There are intimations of artificial telepathy.

radio-ventrilotelepathy

The receivers of these transmissions are quantum computers adept at global pattern recognition. The walls of the conversation compartment are quite cold to the touch. The two conversationalists wear heavy sweaters, and must be very close, face to face, in the tiny compartment. So close that the Arab and the Jew

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are forced to both invade one another’s personal space. The Arab tolerates this slightly better than the Jew, and this shows up on the monitoring screens. The Arab mental model of nature has Allah the endless backdrop of all endeavor, with science a small part of that. The Jew mental model has nature the huge backdrop, and God, Allah, the angel Gabriel and everything else that people think about are elements of nature. Muhammed and Echod are musical mentors of one another. ”The Last Supper” is attended by Echod and twelve mathematicians in a Chi- nese restaurant near Columbia University. Echod has just delivered a talk on his absolutely brilliant but substantially weird new mathematics of consciousness. There was a mathematician in the back row who kept making really ugly faces at Echod during the entire talk. The twelve disciples are a happy-go-lucky bunch of mathematics graduate students, including Achas.

At one point, Echod says, with a laugh but also a catch in his voice, ”One of you will prove this conjecture!”. physical prison chemical prison consciousness engineering

2048

1. Consciousness thugs 2. Mental Sports Page 3. Teachers are the superstars 4. ”A Better Life with Higher Consciousness” is a scam in the headlines

In 2078 there is a small plaque in a cemetery honoring the inventor of consciousness engineering for having improved the world by reducing the sufferings of humanity caused by fundamentalism (Christian, Jewish, Arabic, Scientific, etc.). In 2078 we see a vicious terrorist attack on a large building in a large city. The terrorists are apprehended, and brought to Consciousness Engineering Research Institute (CERI). It seems at first that they are to be brainwashed, but instead are exposed to Sufi’s with electronics, and non-invasively brought to an “Aha!” experience with laughter and heart-rending repentance, but no change of personality. Just a better side of themselves. The utter humiliation of Echod, being treated like a flake by his “peers”, leads to an emotional breakdown. This is the extreme low point of his trajectory. What can consciousness engineering do to ameliorate suffering? This movie is an answer. Visual of his epiphany in Iraq: “seeing” thoughts hurled forward and upward from not-consciousness: black cubes with orderly fractal red flames hurling elements into a fusion leading to writing of words. The academics attack him for setting himself apart from them, by arguing about how Quantum Theory was made, not by a committee, but by an association of geniuses. ”You stuck up putz, what are you, you think you are one of God’s chosen?”

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”Yes.” Echod invents the future of education with emphasis on Acting Class: learning how to act; learning how to be someone; suffering and empathy. Echod matter of factly becomes The Second Coming of Christ. He starts out as an ordinary Hasidic Jew, full tilt, loses his pais, becomes a humiliated genius, endures the loss of Achas, invents consciousness engineering, and dies on a cross. The Hasidic Rebbe declares that Echod Eloim has successfully transcended the belief in God and at the same time the meaninglessness of science. As a teenager Echod wallpapers his bedroom with all the pages of a five volume book on differential geometry, and then annotates the walls with colored marker pens. Muhammed invites Echod to his first jazz club. This is an all-black place, but there is one white musician in the band. Almost everyone is smoking marijuana. On another occasion, Echod brings Muhammed to his first Klezmer concert. There is one black musician in the band. Muhammed is the “Paul” of Echod. Muhammed is Echod’s best friend. When young he was a skinny guy who likes fat girls. He shoots good pool. Goes nuts when Echod is humiliated. Muhammed becomes a druggy derelict pool player. His brain turns into gefilte fish due to a psychotic reaction. Hounded by academia, Echod fails to prove a key theorem, cuts off his pais, burns his yomica, eats pork chops, throws up, and cries for a week. Echod starts as a young orthodox Hasidic Jewish mathematics professor. He meets a black colleague, Muhammed, who introduces Echod to the history of jazz. At the jazz club, Echod with some kindly teasing from Muhammed, smokes some righteous marijuana. Preview of his visual epiphany of thoughts hurled from not-consciousness. Turning Point 2. Innocently thinking he can just buy marajuana in Washington Square Park, he gets arrested and has to go to jail. He meets a real dealer with an attitude. But his best boyhood friend, who shot good pool, and likes fat girls, gets him out of jail after a big lawyer fight. The lawyer is a pothead. The boyhood friend grows old too fast, and looks like a seventy five year old at the age of fifty five. Their two mothers in the same nursing home fail to recognize one another. The footnote became the book. The Book of Ax The Book of Revolution When he chooses, Echod has the conscious thoughts of people he has admired and studied. And when he is like that, and looks into mirrors or windows of shops along his way, he sees the one whose consciousness he has. To others he just looks like his drab self. In a pool game he becomes Willi Mosconi, and wins. He has admired and studied Gurdjieff, and Einstein, and Godel, and Schweitzer and Mandela. Heavyweight neanderthalic colleagues have suspicions about him, and conspire to deny him tenure. They trap him in an alley behind the pool hall. He also admires Muhammed Ali. The Techno Sufi Ninjas have a hidden sinister objective, using Echod for the destruction of Israel. They save The One to destroy the many, and then The

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Suﬁ Ninjas nail him to a cross, and say, “This one isn’t coming back,” and heave the laden cross into a very deep tunnel under Buenos Aires.

2001 01 06 This movie is a science and religious fictional docudrama that was made in 2072. It is to the events depicted in the movie as the synoptic gospels are to the story of Jesus. The tensions in the movie are scientific/religious abstractions personified in the characters and their trajectories. The basic abstractions are Judaism, Christian anti-Judaism, gentile anti-Judaism, secular antisemitism, scientific anti-religionism, and finally, the relientific anti-science and anti-religionism. “Relience” is the fictional religious science that supercedes religion and science.

The reason for 2072 is that 70 years is roughly the interval from the death of Jesus to the writing of the gospels.

Ten minutes of opening credits start with children’s book cartoon depiction of the early history of the Jews. The last cartoon is ﬁctional, fading into a photograph, which fades into the motion picture of Dr. Prof. Milton Misrahi, the father of Achas, in his studio.

The history of the Jews turns into an animation of the Earthly globe with a blue-white lava representing the diaspora of the Jews, with one particular ﬁnger of this lava reaching Buenos Aires. Another ﬁnger reaches to Brooklyn (Hasidic Jews).

The movie consists of ideas and stories. Ideas are referred to in the dialogues of characters in the stories, but are represented by special eﬀects. For example, the idea that human beings are directly aware of an extremely parochial part of Nature, there is a character who is reduced to being able to see exactly one black or white pixel, and who can move a very tiny lever only this way or that way. In other words, this character has one binary input, and one binary output. From the inside, one sees a completely black screen except for one blinking dot. From the outside, one sees a tiny lever ﬂicking back and forth. The outside is what human beings seem like to an anthropomorphized Nature. The inside is to the other characters’ view of her room, as a human with all his instruments is to the totality of Nature.

The title, “The Bent Note” refers to that quality of Jewish music, e.g., the famous cantor Richard Tucker, in which the note sobs. This is a long note punctuated by a sobbing. What the length of the note represents is the endurance of the Jews; the sobbing reﬂects their suﬀering. The loudness of the note represents the overwhelming strength of the Jews. In musical terminology the word

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melismatic - multiple notes per syllable - is applicable to this kind of singing. There is a related musical quality in arabic and sephardic music.

Is there a famous cantor (Richard Tucker?) rendition of the Sh’ma repeated over and over but with variations?

Milton Misrahi is his daughter Achas’ primary teacher. But there is a scene in her grade school class where it is plainly evident to everyone else that Achas has become the teacher’s pet - the chosen child. The reaction of the other children is symbolic of the gentile reaction to the Jews’ claim that they are the Jewish god’s chosen people. This is the earliest depiction in the movie of an abstract idea, gentile anti-Judaism.

Arabic music available in Egypt on 78 RPM records in the late 1930’s, or early 1940’s is played throughout the movie.

Although characters represent abstractions, they are not stereotypes. Each character has many lines, and certain key words are repeated by each character. These key words are tokens of mental models that characterize the characters.

2001 01 14 On a world somewhere, the autonomous complex adaptive systems are almost all transparent. In such a world, biology, not physics, is the easy science. So their mathematics has no problem with wholes being more than the sum of their parts.

“You fuckin Jewish pig.” “You call me a pig? I come from a kosher fuckin family.” “Yeah? and your point is?”

Museum Scene: Achas and Echod visit the computer room and log on to the Museum of Mental Models. They see spectacular animations: 1. inversion of the sphere; 2. Wolfram’s automata; 3. Kauffman’s co-evolving experiments; 4. Crutchfield’s pattern discovery; 5. real-time self-organizing map; 6. gravitational clustering algorithm; Achas: “They all missed the boat, didn’t they?” Echod: “Yes and no. Sure, they didn’t see the landing, but they were out to sea.” The opening credits do not show maps. Show the graph of the map, with labeled circles connected by lines indicating border contact. This is a time- changing graph. The flow of the Jewish lava is along these connections.

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Education Scene: Milton Mizrahi is teaching Achas Mizrahi critical thinking. The exercise is about reasoning fallacies. She has to act out six diﬀerent kinds of fallacy. For example, Failure of Imagination (Argument from Ignorance?), and ”falsus in uno, falsus in omnibus” (one is wrong, hence everything is wrong).

The Sephardic Jews in Buenos Aires retain their legendary airs of superior- ity. The Hasidic Jews in Brooklyn are shown in a frenzy of singing and dancing.

Visually convey mathematical abstractions during technical conversations by exposing plan view of insides of heads, not to show brains, but to show ideas.

Achas: “I don’t care about metaphors, I just want the facts.”

Nicholas: ‘‘Tell me why you believe something." Achas: ‘‘Maybe everybody knows it; or I witnessed it myself; or somebody I trust told it to me; or I saw it on television; or I proved it mathematically; or I can’t think of any other way; or, hey, I just know it."

The conflict is between Judaism and Christianity, at first. The ultimate resolution is the creation of Relience, the union of religion and science, created by Achas and Echod. Achas’ father Milton, a holocaust survivor (number tatoo on arm) becomes obsessed with the history of secular antisemitism. His wife was killed in a camp due to weakness of character hence complicity of his best friend, who converted to Christianity to save his own life. But Milton was nailed for his wife’s death, and fled to Buenos Aires and changed his identity. He is murdered by Israeli war criminal hunters.

Echod dies in abject obscurity. Nobody really knows that he initiated Re- lience, with Achas. Relience is considered to be the creation of a group mind, such as that which created quantum mechanics.

The Arabic Ninjas are ﬁnanced by antisemitic communists.

Semitic Hebrew Jew

First Temple 586 BC Second Temple 70 AD

Torah Bible Talmud Mishnah Gemarah

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Anti-Judaism blood libel usury killed Jesus poison wells conspiracy anti-Zionism antisemitism (race) broken covenant Grace rather than Law holocaust denial

Judaism Rule - circumcision of males Rule - Torah study Belief - one god Belief - covenant (chosen people) Rule - duty to god Rule - duty to fellow Jews Rule - dietary laws Belief - history Belief - matrilineal descent from Judah

Homo messiah An oﬀensive character who asks to borrow, say, a ﬂoppy disk. You take it from a special case. Instead of returning the disk to your hand, or to your desk top if you are absent, he handles your special case and puts it there, right in front of you.

The underlying conﬂict is “man versus man” in the form of conﬂict between the institutions of religion and science.

Milton is Sephardic-Portuguese. His wife is Ashkenazi-Ukraine. He loved her greatly. As they grew older she became physically less involved, and he cavorted a little with younger women. She didn’t know, and was extinguished by the Nazis with the covert complicity of his best friend, and he got blamed for her death. So he has multiple feelings of guilt. His love for Achas is enormous.

Achas ﬁrst hears about Echod’s research on the World Wide Web, and sees his home page.

A Jewish apostate mathematician with a science of consciousness becomes the Messiah (warrior king of mathematics).

In the opening credits, annihilation of Jews is shown by viscous implosions

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of lava, proportional to the magnitude of the annihilation.

Throughout the movie, there is a special eﬀect that ﬁnds faces and animals at random in clouds, wood grain, paint cracks, folds of fabric. This is a subliminal reference to the immense human capacity for seeing pattern where there is none.

Echod’s blackboard of mathematics is compared to Stravinsky music, as over against the conventional music of Palestrina.

There is an analogy of relating Achas to Echod as Mileva relates to Albert Einstein.

Any land occupied for two or more generations turns into a possession worth ﬁghting for to the death.

Module 1 Homework:

1. What if a man and a woman from opposite ends of Judaism collaborate to invent a mathematics that explains and then transcends both religion and science? 2. What if an eleven year old boy sees magical patterns everywhere, in clouds, wood grain, fabric folds, melting snow? 3. What if a band of homosexuals relegated to death by Hitler finds a way to overpower their guards, and save the lives of hundreds of Jews? 4. What if a housewife finds that two pages are missing from a book that her daughter is reading? 5. What if an AIDS researcher learns that his daughter is a lesbian? 6. What if the planets of the sun are balls on a cosmic pool table? 7. What if there is a world of transparent beings, like zebrafish? 8. What if a computer programmer gets a severe anxiety attack because a gay hot shot contractor takes over his main project? 9. What if the family of an African prince is massacred in a civil war while he is a student at MIT? 10. . What if the “black sheep” in a deeply religious family becomes a scientist who identifies the first reliable evidence of extraterrestrial intelligent life? Module 2 Homework: “We looked at your title, how important it is and how easy it can be to come up with one. So your action plan for the next module is to come up with a log line, your favorite idea. Write it down, start to research that idea, think about the story, how you’re going to develop it. Is it worthy of the big screen? Would we want to go and see a film about your subject? If you think the answer is yes, then come back to the next session with your log line.”

What Achas and Echod invent is a mathematical biology that doesn’t use hard- edged symbols anymore. The symbols are gradations and interlockings and transitions, instead of black on white ideograms. Their mathematics is to

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conventional mathematics as cursive writing is to block lettering. With this they are able to draw complex descriptions, e.g., of the zebrafish. They extend this notation to include mental models. They allow for the first time visualiza- tion of fundamentalist and conventional scientific mental models, and because they can show how someone can embed their mental models in a larger mental model, people can change their minds without being brainwashed, without suffering anxiety, or guilt, or impulses for revenge. Instead, they Understand Things Better, and become committed to reducing the suffering of others.

LOGLINE Notes: Religious fundamentalism, genocidal revisionism, and empty science are reversed by a new mathematics of consciousness invented by a Sephardic beauty and a Hasidic apostate.

What if a man and a woman from opposite ends of Judaism collaborate to invent a mathematics that explains and then transcends both religion and science?

The whole movie is a vicious, relentless attack on fundamentalism. The Iraqi terrorists are fundamentalists.

This movie is a science and religious fictional docudrama that was made in 2072. It is to the events depicted in the movie as the synoptic gospels are to the story of Jesus. The tensions in the movie are scientific/religious abstractions personified in the characters and their trajectories. The basic abstractions are Judaism, Christian anti-Judaism, gentile anti-Judaism, secular antisemitism, scientific anti-religionism, and finally, the relientific anti-science and anti-religionism. “Relience” is the fictional religious science that supercedes religion and science.

The conﬂict is between Judaism and Christianity, at ﬁrst. The ultimate resolution is the creation of Relience, the union of religion and science, created by Achas and Echod.

2001 01 15 Is there a pwer law/scaling law regarding extinctions of Jews: many small ones, a few very large ones?

Mileva Mizrahi renames herself Achas when her father is murdered. Abram Zabrisky renames himself Echod after his breakdown.

A twist is that we think Achas is the hero of the movie, but really it is Echod when she dies. His ability to be his heros includes seeing himself as her in a window reﬂection.

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In addition to the Parable of Binary Existence, there is the Parable of the Simple Room: Growing up and living in a simple room, and one day incidentally seeing there is a door, that leads to hallways, other rooms, and the outdoors, sunlight. This is the metaphor for changing one’s mental model and leaving the conﬁnement of fundamentalism.

Echod meets with his Rebbe to declare his apostasy and ambition. The Rebbe stares at him, and dismisses him with a ﬂutter of his ﬁngers. The next day, Echod receives a letter that says in three months he will have money, but that he must leave Brooklyn forever.

The ﬁrst time that we see Abram (Echod) he is working in the electronics repair section behind the counter at 47 Street Photo. He always speaks very rapidly, but in a deep, melodic voice conveying impatience but great skills.

Abram (Echod) gets money by inventing a multiple ﬁngertip operated touchscreen for personal computers.

The destiny of Achas, she realizes, is to transcend the emptiness of science with a new science (cf., the zebraﬁsh). The destiny of Echod, he realizes, is to alter the consciousness/mental models of fundamentalists to see the mansion for the room.

Victor DeLonga is the mathematical adversary who manipulates the department to deny tenure to Abram Zabrisky. He is secretly allied to the Moslem terrorists because both Christians and Moslems fear the development of Abram’s work. Achas completes Echod.

The Bent Note

Religious fundamentalism, genocidal revisionism, and empty science are reversed by a new mathematics of consciousness invented by a Sephardic beauty and a Hasidic apostate.

Mileva Mizrahi as a child is a mathematically precocious Sephardic beauty raised by her father, Milton, in Buenos Aires. Milton survived the Holocaust but Israeli war criminal hunters are seeking him because they mistakenly think he traded collaboration with Nazis in their murder of hundreds of fellow Jews, including his own wife. Mileva grows to become a self-reliant and powerful young woman, and leaves Milton in the care of their trusted housekeeper, Minka, to go and study advanced mathematics with Nicholas Bustamente in Chile. While there she is informed that her father and Minka were murdered. She has found on the world wide web the research of Abram Zabrisky, a Hasidic mathematician in Brooklyn, New York. It seems to her that her interest in pushing science beyond itself towards spirituality is complementary to his mathematical science

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beyond itself towards spirituality is complementary to his mathematical science of consciousness. She commits to her destiny, replaces her name by Achas, and resolves to study with Abram in New York.

In parallel time we meet Abram Zabrisky, as a teenager working in the electronics repair section of 47 Street Electronics in Manhattan. This is a lu- crative business run by Hasidim. He is utterly brilliant, speaking rapidly and always telling about his ideas. He becomes a student in mathematics at the City University of New York Graduate Center. His advanced ideas are politi- cally incorrect, especially to Victor DeLonga, a senior mathematician who is a Christian Fundamentalist. DeLonga perceives that Abram is working towards a means to adjust mental models of fundamentalists to align them with a wider, more inclusive world. Indeed, DeLonga spearheads a clique to deny tenure to Abram. Achas, meanwhile, has become a graduate student too, and the whole mathematics department has lunch at a Japanese restaurant, where Abram declares, “One of you will betray me.”

Abram Zabrisky suﬀers a huge anxiety attack and nervous breakdown. His recovery is ministered by Achas, and he commits to his destiny, replacing his name by Echod. Achas and Echod unite in their determination to advance science and overcome fundamentalism, and create Relience, the fusion of religion and science while visiting the Museum of Mental Models.

Echod’s best friend is Muhammed Cardona, drawn together by their common love of melismatic music - long, wavering, sometimes sobbing notes in the music of Jews and Arabs.

2001 01 17 Climax:

Victor: ‘‘It is your life against my beliefs. Convince me that the bible is not true, and you can go."

Echod: [Shifts into Jesus, fluttering between himself and Jesus] ‘‘I can only show you how much more there is than the bible. You are free to choose whether to stay in one room of the mansion."

2001 01 18 The opening credits are accompanied by the sound of a clock tick for each generation of human beings - that is the time scale.

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Abram and Muhammad meet as teen agers in Brooklyn since they are from proximate neighborhoods. One day as Abram hurries through the streets, avoiding strangers as carefully as he can, carrying technical books, he is beset upon by a small gang of black tough guys. They tear at his clothing, until his shirt is torn off, and we see that this is one well-built Jew. But he is unable single- handedly to fend them off. Suddenly out of nowhere another one appears, and he covers himself as best he can. Before he realizes what is happening, when he peeks up he sees this new fellow, a black young man, peeling off his attackers and he gets up and the two of them obliterate the opposition. Both bloody and panting, they see they are both picking up related technical books.

The Mathematics Department holds a meeting, presided over by the De- partment Head, Victor DeLonga, to determine whether Abram is doing real mathematics.

Milton Misrahi, Nicholas Busamente, and Victor DeLonga happen to attend a mathematics meeting in Buenos Aires when Mileva is about 13 years old. This is where she meets Nicholas. In the background Victor is receiving the applause of his peers for a ﬂawless delivery of an interesting proof.

The touchscreen that Abram invented for gaming becomes the tool Achas and Echod need for their Relience notation.

2001 01 21 LOGLINE A Sephardic beauty and a Hasidic apostate conquer fundamentalism with a new science of consciousness.

Milton teaches the Sh’ma to Mileva, with its esoteric meaning. He repeats it while dying. Minka says it to Mileva over the telephone as the signal that the most feared event has occurred.

Graph of events versus time with “error bars” measuring degree of credibil- ity ranging from “I just know it” to “detailed mathematical proof.”

Nazi Agent [to Milton]: “Your father killed my father. Where’s The List?”

When Mileva departs Milton gives her The List along with the $3,500 in gold coins.

The Cruciﬁction: Victor: ”Prove to me the resurrection never happened!”

Mileva: [age 10] “Whatever that purple vegetable is, I’m not eating it.”

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Achas: Even if the complexity on Earth is the result of realized physical possibilities, that does not mean there are no physical possibilities beyond human capability to apprehend. We may be physical, but there is no known law that the physical can understand all that is physical.

The laws we ﬁnd are the laws that we are able to ﬁnd. The tools we invent to extend our capacity to experience pattern are possible only on the basis of the patterns we can already experience.

The Death of Muhammed: Achas, Abram, Muhammed attend a SantaRia bata drum practice at a loft on the far lower west side (near Canal Street). They are attacked at 3 AM after leaving. The Nazi Agents kill Muhammed. Echod, as Muhammed Ali, kills Milton’s murderer in front of Achas.

Echod: “I loved Muhammed.”

The Israeli Agents show up and capture the remaining Nazi Agents. The Israeli Agents want to use The List, the Nazi Agents want to destroy The List.

Abram is under-appreciated, Echod is a Savior. Theme of The Under- Appreciated Savior.

Milton: ”Achas, listen, if you get a call and you hear the Sh’ma, you must not come here. They will want The List from you. Don’t come back, please, it will be too dangerous.”

Victor: Sitting at his desk and lording it over the Department, his right hand dexterously slips a pen around successive ﬁngers and back.

Humans have a natural tendency to form exclusionary groups. Humans have a natural tendency to dumb down basic ideas.

The Grip of a Mental Model (Ontogeny & Phylogeny p. 168) The Necessity To Have a Mental Model The Conﬂict of Alternative Mental Models

The Impressionability of Children The Acceptance of Easy Answers

Taking someone’s mind back to childhood, just prior to the moment when they are tipped into commitment to the Christian Mental Model, and nudged into the larger mansion.

Commitment to a mental model is like collapse of the wave function.

When Mileva is 4 years of agae, her mother succumbs to Paranoia, Delusions of Nazi Persecution, and is rendered Catatonic. Though Jewish, she has to be cared for by Christian Nuns who quote St. Augustine regarding the Jews. It is

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the loss of her mother, and the nun saying, “Miracles happen, my child,” that compels Achas to create the “miracle” that recovers her mother in the closing scene. Her mother has a Disturbance of Consciousness.

What if in the Zebraﬁsh world it never occurred to anyone to write two dimensionally, because the natural medium of expression has always been based on aero-gel sculptures?

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Four Essences

I. Judaism II. Christianity III. Islam IV. Humanism

Ellis D. Cooper, February 25, 2005

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I. Essence of Judaism

14,000,000 Jews believe the following statements are basically true.

1. There exists exactly one God who is supernatural and eternal. 2. God created human beings in His image, He loves them, they are sacred, and God knows the thoughts and actions of all human beings. 3. Human beings communicate directly with God through prayer. 4. God introduced a covenant with Abraham.1 The covenant stipulated that God would protect His chosen people, the descendants of Abraham, if they would hold Him in awe and obey Him. The physical sign of His chosen people, the Jews, is circumcision of the penis. 5. God gave the Torah with 613 commandments to Moses, including in particular, The Ten Commandments. The Jews fulfill their special covenant with God by lifelong study of the Torah and adherence to its commandments. This is the Covenant of (Mosaic) Law. 6. God promised the land of Israel to the Jews. Jews love God, the Torah, and Israel. 7. God punishes immoral human beings and rewards the moral ones. 8. A King and Deliverer of the Jews – a Messiah – may someday come, but no true Messiah has yet appeared. 9. Human beings may have some possibility of immortality. In general, however, actions – such as obedience to the commandments of God – are more important than such thoughts or beliefs.

This essence is based largely on the web site http://www.jewfaq.org/toc.htm , “Judaism 101”.

II. The Essence of Christianity

2,000,000,000 Christians believe the following statements are basically true.

1 A covenant between two parties is an agreement between the two parties that: if one party fulfills a specified condition, then the other party will fulfill a specified promise. If the condition is not fulfilled, then there is a specified penalty inflicted upon the party that fails to fulfill the condition. If the promise is not fulfilled by the other party, there may or may not be a penalty inflicted upon the party that fails to fulfill the promise. A Divine covenant is a covenant dictated by God making a promise, and a specified set of human beings as the other party that must fulfill a condition. Since there is no possibility that God would fail to fulfill His promise, there is no penalty specified for failure to fulfill the promise.

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1. Beyond the apparent and finite physical universe there exists an infinite supernatural universe occupied by God. 2. God is all powerful, all good and nothing but good. 3. God created the finite physical universe, and in it He created a male and a female human being in his own image, endowing them with the capacity to choose between right and wrong. 4. These first human beings (Adam and Eve) did chose to disobey Him - to do wrong - and all their descendants have inherited the guilt for their Original Sin. Thus, all human beings are intrinsically bad. Nevertheless, God loves all human beings. 5. The supernatural being, God, and the human being, Mary, had a Son named Jesus, who was both God and a human being. 6. God and Jesus are both God, and from them proceeds The Holy Spirit. God the Father, Jesus the Son of God, and The Holy Spirit form The Holy Trinity. 7. The Bible is the Word of God and it reveals that He has a Plan for the Salvation for all human beings, and that Jesus is the sole mediator between human beings and God. 8. Because a human being is born a sinner, a human being is condemned at birth to death. To be saved from death, a human being needs to repent. This means to admit being a sinner, to believe that Jesus sacrificed Himself and died in his or her place, and to have faith that he or she is immortal inasmuch as Jesus was resurrected from the dead, which proves that immortality is possible. If a human being repents then the Holy Spirit enters the human being, and God forgives his or her sins, and grants to him or her immortality. This simple plan for salvation proves that God has mercy and loves all human beings. 9. There was a Divine Covenant of Works between God and Adam, as representative of all human beings forever, in which the condition was that Adam obey God, and the promise was immortality. Adam disobeyed God, and the penalty was physical and spiritual death. 10. There is a Divine Covenant of Grace between God and Jesus, as mediator between God and all human beings designated by God for salvation – the elect – in which the condition is that human beings have faith in Jesus, and the promise is immortality. 11. The Divine Covenant evolved through time, culminating in the Divine Covenant of Grace. The Old Testament is about the Divine Covenant of Works during the “time of law,” which refers to the Jews and their absorption in the Torah. The New Testament is about the Divine Covenant of Grace during the current “time of the gospel.” 12. The Divine Covenant of Grace is superior to and extends the Divine Covenant of Works. First, because Jesus was the Son of God, and Moses a human being was but a servant of God. Second, because the Old Testament emphasizes the external study of the law as given in the Torah only to the Jews, whereas in the New Testament, the Law of God can be inscribed internally on the hearts of human beings – “circumcision of the heart allowing entry of The Holy Spirit” including non-Jews. 13. Jesus will come again – The Second Coming – and judge all human beings, alive and dead: The Last Judgement. All human beings that believe in Jesus, that is, all Christians, will be given immortality in heaven. Jesus will condemn all others, the human beings that reject Him, to hell for eternity. 14. The Bible contains the absolute true word of God and all the information necessary to understand and believe the message of Jesus. In particular, Christian morality is

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centered on the love of God and the love of one’s fellow human beings as God loves them. Specific examples of what “love” entails are to be found in the Bible. 15. Of all religions Christianity is unique because only in Christianity are human beings guranteed salvation, certain of forgiveness of sins, sure of acceptance before God in this world and in the world to come, and know they are immortal, provided they repent, are baptized to allow The Holy Spirit to enter their heart, and accept Jesus as their Savior.

This essence is based on numerous books and web searches.

III. The Essence of Islam

1,200,000,000 Muslims believe the following statements are basically true.

1. There exists a God who is the creator and controller of the universe. 2. Divine revelation affords the clearest and surest testimony of the existence of God. 3. Faith is the acceptance of the truth brought by The Prophet Muhammad, and unbelief is its rejection. 4. Everything created by God evolves towards perfection, and the underlying regularity of the universe in spite of diverse conflicting constraints is evidence of the purpose and wisdom of God. 5. God made human beings to rule the world as his agents. 6. Angels are immaterial beings created from light. Angels have great power and speed. The whole of nature is for the service of human beings. Angels are the controlling powers of nature, so they were made to submit to human beings. Thus, they are intermediaries for Divine control of the universe. 7. Angels convey messages from God to prophets. The prophet not only communicates Divine revelation: he is also a role model for his fellow human beings. 8. There have been many prophets in many nations. The work of each was limited to one nation: they were local prophets. However, the Divine plan is for the uplift and unification of the whole of humanity. Each nation considered itself the chosen people of God, despising the other nations and giving rise, therefore, to jealousy and hatred. This discord contradicted the plan of God for the unification of humanity. God solved this problem by sending the angel Gabriel with the final and perfect Divine revelation, the Qur’an, to The Prophet Muhammad, The Global Prophet. 9. The Prophet Muhammad was a human being who, in a nation mired in vice and degradation, undertook to reform the whole of humanity. 10. The Qur’an deals with all the principles of Islam including the central tenet of the Unity of God. There is neither plurality of gods nor plurality of persons in God. 11. The Unity of God entails that human beings form a single, unified whole about all of whom God cares equally, and advances them through successive conditions leading to their complete perfection. 12. ‘Islam claims a universality to which no other religion can aspire, and lays the foundation of a brotherhood as vast as humanity itself.’ The Qur’an gives prominence

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to the great prophecy of the triumph of Islam, that it would ultimately be triumphant over the religions of the world. 13. ‘A Muslim believes in all the books of all the prophets of God, because every nation had a prophet and every prophet had a book. Divine revelation attained perfection in the Qur’an, illuminating all religious problems once and for all, thereby bringing religion to perfection, and hence The Prophet Muhammad is the last prophet. Only the Qur’an advances the claim that it has brought religion to perfection by eliminating the defects of all earlier Divine revelations.’ 14. Physical passions are necessary for the physical life of a human being, but they impede advancement to a higher life when they are out of control. 15. Human beings must control both nature and their physical passions. For the former, angels were made to submit to human beings, since they are the controlling powers of nature. But greater perfection requires conquest of the inner self, and this is possible only with closer connection to God. It was to make this spiritual perfection possible that Divine revelation was necessary. 16. The process of perfection in a human being is not limited to this life. The day of personal resurrection is the great day when the consequences of all good and evil deeds during physical life are made fully manifest. 17. A faith in a life after death – a higher form of life - is the last of the basic principles of Islam. That every deed must be requited is brought before the mind of the Muslim continually. Every deed shall find its full reward. Therefore this belief is both the greatest impetus towards good and noble, and the greatest restraint upon evil or irresponsible deeds. It makes a human being work with the most selfless of motives, to seek no reward for what is done, to work only for higher and nobler ends relating to the life beyond the grave. Therefore, personal resurrection is necessary to give meaning to the physical life of a human being. 18. Life after death takes two forms: a life in Paradise for those in whom the good exceeds the evil and a life in Hell for those in whom the evil exceeds the good. The ultimate objective of a human being is to meet God and live in His service. However, only in the life after death can this objective be achieved. The physical life of a human being lived in sin debars him or her from the Divine presence. But, purified by fire in Hell, the human being may be made fit for Divine service. Thus, Hell is a remedial chastisement. All those in Hell shall ultimately, when they are fit for a new life, be released from it.

This essence is based on The Religion of Islam, Maulana Huhammad Ali, The Ahmadiyya Anjuman Isha’at at Islam (Lahore), U.S.A., 1990.

IV. The Essence of Humanism

Humanists believe the following statements are basically true.

1. "Humanism believes in a naturalistic metaphysics or attitude toward the universe that considers all forms of the supernatural as myth; and that regards Nature as the totality

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of being and as a constantly changing system of matter and energy which exists independently of any mind or consciousness." 2. "Humanism, drawing especially upon the laws and facts of science. holds that man is an evolutionary product of the Nature of which he is part, that his mind is indivisibly conjoined with the functioning of his brain: and that as an inseparable unity of body and personality he can have no conscious survival after death." 3. "Humanism, having its ultimate faith in man, believes that human beings possess the power or potentiality of solving their own problems, through reliance primarily upon reason and scientific method applied with courage and vision." 4. "Humanism, in opposition to all theories of universal determinism, fatalism, or predestination, believes that human beings, while conditioned by the past, possess genuine freedom of creative choice and action, and are, within certain objective limits, the masters of their own destiny." 5. "Humanism believes in an ethics or morality that grounds all human values in this earthly experiences and relationships and that holds as its highest goal the this- worldly happiness, freedom, and progress -- economic, cultural, and ethical -- of all mankind, irrespective of nation, race, or religion." 6. "Humanism believes that the individual attains the good life by harmoniously combining personal satisfactions and continuous self-development with significant work and other activities that contribute to the welfare of the community." 7. "Humanism believes in the widest possible development of art and the awareness of beauty, including the appreciation of Nature's loveliness and splendor, so that the aesthetic experience may become a pervasive reality in the life of human beings." 8. "Humanism believes in a far-reaching social program that stands for the establishment throughout the world of democracy, peace, and a high standard of living on the foundations of a flourishing economic order, both national and international." 9. "Humanism believes in the complete social implementation of reason and scientific method; and thereby in the use of democratic procedures, including full freedom of expression and civil liberties, throughout all areas of economic, political, and cultural life." 10. "Humanism, in accordance with scientific method, believes in the unending questioning of basic assumptions and convictions, including its own. Humanism is not a new dogma, but is a developing philosophy ever open to experimental testing, newly discovered facts, and more rigorous reasoning."

This essence is quoted from The Philosophy of Humanism, Corliss Lamont, Frederick Ungar Publishing Co., New York, 1965. It is available gratis on the web at http://www.corliss-lamont.org/philos8.htm .

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Ellis D. Cooper . Should the United States Televising the great government revise its electoral system? . Should the United States keep American debate its word to the Palestinian people in spite of the Israel Lobby? A Resolution is a declarative . Is The United States of America statement that responds to the Central breeding a politics of cultural blinders, Question in a controversy. Each week bickering about the Bible, using the on television two teams called military to proselytize, and seeking to Humanists and Abrahamists debate for fulfill the Book of Revelation? one hour a resolution corresponding to a . Should the Unites States Central Question. Here are examples of government change focus from values Central Questions facing The United to problem-solving that reduces States of America in the Twenty-First suffering? Century. . Can science ideally be free of . Is the World Coming to an end, the personal values of scientists? so governments should operate in . Do most Americans vote while anticipation of the return of Jesus? keeping in mind the virgin birth of Jesus, . Is America a Christian Nation, or His death on the Cross for our sins, His a nation including many Christians? physical resurrection, and a personal . Is America a declining Empire, or relationship with Him? can a Faith-Based (social relief, war, . Should The United States of science, education, and medicine) America impose God’s will on other Nation compete against an Asian people, not tolerate dissent, and rely on Empire? inerrant scripture for ideology and . If God exists, why do humans authority? (and animals) suffer, and is the love of . Should The United States of one’s God more important than concern America separate into two nations, The for human (and animal) suffering? Secular United States and The . Are the major perils facing The Religious United States? United States over-reliance on (A) Each team has one Speaker and shrinking oil supplies, (B) radical eight Researchers. Teams face each religion, and (C) borrowed money; or, other on-stage in front of a Studio immoral (a) biological theory and Audience. Televised auditions are held research, such as evolution and stem to insure Studio Audience ideological cell research, (b) medical practice, such balance. Speakers are prominently as abortion. positioned in front of their Researchers. . Is The United States of America Each Researcher has a an exceptional nation, its people God’s Research Computer with World Wide chosen people, a nation whose God- Web access. Giant screens show the given mission is to enlighten, civilize, Researcher screens, both to the Studio bring order, freedom and democracy to Audience and to the Television the rest of the world, only using force as Audience. Each member of the Studio a last resort? Audience has voting buttons. An on- . Can The United States of stage Arbiter ensures that Speakers America publicly admit foreign policy conform to The Rules of Debate. mistakes without offering succor to its Speakers may request Research at any enemies? time. The other team, the Studio and . Is the United States Military Television Audiences see the request. A becoming a Global Oil-Protection web site, synchronized with the Service? television show, also displays all

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Researcher screens. There is a distinguished Panel of Rhetoric Judges who render critical opinions on the nature of each team’s arguments. They note, for example, ad hominem argument, bandwagon effect, begging the question or circular reasoning, fallacy of composition or division, hasty generalization, non sequitur, post hoc fallacy, red herring, self-sealing argument, slippery slope argument, and straw man argument. America Votes to eliminate Researchers. For one hour after each debate, members of the Television Audience call in votes for their favorite team. Technology is used to ensure one-person-one-vote. The losing team must choose a Researcher to go home. The Studio Audience expresses its preference by shouting, applauding, and screaming, and using the voting buttons, but the final decision is made by the losing team. The Debate Winner is the team retaining the most Researchers after all the Central Questions have been addressed. In case of a tie, the Speakers must debate one further question, "What is the Meaning of American Life?" and the Television Audience must vote again. The Great American Debate Prize is one hour of commercial-free, prime television time to express its Twenty-First Century Agenda for America.

Ellis D. Cooper is a retired inventor and mathematician.

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Science on a Napkin: Virtual Drawing is to Drawing as the Telephone is to Speaking

Ellis D. Cooper, Ph.D. December 15, 2007

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1 Executive Summary

A group of physically separated people can work on the same big picture. All this requires is low-cost consumer video and computer technology and Extended Messaging and Presence Protocol (XMPP), i.e., instant messaging. The hardware platform for a new 1 2 2 -dimensional instant messaging algorithm is a digital video camera mounted on a vertical bracket so that it views a computer screen laid ﬂat. Details of a user interface are described.

Assume persons A, B, C are on-line and equipped with virtual drawing hardware: a digital video camera affixed to a vertical camera bracket, computer, and screen laid flat beneath the camera. A protective transparent plastic sheet called the “napkin” is laid on top of the screen. Starting with a blank napkin, suppose A draws a mathematical diagram. Software for camera-A stores it as a bitmap image file, displays it on screen-A, and transmits the file to B and C. At B and C the file is stored and its image is displayed on their screens. At this point all three view the same picture. Suppose C adds an equation to the diagram on his napkin, and pushes the “send” button. Then the C-screen is momentarily blanked while the C-camera photographs just the addition. The result is converted to a bitmap image file, stored, and transmitted, exactly as before. Soft- ware at each site overlays the equation image with the preceding image, and each screen displays the result. At this point, B has nothing added to his napkin and views both diagram and equation, A has the overlay of the equation image added to the screen display of the original diagram, and likewise for C. Nothing is lost if both A and C choose to erase their napkins. For – at this point in time – A sees the diagram superimposed on itself since the overlay is beneath the diagram on the A napkin. Similarly for C except the equation is redundant. The process described may be repeated indefinitely. Every transmission is an incremental addition to the virtual drawing shared by everyone, and each person has an accumulation of the additions saved as bitmap image files, as well as a currently visible overlay image of all additions so far. The user illusion is a virtual napkin on which the people are collaborating.

2 Introduction

Collaborative development environments will be the next “killer app” in software development. ... Just like the goal of the open-source community, where code is freely available to anyone to use, redistribute and make better, the use of open standards in creating a collaborative development environment opens the door to a community of goodwill and support to improve the experience. Grady Booch In the old days when friends and relatives visited, everyone had fun looking at your photograph album and listening to your new records. Food was served, and some people would enjoy talking while others might listen to the radio or, later, watch television

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together. Nowadays that basic personal and communal activity has evolved into “social networking,” and web-sites allow tens of millions of individuals to publicly share photographs, music, and videos. As it says at the Wikipedia web-site: A social network is a social structure made of nodes (which are generally individuals or organizations) that are tied by one or more specific types of interdependency, such as values, visions, idea, financial exchange, friends, kin- ship, dislike, conflict, trade, web links, sexual relations, disease transmission (epidemiology), or airline routes. Parallel to the thick layer of ordinary social life there is a thin layer of scientific social life. Mathematicians and scientists and engineers are for the most part quite sociable folks. There are exceptional loners, but most scientists enjoy productive collaborations. They will travel thousands of miles to attend conferences in their specialty. There they will attend presentations on new research, look at posters generated by up-and-coming students, ask questions, and engage in discussion with the presenter. They will also say, “Let’s have a beer and dinner and talk about this.” Thus, in a bar or restaurant old and new scientific friends relax and talk. More often than not, an idea is best discussed in terms of symbols, a calculation or a diagram. Then we get science on a napkin: not having brought a pad of paper, and anxious to jot down the idea, a scientist grabs a paper napkin to write and draw. Before you know it, people are huddled over that napkin, each contributing a correction or nuance. At some point they sit back and say, “We have to write a paper on this.”

3 The Vision

The idea for virtual drawing emerged in consideration of the needs of sociable mathematicians to collaborate by writing and drawing on a napkin in a bar or restaurant. But a virtual napkin could also be used by business people who attend virtual meetings (“teleconfer- encing”), and by educators virtually joined together with students (“distance learning”). To be sure, if you omit the need to place printed material on the napkin for all to see, then restricting the virtual napkin to a surface on which people just write and draw is functionality available now. Collaborative software on the internet is diverse, including screen sharing/desktop sharing/application sharing, where participants can view anything the presenter is displaying, and can manipulate the presenter’s screen. Thus, people using a mouse or pen on a tablet peripheral may annotate a slide show, word processor document, or spreadsheet. Browser- based commercial products along these lines include Adobe Acro- bat Connect, Central Desktop, DimDim, Glance, GoToMeeting, Mi- crosoft Live Meeting, Ready Talk, WebEx MeetMeNow, and Yugma. Web-sites such as Graﬃti at http://graﬃti.playdo.com/, Imagina- tion at work at http://www.imaginationcubed.com/LaunchPage, and Thinkature at http://thinkature.com/ invite people to use mice

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to write and draw on the “same” surface in a browser window. Fur- thermore, there is a patented System and method for distributed collaborative drawing, U.S. Patent Number 5, 940, 082: “The present invention pertains to a system and method for managing a real-time distributed collaborative drawing activity. A community of collaborators associated with client computers are connected via a communications link with a server computer. Each collaborator contributes to the real-time design of a shared drawing that is displayed in each collaborator’s virtual whiteboard. Contri- butions to the shared drawing as well as communications between the collaborators are transmitted to the server computer. The server computer broadcasts the communication to each of the collaborators engaged in the design activity. Each user has a set of drawing tools that are used to generate a variety of drawing strokes that edit the shared drawing. A curve generating procedure is used to capture the manner in which the user draws the stroke in order to emulate a selected drawing tool. A compact representation of the user’s stroke is generated and transmitted to each collaborator.” The virtual drawing idea is largely consistent with this patent, except that the key element “curve generating procedure” is replaced by a digital video camera. A serious advantage of the above invention is the “compact representation” which substantially minimizes network transmission time of drawing strokes. Sending a ﬁle with a snap (bitmap image) would not be as eﬃcient. However, in compensation we get the increased dimensionality of a virtual drawing (see below).

Some mathematicians have contrived other means to approximate a virtual whiteboard. For example, one mathematician writes mathematical text in LATEX, compiles it to notations and diagrams on the screen, and uses a bitmap image capture utility to take a snap that is then pasted on a virtual whiteboard. A more integrated attempt at helping mathematicians to collaborate is the MathChat project, described by David Chiu: “MathChat is a protocol to support web-based chat sessions to simulate a classroom-type environment enabled for mathematics education. With MeML support, Math- Chat allows mathematics computation and display, which empowers mathematics educators to hold lectures and discussions outside the physical classroom. An online educational framework oﬀers such advantages as more ﬂexibility in scheduling and less apprehension to ”shy” students.” The virtual drawing idea replaces “MeML” – or other schemes for encoding mathematical notation – by old-fashioned handwritten notations and diagrams. This is important because one of the hall- marks of science is the creation of entirely new notations and diagram conventions. For many people the pen is mightier than the

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mouse.

Proliferation of such inventions and ideas, combined with my personal experience, is evidence that people do want to draw as well as write during collaboration. Obviously, mice and drawing tablets are inadequate for implementing the larger vision of virtual drawing that encompasses sharing of printed material or specimens on top of the virtual drawing. A solution to that problem uses a camera supported by a vertical camera bracket to point downward upon a surface. A still camera affixed to a vertical camera bracket is called a document camera. If the camera is a (sufficiently high resolution) digital camera then the resulting photograph stored in a bitmap image file is equivalent to the output of a scanner. If instead of an analog (or digital) still camera one uses a digital video camera then the video signal from the camera may appear in real-time as an image projected on a wall-size screen for an audience. This arrangement is called a visual presenter, and there are many varieties of commercial products of this nature from, e.g., Elmo and AverMedia. If the video signal is received by a computer, then software interposed between the camera and the projector offers more options. Material viewed by the camera may be presented in one window, while simulation or other educational software runs in a nearby window. Finally – and this is the point – the combination of vertical camera bracket, digital video camera, and computer may be used to implement a virtual drawing upon which mathematicians, engineers, scientists, educators and designers of all kinds, may write equations, draw diagrams, run educational applets, and display printed text and specimens.

In this paper I introduce the concept of virtual napkin and describe a plausible implementation. A virtual object is an illusion: an appearance that is in reality induced by invisible actions behind the scene. More precisely, a virtual object is that which exists only by virtue of a computer program executing its instructions. A virtual object “supervenes” on a program.

A virtual napkin provides computer and internet technology to members of a group – however widely distributed they may be in space – the means to write and draw on the same “napkin”: whatever one writes or draws is seen by all, and “on the same page” gets a new meaning. Granted, they are not rubbing shoulders as in a bar or restaurant, but they can at least share ideas in real time. Also, travel expenses are nil. And the illusion is improved by more technology. For example, technology for real time audio and face-to-face video and text exchanges already exists: we call these “telephone”, “webcam” and “instant messaging” (IM).

4 Person and Presence

I have tried a variety of words to denote a person who uses a virtual napkin for collaboration. From the generic “user” or “participant” or “collaborator” to the more speciﬁc “mathematician” or “scien-

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tist” or “artist”, I have even invented words such as “virtist” or even worse, a “drawer” being someone who draws. At one point I was very proud of saying that “drawing people together” is a triple pun: you could have a drawing of people together, or “drawing people” get together, or people are drawn to one another. These interpretations correspond to alternative uses of parentheses: “drawing (people together)”, “(drawing people) together”, and “(drawing people together)”. Enough. Let us agree that in this paper by person I mean someone who is using a virtual napkin for collaboration. Also, deﬁne a virtual drawing to be what the people see when they look at what is on the virtual napkin. The virtual drawing is the big picture.

Presence information is taken for granted when people are gathered at a real napkin in, say, a restaurant. There, visual and audible cues about who is involved, their attention and mood are constantly available. The concept of “presence” is as central to the virtual napkin as it is to IM: although a virtual napkin is always available, not every person is present at a given time. Thus, each person may look on their computer screen for information about the presence of other people. Common IM indicators are “Online”, “Free for chat”, “Low attention, busy”, “Online but elsewhere”, “away”, “Extended away”, “Will return on ...”, “Do not disturb”, etc. Ad- ditional presence attributes may even include information about a person’s health or mood. The virtual napkin endeavors to provide an illusion of presence with IM-style indicators.

5 Using a Virtual Napkin

A person working on a virtual napkin is at a computer with the usual keyboard and mouse, and with an added peripheral, a writing surface called napkin for symbols, calculations, and diagrams. One or more virtual drawings can be available one at a time on the napkin, each with its own set of other people who are collaborating. Also a manuscript or an open book or a specimen may be placed on a napkin. In any case, all the other people collaborating with the person see the virtual drawing. Additional peripherals might be attached to the computer to enhance collaboration, such as a webcam directed at the person, or a headset with earpieces and microphone. At the least, however, the computer screen should include a text messaging window, so that the collaborating people can enjoy text- based IM along with writing and drawing on a virtual napkin.

Physically, the napkin is a clear plastic sheet covering a computer screen (e.g., an LCD monitor) laid ﬂat. Each person may put a note or notes on their napkin. On the screen beneath the napkin the person also sees a patch of the virtual drawing. A note is handwritten on the napkin and as such may be any combination of handwriting and drawing. A person may also place an object, or more than one object, such as printed matter or a biological specimen, on the napkin. Thus, screen and napkin must be in closely

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spaced parallel planes to avoid visual parallax problems. Also, a person may drag or paste an image such as a bitmap image, e.g., of LATEX output, an animated .GIF, or even an interactive applet, or several of these, on the patch of a picture. The camera view, therefore, is a layered composite of patch, images, notes, and objects, and we may write

screen0 = patch0 + images0 (1)

napkin0 = notes0 + objects0 (2)

view0 = screen0 + napkin0 (3)

Let us deﬁne the images, notes, and objects to be the current addition by the person:

addition0 = images0 + napkin0. (4) A snap is the new image obtained by (momentarily) blanking the patch in view (patch0 = patch(view0)) and photographing just the addition in view (addition0 = addition(view0)). In other words, a snap is initiated by pushing a button, and is comprised only of the material to be added by the person:

snap0 ← photo(addition0). (5) A person may edit a snap. Edit operations on the new image include scaling, rotating, and cropping:

snap1 ← edit(snap0), (6) where edit stands for a ﬁnite sequence of edit operations. When the snap button is pressed the person’s view is updated by the overlay operation (see more detailed deﬁnition below) of the new snap upon the patch in view:

patch1 ← patch(view0) OR snap0. (7) It is important to observe that at this point, the view is redundant:

view1 = (patch1 + images0) + (notes0 + objects0), (8) because the images, notes, and objects of the addition are still where the person put them, but the patch now includes an image of them. No information is lost if at this point the person removes the additional elements (e.g., by wiping the napkin), so that we have

view1 = patch1. (9) The user interface separates the viewing options of the big picture, such as zooming in or out, from the scaling operations on a snap. The choice is implemented with an indicating toggle button tied to, for example, the mousewheel. Motion of the picture with respect to the addition may be perceived either as sliding the picture

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on the screen beneath the fixed addition, or as gliding of the addition across a fixed picture. The choice between these perceptions is controlled by how the “grab hand” works. If the addition is selected and the grab hand moves to the right, for example, then the perception is that the addition is grabbed and glides to the right across the picture. If the picture is selected then the picture is grabbed and slides to the right beneath the addition. Another way to place a snap at a location in the picture is to go to that location and push a button to bring over the latest snap. Or, one may store the coordinates of the location (optionally with an assigned identifier), and at some later time select a saved snap and a saved location for placing the snap there.

Once editing of the new addition is complete, the person may push the send button. The file that is sent consists of a snap, a unique identifier assigned to the snap by the person, the name (or other unique identifier) of the person, the date and time of the snap, the location of the view in picture coordinates, and the sequence of cropping, rotating and scaling edit operations performed on the snap:

file1 = (snap1, identifier1, person, timestamp, location, edit). (10) (To re-edit the snap later, the originator of the snap need send only the revised sequence of edit operations, not the snap itself.) All persons working on the picture, including the originator of the file, receive the same file. If person A pushes the send button then each other person X working on the same picture receives the file:

pictureX ← (pictureX , file1). (11)

For each person X, pictureX is stored in a separate directory assigned to the picture, and contains all the files generated by all the people working on the picture. Whatever the patch in view, the image presented is calculated using all the files at the picture coordinates of the patch. This calculation, called the overlay operation, performs the edit operations in the files on their snaps, renders transparent the background of each snap, and then OR’s the snaps together, sorted upward in order of increasing timestamp of creation. (The overlay operation is performed just-in-time and the result cached so that gliding across the picture is as smooth as possible.) Everyone now sees (some patch of) the same picture.

A person may elect to prepare more than one snap before sending them in a batch. In any case, when the file is received by a person’s system, an indicator turns on to alert the person that a file has been received. Attributes of the file may be displayed according to preferences set by the receiving person. A person who views the new snap may acknowledge receipt by pushing a button, or in a text message.

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The user interface provides a pop-up chart displaying the viewed- versus-pending status of all snaps of all workers on the picture. De- tails include a list of all incoming snaps not yet viewed, and a list of all snaps not yet sent.

Overall, the user interface has the observation window, a control panel window, and a text IM window. The control panel includes controls and indicators, such as the snap button, the zoom-versus- scaling indicating toggle button, the send button, the received-ﬁle indicator, and acknowledge button already mentioned. The control panel title bar indicates the name of the current virtual drawing, and has the usual window minimize/maximize/close controls. There is a control menu with the usual new/load/save/save as/close choices for virtual drawings, and where applicable, another menu for operating voice-over-IP communication and for operating webcam face-to-face views in a separate window.

1 6 22-Dimensional Instant Messaging Instead of one little napkin, imagine that the napkin is indeﬁnitely large and new images, notes, and objects can be added to any open area. At any given time a person is not seeing the entire picture: one sees instead just a portion of it through an observation window. One may re-position one’s addition to the picture by sliding it along the picture either to align with some other person’s work, or to isolate it at a new region. If text-messaging is one-dimensional and virtual reality is three-dimensional, we may say a virtual drawing is 1 a layered or “2 2 -dimensional” structure.

Implicit in the idea of an infinite virtual drawing is the requirement that the user interface provide means to move one’s observation window across the virtual drawing, or even jump to a region (specified by an identifier associated to coordinates). And there is a button to move one’s additional material to the region one is currently viewing. An indicator alerts people that new material has arrived at that region. After all, at any given time each person may be viewing a different patch of the virtual drawing. However, if a person B does view the recently arrived material sent by A, a confirmation signal identifying B – and an identifier of the patch viewed – is displayed in the user interface of A. These are some of 1 the implications of 2 2 -dimensional instant messaging.

The view of the napkin seen by a person is layered: at bottom is a patch of the big picture, next are one or more images, then there is writing and drawing, and on top possibly one or more physical objects. But the big picture is itself layered in another sense. Each collaborator has contributed a set of one or more snaps at various locations, so we may consider the picture to be composed of personal layers. Moreover, the snaps in a personal layer are “mutable” in the sense that a person may choose to select and edit a snap and then send the edit operations (or even replace a snap by a new one with

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the same name and location). Of course all people working on the picture need to be informed of the change, or perhaps discuss the change before it is sent.

Beyond the personal layers of the big picture there may be a group layer on which may be drawn material that spans across multiple snaps. Thus, the user interface provides a tool box in the control panel for global drawing (arrows, circles, rectangles, etc.). Also on the group layer people may place “hot buttons” that jump immediately to picture locations. A person may place a virtual laser pointer dot of a speciﬁed color at a location named so that it can be referred to in a text message. A dropdown listbox of laser point names allows select-and-jump to any laser dot.

6.1 Calibration for Registration One problem is that commercial camcorders are not intended for close photography: distortion is introduced. Fortunately, there are standard algorithms that can compensate for “ballooning” of the image, for example. A more difficult problem is registration. What guarantees that an annotation by C of a diagram by A will appear at the correct location on screen-A? After all, different LCD monitors may have different pixel shapes or monitor adjustments. A solution to this registration problem is that to start a new virtual drawing, each person is required to place on their napkin two dots at opposite diagonal corners, say upper left and lower right. These calibration dot images are transmitted with identifiers to all the persons who intend to work on the picture, and software at each site uses the coordinates of the received dots to calculate correction transformations associated with each other site. That is, calibration merges the dots for each person. For a person the pixel color surrounding the locations of the two registration dots should be the same. This color will be considered the default background color of that person’s napkin, and thus is the color that is automatically made transparent when that person’s file is an input for the overlay operation at receiving computers.

6.2 Foundation for Implementation In this paper features of a system for implementing virtual drawing are described in the present tense. This should not be interpreted as implying that a complete system actually exists. However, a working implementation of some of these features is described here. There is a program for virtual drawing in C] using MicroSoft Visual Studio.NET 2005. Video is handled by the DirectShow.NET library, and bitmap image .JPG transmission is in care of the XMPP Jabber.NET library. There is an “instant messenger” style main window, and show/hide toggles of the napkin and incoming bitmap image windows. There is a server-side contact list, and settings for saving ﬁles on the client or server side. There are pan/zoom controls for the incoming bitmap image operated either by drawing a box on the image or by using “shift-mousewheel up/down.” The

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software communicates with a “jabber” server. On this foundation a complete implementation of virtual drawing as described in this paper would probably take less than six weeks of full-time work by an experienced Visual Studio C] programmer. How to adapt the virtual drawing algorithm to cell phones and their cameras is an open problem.

A web-site dedicated to educators, students, mathematicians, engineers, scientists, artists, architects – designers of all kinds – will be helpful for maintaining the database of people working on different virtual drawings. No virtual drawing files would be stored on the server: all such files are stored only on the local computers of the people registered for the virtual drawing. Likewise, the IM server is only a conduit and never a store of virtual drawing files. A virtual drawing viewer will be a free download available to any visitor who would then use it to view – but not edit – published virtual drawings. Construction of such a web-site would be a job for an experienced web-site builder.

7 Conclusion

There is a signiﬁcant need for collaboration on pictures in the worlds of education, research and design. A virtual drawing is globally “the big picture” assembled locally by a number of people. Each person contributes additions at locations in the picture. Finally, high speed and high resolution computer and video technology is available at such low cost that there is a clear and present opportunity to implement “science on a napkin.”

8 Acknowledgment

Key ideas for virtual drawing arose during discussions with Dr. Flo- rian Lengyel and with Dr. Robert Reasenberg.

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RE “Being good for God’sake” (Page A15, Nov. 26) Jeff Jacoby, apoplectically reacting to a purportedly anti-religion comedic film, writes “what so many secularists fail to understand about religion” is that “at its best” religion elevates people by “prompting them to do good” out of obligation to act “in harmony with God’s transcendent” – and one would hope intelligent – “design.” What he fails to understand is that religion based on human interpretation of texts written by humans claiming to know God’s design for people is bound to elevate obligation to God over relieving others from suffering. This inversion of priorities inadvertently opens the way for religious extremists to inflict suffering in the name of God. At its worst, religion brings “joy” to the families proud of their “ennobled” martyrs. Jacoby would be wiser to acknowledge a secular definition of evil as the unreasonable infliction of involuntary suffering. So, doing good is diametrically opposed to evil, and in these terms good and evil have nothing to do with religion, or God.

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Dear (not really) Christianity and Islam,

With honest high regard for your considerable accomplishments, it still remains that your insufferable bigotry is testamentally institutionalized.

Judaism rejected Christianity; Judaism and Christianity rejected Islam. So twicefold has Judaism suffered the scourge of “true believers” belonging to younger religions.

An infinitesimal percentage of your testaments infinitely demonizes Judaism for not accepting the claims of your beliefs. Jewish and Christian dhimmi were the “niggers” belittled and bowed by Muslims for too many centuries. Your disproportionate reaction to mere rejection has been dumbed down for mass consumption by your most articulate believers, and militarily amplified up to rape and murder, massacres, pogroms, and The Holocaust.

By systematically dispersing Jews across the world, Christians and Muslims artificially selected them for cultural assimilation and cunning survival skills in science and finance. Ironically, the cauldron of your hatred fired by a few burning words of your testaments has weeded out the weakest, leaving to survive and propagate better and smarter Jews.

Another irony is that only those not holding extreme beliefs can initiate the absolutely necessary reforms of Judaism, Christianity, and Islam. Moderates are powerless against extremists because only extreme acts can thwart extreme beliefs.

Jews need to forego claims of being “chosen” by god; Christians need to forego once and for all claims to a covenant with god that supersedes the supposed covenant of god and the Jews; Muslims need to forego claims of being sole owners of ultimate truth. None of these seemingly core beliefs, nor implicit hatred of “the other” – let alone specific rituals and conventional practices – is absolutely crucial to a fully rewarding spiritual life.

What is suffering? Suffering is the interference with purpose, and the purpose of life is to survive. If freedom means anything, it means freedom from suffering. Evil is the unreasonable deliberate infliction of involuntary suffering. Islam and Christianity invert ultimate priorities by raising their beliefs above freedom.

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As a chain is only as strong as its weakest link; as the road is only as safe as the worst driver; so any religion must be judged not by the “serenity and peace” it claims to foster, but by the suffering inflicted in its name by some believers.

Hatred is nurtured, not natural. You must carefully purge explicit and implicit hatred of Jews – and all other non-believers – from your testaments, and likewise thoroughly scour your schoolbooks. No child should be raised to hate.

Israel “giving up land for peace” is meaningless unless peace guarantees removal of hatred of Jews from the press and the schools of Islam. A “two-state solution” is pointless if the children of either are raised to hate the other.

Enjoy the vast lands and resources already under your dominion, and capitalize on intellectual, cultural, and commercial trade with Judaism and the rest of the world, neither of which will ever bow to your ill-conceived and ultimately evil claims of uniqueness and supremacy.

Yours, not really, Your most obedient servant, never again, Respectfully, but only if reciprocally, But Sincerely,

A Jew

REFERENCES

The Religion of Islam, Maulana Muhammad Ali, The Ahmadiyya Anjuman Isha’at Islam, Lahore, 1990. Constantine’s Sword, The Church and the Jews, James Carroll, Houghton Mifflin Company, Boston, 2001. Living Letters of the Law, Ideas of the Jew in Medieval Christianity, Jeremy Cohen, University of California Press,1999. Christian Antisemitism, A History of Hate, William Nicholls, Jason Aronson Inc., Northvale, New Jersey, Jerusalem, 1995. The Jewish Mind, Raphael Patai, Wayne State University Press, 1977. The Arab Mind, Raphael Patai, Charles Scribner’s Sons, New York, 1983. From Time Immemorial, The Origins of the Arab-Jewish Conflict over Palestine, Joan Peters, JKAP Publications, U.S.A., 1988.

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QURAN (4:47) There are some among the Jews who pervert words from their proper places. And they say, ‘We hear and we disobey,’ and ‘hear thou without being heard,’ and ‘Ra‘ina,’ screening with their tongues what is in their minds and seeking to injure the Faith. And if they had said, ‘We hear and we obey,’ and ‘hear thou,’ and ‘Unzurna,’ it would have been better for them and more upright. But Allah has cursed them for their disbelief; so they believe but little.

(5:42) O Messenger! let not those grieve thee who hastily fall into disbelief — those who say with their mouths, ‘We believe,’ but their hearts believe not. And among the Jews too are those who would fondly listen to any lie — who listen for conveying it to another people who have not come to thee. They pervert words after their being put in their right places, and say, ‘If you are given this, then accept it, but if you are not given this, then beware!’ And as for him whom Allah desires to try, thou shalt not avail him aught against Allah. These are they whose hearts Allah has not been pleased to purify; they shall have disgrace in this world, and in the Hereafter they shall have a severe punishment.

(5:52) O ye who believe! take not the Jews and the Christians for friends. They are friends one to another. And whoso among you takes them for friends is indeed one of them. Verily, Allah guides not the unjust people.

(5:83) Thou shalt certainly find the Jews and those who associate partners with God to be the most vehement of men in enmity against the believers. And thou shalt assuredly find those who say, ‘We are Christians,’ to be the nearest of them in love to the believers. That is because amongst them are savants and monks and because they are not proud.

(9:30) And the Jews say, Ezra is the son of Allah, and the Christians say, the Messiah is the son of Allah; that is what they say with their mouths. They imitate the saying of those who disbelieved before them. Allah’s curse be on them! How are they turned away!

(62:7) Say, ‘O ye who are Jews, if you think you are the friends of Allah to the exclusion of all other people, then wish for Death, if indeed you are truthful.

NEW TESTAMENT Matthew 27:20 to 23: "The chief priests and the elders persuaded the crowd to ask for Barabbas [to be released] and to have Jesus executed...'What shall I do, then, with Jesus who is called Christ?', Pilate asked. They all answered 'Crucify him.'...they shouted all the louder. 'Crucify him'...Pilate...took water and washed his hands in front of the crowd. 'I am innocent of this man's blood,' he said. 'It is your responsibility!' All the people answered, 'Let his blood be on us and on our children!' "

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Me, And Benoit

2007 10 13 Re getting pictures hung more perfectly: BMAN “I would tell my students, don’t try to get it so perfect if you need to meet a deadline – fine tuning is for later.”

Re Hopperesque painting to be in bedroom maybe opposite wall at foot of bed: ECOO “Even if you are lying in bed for a few minutes in the morning, is this what you like to see?” BMAN “That is the right question.”

Re my choice of tea, as offered by AMAN: BMAN “I prefer Russian tea, it is stronger. They rot the leaves a bit.” AMAN “Sugar? Milk?” ECOO “No, thank you. For the same reason that if I did drink coffee, I would not add milk. I like pure things.” BMAN “Yes.”

Architect of apartment building: BMAN “Moshe Safdie. Not top of the line.”

Re rug hung on wall at head of bed: BMAN “It is a family heirloom. From the Caucasus. Very, very old. As you see, the design is more complicated than that rug on the floor.”

BMAN was seated on the couch at one end of the living room, facing the wall on which I was hanging a “triptych” of IBM fractal posters. He could see the Longfellow Bridge and observed that it is quite rusty underneath. I said that it is surprising they had not used that orange-colored rust preventive paint. Then he related that a nephew of AMAN had become a hero in New York because he undertook to solve the problem of rust – “that four letter word,” BMAN said – by paint. After a battle to get it done, the nephew had New York covered against rusting infrastructure.

Among books and papers BMAN encouraged me to take away are not so old issues of the New York Review of Books. One time I notice he had placed an issue upon my rolling cart cum toolkit. He said the article about Gertrude Bell is pretty interesting, and said she was instrumental in the creation of Iraq. I said that I am pretty interested in that kind of thing, and rattled off “Sykes-Picot, 1928,” to show how smart I am. It was 1920, by the way. I said Sykes was the British counterpart, and BMAN said, yes, she worked with him and it was her idea to mix the Shia with Sunnis and Kurds.

Questions for BMAN: About My Work

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1. Isn’t distance drawing a potentially terrific contribution to mathematical collaboration? 2. Would you care to contribute to selling the T_Rig by recording a “lecture” and having the movie appear on YouTube (e.g., explain “cartoon”)? About His Work (M1997E?) 1. If one had a record of numbers of cars moving past a certain point of a highway near a city (e.g., 128 north of Boston), the data would have a periodicity only explainable exogenously. Why are stock prices any different? 2. Is there a glossary of all of your technical terms (cross-linked, and keyed to publications)? 3. Relate your work to Wolfram re algorithms and philosophy of looking at their outputs. 4. Possible multifractal application to Gould-Eldridge punctuated equilibria? 5. What about infinities in QED? 6. What is meant by generalized statistical thermodynamics?

My Fantasies About Relating to Benoit 1. Friendship, humor, companionship NOOOO 2. Scientific collaboration NOOOO 3. Intellectual conversation, e.g., “virtuality” NOOO 4. Technical tutoring me NOOO 5. Imprimatur, encomium, endorsement of T_Rig upon demo NOO 6. Write glowing on distance drawing NOOOO 7. Computer programming for research ($80/HR) NOOOO 8. LaTeX composition for BMAN book ($75/HR) ? 9. Web-site management ($60/HR) ? 10. Administrative Assistant/Memoir typist YES 11. Hanging of art in apartment DONE 12. Mover of boxes of books and files DONE

More than once AMAN would say, “You are an angel to us,” presumably because they did not know before me how they were going to get so many things done. The basic thought BMAN had was to repeat what he had done before, say, in Scarsdale, where he was able to hire local high school boys to move stuff.

ECOO “Do you know about Ralph Abraham?” BMAN “Yes, a maverick. At Santa Cruz, yes?” ECOO “Right.”

BMAN related how John Backus, inventor of FORTRAN, was a maverick graduate of the Columbia University School of General Studies (I said, “So am I.”), had been a good friend, and died last year (2006). BMAN said Backus was brilliant, did his great work as a young man (26?) in a short amount of time (6 months?), and later in life devoted himself to helping others (how? Offering aid to disadvantaged children?). In BMAN's home office it was necessary for me to raise his work desk by rotating four bricks

Page 72 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 3/66 supporting the legs by 90 degrees (ECOO “So, we multiply the bricks by i,” BMAN “Or, by –i”). Then BMAN told me that was the desk on which Backus invented FORTRAN, and also the desk at which BMAN created fractals.

BMAN spoke of the inevitable decline of functionality during approach to death. He mentioned how consultation with others led him to the plan of ending up with nothing to do but re-read favorite books.

At one point in speaking of his two sons (one now 50 years old) that when he and AMAN look at photographs of the boys as children they have trouble telling the two apart.

I asked him if he knew Eilenberg. He said he was a friend of a friend. BMAN is anti- Bourbaki, but when I showed him Notices of the AMS with reviews of books about Bourbaki he said he would have to get a copy.

A theme repeated by BMAN is that he had experienced so much rejection (for forty years (?)) that he is completely hardened against criticism.

When I was emptying his filing cabinets in the Batelle office (16 boxes were filled from them), I came across a box holding papers that was formerly a carton for a bottle of Armagnac. That reminded me of how Gurdjieff favored Armagnac when he was recovering from gunshot wounds suffered during his travels in Russia (?). Also in the files was a very clear, well-preserved group photograph of bearded men, with the name Szolem Mandelbrot, BMAN’s important uncle, in the caption. There are many (duplicates of) reprints among those files, including Fama. BMAN said Fama had been his student, but broke with him. He asked me if I knew who Fama is, I said, I have a book by him. “On what?” I said, “Portfolio theory.” BMAN nodded.

In his home office BMAN has two nicely made wooden table-top leaves. He uses them in various arrangements to support levels of work. He said that one time he was in a shop where there was a whole pile of these well-made objects. He asked how much would two of them cost. The proprietor said, “$100, but you can have the whole pile for $20.” BMAN bought them all.

In the hallway to his apartment while pushing a full load of his boxes of books, I asked, “What is the most you ever paid for a book?” (I had in mind that I have spent around $150 for a book more than once.) He said he did not know.

He gave me a whole set of old calendars with fractal illustrations.

We agreed on the estimate that there were about 170 boxes to move from the Batelle office. One set for the new IBM office, one set for the apartment (and its sub-basement), and one set for the Library of Congress.

At one point he said that AMAN does not cook, but that “I knew that when we married.”

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While hanging art in the apartment I witnessed testy (but loving) exchanges between AMAN and BMAN. He is quite precise and sensitive to symmetry, while she is more like, anything goes.

BMAN was relating how he and AMAN made several trans-Atlantic crossings by ship before airplanes became the normal travel method. He said it took five days. I said, so you got some work done? He said they were throwing up all the time.

When I was installing the bookshelves at one point I had them arranged on the floor and was eyeballing along what would be the bottom edge of five supports to see if they were collinear. So I said, “Now I get a chance to use a postulate of Euclid.” BMAN launched into a story about how there was a theory that “Euclid” never existed. Sort of an ancient Bourbaki. That, he said, in fact, there is no biographical information about Euclid, and text analysis indicates that parts of it showed clear evidence of multiple authorship. He said that on these shelves he would be displaying “with shameless pride” all the books by him, and others, about fractals. There are hundreds.

When I first met BMAN – at the insistence of Carolyn, who dragged me away from a fairly interesting conversation with a Sun Systems technologist, “You have to come with me. Now!” – he related how he had an immediately forthcoming need for help in moving boxes. He also spoke of a need for an assistant in connection with his memoirs, and also another mathematical work. I said, immediately, “I can do these things for you.” Here are my motivations for saying that:

1. My mother’s business (ceramic art supplies for hobbyists) brought her into contact with a wide variety of people. From time to time she even met engineers and the like. Whenever she encountered such a person she would encourage them to meet me. So, for example, one very generous man spent some months attempting to teach me calculus. But one time she said that I could get to meet with a relative of Niels Bohr (Brother? Son? Nephew?). I quickly declined, and she wanted to know why. I said, “Because, I don’t have anything to say.” Thus, now, at the age of 65, I shall not repeat that mistake. 2. I do have a generous streak in my soul, I could tell BMAN needed help. 3. Since he was moving to IBM, I saw this as an opportunity to get my foot in the door, who knows, I thought, how that might be a good thing, say, for the T_Rig distance drawing idea. 4. I am curious about genius, and have speculated about correctly defining it. Certainly, I am not one, but this might be some kind of chance to measure myself against it. 5. Gee whiz, maybe I could even be interesting to BMAN. 6. A much deeper motivation is the recognition of an opportunity to plumb depths of any remaining intellectual insecurities I might have. Who better to measure myself against, than someone without them?

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BMAN told a story about some people who intended to stay overnight in a certain castle. But they left early: “The art is too scary.”

BMAN collected works by a woman Russian artist, Karaskaya. He emphasized the strength of her personality. It seems that she was a housewife, but did not know what to do when her husband died. She had an opportunity to buy a cartload of scrap leather, rope, and other junk so for $20 she had her son get the stuff. She proceeded to fabricate these intricate, messy, heavy wall hanging sculptures. I mounted two of them to walls in the BMAN apartment. He said that in those years Picasso was an A-list artist, and Karaskaya was B-list. Her popularity has waned, though. The second hanging involves a horizontal wooden support piece near the top. He said, “It better not look like a crucifixion.” I said, “God forbid.” Or, at least, I wish I had said that. After hanging this unwieldy piece, I said, “That was a bit of work.” He said, “Physically and mentally.”

I typed up corrections to about 40 pages of his memoirs, sort of as a “puzzle,” he said. This was to test my ability to decipher his handwriting, idiosyncratic copyediting symbols, and re-organization symbolic notations. On examining my work he said, “Acceptable.” Not good.

AMAN said, “Do you want to hang the rug first, or the three framed works in the living room?” I said, “Let’s hang the rug. As a rule I do the hard thing first, because I don’t like procrastination.”

AMAN offered lunch with her and BMAN. I declined, noting that BMAN does not take lunch. She said, “This would be an exception.” I said, “Let us take a rain-check on that.” On another occasion I declined her offer to drive me to the train station. In the back of my mind I am thinking, “She knows that it could not be just for pay that I am doing this rather strenuous physical stuff. She feels a kind of guilt. She wants to make herself feel better by offering lunch. No. I will not accept this, I will hold out for what I really want, which is intellectual recognition. Not lunch.”

2007 10 23 BMAN offered to give me books and other printed material that is no longer of use to him. He mentioned something about New York and book reviews, and I said, “Since I left New York I no longer get The New York Times and its book reviews.” He said, “No, I said the New York Review of Books.” I felt ashamed of myself. When he asked what sort of books I might be interested in. I said, “Classical thermodynamics, statistical thermodynamics, quantum statistical thermodynamics.” Then I felt better.

BMAN said, “I was in a Nobel prizewinner’s house near Harvard. His trophy certificates were on the walls next to the washing machine room. The Nobel prize certificate was next to the downstairs bathroom.” I said, “My mother had a trophy room.” He said nothing. But AMAN said, “What did she compete in?” I said, “Israeli Maccabiah 1932.” “What events?” “Track and field. Broad jump, high jump, hurdles.” I am thinking of offering her a copy of SCOO 4 page leaflet about “Syd Koff” and life.

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2007 10 24 AMAN held a circular work of art made of stained glass wedges of different colors. She went to a wall area perpendicular to and at the end of the broad windows facing the Charles River, and held the art up against the wall. “What do you think?” she asked. It so happens that wall area has a small number of mechanical plates relating to functions of the apartment. I said, “No, that would cheapen the piece.”

BMAN handed me another issue of Daedulus, and a New York Review of Books. Putting them into my backpack, I pulled out a book on thermodynamics that he had also given to me, and I said it is my favorite book right now. But I also showed my copy of Thermodynamics by Enrico Fermi, which I said I love. He said, “You know, there is a mistake in that book. Fermi has an error in a formula.” He went on to tell the story of Don Glaser, who found the error in connection with the theory of bubbles. He spoke of Glaser in a bar showing off the bubbles in a glass of beer. This was at a time when Fermi was thinking about how a bubble chamber should work. BMAN said that Glaser was a superb instrument maker, and had been criticized for not publishing enough. Nevertheless, Glaser did get a Nobel prize. BMAN said that Glaser was the most relaxed Nobelist he knew. Then he said, “Feynman was pretty relaxed too.”

“The 1960 Nobel Prize in Physics was awarded to Donald Glaser for his invention of the bubble chamber. "Glaser first conceived of the bubble chamber in 1952, at the age of 25, while a faculty member at the University of Michigan. According to scientific lore, Glaser was enjoying a cold beer when he observed the stream of bubbles in his brew. It was a moment of saloon science that inspired a tool second only in importance to the cyclotron for atomic physicists.” http://www.osti.gov/accomplishments/glaser.html

I probably moved on the order of 200 boxes of his books and files. Dozens of boxes of files. In his new office I said to BMAN, after unpacking several boxes, “My entire scientific output could probably fit into two of these boxes.” He said, “That’s more than some.”

I was arranging BMAN’s reference works, including one copy each of his “selecta” books, and his dictionaries and encyclopedias (American Heritage Dictionary; Merriam- Webster Collegiate Encyclopedia is a favorite). When I picked up Eric Partridge’s “Origins” I exclaimed that it is one of my favorite works. He concurred, and told me about how Partridge was quite a loner, and did his work in a place set aside for him in the library.

BMAN is quite anxious to finish his memoirs. His agent has obtained a publisher (Knopf- Pantheum(?)), and IBM is providing office space and support to give BMAN opportunity, so I gather, to compensate for a “mistake.” I do not know what was the mistake. But he did use that word, and it was in the context of his obligation to IBM.

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Dozens of boxes of papers are destined for archival storage. The leading candidate for the repository has been the Library of Congress. But BMAN related that Gould’s (second) wife – “Do you know who is Gould?” “Stephen Jay Gould?” “Right.” – was an artist of a strange kind, they had a huge place in New York south of Central Park West, building up the garment district. She urged Stanford, even though BMAN has no connection whatever with Stanford. She said that Stanford is all set up to handle archival storage, they did Gould. BMAN also speaks of Yale, where he has had a very strong relationship, and of Duke, where evidently many economists’ works are stored.

He said that all of this moving and so on has interfered with getting on with his book, and he also said that he wants to continue research. He said a programmer is needed, and that he had been thinking of hiring a high school student. “Do you know programming?” he asked. I said, yes, I have been programming for forty years, it’s in my blood. He asked, “What language?” I said, “Several microprocessor assembly languages, FORTRAN, BASIC, FORTH” and he said, “I heard about FORTH. What is special about FORTH?” I said, “I’ll come back to that after saying the rest of the languages I have programmed. Then I continued with a little C, object-oriented language, such as JAVA, APL, Mathematica, and LabVIEW.” He said he has a particular project to continue. I asked where is the code. He hesitated, and indicated that we would start from scratch. Then I told him about how FORTH is more than a language, it is an operating system that allows one to come from nothing except a CPU with electrical power up to a full-fledged system with application programs. We discussed the disadvantages and advantages of Mathematica and C, and assembly language, for computationally intensive graphics programming. Maybe I will be Benoit Mandelbrot’s last programmer assistant. He said without a project he would go crazy.

ECOO: “So for what would you prefer to get a Nobel Prize?” BMAN provided, as expected, a lengthy reply. He began by recapitulating how the “Nobel Prize in Economics” is not exactly a Nobel Prize. I knew this. He thinks it unlikely that they would select him for that prize, seeing as how his work pretty much demolishes most prior economic theory. He said his work proves that economists are a bunch of fools. At one point he said – I think in reference to the selection personnel – that “If it were not known that these are brilliant people, they would all be on probation.” (I think he did say he was in Samuelson’s office when the latter’s Nobel phone call came in.) As for physics, he said that physics is having a tough time these days. He said he might get a Nobel when they realize he is very old and in failing health.

I make stupid arithmetic errors repeatedly, even with my time-put-in-spreadsheet for billing BMAN, so I feel really embarrassed and beat myself over the head constantly. However, I sort of knew this kind of thing would happen, and am consoling myself with the idea that watching my reaction and thought process could be useful for understanding myself. Then again, I am taking dictation for his email replies and on occasion I catch minor writing errors. Once in a while he mentions how this or that person is “the most interesting person I ever met” or “brilliant.” If he ever were to mention me he would say, “Not too bright,” as in the movie, The Matrix, when the old lady savant mystic meets Neo.

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A major part of my responsibility, as mentioned, is to review emails for BMAN and take dictation of replies. Sometimes the incoming email is a tad weird. A writer said that he was “bored” and decided to write a journal to share with BMAN, who said, “He is bored with himself, now he wants me to be bored.” My barking guffaw rather startled BMAN. So now I have to worry that BMAN will prefer not to make jokes to avoid my boorish outbursts.

BMAN returned from a trip to Europe and had taken from an Italian airplane a page from its flight magazine. The article had an artist’s rendering of a New Egyptian Museum to be in Giza. It showed a huge slanted triangular façade of the building, adorned with a pattern of diminishing nested triangles. He wanted me to get him the email address of the architects, a firm in Dublin.

I blurted out that the pattern is from Wolfram, at which BMAN very quickly declared that no, it is based on “The Sierpinski Gasket.” He proceeded to relate a brief biography of Wolfram, including remarks to the effect that being a child prodigy is certainly no indicator of future accomplishment, and in fact that he has seen many cases of intellectual sterility. He said the only new thing in Wolfram’s book (“A New Kind of Science”) is a result proved by an employee. BMAN implied that all reputable scientists have turned their backs on Wolfram. He had me type a letter to the architects, and included an introduction of himself as “the father of fractals,” and the creator of the name “Sierpinski Gasket” for that pattern. I said, “You actually named it?” to which he replied by recounting how he had not actually known Sierpinski, that Sierpinski had published that pattern first, and how Sierpinski had influenced BMAN’s family move from Poland to France.

An email correspondent asks BMAN for what the middle initial in “Benoit B. Mandelbrot” stands. BMAN tells the story that when he first came to the United States, Harry Truman was President, and that Truman did not have a middle name. However, Truman invented the middle name “S” for himself, and was angry if someone put a period after it, because S was his one-letter middle name. So, Benoit gave himself the one-letter middle name, B, but Benoit says he is more patient with people than Truman, and does not mind if someone adds a period.

BMAN has no arrogance, although he is rightly interested in being sure that his work is appreciated. He is both a gentle man and a gentleman. A fair amount of my work for him is turning towards cleaning up the huge mass of material that will become the archive for future researchers. He wants his web-site to be a thorough, well-organized repository not only of his papers, but also reviews of his books, etc.

I said to BMAN, “All the pictures I see associated with your work fall into two distinct types: things like the Mandelbrot set are always smooth and curvy, but things like financial time series or the Sierpinski gasket are always very jerky and rough.” He said something like, “The curvy things are really rough too.”

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Returning from work with BMAN, to my right on a waiting bench at North Station sits a heavy set pasty-faced mother with a small boy on her lap. His older brother kneels on the floor and is examining the puzzle page of a newspaper spread on the bench between the mother and me. There are two sudoku, and a crossword puzzle to which he points and asks, “How do you play this game, Mom?” She barely glances at him and says, “You don’t know how to do that.” He looks at the sudoku grid, and I give him the pen from my back pocket. He hesitates shyly, clumsily opens it, and writes a number 7 in a square. I say to her, “Sure he does,” and to him, “Make sure you put a number in every square.”

BMAN received a beautiful French paperback book, “Les Fractales”. It is a lavishly produced colorful glossy print job. It has articles referring to Dürer, Dali, and Sierpinski, among many others. It seems intended for the popular audience. He asks whether he should contact the publisher regarding English translation for Americans. I said, “What a coincidence, last night I was thinking there should be a biography of you – including your work – targeted to high school students.” He referred to the memoir his is writing. I said, “Look, how do you want to be remembered 100 years from now? There will be plenty of work between now and then. Do you want to still be on the table?” He said, “Yes, I want to still be on the table.” After further discussion he dictated a letter of inquiry to the publisher.

At the prospect of grave risk of cheerless – nay, morbid – thoughts and words, let me state at the outset that my final project is the plan for my death. But, if – as I endeavor to – you consider projects to be life-affirming challenges, then fun can be had by all. The manner of one’s death depends on ultimate truths to which we humans are not – and will never be – privy to. So, the idea of “planning” my death is ludicrous: LOL. The opposite of planning, however, is carpe diem, and let the cookie crumble, whatever.

Sometimes while dictating an email – specifically to some extremely distinguished person – BMAN will pause to ponder, “How to say this in a nice way.” He thinks in units and then re-arranges the order of the units.

At one point we were talking about IBM hot-shot intellectual power houses, and I said, “Chaitin?” To which BMAN replied by implying to me that no, Chaitin was lower echelon; that BMAN hired Chaitin. BMAN said he fought hard to bring Chaitin to IBM. He said Chaitin at age 40 wrote a book that sounded like a 15 year old. I piped up with, “I haven’t thoroughly read all of them, but to me he seems to be writing the same book over and over again.” BMAN nodded, I think.

BMAN has strong opinions about Bourbaki. I asked, “Did you know Dieudonné?” Yes, he said, from a wealthy family, but down to Earth, unlike Weil, also wealthy, but … . Around 1945 BMAN was giving a talk and Dieudonné interrupted, “Shut up or sharpen up!” regarding BMAN discussion of Peano Curve. So, BMAN went over the proof much more slowly, and half way through, Diedonné said, “Okay, stop, I get it.”

I am quite aware of having committed almost a dozen prodigious erros for employers or clients. So I live in fear of failing again, except in front of BMAN.

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I asked BMAN, “Are multifractals to self-affinity as fractals are to self-similarity?” No, he said. He muttered something about spectra of singularities, and about a very big paper on multifractals and turbulence.

We left the building together. A large storm had recently passed. I told him about the foot high mound of snow that had accumulated on the round table in my backyard in Rockport. He started saying how surprising is the variablity of weather. I said, “You! Of all people, marveling at the variability of weather!” He said, “Yes – me. Of all people. Maybe there should be a book just about how surprising variability can be.” I said, “Well, its all about thermodynamics.” He said, “Not quite.”

I was working in “my” office and an IBM mathematician came by. He commented that he had overheard me working with BMAN. “You are so patient with him, about computers. I wish I could hire you to work for my mother.”

BMAN and I necessarily come across many fractal images. At one I said, “Correct me if I am wrong, but the image is only an approximation – in fact, you can never see a fractal.” He draws a line on a sheet of paper and says it too is only an approximation. Arguing, I said, “Yes, but for a line I can write a finite expression. That is not true for a fractal.” He waved me away.

We were examining the list of participants at a conference in which BMAN is interested. The name Jean-Pierre Serre appeared in the list. I exclaimed, “Wow, one of the great men of mathematics,” having in mind the Serre Spectral Sequence and the Serre Conjectures. I said, “I think his number theory conjectures have been proven.” BMAN said Serre also worked in topology. He went on to declare, “He is as dry as paper. He is my contemporary. I was in the class of 1944, he in 1945. He has a very narrow view of mathematics, even saying what I do is not mathematics. Milnor had accepted my work as being mathematics, but at some point he stopped accepting it.”

I would ask, “Are there fractals that are only finitely deep, so exactly representable?” I would ask. “Is there a replacement for the Black-Scholes Option Valuation Formula that is based on fat tails arising in fractal mathematics? Wouldn’t the predicted premium be higher, since it would allow for long-term correspondences and greater risk?”

BMAN consistently seeks the shortest path, whether it is walking from his office to the elevator, or editing the text of his memoirs. (I told him he has a “geodesic mind.”) For example, to change “However, …” to “Historically, however …” he draws

However,…

istorically, h

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BMAN and I were viewing a web-site of an upcoming Canadian conference on fracture. He studied the names of the speakers of which many were apparently of Middle Eastern type. I said, “What are you complaining about?” He said, “Not complaining [you friggin’ moron], observing.” He went on to note how much engineering studies have declined in Canada and here. I piped up with an opinion about American anti-intellectualism, post World War II, Sputnik, funding for science and technology, and I went on and on about how even the high school where I taught cancelled courses for bright students but provides extra support for cognitively challenged “special education” students. He left the room. I took out a loaded gun and shot myself in the brain through my mouth.

On the bottom of the title page of his vitae there are a few links to web pages of unauthorized biographies of Mandelbrot. I did a Google search on “biography “Mandelbrot”” and it gave over 40,000 results. He said, “Maybe I have the right to be an arrogant bastard!” I said, “You and I both know that is impossible.” But he had me print 4 pages of the results to show Aliette.

We have spent a couple of hours in Microsoft FrontPage on the re-design of his web-site. He is acutely sensitive to design “balance.” I am kind of excited about re-doing his web- site. It is a chance to learn more current stuff and to exercise my taste.

BMAN asked about how my business is going. I briefly outlined my expectation that I will be cancelling it. I went on to explain how there seems to be no interest in it. He sympathized by saying something about how his father had tried to get involved with some inventions.

I have been using Mathematica to play with generating the Mandelbrot set. I kind of get it. Two things: I picked out a couple of dozen colors in a nice photograph that he has already said would be good for his home page, and have used them as the colors in a Mandelbrot set. That might look good on his home page.

The second thing is that I observe that the Mandelbrot set associates with each point of the plane an infinite sequence of polynomial evaluations – evaluations which may or may not leave the magic circle of radius 2. So, this art generation depends completely on the arithmetic of complex numbers. Simultaneously, I observe that 2 dimensional cellular automata depend on the standard topology of the Euclidean plane. Topology, arithmetic, topology, arithmetic,… Hmm. Maybe there is a terrific algorithm that combines Wolfram generative art with Mandelbrot generative art. We will call it MandelWolf art.

With BMAN’s stories in mind about encounters with the likes of Serre and Diedonné who did not care for his mathematics, I brought to him a printout of Terry Tao’s recent Bulletin of the American Mathematical Society article, “What is Good Mathematics.” BMAN said he had thrown away that issue, but he took the copy with him anyway.

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The other day BMAN handed me a printout of a few lines in French, and asked me to translate into English. So, I did, somewhere around the 80% level. He said I did OK. But today he gave me a handwritten letter to a dear friend, written in French, and with his characteristically convoluted editing instructions. He asked me to type it up in preparation for sending it as an e-mail. So, I did it, including using accents, circonflex, and cedila notations. After a couple of iterations between us it was done. He gave me a good mark for my work, and added that sometimes previous people who worked with him did not care for doing things in French. I said that I love French, that I really enjoyed it in High School and College. I added that something happened to my mouth as I aged, and that I can no longer whistle, or pronounce French as well as I did in my youth. I even said that my family shuts me down when I attempt to read French poetry out loud to them.

All along I have been somewhat subliminally aware of the following. In any century just a few people happen by genetic endowment, opportunity, and luck to contribute a sea change to the history of science. Not me, of course. But, actually working side by side with one who has done that provides a gigantic vicarious boost.

I have a draft revision of BMAN’s home page and revealed it to him today. He immediately suggested small revisions about position. Of course I had put a Mandelbrot set in a square region, and a photograph of him in another square region. I calculated the set using Mathematica (“Wolfram, that bastard.”), and he wanted me to change one of the colors (which I had deliberately extracted from his photograph). So, I had this chance to show some Mathematica “chops” to the inventor of the Mandelbrot set. For example, he wanted a region that I had colored black to be white. So, I did that. In the few minutes it took to recalculate a 512 x 512 image, he mentioned how it used to take him all night for such things, and that the printouts were “spotty” images at best. I said I could barely imagine the fortitude it demanded of him, and that he must have really wanted the images. He clearly concurred.

So, now we get back to me, and the BMAN. I am now engaged with implementing his corrections to LaTeX manuscripts of two joint papers, one with a collaborator at Yale, and another in France. I do enjoy exercising my LaTeX skills, although I am undergoing a medical investigation (e.g., blood tests and X-rays) to determine what may be the cause of increasingly annoying pains in my right arm – after all, I am operating a mouse over twenty hours a week. (I also have pains under certain conditions in my knees.) Anyhow, if I am in denial about my intellectual incapacities then a recent interaction with the BMAN proves it. A paper in preparation for publication in the American Mathematical Monthly is about negative dimensions. At one point he has a “Moran Equation,” 11DD rrDD1, and that is followed by an example, something like 31. 1 n 24 in D Then, later, he has the general equation written as ri 1. So, since I print out i 1 preliminary changes for him to review, I added in the margin – in red, no less – the

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in D change of the latter equation to Nrii 1. But rather abruptly he said, “No, it is correct i 1 as is; I will think about it.” When the manuscript came back to me he had crossed out my marginal annotation, and wrote, “Thnx.” He did, however, change the text associated 1 1 with the example to include the equations r r r and r . For three days I 1 2 3 2 4 4 have been beating myself over the head, feeling truly stupid and in a swirl of mental reviews of all the stupid things I have ever done. I even woke up last night at 2:30 AM thinking very dark thoughts about myself. So, I made and ate a bowl of hot oatmeal with a little salt, and went back to bed after reading in front of a fire I started in the wood stove. I had thought he considered my change to the general equation to be completely wrong. But just this moment – after a search on Google regarding “Moran Equation” – I realized that I really did help him correct his manuscript! What I had done was assume that he would want to write the general equation with distinct values ri . Of course, if that is not assumed then his equation is correct. On the other hand, by inserting the equations about repeated values he did take my advice in connection with clarifying his example. It is nice to work with a genius, because he did understand what bothered me, and found the right way to help future readers.

We had just completed editing the LaTeX of the joint paper for the Monthly. After we sent an e-mail to his collaborator about re-submitting the paper, he explained that previously it had been rejected – one reviewer was happy with it, the other was “insulting.” I asked BMAN, “Have you ever been called a ‘flake’.” I had in mind that an article I submitted long ago about timing machinery had been rejected because – among other comments – the reviewer could tell just from the title that I am a “flake.” BMAN replied, “I have been called everything.”

About what am I in denial? Most importantly, I never fully acknowledged that in my ambition to be a great contributor to science – for example, like the BMAN – I have repeatedly challenged myself with projects that are demanding of talent, skills, and luck that are considerably beyond my resources. But if I were judged on the basis of heroic effort, I would be as great as I wish I were. A possible corollary to that denial is that my lifelong bad temper – almost entirely under control – is consistent with someone who, underneath it all, is utterly angry at himself about failing to achieve greatness.

I told BMAN I needed to take a course in fractal geometry so that I could talk to him. With his memoirs constantly in front of me, I am well aware of his satisfaction with his “long life.” I said to him, “In my medium length life, I have been exposed to a great deal of mathematics. I won’t say I have absorbed it all, but I certainly am heavily exposed. Your work is very, very different from most of the mathematics I have seen.” He concurred. An e-mail arrived with a draft of the Carus monograph that his collaborator at Yale is co-writing with him. When I saw it I indicated that it appears to be a good book for the likes of me, since I find the Selecta to be very tough going. He agreed, and even said that sales of the first two were okay, but not the last two, so he stopped doing Selecta. But, he said, The Fractal Geometry of Nature sold extremely well. Indeed, he

Page 83 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 14/66 confided, it put both his sons through college, and his younger son through medical school as well. I haven’t quite told it to him outright, but I have extremely high regard for his English writing style in general, and that of the memoirs in particular. His mathematical style, however, is quite obtuse. This is not tremendously surprising; in fact it is probably inevitable for someone inventing a new kind of mathematics. I would say that it might take a couple of decades for his insights to be boiled down to some kind of “algebraic fractal geometry.” I wish I had the talent for creating that, say, by finding the right categorical framework for fractal geometry. Life is too short. Many chapters of the memoirs start with an engaging quotation, usually from someone BMAN met. But a chapter on financial prices needed a quotation from a letter to the editor of Scientific American in 1999. BMAN had published an article on multifractals and finance, and the response was vehement. It seems that a large number of readers viewed his ideas as plagiarism from a “technical analysis” methodology created by someone named Elliot fifty years before. Modern surfers of the “Elliot Wave” angrily demanded that Scientific American publish a rebuttal article by a person named Sprechter. The editors selected only two of the many letters for publication, one by Sprechter, and another by someone whose name BMAN could not recall. But that is the one BMAN wanted to quote from. He was quite pleased when I used Google to actually find the text of the letter from a Mr. Lavanne.

I asked BMAN which of his many contributions – to finance, to geometry, to the theory of turbulence, and so on – would he prefer to survive if only one of all of them were permitted to subsist. He did not want to pick one, although for my part I suspect that his contribution to the theory of turbulence may be the most important of all. He said only one thing, his set, is named after him, and that he is proud it is used in high schools. Then he expressed satisfaction that his conjecture to do with “uniform compactness” or something like that, of the Mandelbrot set, has been open for decades. He also said that the problem is so hard that students are no longer advised to undertake its solution, since even the best minds are stumped. Trying and failing would not get one a doctorate, after all. Then there was a lengthy discussion of mathematicians in the twentieth century. Mentioned were André Weil (evidently a horrible person, and whose sister starved herself to death in London even though there was an easily accessible underground trade in French food), Laurent Schwartz (who merely formulated the idea of distributions which was already in the air at the time), James Lighthill (with whom BMAN had dinner – I said, “The swimmer?” and BMAN said yes, and I said yes, he swam the English channel numerous times (BMAN said he killed himself that way; I knew that), he wrote extensively on hydrodynamics, and ran the Royal Air Force during the Second World War – I kind of know about this guy), Poincaré and Hilbert (who were real giants of mathematics compared to everyone else), von Neumann, and I added I thought very highly of Grothendieck. BMAN basically concurred.

Time for some hubris. I am to virtuality as BMAN is to fractality. The only difference is that he is a genius – a mathematically very gifted person who, by his own admission, was often in the right place at the right time with the right genes. Examine not what the differences are, but instead the similarities. For example, BMAN is hard put to define what “fractal” means. I have struggled for many years to define “virtual.” There are other

Page 84 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 15/66 concepts in mathematics that have been difficult to define. For example, “higher- dimensional category” is currently hard to define. The Feynman integral has been hard to define. Examples of things easy to define: topological space, ring of operators, category, functor and natural transformation. Reverting to similarities: BMAN drives home the point that nature and art are shot through with roughness – and that he is the apostle of the scale invariant kind of roughnesses. He insists that we recognize the fact that far more is rough than is smooth. I insist that far more is virtual than is real, and that nature and art are shot through with virtuality, whether known by that word, or not. Here, for the record, is what I did today, a Saturday. These items are not in chronological order. I shipped two boxes to my son at college (a battery powered dustbuster, and a set of books); performed the recycling chore at the local recycling center; and dropped off and later picked up my wife for her weekly visit with her mother. I vacuum cleaned the “family room” floor, and cleared out a workshop area in preparation for discarding junk and sorting the items for storage. I did a bit of carpentry towards installing new baseboards behind bookshelves in the living room. This entailed trying out my new 5 ½” battery powered circular saw, and using my professional grade mitre saw. Some peculiar angles were required due to the not quite rectangular nature of the house. I copied the “favorites” folders from the three computers that I use on a daily basis onto an 8 gigabyte “jump stick.” Most importantly, of course, I wrote up my algebraic proof of a theorem on an abstract version of the Legendre Transform, and finished by proving the equivalence of the Lagrange and Hamilton equation systems for classical mechanics. That was the goal of Chapter 2 of my book on virtuality. I began Chapter 3 by articulating a system of axioms for thermometrics that do not depend on an intuitive idea of “the sensation of heat,” but instead depend on the phenomena of bodies changing volume upon contact.

Some BMAN annotations for editing his memoirs can get a tad complicated. He will take a printout, cut it to pieces with scissors, and use tape to rearrange them. Brackets and labeled arrows are used to indicate source and target locations, often with his handwritten renumbering of pages. Words to delete are highlighted in colors, but there is no particular color code. Rescinding a highlighted deletion is indicated by a row of dots beneath the material to restore. Tiny little marks indicate punctuation changes. Rarely – but today it was necessary – do we sit together with BMAN instructing me to carry out the moves I made that were not consistent with his intentions. He said, “Sorry that my instructions are so incomprehensible.” I said, “I’m sorry I don’t catch on quicker.” We both chuckled over that.

BMAN often needs the e-mail address, sometimes the phone number, of someone. I am fairly adept at using Google, so it is not difficult to find the contact information. Today we did the search for a rather accomplished individual, and we pulled up his vita. I said, “This guy is going to give you an inferiority complex, his vita is 76 pages long and yours is only 56 pages.” Looking at the top page of it, BMAN said, “But look at all that white space.” I said, “Right.” Then he said, “Pick any of his pages at random, go ahead.” So I did, and we were looking at a page densely packed with citations to articles by the individual. BMAN said, “See, he uses up a lot of space by repeating his name.” Then he

Page 85 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 16/66 said, “It’s not the weight of a vita that matters.” I said, “He should learn how to use ditto marks.”

This is reminiscent of Michel de Montaigne’s aphorism, “The value of life lies not in the length of days, but in the use we make of it.”

The Cooper Set is like the Mandelbrot Set, except it is a three dimensional nesting of infinitely decreasing rectangular fractal solids; and as you zoom, you see tortured, crying faces as well as all kinds faces with tears of happiness, as though flying through clouds of human condition; and your overwhelming emotional reaction – more severe than your response to any music or painting or architectural wonder – forces you to avert your eyes after but a few moments, so that you can catch your breath.

BMAN received an e-mail from a would-be scientist with a new theory about the physics of time, something about cyclical time. Information was attached to support the legitimacy of the theory. For example, indications that a major publisher is interested in the work; a copy of several chapters of the work; copies of e-mails between other parties expressing interest in the work; a link to a Richard Dawkins web-site; and a half-page review of the work by a physicist. The wannabe is soliciting BMAN’s help, since evidently the fellow has no standard credentials. BMAN had me print a web page on the Richard Dawkins web-site wherein it is argued that God does not exist because time is cyclical. BMAN wants to read the comments appended to that argument in preparation for a possible reply to the guy’s e-mail. BMAN had me print out the first couple of pages of the attached chapters, and when he saw them he immediately said, “See, this is how I know the man is incompetent,” and he thrust a page for me to look. I did not catch on quickly enough, even though I know what BMAN wants a page to look like: the type has to be sufficiently large enough to read easily, and above all else, the width of the lines should be small so that the eye can scan the line without losing track of it in the vertical direction. In these regards the printout is deplorable.

The next thing I know, BMAN is talking about how his father was a failed inventor – who went on to manufacture clothing for children – and he ever so kindly touched me on the arm, saying, “I hope you don’t mind, I know your invention didn’t work out.” I have no emotional problem with thousands of dollars and two and half years of effort going down the toilet, but I definitely was touched by BMAN’s concern for my self-esteem. He proceeded to discourse on how he was genetically advantaged, on both sides of his parentage; how a year younger or a year older, or a mile this way or that, and things might very well have turned out differently for him; how surviving through the Second World War “hardened” him; how he did tend to make risky choices; and how life offers a “narrow slit” that one may be lucky enough to fit through.

BMAN received by e-mail further information about the architecture of the new Egyptian Museum in Giza, including quite detailed engineering specifications of what will undoubtedly be larger – and last longer – than the pyramids. When I observed that evidently they were planning to move the 3200 year old statue of Ramesses II to the entrance of the Museum, BMAN said, “They should also put a statue of me there.”

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Commencing with the usual apology for my temerity, and with acknowledging his prodigious memory, I said to BMAN that in connection with bringing his memoirs – currently a manuscript of over 800 pages – down to a size more acceptable to his publisher, he could omit some parts about the history of the world and focus more on his own story. He said, “Congratulations, that is just what my friends have been telling me.” Into his memoirs BMAN put Rugositatem regunt numeri, meaning “roughness is ruled by numbers.” He said this is “kitchen Latin,” and asked if I know what that means. I said I guess it means “made up Latin.” He said, no, it could be “High Latin” or “Low Latin” that is imitated. So I said, “Okay, okay, it’s made up Low Latin.” Then I said I know how to say Occam’s Razor in Latin, and he said, “There are many alternative formulations.” So I said, “Entia non sunt multiplicandum praeter necessitatum.” He replied, “That is one of them.”

BMAN has tentatively introduced a title — for a part of his memoirs — including the phrase “Father of rugometry?” I objected to “rugometry” saying people will think it is the study of rugs (I should have said, measurement of rugs), and that the word is certainly not euphonious. I intend to to suggest the new word, fractometry. Actually, “fractometry” (2490 hits) seems already taken in the realm of gas chromatrography. Or, fractametry, already introduced by http://www.fractalus.ch/chaos/chaos_e.html (among 2 hits).

BMAN was incertain about the date, I said February 14, St. Valentine’s Day. He said, referring no doubt to Aliette, “For us, every day is St. Valentine’s Day.”

I handed BMAN a set of questions. “Is every rational number the fractal dimension of a rough visual object?” He said yes, and even every real number. He described making a (Cantor) dust by subdividing an interal into n subintervals, and discarding any r of them, and then repeating this operation on the remaining pieces. I asked, “If so, is there an algorithm?” Of course, he just gave one. I asked, “Do all rough visual objects with the same fractal dimension have a similar appearance?” He said, “Yes and no.” I asked, “Do the rational approximations corresponding to a real number correspond to rough visual objects that appear to converge?” He said, “That is an interesting question.” Then he gestured with his hands indicating a set of leaves increasing in number and in the limit merging to a closed region, and said this is about Siegel numbers, Carl Ludwig Siegel.

There is no doubt in my mind that BMAN has brought some order to recognizing and measuring roughness in nature. Only a tiny part of the statistical regularities in nature are Gaussian. Most are “power law.” The observations and calculations he has made about financial time series, and specifically his algorithms for generating artificial price series may be used as follows to make a fortune. I mentioned to BMAN that Feynman introduced the idea of a Brownian motor, a motor that rectifies non-directional random motion to produce a net directional motion — an idea that is now central to biological molecular motor models. He was too tired to comment. The first step in my process to make a fortune is to agree that having BMAN’s artificial price series generator is perfect for testing any trading algorithm. It would be natural to compare the performace of a

Page 87 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 18/66 trading algorithm when it is subjected to, say, a Brownian price model versus a BMAN price model. He would would win that contest because, he has voluminously argued, the Brownian-Bachelier price model is woefully inadequate. Okay, the next step on my way to a fortune is to observe that all financial trading instruments and associated costs can be described by a single, finite formula. In fact, I did something like that using a variant of Dexter Kozen’s probabilistic dynamic logic in the 1980’s (to no acclaim whatsoever). A trading algorithm is a choice of formula for using financial trading instruments. Okay, so the game is to solve an equation which “rectifies” the BMAN artificial price movement to yield profit based on the solution of the equation. In other words, the equation unknown is a trading algorithm, the coefficients in the equation are the financial instruments, and the given input is a BMAN price model. The one and only question is, does the equation have a constructible solution? For, if it does have such a solution, that is how to make a fortune. So, we need a theorem proving that the equation has a constructible solution. Why should we believe the equation has as solution? Because the world of economics has a built-in bias: it can’t go lower than 0, but it can go arbitrarily high. Since financial instruments include both put and call options, the intrinsic bias can be translated into rectification. Quod errat demonstratum.

I told BMAN I needed a good technical reference on “multifractal measures,” and that it seems a book by Falconer might do. He said mathematicians haven’t done such a good job on defining these things. I said, “That’s why I’m interested. I am compexity-tropic.”

Sometimes I tell people, arrogant young people, that you have to ask yourself, am I good enough to clean Bill Gates’ toilets? The point being, that Bill Gates can afford the most professional, meticulous person for absolutely anything that he needs done. So, if you think you are such a hot shot … I, for one, am good enough to type BMAN’s memoirs.

Imaginary Time: In my high school calculus class I deviated from the conventional high school approach to the subject by introducing the students to hyperreal numbers — the algebra of reals, infinities, and infinitesimals. In particular, infinitesimals were presented as “numbers” that converge in imaginary time to reals. Thus, the “shadow” of a finite hyperreal is the real number to which it converges in imaginary time. My diagram to show this was based on a vertical line to represent the real numbers, with 0 in the middle and 1 a short distance above it. Then a finite hyperreal was shown as a wavy graph from left to right that asymptotically approached a horizontal line drawn through a real number on the vertical line. In this way I was able to discuss the “cloud” of hyperreals that are “infinitely close” to a given real number.

Recently I encountered the idea of imaginary time again while studying classical mechanics. There is a proof involving imaginary paths of a particle motion nearby to its actual motion. So a square is drawn with the time coordinate horizontal, and an imaginary time coordinate vertical. The left and right sides of the square map to the start and end points of the particle motion, while horizontal lines across the square map to variations of the actual path, which is the image of the bottom edge.

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How does this relate to BMAN? Well, I was just reading his enormously enlightening book, The (Mis)Behavior of Markets, and learning about his “cartoons” constructed to model various categories of financial time series, from “mild” to “wild.” Now, these time series by definition evolve in time. But his constructions are done “sideways” in the sense that to construct the whole graph of the time series he sub-divides the time interval — the “initiator” — and replaces it by a “generator,” a zig-zag. Then he repeats this move upon the sub-intervals of the interval, ad infinitum, but finitely for an approximation. My point is that his construction is performed in imaginary time.

This is off-topic but it is indicative of my environment. My wife wrote on the kitchen whiteboard, “Heaven is the consciousness of the spacetime continuum.” In reply, using a phrase that I recently noticed has entered the American zeitgeist, I wrote, “That is more exciting than a ferret on crack cocaine.”

Closer to the topic, mathematics provides notation for geometry. I am beginning to realize that algebra is the geometry of notation. BMAN tells — with justifiable relish — that he astounded an early teacher by mentally calculating a multiple integral that represented the volume of a sphere, except the integrand was in a weird coordinate system. How could he have recognized that except by recognizing and possibly performing mental algebraic manipulation of the integrand? [I asked him about this, and evidently it was not a formal integration problem, but a problem described geometrically. So this is not an example of “algebra as geometry of notation.”] But my claim is that mental algebraic manipulation is a mental image of symbol re-arrangements and/or substitutions. Indeed, for many algebraic calculations these manipulations are performed in an imaginary plane — like a mental piece of paper. To be sure, there are calculations that are in three dimensions. For example, some diagrams in category theory are arrows along edges of cubes or prisms, and proofs about these arrows involve changing the shape of the diagram, or inserting additional arrows in faces of the shape.

What is ironic is that BMAN is above all a geometer, and his relationship to algebraists in Bourbaki, and at the University of Chicago, seems to have not been cordial.

Someone sent BMAN links to their blogs. One described an experience in Second Life, and displayed a picture of the blogger’s avatar. BMAN asked, “What is that?” and I said, “Second Life is a virtual world that people join to interact using representations that are cartoons of themselves called avatars.” He said, “Very good.”

Reasons I might reject a scientist’s theory: 1. the theory contradicts my (dogmatically held) beliefs; 2. the theory is based on the wrong data; 3. the theory is based on incorrect modifications of the right data; 4. the theory interprets the right data in a mistaken conceptual framework; 5. if I accepted the theory then I would lose money (or power, or prestige,…); 6. the theory is too complicated and so no one can understand it, or at least I cannot understand it; 7. there is unassailable evidence or data that contradicts the theory;

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8. there is a better theory (that explains more with fewer assumptions).

I am guessing that some financial wizards use (1), (5) and (6) to reject BMAN’s theory of finance.

A married pair of mathematicians was presenting a proof of one of BMAN’s conjectures to him. One held a cigarette while the other used chalk at the blackboard. Then they would switch. BMAN was much delighted with this exhibition.

Because for a month the powers-that-be at IBM cannot seem to make my entry card function — beaurocratic erroneous data entry, incorrect attribution of the problem, travel delays — I said to BMAN, “They treat me like a dhimmi here.” “ — What’s that?” “You know, like the Muslims treated the Jews in Spain.” “—Yes, I know, I also am a dhimmi.”

Typically, BMAN starts our workday with a stack of papers in hand: prodigiously annotated printouts for the memoirs; email printouts for dictating replies; newspaper or magazine clippings for web-searches. He said, “Oh bother. I’m getting so old I can’t keep track of all these things.” “— But also, the older you are the more things you have to deal with.” “— Yes.”

I put to BMAN the idea that the memoirs would be a terrific foundation for writing a screenplay. He said we have to finish the memoirs. Of course, I said. But such a movie would finally let the world know how exciting, difficult, romantic a mathematician’s life can be.

But I can’t get the idea out of my mind. Like I don’t already have a huge project: the LaTeXing of my notes for a book on Virtuality, including detailed derivations of fundamental equations for particle and continuum mechanics. (And, I am proud to say, a Mathematica-based display of all 81 components of the four-dimensional isotropic stress- strain tensor, which is crucial to the Navier Equation for an elastic body, and the Navier- Stokes Equation for a fluid body.) Then again, I got a postcard advertising the upgrade to Movie Magic Screenwriter for $99. Like, I can afford that on a whim.

Why I think such a movie would be a natural winner: BMAN’s geometry is exquisitely, beautifully visual; he is a complex, driven character whose life has been lucky, but driven from poverty to success, and then to “fossil”hood; he recounts numerous ferociously dramatic encounters; the abstract tension between algebra of the Gottingen school and the geometry of BMAN is an exquisite pain to be assuaged only by me and my slogan, “algebra is the geometry of notation.” I can envision enormous visuals of algebraic machinery grinding away in parallel with beautiful geometric counterparts. People need to know this! I an perfectly, uniquely qualified to make this happen: I have experience writing a screenplay, I love mathematics — be it algebra or be it geometry —, I have an enormous visual imagination, I have reasonably good health, I want to succeed, and I have a lifelong desire to impact high school education. I want to write the greatest mathematical screenplay in the history of humanity.

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BMAN is competitive, even on the smallest things. We have had an inconvenient problem with sending (not receiving) emails. But somehow today everything works. So he says, “A Miracle on … street.” I say, “First Street, in our case.” He says, “No, Rogers Street.” Well, our IBM building entrance is on First Street, but the windows of our office face out on Rogers Street. This raises the question, in the movie Miracle on 34th Street, to which I suppose he was referring, is the “miracle” literally on the street, or in a building on the street; and if in a room of the building, and if the room has windows, what street do the windows face?

BMAN needed airline flight information for travels from Europe to Mexico. I put before us a chart at an airline, and during the discussion about interpreting it, especially with regard to multiple appearances of the same flight number at different times on different days, he said to me, “Do you get my logic?” Now, in fact, I didn’t get it because I didn’t agree with what I thought he was saying. At the same time, and for the first time, he made me feel stupid. I severely hate that feeling. But, I recover, and so it goes.

Another fact is that the more I deal with his memoirs — as well as ancillary reading about the history of Bourbaki in connection with research for my never-to-be-written screenplay — the more I appreciate the wonder that is the BMAN. In fact, last night, after consuming 0.375 pint of Captain Morgan’s spiced rum, I told the BW that “I love the BMAN.”

He is – if I haven’t already said so – one great storyteller. The IBM Research building and neighborhood had a power outage this afternoon, so the researchers had to consume ice cream from the refrigerator. The BMAN took on the responsibility for a piece of cake, a large chocolate covered ice cream sandwich, and a large handful of cookies. I had a little strawberry shortcake thing. We sat on a couch in the lounge.

I started out by saying I had just read in the Wall Street Journal that the recent upheaval in the markets is a “10 sigma” event, but that the market crash in 1987 was only a 4 sigma event. I am particularly sensitive to this information these days because I am reading Nassim Nicholas Taleb’s book, The Black Swan, which is precisely about these extremely far out probabilities.

ME: “These high spikes in the market seem to me to be the result of a regenerative feedback effect. Some people see other people selling off, so they jump on the negative bandwagon and sell off too, and so it goes.”

BMAN: “Maybe its like that. But these things happen almost instantaneously – one morning the traders wake up to see that nobody is buying and nobody is selling so they don’t know what the price is …”

Then he told a story. It seems there was a highly placed French beaurocrat named Trichet (the BMAN said this sounds like the word tricher which means to cheat). The BMAN sought to meet with Trichet through a friend of a friend. An early morning meeting was arranged in Paris, and Trichet said to the BMAN that all of these mathematical

Page 91 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 22/66 economists didn’t really know what is going on, they were too much in love with the Gaussian distribution view of prices. The BMAN informed Trichet, I am sure it was charmingly, that he was quite aware of this, and had graphs to show it. The next time they met, again early in the morning because Trichet was a heavily-booked fellow, he remembered absolutely everything the BMAN had told him, and was quite interested in the graphs. The BMAN said this Trichet is a completely brilliant fellow, with a phenomenal memory. In fact, when the friend of the friend learned that the BMAN was having such good relations with Trichet, he asked if he could come along at the next visit. When Trichet saw this tag-along he told him, “I know you. You and I once marched together in for the Christian Socialists [or some Christian organization, whatever].” Trichet continued on to provide a detailed recollection of that occasion. The friend-twice- removed later told the BMAN that everything Trichet said was correct.

On Googling “Trichet” one finds that Jean-Claude Trichet is President of the European Central Bank, and previously he was Governor of Banque de France. This explains why his time is at such a high premium. The BMAN tells that he met with Trichet in Frankfurt and thanked Trichet very much for making time available for their meeting. Trichet said, “Sir, two hundred years ago I would have been as grateful to meet with Laplace.”

The BMAN took delivery — the final lot from his former Yorktown office — of half a dozen filled 4 drawer wide-style filing cabinets, and six large boxes of additonal filled file folders. These very heavy cabinets had to be re-arranged in his IBM Research office. I drove down with my trusty hand-truck, and using it with a huge exertion of energy and cunning leverage, managed to achieve his objective. At the end he exhibited to me a poster he had obtained in Paris at the Mineralogy collection of the Muséum National d'Histoire Naturelle in the Jardin des Plantes. The poster shows a fabulous piece of gold ore in a quartz matrix. What caught his eye — and mine as well — is the intricately fractal structure of the gold. He commented specifically that the holes appeared to be of many different sizes. So, I said, “What would you say is the dimension of this fractal?” To which he replied, “Well, it is a three dimensional object, so the photographer made a choice among many possible views.” “Okay, so, just this two dimensional projection of the object, what would you say is the dimension?” Pause. “About 1.82, 1.80. In that range.”

In the car on the way home I said to Carolyn, “When he asks me what I want I know two things I want from him.” “How do you know he will ask you what you want?” “I don’t, it’s just a fantasy.” “So, what do you want?” “I want the exclusive rights to write a screenplay based on his memoirs, and I want an introduction to his friend, Nassim Nicholas Taleb.” “Who’s that?” “The author of the New York Times bestseller, The Black Swan, which is dedicated to the BMAN.” “Why him?” “He is an enormously successful philosopher-trader.”

The BMAN kindly gave me the current issue of Daedulus. I read a great short story about a fictional Coleman Hawkins, the master jazz musician. I said to the BMAN, “I was disappointed to learn that something Carolyn and I always agreed to -- that musicians are the nicest people -- evidently is not true: they are just as competitive as every other

Page 92 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 23/66 group. The BMAN related that he always knew this. For example, he had a woman friend (Ruth …) who was an excellent professional opera singer, but perhaps due to a certain arrogance and idiosyncrasy of style she did not thrive for too long in the competitive world of opera. Oh well, there is just no oasis of nice.

We are probably some number of weeks away from submitting his memoirs to the publisher, but the BMAN repeatedly asks for my opinion on the length problem. A very long conversation recently brought him to mention Wiener and von Neumann — people with whom he studied — as having made contributions in many different fields. He strongly insisted that somehow the memoirs must convey that he too has made his mark in many scientific areas, but that there is a serious problem about how to do that within the length limits agreed to by the publisher. I suggested that there be an extension of the section of pictures — which he expects is the part most people will spend the most time looking at — to include, say, one page on each area of contribution with a representative picture and technical but non-mathematical description. He decided to take a day off to think about the problem.

In connection with reducing the page count, I suggested to the BMAN the following algorithm:

1. Establish criteria for contribution to the memoirs: a. Diversity of BMAN contributions to science b. Mathematics education c. The BMAN is alive and kicking d. The BMAN is a unique storyteller 2. Assign to each section of the memoirs a “grade” from 0 to 5 for its contribution to the memoirs for each of the criteria 3. Calculate for each section its expected contribution by multiplying each grade by a pre-assigned weighting per criterion, then sum 4. Sort the sections in order of decreasing expected contribution 5. Drop sections starting from the bottom until the total number of pages reaches a pre-determined threshold maximum 6. Touch up the memoirs to account for continuity problems arising from the deletions of sections.

He didn’t say no to this idea.

We went to a Basque web-site recommended to the BMAN in an email. I said, “Wow, they even show some mathematics.” “The Basque are strange people.”

The BMAN has a foot high collection of printouts, and a CD of IBM “SCRIPT” files for a new book with working title, “Son of Fractals”. This would be the second edition of “The Fractal Geometry of Nature”. I have been organizing this material, including generation of a LaTeX table of contents.

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Starting by saying that “I am a big picture sort of person,” I suggested to him the idea that the great variety of fractals might be represented by a hierarchical chart, an inverted tree, with the word “fractals” at the top, and then branching into various sub-categories, etc. At each level, I added, there could be a “miscellaneous bin.” I was quick to add that since he is deliberately insisting on keeping the notion of fractal fluid and not rigidly defined. I also suggested that the nodes in such a chart could have little thumbnail pictures of fractals of each type.

His response was that he would not care to have such a chart in the middle of the book, where he is planning to place the large gallery of pictures. He said maybe it could be in the back of the book. He insisted that “Son of Fractals” is not intended to to explain things from the top down, but that it should be a reference work for people to find tricks and gadgets for applying fractals.

I was also excited to tell him about what I am learning about the possibilities for presentations on the web using linked PDFs generated from LaTeX (hyperref, acrotex). I was so bold to suggest that since his technical writing is so non-linear that a printed book might not be the best way to convey his material: I suggested selling access to linked PDFs on the world wide web. He quite bluntly said that “I am not interested in being a pioneer.” Hmm. That is just what I want to be!

Even on small things — choice of word, how to type an edit — the BMAN is competitive. So, I said to him, “You’re very competitive.” He replied, “I was selected for that.”

Last evening Carolyn and I attended the annual IBM CUE Research social gathering (at Harvard’s Peabody Museum of Archaeology and Ethnology). A year ago is where I met the BMAN and offered to help him out. I had decided ahead of time not to speak with him (thinking that (a) I’d rather not make small talk with him, (b) I get plenty of time to talk with him, (c) other people should have every chance to speak with him), but I rehearsed what I wanted to say to Aliette. So, while he was seated at a round table with several other people, and Aliette, I went up to her and kneeled on the floor beside her, and said, “C’est très bien de vous voir, ce soir.” Then I said in English I hoped my French was okay. She said it is, ever gracious person that she is. (According to http://babelfish.altavista.com/tr I might have said, “Il est très bon de vous voir.” I’m just glad I got the “de vous voir” part right — that is purely by my French language instinct.)

A series of emails to “Dear Benoit” from a stranger exhibits this individual’s geometric constructions that combine Klein bottles into an infinite nested sequence of smaller and smaller bottles intertwined one with the next. He has also offered an infinite nested sequence of Möbius strips. He asks, “Are these fractals?” The BMAN has replied that they are not. We had a discussion about these examples, in which I started by saying that most of the BMAN’s examples are subsets of 1-dimensional or 2-dimensional Euclidean space. He was quick to say that galaxy clusters are in 3-dimensional space, and I just reiterated that I was referring to “most” of his examples. I went on to say that the Klein bottle examples are sort of like Alexander’s Horned Sphere, and perhaps only fractal at

Page 94 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 25/66 the limit. The BMAN said that some of his examples are at the limiting ends of trees. I observed that his correspondent’s Möbius example might qualify as a true fractal, since it is indeed a binary tree whose nodes are strips that bridge to the next higher up strip. I said, “He seems quite obsessed.” The BMAN replied, “Unemployed and bored.”

For the record, I have heard the BMAN use the words “idiot” and “shit” at most twice in about a year.

I asked the BMAN whether associating the names Cootner with finance, Simon with economics should be joined by André Weil with Bourbaki as representative of the great controversies in his life. He agreed, and proceeded on over an hour of mentioning names of very famous people, and some comments on them, based on first-hand experience: André Weil (arrogant, made claims about solving the Riemann Hypothesis); Herbert Simon (knew no mathematics; effort in artificial intelligence a bust; egotist); Jerome Weisner (at 23 already a leader – not a great scientist, but extraordinary understanding of human beings); Norbert Wiener (whacky); John von Neumann; Eugene Wigner (kindly); E. O. Wilson (I mentioned having read his autobiographical Naturalist because we were discussing, again, the length of memoirs); Saunders Mac Lane (a nasty man; very poor autobiography); Poincaré; Hadamard; Kenneth Arrow; Debreu (mathematician writing on abstract economics with no contact with data); Tobin; Friedman; Greenspan; Bok; Summer; Hockfield (I told him that through Carolyn’s best friend Barbara Hockfield that we had socialized with Susan Hockfield; the BMAN finds that she is not very street smart). A couple of times the BMAN circled his index finger next to his temple. Once he circled each index finger at a temple. The BMAN does not appear to have eyebrows, and he has very large ears.

The building management of the building where the BMAN had his office when I first started moving boxes for him recently required that he finally remove a set of boxes that had been stored away. So a couple of weeks ago the BMAN asked if I knew of someone who could help make the final move. I recalled seeing on the order of a couple of dozen boxes.I said I could only ask Henry, since the job might fit into his end-of-year schedule of finals. We went ahead with that plan, and I quoted $15 per hour for Henry’s time.

But then the BMAN said the management could not permit non-union people do the job, so it was canceled. Then a few days later the BMAN said he had found out that we could do the job after all. So we planned it for a Saturday. But then the BMAN was extremely tired after giving a talk in New York, so we had to cancel the Saturday move. I studied Henry’s calendar and realized that the following Wednesday could actually fit. (Henry came down on the Saturday anyway, and we had a terrific day walking in the park and visiting the new Apple store in Boston.)

After the on-again off-again history of the job, by Wednesday morning I was thinking that the “$15 per hour” quote was wrong, and that Henry should get paid a flat rate for doing the job, and I had in mind $100. However, I didn’t want to be way off base – I supposed that the BMAN had gotten a quote from a professional moving company.

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So, right after picking Henry up at North Station we went to the Best Buy next to the IBM Research office building, got on-line, and obtained the phone number of a local moving company. I told them about two dozen boxes of books and papers. They quoted $200 for two men, one truck. When the BMAN and Henry and I went over to the old office building we saw that in fact there were over 100 boxes — a large set had been moved to the room after the last time I was over there.

So, all day I agonized — to myself — about how it would go down at the end of the day when I would say $100 for Henry. We did the entire move. When I said,”100 for Henry,” the BMAN went, surprised, “Hmm, 100 dollars.” I was prepared to launch my pent up rant about how this whole thing had gone down, but resisted the impulse. The BMAN asked Henry how much the train fare was. “30 dollars, round trip.” The BMAN said, “Okay, make it 150 dollars.”

Later I said to Henry, “Now, that is a mensch.”

On seeing several fractals from a manuscript of a book one of the BMAN’s collaborators is working on, Carolyn said, “There should be a periodic table of fractals.” The T_Rig system could be used to quickly capture bitmaps of a comprehensive set of fractal images drawn from works by the BMAN.

We were web-searching in connection with Lévy stable distributions and came across movies about fractals. I said to the BMAN, “I told Carolyn I think you probably never saw more than three movies in your life.” “ — More.” “ — Oh, so four?” Then he signified that back in the day he went to see lots of movies. But not lately.

JOB DESCRIPTION 25% EMAIL o download emails daily o make judgement to print selected emails o take dictation (occasionally in French) to reply to emails o track minor writing gaffs o provide my name as cover to certain correspondents (wack jobs) 15% WEB-SITE o editing files on web-site 40% MEMOIRS o edit WORD document according to handwritten annotations on printouts o provide counsel on organization/content 15% TECHNICAL PAPERS o LaTeX file editing o Track minor mathematical gaffs o Document keyword searching (dtSearch indexing) 5% PHYSICAL LABOR o movement of hundreds of boxes of archived papers and books between offices and home apartment o re-arrangement of files and filing cabinets

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o installing book shelves and hanging art in home apartment

FORTHCOMING JOB DESCRIPTION Web-site redesign (e.g., Memoirs spillover) “Son of Fractals” LaTeX book creation/editing

The BMAN is extremely concerned to be sure that his web-site provides a comprehensive representation of his work, and especially reviews of his books and books about fractals. So, we spend a fair amount of time searching the web. When we had a particularly large set of titles of books about fractals and brief review/abstract information, he wanted to have a list made of just the titles, and said, “Can’t you just form a list of those titles in blue?” I explained that yes, I could do that, it means going into the HTML file and searching for the hex color string, and I showed that to him. And I showed him how the search would need to identify the start and end tags of the title. He became impatient and wanted to know why it couldn’t be done quickly. I started to say it might be a couple of days to get it all correct, and I told him no programmer will tell him it will take less time than he thinks it should take. His impatience unabated, I lost mine a bit and said, “You have me so riled up I could go do it at home tonight and not charge you for it.” He demurred.

The BMAN suggested that some former helpers were helpless when it came to reformatting articles for inclusion in his “scrapbooks,” whereas others performed the work quite rapidly. I took that as demeaning my skills, and I got all defensive, my weakest kind of moment.

I said to the BMAN it would be interesting to put an iterated function system type of fractal on a Penrose tiling (I had in the back of my mind something to do with a hyperbolic mapping). He sort of said that would be interesting. Then he went on to tell me about Roger Penrose, how he mumbles his talks, and has that peculiar British elitist attitude of the wealthy. Then the BMAN talked about Haldane, a friend, and about “Bertie” Russell, and in one long story spoke of Frisch the Nobelist, and Aristotle in regard to horse versus donkey versus mule.

I have been friends with my graduate supervisor, Barry Miller Mitchell, for over thirty years. Yesterday I suggested that he write his memoirs. I said that although the BMAN is a fabulous storyteller, he is short on humor – not in person, just in his memoirs. I said that Barry would write memoir filled with humor. He declined by saying, “To write memoirs, or any book for that matter, one must be fascinated by the subject. After two or three pages about myself I would lose interest.”

Once again the BMAN asked what I think about his idea about reducing the 25 page “Foreword” part of his memoirs – appearing ahead of the basically chronological biographical material – to 5 pages. He explained that he wants the start of the book to really introduce the reader to the unusual breadth of his contributions to science, and thus to motivate the reader to read the memoirs. He said he is asking several people about this. My reply, in brief, was that he should select 24 representative graphics of fractals and

Page 97 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 28/66 multifractals, give them captions, and integrate them into a brief, focused image of his accomplishments. I showed him a web-site with multifractals and said, “It blew my mind.”

In recent weeks there is much less work on the memoirs. The BMAN focuses instead on surfing the web for books about fractals, and reviews of books authored by himself or translations of them. We spend hours searching for these things. At the New York Public Library last weekend I learned about worldcat.org, and turned up with little effort a list of thousands of books with “fractal” in the title (one gets books in some other languages using “fraktal”). He is fascinated with this stuff, taking great delight in finding some nugget he didn’t know about. The dates of some of these finds are weird, for example, one is dated 1893. I said, “So, you are an intrepid time-traveler too.” He said, “I had lunch with Moses, and I was at the crucifixion.”

In another alley of the search we spent some time in Cheltenham, England. We found CD’s of fractal art, a fractal astrology web-site, and even more fringy new-age web-sites. He said, “Ommm.”

In the memoirs he introduced a change of chapter title, ending with “A FIRST DASH THROUGH.” I said, “That is ending with a preposition. But that’s allowed.” He said, “It’s allowed.”

Combining his great invention, fractal geometry, with my great invention, timing machinery, I think of a “fractal timing machine” in which each state is a timing machine, and each state of each of those timing machines is a timing machine, and so on, all the way down forever.

The BMAN and I continue to refine a means to obtain a list of all books with the word “fractal” in the title (and “fractals” or “fraktal” or “fraktales,” etc.). He wants to be sure that his web-site has a thorough list, without duplications or fluff. We found that sometimes a dissertation appears in a book list, and that is to be avoided as well. This led to the BMAN relating a “confession.” He said that he was visiting the library at the Sorbonne, and examined a copy of his own dissertation, only to observe that it was an incorrect early version. He said, “So, what did I do?” and waited for me to reply. I said, ”You told someone about it.” – “No.” “You replaced it with a known good copy.” – “No.” “You wrote in the corrections.” – “No.” “You stole the copy from the library.” – “No.” “Okay, that is four answers, I give up.” He said, “I tore up the catalog card for it.” – I said, “But that is illegal.” – “Yes, I confess.”

Recently I learned that software is available for the hobbyist to turn a webcam into an eye-tracker: a device that “knows” where one is looking, say, on a computer screen. I got excited about that because I know a lot of interesting psychology experiments depend on that kind of instrumentation, such as experiments on attention. But I also thought that it would be very interesting to hook up an eye-tracker to a Mandelbrot set generator that zooms in at the point where the eye is gazing. I think that might be a rather unsettling experience.

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The BMAN had received a DVD of fractal art, including incredibly deep zooms into the Mandelbrot set. I asked him, “How did you feel the first time you saw a copy of the whole thing deep down inside?” He said, “It was a high point of my life.”

The BMAN almost endlessly tinkers with his writing, but he will eventually say, “Enough.” He said, “The world is imperfect simply because God tried very hard to make it perfect, but after a while he had to throw up his hands, saying “Enough.””

In connection with our search for all books about fractals I asked my friend Florian, Director of Research Computing at the Graduate Center of the City University of New York, whether he could bring our query to libraries there.

This request for a favor came after a very prolonged interruption of our friendship arising from a misunderstanding about our roles in a business venture of mine. Nevertheless, we seemed by telephone to resume at the least a cordial relationship. Florian sent to me by email the following story about the BMAN and CUNY.

I've heard stories about Mandebrot and the Graduate Center; the following story was recounted to me several times by several professors on the GC faculty. Once Mandelbrot was being considered for a professorship at the GC. Mandelbrot informed the executive officer, Eldon Dyer, that he wanted to be paid one dollar more than anyone else in the department, as a symbolic gesture, since he was better than everyone else. Dyer said that he Dyer's decision about his compensation. Dyer said that he had thought about it and decided that although he could arrange it, he would not do it. Mandebrot wanted to know why. Dyer replied, "because the reason I would be paying you one dollar more than anyone else on the faculty isn't true."

I should add that it was a mistake for the Graduate Center not to have hired Mandelbrot. He's famous, and would have added immeasurably to the prestige of the program. Someone like Mandelbrot could decide he wanted to have lunch with the mayor, and it would be done. Not so for Sullivan, for example.

Maybe he was once that egotistical. But from my experience, even though he had a Keplerians-sized ambition, he is almost totally devoid of intellectual snootiness. Sure, once in a while he will declare this one or that one is an “idiot,” but I am fairly sure he is usually right.

The BMAN is quite frank about his mortality and speaks freely about such things as a move from the current very nice apartment overlooking the Charles River to a final place for retired faculty and the like near Harvard. He is increasingly tired (“I’m pooped.”) and

Page 99 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 30/66 our work week is no longer five days. More like three or even two. He knows that his body is breaking down, yet has plans for a couple of more books. This even though he acknowledges that maybe he has to take a vacation (!) before finishing the memoirs – which are due at the end of the month. He says he does not have even remotely the stamina he once enjoyed. I said, “I know that – I have seen from the evidence of your web-site how prodigious was your output. I don’t know how you did that. Then again, in my circle of people, I am that kind of productive guy. So I am just in the middle.”

The BMAN asked me what I think will happen in Georgia. So I spouted what I understood from recent newspaper articles, and he said they will get rid of Shakravili (the president is who I mean). I said, “Do they have an impeachment process in Georgia?” He held up his forefinger and pointed with his other hand at the end joint of the upheld finger and said, “Yes, a piece of metal this big..”

I asked the BMAN whether for any positive integer N there exists a point outside the Mandelbrot set that require more than N iterations to leave the circle of radius 2. He seems to have replied that, yes, because the closer one approaches the Mandelbrot set from outside, the more iterations one will need. Personally, I would like to see the proof.

September 12, 2008 Roughly a year has passed since I began my work helping the BMAN. The greatest development for that we emailed “the bulk” of his memoirs manuscript to the publisher.

In the last several weeks for one reason or another the number of work days has diminished to four, then three, and recently two days per week.

The BMAN is seeking funding for a “collaborator” on “Son of Fractals,” the revision of The Fractal Geometry of Nature. He is also seeking the person to be funded. I have sent out emails to more than one potential funding source. The BMAN is, in my opinion, a polished supplicant. He is also sending cold emails to professors and IBM company managers for help in finding a “collaborator.” He asked me to advise him on the qualifications for such a person. Naturally, I describe myself. I, however, according to him, not being a member of an institution, would not be a suitable candidate. I printed for him this:

Qualifications of person appropriate for helping with “Son of Fractals” (temporary working title), a proposed refreshing update to The Fractal Geometry of Nature

1. Experienced computer user with LaTeX expertise. 2. Specifically, familiar with modern computer graphics and its programming. 3. Some knowledge of analysis, geometry, and probability.

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4. Experienced writer in English, with strong organizational skills. 5. Resourceful and pro-active worker with “a good eye.”

I should have added that the person would need – probably by writing a computer program – to translate a foot-high pile of manuscripts – composed using the defunct IBM SCRIPT language – into LaTeX.

Putting aside my possible hurt feeling that he has not come right out and say that he wishes I could be the person to help on this project, I have to say the following.

This is a very large project, at the end of a life when his energy is diminishing. In my opinion his life is in some ways a sad one, and I think that he wishes to recapture the success of his one very good book by revising it. In my opinion his most useful contribution at this point would be to focus entirely on having me re-create his web-site according to my very advanced ideas for it. He won’t do that, of course, for the following reasons. First, he barely understands how the web works, but knows extremely well how to write and get a book published. One must do what one knows how to do. Second, he never works on only one project, and “Son of Fractals” was worked on quite a bit and back-burnered only because of the memoirs.

2008 08 17 We are in sort of a holding pattern: the (bulk of the) memoirs having been sent to the publisher, my main activity has been to perform copyediting on the manuscript. I am reading from back to front and sticking postits on pages where I note a typo or grammatico.

Thus, apart from checking the email and taking dictation for responses, there is little work to do. We are meeting but twice or thrice per week, so my income is diminished.

A couple of days ago he brought his usual pile of printouts of items to deal with, including printouts of emails to be answered, web pages to revisit, advertisements or notices of interest to him, magazines or journals to give to me, and so on. This time was included a printout of the email from Florian Lengyel that I had forwarded, since Florian visited the CUNY library and sent the list of 143 books among the library holdings with the word “fractal” in the title. The BMAN is keen to have on his web-site as complete a listing as possible.

We had previously discussed what could be done to merge the WorldCat listing with his WORD document listing, and I had briefly outlined how I would do that – automation of that task requires text-oriented programming.

So, now we have his WORD list, the WorldCat list, and Florian’s email. Since he has printouts of these things in front of him, he gives every impression of thinking that it

Page 101 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 32/66 shouldn’t be difficult to extract the items needed to update the web-site. So, I outlined in some detail how I would go about doing it.

I said that all three should be translated into Excel spreadsheet form with the same column headings. Then they should be combined into one spreadsheet, and let Excel sort them, say, first by author, then by title. Anomalies, like nearly same title but treated differently by the search engine that produced the list, would have to be managed by hand. But, once that is done, duplicates would be removed automatically by Excel, since I have done such things many times. I explained that having a spreadsheet database of fractal books has the additional advantage that it would be an excellent way to store information for presentation on the web-site, since it allows separation of content from form.

The remaining “challenge” is how to translate these three sources into spreadsheets. I told him that converting a Microsoft WORD document into a Microsoft EXCEL spreadsheet is very easy: it is merely a matter of creating a tab-delimited file.

Converting the WorldCat file – once the remaining 25% of its entries have been brought down from the web – into a spreadsheet is a similar bit of work. The real problem there is to delete WorldCat fractal entries that were not published as books, in other words, to eliminate doctoral theses, internal reports, and the like. I suggested that we would compile a list of University press publishers and regular publishers, and use Excel to match WorldCat publisher listings against the publisher lists. Any item not associated with a bonafide publisher would be discarded.

Converting Florian’s list is a different challenge, since his email includes the 143 books in HTML format, including book cover images, I would have to write a macro to search for the HTML tags, delete unwanted information, permute items into the desired column order, and output a .csv.

I said, “Does that help?” He didn’t say yes. He said that he usually understands what the program will be doing, but not exactly in this case. I said, “But have you ever worked on text-oriented programming?” He said no. I said, “Well, we have discussed this kind of thing a few times now, and you still don’t seem convinced. What would convince you?” Instead of answering, he picked out a couple of books in Florian’s list and asked me to find them on the web, which I did, basically at Amazon.com. After doing some more of that he said, “It is a big job.” I said, “Now that you are convinced it is a big job, what would convince you to actually have it done?” He didn’t answer.

On another topic, his search for a collaborator on “Son of Fractals” continues, with my help. I continue to be slightly hurt that he gives no indication of trying to find funding for me to do it. Instead, me has me print out emails and CV’s of candidates, even one from Burlington, Vermont. If he actually finds someone to work with him in Cambridge, I wonder whether he wouldn’t just as well have them handle his email for ten minutes a day.

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2008 09 18 The BMAN – proud of being a “maverick” who picked low hanging fruit in an orchard few knew about – also had the ambition of being a Kepler revered for mainstream success.

2008 09 29 The BMAN and Nassim Taleb are linking up to pay attention to the black swan that has crapped on the US economy.

I explained to the BMAN that I had gotten in contact with World Scientific because of Ralph Abraham. He asked me where Ralph is, and I said I am not sure – that he had been working with an elite school in Long Island, and that he travels a lot. Or at least he did when he was sort of a public intellectual. The BMAN considers Ralph to be a bit off to the side; I said, well, he was a student of Steve Smale. The BMAN launched into statements like Smale is actually a sad man. Then he spoke of René Thom, and somehow things went to the Dreyfuss Affair. The BMAN informed me that Dreyfuss was a wealthy man and also some kind of agent moving between France and Alsace. This led to remarks about the difference between the South and North Rhineland, and on to Hadamard, who also was Jewish.

I had supplied the BMAN with four pages of my (38 page un-published) paper on Timing Machinery, along with a copy of the draft World Scientific contract. He opined that it is a standard looking, simple contract. He also thought maybe the four pages are from the book, and were written “concisely” like a paper, not like a book. He said writing for a book must be much more motivating.

The BMAN said he wanted to know where occurred the expression “awesome symmetry,” and was guessing maybe Donne or Milton as the source. We googled around a bit, e.g., I was thinking seventeenth century and he was thinking sixteenth. Then, suddently, into my head popped “fearful symmetry” was what he really wanted, not “awesome symmetry.” We found that quickly in William Blake’s poem, The Tyger. I cheered myself, and he said, “Congratulations.”

2008 10 02

I had on my desk a book, “Computational Finance Using C and C#” (where “C#” is pronounced “C sharp”). The BMAN read the title out loud as, “computational finance using C and C+” and said, “It isn’t even D and D+.”

The BMAN is tense about getting the memoirs done. He is mumbling about having one or two “helpers,” and said words to the effect, if I understand him correctly, that he definitely would have us cut back our meetings by one day a week, like Wednesdays. Nothing is being said about my working on the web-site, but I think that may be forthcoming. Certainly, nothing has been said about “Son of Fractals” and me becoming the BMAN’s last collaborator. I would not charge him for that. Then again, I don’t think I

Page 103 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 34/66 could really be creating my own book, and collaborating on writing a mathematics book with the BMAN at the same time. Frankly, I need to start thinking post-Benoit.

2008 10 03 I had received the URL of “The world’s 23 toughest math questions” as perceived by the Defense Advanced Research Projects Agency (DARPA). I showed it to the BMAN and he singled out the one on “Stochasticity” (I said, “I think that is in New Jersey.”) explained in a talk by David Mumford on a “call for new mathematics for the 21st century.” The BMAN said that Mumford is a very rich man, but despised in some quarters. I said, “But why?” because I know Mumford got a Fields Medal in algebraic geometry. (I own a preliminary edition of the first 3 chapters of his “Introduction to Algebraic Geometry” that must be about 40 years old.) The BMAN said that it’s his fault: he got Mumford so interested in pictures that he left pure mathematics for research in vision and so on.

I asked the BMAN if there isn’t an analogue of stochastic differential equations involving “wild” randomness. He said that fractional Brownian motion is a solution of a generalized Langevin equation, in which ordinary differentials are replaced by fractional differentials (or integrals). I asked if there is an analogue of Ito’s Calculus, and he said it doesn’t exist, and I asked whether it should and he said of course it should but it is a very hard problem, that there are only a couple of papers on it. He spoke of the French word, topinage .

2008 10 09 Almost every day the BMAN starts by saying, “I’m tired.” But in order to collaborate by multiple cross-Atlantic emails with his co-author, Richard L. Hudson, on the (Mis)Behavior of markets book – it is being re-issued with a new foreword reflecting the current world financial and economic crisis (and a corrected Figure) – we did have two full 9 – 4 days.

At an opportune moment I said I looked forward to the discussion – after we wind down from the memoirs – of my work for him in the future. He said, “There will be plenty to do.” Of course, he has a long history of carrying out multiple projects, and he is not going to change. But, if he does hire a collaborator for “Son of Fractals,” say, and I work on modernizing his web-site, and he can just about hang in there for a half-day…one wonders about whether completion of either big project might be more likely with a more single-minded focus. Then again, I am still capable of a full day of work, day after day, but wonder how I could possibly pull off working at the present pace for him, and at the same time actually be writing my own book.

2008 10 18 A while back the BMAN was interviewed for a Public Broadcasting System NOVA show, and the transcript was submitted to him for revision. It is eighteen WORD pages. I got a big kick out of being given the responsibility to edit the document: remove repetitiousness, clean up the language, try to shorten it. I got it down to sixteen pages and handed him the printout. Immediately he started micro-editing it, but the end of the work

Page 104 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 35/66 day was near, and he decided to work on it over the weekend. Anyhow, I enjoyed doing that tremendously.

2008 10 19 I came in on Sunday to implement the anticipated editing.

Before starting up I informed the BMAN that I am going to need two or three months to make a big push on my book, so that I have a solid first draft by May-June, and an essentially complete draft by October, 2009 – a year from now.

I knew this information would give him a feeling of uncertainty about the final work on his memoirs, so I was quick to insist that my highest priority over the next four to six weeks is to meet his needs on the memoirs. He seemed relieved. I said that since he is interested in getting a laptop for home use, maybe Aliette could help with handling his email. He said that was tried once and it didn’t work out, but that he would probably be able to get someone at IBM Research to put in half an hour a day to help him out.

I said that since I am in principle a contemplative ascetic, it would be great if I had a room somewhere in Cambridge where I would hole up to do my work, help him out, and visit Carolyn on weekends, since she is utterly dependent on me for shopping and visiting her mother. He muttered something about there are all kinds of ascetics, and I said for sure I am no Grothendieck.

He asked me what my book is about and why it is unique. He emphasized the importance of uniqueness. I said that it is for college students and that also high school physics and mathematics teachers should read it, because they should know more than what is needed for their students. I told him it is a book about mathematical mechanics, covering the territory from elementary particle mechanics to the nonequilibrium statistical mechanics and chemical dynamics of muscles. There is no book with that range of material.

I told him this project is my swan song, that with a wedding coming up, and the fact that I am not even within visual distance of the financial situations of, say, the people who work at IBM Research, I need to do this book.

When I turn on the computer that we use at IBM Research I have to enter a password. Currently, I said to him, the password is langevin. He said he knew the aged Langevin, and told me a story. He said that Langevin came from a very poor background, had a great success as a scientist, and was given an apartment in Paris (?). But then he fell out of favor in his old age because he insisted on keeping the apartment long after his eligibility for it.

2008 10 23 The other morning the BMAN called – excited – to say that in the evening there would be a MacNeil/Lehrer Reports program segment with him and Nassim Taleb. I am an admirer of Nassim Taleb, having read two books by him. He in turn is an admirer of the BMAN: he dedicated the second of those two books, The Black Swan, to him.

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I eagerly waited for the segment, and sure enough, there they were. I was shocked, I tell you, shocked, to witness Taleb’s crass self-interest. His main point was that the current financial/economic crisis is the worst crisis since – not The Great Depression – but since The American Revolution. This is pure and simple fear-mongering, which if taken to heart by the innocent viewer/investor would lead him or her to (compete to) sell all their stocks immediately so as to lose no more than has already been lost. Then Taleb very, very quickly repeats that he hopes he is wrong, that he goes sleepless with hope that he is wrong. Why would he say these two things so vehemently? Because, if people are afraid enough to sell low, he can buy low, and if he is wrong, then later he can sell high. So, I am morally pissed at Taleb.

The other thing that bothered me is that the BMAN was fairly reticent, but when asked what could be done, he mumbled something about needing more data. Excuse me? This was a “teachable moment” where a brief case could have been made for mathematics education and scientific understanding.

2008 10 27 The BMAN stores his cell phone in a shirt pocket. Since he does not quite have a well- established habit of keeping the thing fully-charged, during the day it emits a warning beep signal once in a while. Sometimes this annoys him, a lot. But today, in spite of saying right away, when I asked, “How are you?” that he is “fatigued,” when the thing beeped he made me laugh a lot by petting it affectionately, saying, “There, there, everything will be alright.”

The BMAN is tremendously excited about tomorrow’s television program on NOVA devoted to him and fractal geometry. I sent out an announcement – on my own – to my friends and relatives, that started by saying I have been the BMAN’s helper for a year. Ralph Abraham received this, and said he did not know about my being the helper. I guess I forgot to tell him. Anyhow, Ralph himself appears on the show. So, I have two completely independent connections to the show. After all, in 1968 I was Ralph’s A+ student (I wrote the notes for his course, “Differential Geometry and General Relativity”). And, of course, I was part of Eagle Mathematics.

2008 10 30 The BMAN has received numerous honorary doctorates, and meticulously he lists them in his Vita. Recently he recounted two occasions – evidently among several – where he did not get an honorary degree. For example, he was invited to receive one, but then they called to cancel because they had not realized that it was necessary for it to be approved by the Congress of the country. For another example, he was invited to receive one, but it turned out there was a local candidate who for political reasons had to be preferred. So, he said there should be a list in his Vita of the Non-Honorary Doctorates.

Many emails were received in response to his show on NOVA. A number of them were from “flakes” and other wack jobs. One sent us to visit a web-site about unifying science and religion. I said, “Frankly, that is a waste of time.” He said, “Yes.”

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Dear Carolyn,

Aliette called this morning to say that last evening the concierge at Benoit's building saw two young men bring Benoit through the door. He called 911 immediately, and so the commotion you told me that Jeff saw must have been the team that came to bring Benoit to the hospital.

She said Benoit feels perfectly fine, but that once someone is brought in, the hospital has to do all kinds of tests. She said he plans to be back at the office tomorrow. All the excitement about the NOVA show was just too much for him, she said.

Your hubby, Ellis

2008 11 03 The BMAN and I had a normal workday following the above scare.

2008 11 05 The BMAN wrote in an email about the NOVA show, “One thing I noted: the stock market was never mentioned. However, a few days before the Jim Lehrer Report had a little piece on my contribution to that topic. Several people reported that I bothered them and led them to get drunk.” I said, “That is what your middle initial stands for – your whole name is Benoit Bacchus Mandelbrot.” He said, “Excellent.”

I asked the BMAN what is the difference between a Markov process and a martingale. He replied that the latter has all the information from the past built into the present value, whereas in the former, the probability of the present value depends only on the previous value (previous n values). I don’t really understand this, but sooner or later I have to understand a generic example of each.

2008 11 12 I said to the BMAN that I am reading an article in the current Physics Today about networks and that it exhibits a power law graph. Since we had recently discussed the power law associated with Pareto as in (Mis)behavior of Markets, I asked him whether every power law is necessarily associated with a probability distribution. Quickly he pointed out that for example, the coastline length observation of a power law due to scaling has no associated probability distribution. He went on to say that of course Kolmogorov and Onsager had discussed power laws prior to him. And he mentioned also Bridgman in that regard – including stating that Bridgman was a mediocre scientist but got a Nobel after World War II.

2008 11 14 A day ago BMAN and I were leaving together. Next to the elevator he suddenly wobbled and appeared to be going faint. He recovered quickly, however, and I helped to support him for the walk to his apartment. Today I said to him that such episodes of imbalance are in themselves possibly not a big problem, but that for aged people a subsequent fall

Page 107 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 38/66 could be much more of a problem. He said that the recent episode was a consequence of not having slept well the night before.

2008 12 02 We have not worked on the memoirs for weeks. The BMAN has been caught up in a lot of emails, especially in relation to the re-printing of the (Mis)behavior book, and most recently getting him nominated for the National Medal of Science. For some reason not entirely clear to me, he felt it extremely important to write his own nomination. The pleasure for me was that he gave me three examples of three prior nominations that he had obtained from IBM Yorktown – two that succeeded (Wolf and Japan Prizes) and one that failed (last years NMS) – and asked me to draft his nomination. So, I did that, and it did serve as a significant foundation for the document that we prepared after two hard days of work.

He has me performing a lot of actions to keep his Vita up-to-date and free of minute errors. He is also very concerned that all web-based biographies are scrupulously correct, so for example he wants to have me correct his wikipedia entry. He would explain his concerns in terms of, like, friends informing him of information that they think is wrong. Since this journal is really more about me than it is about him, I have to reveal that his self-aggrandizing obsession with what the world wide web says about him is wearing me a little thin around the edges. A major part of my being ticked off is that he appears – still – to be uncertain about how to finish off the memoirs. This in spite of repeatedly asking me how to handle the editor’s remarks, and informing me how they have conferred by telephone since the email in which (I read) the editor’s primary concern.

I get a lot of pleasure when we have to compose a letter in French. He will explain, for example, that repetition of a phrase in French is not so deadly as repetition in English.

He told me a story about how proud he was when a mathematical question relating to the “unicity problem” – which he said, I think, has to do with the fact that the same function can have more than one Fourier representations – was shown by him to have a very brief solution in place of a forty page publication.

He can be quite critical of other people. For example, he said that some people are very organized, and others seem to have “rigor mortis from birth.”

Since the number of hours I am putting in for the BMAN are diminished, and since a number of high school teachers email him with requests for someone to come and give a talk on his work, I came up with the idea that I could be his Ambassador on Fractal Geometry. For a fee (I am thinking $500 for one hour) I would travel and do a bang up job. Carolyn and I would create one of our masterful powerpoint presentations, and it would include a brief introductory video showing that I actually associate with the Great Man. I am trying to figure out what a multifractal really is.

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2008 12 04 The BMAN adds to his doubts about the existence of someone named Euclid also doubts about the existence of Shakespeare. “He was just an actor.”

I had to tell the BMAN that Carolyn and I have a money problem – in the thousands of dollars – with the town we live in, that this is a new problem at a bad time, that a lawyer will be involved and we need to sue the town. I told him I have four options: to get a job in a depressions that matches my skills but likely not in engineering; to get paid more per hour by him; to put in more hours with him; or to seriously develop with his cooperation a powerpoint presentation for high school and other audiences. He sympathized by referral to hard times that he had lived through, and I said I felt embarassed mentioning to him my little life. He said more hours are hard to arrange because ofhis health and travel plans. But, he said income from new publications in the future could lead to more pay. And, finally, he is quite enthusiastic about creating a presentation, and, indeed he might be able to use it himself. This would require collaboration.

2008 12 14 We came across a web-site where a blogger opined that the BMAN should get a Nobel prize. I told the BMAN the reason he will not get a Nobel prize is that there is no Nobel for a Great Scientist. I mentioned Newton and Poincaré as generalists who would not have gotten a Nobel for the entirety of their work. I also have in mind that Einstein’s Nobel was limited to his work on the photoelectric effect.

The BMAN has a seemingly self-aggrandizing obsession with knowing about and referencing on his web-site all books on fractals.

A conversation came up about whether the BMAN should write another article for Scientific American. I jumped on that, saying I would be perfect for doing the illustrations. Subsequently I have composed an outline. I am toying with the idea of creating a very rough draft and submitting it to him. I would include a careful response to the Elliot Wave people. But for that I guess I have to get the Elliot Wave book. My basic thought is that Elliot in the 1930s was one among many who had the idea of a recursive self-similar construction (Koch, Weierstrass,…). Furthermore, and this is what I have to check, the Elliot Wave people don’t get that the BMAN’s construction is a multifractal and hence has intrinsically a probability distribution and its attendant parameters for modeling price series.

2008 12 15 The BMAN increasingly says he needs a vacation. His emails to some people include apologies for delays based on his being an old man. He said that when he can’t work anymore he looks forward to reading, listening to music, and taking walks.

He went to a concert for elderly people, and gestured about folks using walkers. An old lady fainted and a doctor had to be called. The BMAN indicated this kind of thing could be in his own future.

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2008 12 26 The BMAN has been offered an opportunity by Yale University to receive a grant to help defray the costs of finishing his memoirs. This includes the money has paid me out of his own pocket for the last year or so. I have been instrumental in drawing up the budget spreadsheet and the composition of emails to move the process along. At one point he mentioned how reasonable was my offer at the beginning of my offer to help him out. I said, “I have an ulterior motive which seems to be coming into fruition.” I said that it is not now time to discuss this, and he agreed. Part of what I have in mind, and we have discussed this, is the possibility of me creating a powerpoint presentation on his work for high school students, and getting paid to deliver it. This is fine with him, but I have indicated that I would need access to him for preparing that – and he has been enthusiastic. He also knows I would like to write articles about him and his work and get published. I have made it clear that I need more money.

Today by telephone he told me that he has finally come to a plan for finishing the memoirs, and said that when it is all done there will be a party. The other day he mentioned this, and that there will be a gift for me.

But another thing on my mind that has not been mentioned is along the lines of asking him to write a Foreword for my book on mathematical mechanics. He is aware of my resolution to write this book and is always happy to hear that – when not working with him – I am working on it.

The other day we came across a web page about a recent winner of the Abel Prize. He said the recipient is either bottom of the 2nd tier or top of the 3rd tier of mathematicians. Then he went on to ask me if I know the difference between the Swedish language and the Danish language. I said no. He went on to explain that Swedish, Danish, Norwegian all come from the same foundation, but that Swedish is like singing, and he goes, “La la laa,” waving his hand in the air, and that Danish is more like, and then he makes a very funny face and goes, “Gr hag glgsh.” I laughed a lot.

The BMAN has opinions about a great many people he has known. He was particularly shaken to hear of the death of his good friend Carleton Gajdusek.

Wikipedia: “Daniel Carleton Gajdusek (September 9, 1923 – December 12, 2008) was an American physician and medical researcher who was the co-recipient (with Baruch S. Blumberg) of the Nobel Prize in Physiology or Medicine in 1976 for work on kuru, the first prion disease ever described. His later life was marred by a criminal conviction for child molestation.”

From his memoirs I know that the BMAN and Gajdusek became friends at CalTech. At one point I said to the BMAN, “He had a hard life.” As usual, the BMAN corrects me, “No, a complicated life.” I asked him whether the concept of “charisma” is of any use to him in his opinions about people. He said charisma can be learned, that some very dumb princes were given lessons on how to appear charismatic. As for real charisma, he mentioned Robert Oppenheimer and Pope Jean Paul, both of whom he met.

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The BMAN has me type some emails in French. (He knows I enjoy that.) Recently he commenced such a letter with “Mesdames, Messieurs.” I said, “Ah, that is logical.” He said, “Well, it’s medieval.”

2009 01 05

In connection with a major – and belated – project to get the preparation of the memoirs funded by a foundation, I was requested to supply by email a brief description of my responsibilities. Here is the body of my response.

My doctorate, in mathematics, was granted in 1973. However, my primary vocation was in computer engineering. From 2005 to my retirement in 2007 I taught mathematics in high school. I am preparing to deliver educational presentations on Professor Mandelbrot at high schools based on his life and work.

I.support him on the preparation of his memoirs. One of his key hopes is to awaken the interests of young students in mathematics, science, and also art. My teaching experience is of direct value in this task.

I maintain the computer directory of successive versions and implement his innumerable handwritten or dictated corrections. I perform web-based research on details such as important dates. I offer suggestions on style. More generally I am one of several readers who provide feedback to him.

A severe challenge is to stay within the length limits imposed by the publisher. Therefore, the memoirs, especially the notes towards the end, must be supplemented by elaborations provided on his comprehensive web-site. I am responsible for re-designing, extending, and updating it for this purpose.

2009 01 09

Sometimes I get a tad turned off by the BMAN for the great care – and great lengths to which he goes – to advance his potential rewards and publicity. Then again, what the heck, his young self gave his old self a lot of great material to work with.

He once wondered what was the genetic origin of his gift. “Was it a great grandfather?”

2009 01 20

The BMAN occasionally gets what I consider to be “wacko” emails – mathematical wannabees who write directly to scientists for approbation. So I open my mouth to him expressing my derision of these people who seek to circumvent the standard peer review process. This gets him going. He has great disdain for the peer review process, especially

Page 111 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 42/66 in physics; perhaps less so in mathematics. He thinks that in physics the process is about 50% wrong, and goes on about examples of trash that gets accepted in spite of reviews.

I would like to invent a “Turing Test” for market models: the winner is the shortest algorithm that generates price series that fool the eye the best.

I am consciously trying to be the me that I would want me to be if I were him.

2009 01 22

Occasionally while working shoulder to shoulder with the BMAN some kind of resentment wells up within me. It is a mixture of envy and impatience. It is a rare feeling. Less rare is affection and occasionally, a laugh within at re-reading some funny story he tells in his memoirs – bear in mind, I have been through the memoirs a number of times. All in all, dare I say it, this is a lovable man. But for me it is not the same as the unconditional sort that I feel for Eilenberg. When I think of Eilenberg, even now years after his death, my breath shortens. Ah well, sipping Glennfiddich while I write may have something to do with it.

2009 01 23 I mentioned to the BMAN Freeman Dyson’s article on “birds” and “frogs” in the current Notices of the American Mathematical Society. I related that a “bird” is a mathematician who defines an over-arching mathematical research program, while a “frog” is a mathematician who burrows deeply into solving individual problems. I said Dyson calls himself a frog; that Witten is a bird; than von Neumann was a frog. So I said to the BMAN, what would you be, a hybrid, a “brog”? He didn’t bite on that one, but instead reviewed his relationship to and opinion of Dyson.

He said that Dyson is a terrific writer, that they are contemporaries, and met when Dyson had been made a professor on a whim of Oppenheimer and the BMAN was von Neumann’s post-doc. He said that Dyson has never been comfortable in his own skin, and so is a strange man. He said that Dyson was influenced by Tomanaga in connection with seeing the equivalence of Feynman and Schwinger’s work [on QED], but was himself not in the same class with those people.

The BMAN said that Dyson no longer works on science, but is involved in the JASON group, to which the BMAN’s IBM friend Garwin also belonged. The BMAN does not care much for what JASON is up to, but that maybe the Democrat’s will pay attention to them.

2009 01 25 Here is a prime example of the BMAN’s poor pedagogy and competitiveness leading to a writing failure. He has multiple projects going on at the same time, with the memoirs as highest priority, of course. But from time to time he needs to edit or even revise his own previously written material, or interviews with him transcribed by others. In particular

Page 112 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 43/66 there is a previous piece which is a chapter in a book. I am responsible for implementing his edits and revisions. Anyhow, in this piece he offers the following rough definition.

“A fractal is a geometric shape, one that can separate into parts, each of which is a reduced-scale version of the whole.”

Noting to myself that a square divided into quadrants satisfies this condition is not a fractal, I took it upon myself to revise his writing, yielding

“A fractal is a geometric shape with the property that it can be separated into parts, each of which is a reduced-scale version of the whole and has that same property itself.”

Of course I alerted him to my change. Now, sometimes he accepts small changes that I suggest without comment. Once in a while he rejects my change, pointing out why he had it right to begin with. A lot of the time he replaces my change with a better one, and I have to agree that he is right, and suffer the ignominy of having been bested by a genius. But this time he accepted my change, but his competitiveness led him to add a parenthetical “improvement”:

“A fractal is a geometric shape with the property that it can be separated into parts, each of which is a reduced-scale version of the whole and has that same property itself. (The trouble is that by this definition an interval or square are fractals. This problem is familiar in a classical context, insofar as the definition of complex numbers does not prevent them from being real.)”

On reading that I realized the underlying problem with all three definitions. I knew he was anxious to get this piece emailed away under a demanding deadline, so I said to him, “Well okay, you don’t have the time to go over this. But I know what the real problem is.” He urged me to go ahead and email the thing as is.

The underlying problem is a failure to distinguish between process and product. A fractal is a geometric shape, all right, but it is the product of a process, and backing out the process from the product involves arbitrary choices of subdivision. In other words, a “fractal” cannot be defined without mentioning the process by which it is generated.

2009 02 01

As we approach conclusion of the memoirs project – I call them “the Benoirs” – my thoughts turn to what comes next, especially to me.

1. The BMAN and I collaborate on a powerpoint presentation that I would charge money to deliver.

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2. The BMAN writes a foreword to my book.

3. The BMAN writes a letter of the “To Whom It May Concern” type like the one that Edmund C. Berkeley wrote for me.

4. Give me a raise from $25 an hour to $30 an hour.

5. Or, categorize my rate depending on the task: $25 an hour for email, editing, and web-search; $40 an hour for web-site design and LaTeX work.

6. Suggest readers for the draft of my book.

2009 02 02

At my BMAN desk I sometimes have this (very well-hidden) “chip-on-the- shoulder” feeling. Definitely it is connected to the fact that I give the BMAN a fantastic one-stop-shopping deal. I should also add that his enormous self- centered obsession with himself grates on me. Pouting, I say I’m a prolific genius too, except I haven’t published, haven’t changed the understanding of the world, and haven’t impressed anybody else much at all.

2009 02 07

Everybody knows the expression for the big shot in a company – “the big cheese.” But the other day the BMAN was expressing an opinion about the ineffectual but self-importance of a vice president at a publishing company. He said, “In French he would be called ‘une grosse legume’.” That is, “a big vegetable.”

2009 02 24

Ah, finally, I caught the BMAN in a geometry error. I don’t remember the details, but we were resizing an image in a document to somehow coordinate with the width of a column of text. The image is a rectangle and I was dragging the lower right corner and he said a certain angle should be 45 degrees. I said that would be okay only if the image was square. He acknowledged his error.

2009 03 03

The BMAN is composing (yet another) brief article on his theory of price behavior, and making (yet again) his point about price discontinuity – contrary to Bachelier’s assumption of continuity – being an intrinsic feature of price series. I said, “Continuity would be a consequence of the Boltzmann distribution, right?” He said, “No. Of inertia. Little hits can’t change momentum [of the big particle] much.”

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In response to an email asking about the relation of statistics to self-similarity, I said, “So he wants to know about multifractals, right?” The BMAN said, “Not necessarily. Look at a coastline.” I replied, “But if you raise the water level by six inches you get a completely new coastline, so what really matters is the mountain – and that’s not random.” Thinking now as I write that, I know it’s wrong. (In fact, excellent fake mountains have been generated using multifractals, which to the best of my understanding do involve random choices.) But he said something to the point that science deals with simplified projections of reality.

2009 03 05

Aliette has become an Apple MacBook user, primarily for email communication with relatives and friends. Along with other people she consulted, I had suggested that choice of platform. I also volunteered to help with hooking it up and arranging printer, modem, etc. for wireless use in the Mandelbrot apartment.

On the way from the office at IBM over to their apartment to do this, on a clear not-too-cold brilliant afternoon, I asked the BMAN, “In your long life, did you ever suffer? Were you ever so low emotionally and psychologically that you were crying by yourself?” He said no. I persisted, trying to determine what emotion he had had at his lowest point. He said, “Fear.” See if you can find out why he experienced fear by reading the Benoirs.

2009 03 20

I said to the BMAN, “I really need to be sure about something. Pictures of the M- set are not pictures of the M-set – they are pictures of points near it. The M-set is actually all black.” The BMAN said something that qualifies that assertion, but definitely agreed that most pictures are not pictures of the M-set.

2009 03 22

My full evaluation of the BMAN must be postponed until the “day of reckoning,” the day he declares that we have finished preparing the Benoirs for publication. But already I say this.

One hears often, “The rich get richer and the poor get poorer.” Those who deplore this as an extremely unfortunate if not immoral inequity are excoriated for advocating redistribution: they are called “socialists.”

As one at the low end and as possibly the last “helper” to someone at the high end, I argue that the anti-redistributionists are failing to recognize an overwhelmingly important ingredient of success: good luck.

The BMAN had the good luck of a genetic endowment, his “freakish [mathematical] gift,” as he himself calls it. And time and again, he had some good luck in his career, for example joing IBM just when it was ramping up for its “Golden Age of Science,” as he calls it.

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Now, these notes are not called “Me and Benoit” without reason. They are about me, then about him. I have had some great good luck from time to time, to be sure. But the fact is, I do believe his good luck in encountering me is a far greater boon to him than my good luck in encountering him. My gripe is that that is not fair.

I am his perfectly resourceful, reliable and efficient helper. I provide to him several hundred dollars worth of hardware and software tools gratis, because they make my life easier. My extraordinary wide range of skills (including familiarity with the French language), and physical strength (including mechanical skills in creating mountings for sculptures and attaching shelving to walls), have provided him with low-cost “one-stop shopping” for help in a large number of areas (web searching, web-design, text and photo-editing, English usage, and so on – including installation of the wireless MacBook Pro system with printer for Aliette; the use of my laptops when we work at his apartment). He borrows books from me, such as memoirs of E. O. Wilson, and the thesis of L. Bachelier. He took it for granted – and I complied – that he could have me use my name to front for him in reply to somewhat flaky email correspondence. Today he asked me – and I complied – to telephone a non-certified person who asked for his permission to transmit the new foreword of the re-published (Mis)Behavior book to the Obama administration. He asked me to be polite, not knowing that I have fifty years of experience being polite on the telephone with vendors, salespeople, technical support personnel, and so on. Recall, Aliette called me an “angel” to them. So, what’s the beef? Simple. Recently I had to borrow money at large interest rates just to cover the recent mortgage payment.

Like I say, I make his life a lot easier than he makes mine, and there must be a day of reckoning. After the Benoirs are done.

The BMAN received yet another invitation to give a talk. He said that in the old days he could do three things at a time, but now he is down to two at a time.

2009 03 23

The BMAN repeatedly comes back to elicit my opinion on how to organize chapters of the Benoirs to cover iteration, finance, and his work in science generally. He mentioned in particular that he wants to include the “mountains.” I said that in my opinion the mountains should be a paragraph in the chronology where multifractals were introduced, no more, but that there should be chapters on Iteration, Finance, and Physics, with the latter broken down into geology, hydrology, turbulence, and so on. Today I also told him that I did not expect my suggestion to see the light of day, because I know he is very competitive and no matter what I say he will come up with something better. He replied that he has a special perspective on the book.

I asked Carolyn to explain what made her tug on my arm – at that IBM social function – to introduce me to the BMAN. She said that she was talking to him

Page 116 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 47/66 something about powerpoints and how he found it incovenient to be turning back and forth between looking up at the projection and looking at the audience. So, she told him right away about my T_Rig product, which, of course, obviates that twisting and turning. I have to say that when I showed it to him some weeks later he had no interest other than observing that his father had been an inventor whose projects in that direction did not make much headway.

Then I asked Carolyn whether, when she met him, did she know about fractals and his name. She said that of course she did. It was part of her general education to know about fractals.

2009 03 25

The BMAN receives notifications of physics developments called “This week in Physics.” One arrives with announcements of two articles, “Order out of noise” about amplifying a signal by adding noise (!), and the other “Turning one way” about a dynamical symmetry of a nanoscale molecular simulation. He asked me to print the abstract of the latter article. I said, “Why that one?” He said, “I am curious.” I said, “You’re not curious about the other one?” He said, “Well, okay, print that one too.” We laughed heartily.

2009 0 401

I catch the BMAN using double negatives. Today he used an elaborate one in an email to an intellectually giant friend. The BMAN said, “He likes quadruple negatives.”

2009 04 02

In his seminal paper on multifractals (“Intermittent turbulence in self-similar cascades”) the BMAN acknowledges the people who carried out the computer simulations. In my zeal to understand multifractals I hoped the paper would display the results of these computer simulations, but it does not. I said to the BMAN, “But you’re the picture man!” He replied, “It didn’t need pictures.” What a thing for an iconodule to say.

2009 04 03

In an email the BMAN pointed out the name “Guerra” and said, “He should change his name to “Pace.”” I laughed out loud and said, “He should change it to Tolstoy and get both at once.”

Late Friday afternoon, I walked the BMAN from IBM Research back towards his apartment. I said, “My greatest pleasure this week was finding an ordinary differential equation relating specific heat to thermal conductivity. This connects Fourier theory of heat to Carnot theory of heat, and I haven’t seen this equation in any of the textbooks.” In truth, I have did a web search on measuring specific heat and thermal conductivity and turned up a patent for a slick device for measuring

Page 117 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 48/66 them simultaneously. But I skimmed the text and didn’t understand the equations. Anyhow, I worked it out for myself and then went back and saw that the inventor really does appear to be taking advantage of the same identity I had found.

I asked the BMAN if he might put me in touch with a very patient expert on thermodynamics in the Cambridge area. He said that he is always interested in thermodynamics. I said I could not understand anything he has published on statistical thermodynamics. He proceeded to say the only person he knows is over 100 years old and hospital-ridden, namely, Laszlo Tisza. He said he will tell me a story about him – it seems the guy had a tough time as a scientist.

2009 04 06

When an article about him and his work appears in print – especially in these financially critical times – the BMAN anxiously has me reformat the article in a WORD document so that he can email it to selected parties. When I suggest that merely sending the web URL, he insists that he doesn’t care for the web layouts because the ancillary material surrounding the article might distract the reader from him.

2009 04 13

The BMAN encountered an article about a mathematician named Gromov receiving the Abel Prize, so he wanted to visit the web site of the announcement. He said, “We got the Wolf Prize together, he in mathematics, I in physics.” Referring to a story he tells in the Benoirs about how the physics community did not give him “props” for his achievement, I said, “I hope he got more approbation from the Mathematical Community than you got from the Physics Community.” He said, “Well, at Yale the Physics Department didn’t throw me a party. But the Mathematics Department did.”

2009 04 16

Today is Thursday, and the deadline for delivering four chapters of the Benoirs to the published is next Tuesday. The BMAN expresses how much he hates this pressure, but we plunge on: he revises and tweeks, I implement and print.

2009 05 07

I alerted the BMAN to my dependency on working for him to get by. I hinted that when the book is done my finances are going to be taking a hit.

I asked the BMAN whether he has notes from the course on statistical mechanics by Tolman that he attended at CalTech. He just pointed at his head. He went on to tell that he had belonged to a music listening group that included John McCarthy, and that McCarthy introduced him to Mahler.

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By the way, at one point the BMAN said he is self-conscious when he is being photographed. He said he has a reflexive move to hide his double-chin.

2009 05 11

The BMAN is constantly talking to publishers in Germany and US to push back the delivery of the Benoirs. One sticking point with US publisher is the front matter. There seems to be agreement to have two pages with pictures in color and black-and-white, with a foreword in which the BMAN urges the reader to examine the pictures there, and within the book, as soon as possible. Another sticking point is that the BMAN wants to add entirely new chapters at the end of the book.

I insisted on understanding why he wants to add these chapters, at this late date. The small reason is that he feels it is needed for young people to have a better idea of what a self-education is like. The big reason is that – evidently – friends who want to nominate him for awards need more information because he is an “inclassifiable” scientist.

I believe the BMAN’s main concerns are that he has a wide circle of friends who he feels must be made more aware of the full scope of his work; that he wishes to provide a financial legacy for his grandchildren; and that he is increasingly close to being dead.

The BMAN is resigned to not getting a Nobel. He often mentions how it is not on his mind.

He is less concerned about being well-known to the general public for any particular achievement, than that he is anxious that a certain few people are fully aware of the breadth of his work. It is almost as though he is writing the book for his friends who need the story to prepare nominations for more prizes and awards. I asked him why, after the book is seemingly done, he needs to add new chapters. He said, “Because that is the gold. All the rest is silver and copper.” (By the way, a persistent metaphor in his book is that of an “alloy.”)

In adding new material regarding the current World Wide Financial Crisis and how his work on prices relates to it, he makes it clear how much he was hurt by Cootner, even more than he was hurt by the Simon broohaha.

2009 05 21

The BMAN has now missed several days. His basic explanation is that because the April 21 deadline was missed – in spite of a massive effort – the deadline for publication has been pushed back for a couple of months. Consequently, he says, he is suffering the after-effect of an enormous release of pressure. Add to this that

Page 119 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 50/66 his sister-in-law in France has fallen ill, and the man has had the wind taken from his sails.

2009 05 26

The BMAN affirms he is proud, not arrogant, not self-centered. “I’m a freak, but I can’t say that.” “I never wanted to have power. There is the moral power of my parents’ families, but no political power.” “The College de France is a place of thinkers, but I turned down the job.”

The American publisher’s October deadline for the benoirs has been moved back to February.

“I must go beyond a standard memoir.” “Pantheon has cancelled half their books.” “Old men become indecisive. I need a break.”

2009 06 04

For almost 3 weeks now, the BMAN has cancelled my trip to Cambridge for work with him. Aliette seems to be doing okay handling his Yale WEB MAIL. I sent him this email recently:

Dear Benoit,

I hope that when we reconvene there will be time for

(a) a check; (b) a signed copy of (Mis)Behavior of Markets for my friend Seth Yorra ; (c) starting discussion of how we might create a separate payment arrangement for re-vitalizing your web-site.

I had the wild idea you might find it useful for me to tape record your answers to provocative questions such as

(1) Are you the only fractalist? (2) What do you think of Andre Weil the man? The mathematician? (3) Do you feel differently about the mathematical community than about other scientific communities? (4) What do some people not understand about you as a person? As a scientist? (5) Did you realize your dream? What remains for others? (6) What is the future of the physics profession? (7) What might you have done better? (8) Of what in your legacy are you most proud? (9) Are the squabbles among scientists more vicious than those among other intellectuals?

I would gladly type the transcript if you think that would be useful in some way.

Sincerely, Ellis

He sent the following email to his German publisher:

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Dear Britta,

The last 10% of my book--alas--keeps changing and is not finished yet. But it proves very hard to organize.

Why not simply cut it off? Why not keep those pages for another occasion? Because I feel the world situation makes them simply essential.

This delay greatly embarrasses and disappoints me with myself. These are not just shallow words. My high standards and ambitions are unchanged but unavoidably somewhat slowed down by age.

This letter was delayed because I hoped to attach those essential missing pages. But they are taking far more time than I thought and I am still incapable of giving a firm date.

Pantheon has postponed the US publication to February.

I am infinitely sorry to create such complications.

How do you wish to proceed?

With apologies and best wishes,

Yours very sincerly,

Benoit

May, 2009

Ellis to Carolyn: “You are way too hyper.” “I’m just very sensitive and aware of sounds. Did you hear about the woman with lime disease?” “What happened?” “She wasn’t diagnosed early enough and was in excruciating pain from every sound. She begged the doctor to sever her ear drums. He wouldn’t do it. So she killed herself.” “Oh. Did you hear about the actress in Paris who hung herself?” “No. But women usually take pills or leap off of a building. I wouldn’t commit suicide by jumping off a building because I am afraid of heights.”

Television commercial for a tiny bug-like car. Carolyn: “Great. A car and a coffin all in one.”

2009 06 12

I haven’t seen the BMAN for about a month. Today I had to visit with him to get a check for work previously done, and to feel him out about future work. I told him and Aliette that my physical health is very good, but my financial health is terrible due to several things going on at the same time.

He appears to be very tired and was heading for a midday nap, and said he hasn’t done any work on the benoirs for a week; he is paying the man Joel Segel for consultation on completing the book. He said, “I detest this book.”

I wanted to promote the idea of revising his web-site, and he is open to doing so – as previously said – especially to present material that is a spillover from the limited space in the benoirs. He asked me what I have in mind. I told him $40 per hour and 120 hours ought to do it. He wanted to know how I would go about it. I

Page 121 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 52/66 said that he and I would sit together and look at the web sites of scientists and mathematicians and get his judgements. Then I would design a new web-site and we would iterate towards his preferences. We actually visited mathematics faculty web-sites maintained by Yale, and saw that perhaps one in three information pages of faculty members actually have links to personal web pages. None of those pages had even remotely good design or as much stuff as he has. He argued against rebuilding his web-site from scratch, saying he much preferred revising it step by step. I argued that that can be done, but that since the site was poorly done to begin with (he said it was made by a bunch of teenage students found around Yale) it could really take longer that way than starting from scratch. I am sure he remains unconvinced about that. In any case, we went on to discuss how I would get paid. He certaintly would prefer the money not to come out of his own pocket, and the main question has to do with taxes. My task is to find out what is the income threshhold over which my earned income would be taxed, in view of my status as a retired person receiving social security payments.

I also have the wacky idea of forming a non-profit corporation in Massachusetts whose purpose is to advance public understanding of mathematics, and accepting charitable donations on my web-site.

He also spoke of some roundabout method of payment involving him paying me, me paying him back, and then me being compensated by Yale.

I gave him a printout of an article from wilmott.com entitled, “Mandelbrot’s Extremism.” He studied the abstract of it, but said nothing.

For good news I showed him Henry’s 2 Fs 2 As for Fall 2008, and 5 As for Spring 2009. He said, “Excellent. What is his major?” I said, “Philosophy, and possibly another major in Anthropology.”

2009 06 13

There are laws against a 501.3.c non-profit organization using funds for “private interests.”

A section 501(c)(3) organization must not be organized or operated for the benefit of private interests, such as the creator or the creator's family, shareholders of the organization, other designated individuals, or persons controlled directly or indirectly by such private interests. No part of the net earnings of a section 501(c)(3) organization may inure to the benefit of any private shareholder or individual. A private shareholder or individual is a person having a personal and private interest in the activities of the organization.

I forget to mention that yesterday I also configured Aliette’s MacBook with Benoit’s Yale email account in her MAIL program. This pretty much completes the transfer of my responsibility for handling his emails to her.

If I can steal some time away from working on my book – I am currently very deeply involved in straightening out a Kolmogorov probability space interpretation of the statistical mechanics argument for calculating the probability that a system of an

Page 122 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 53/66 ensemble is at a specified energy level – then I might go ahead and use my beautiful program 1nce to generate the HTML page for a subset of the links in the BMAN’s 56 page CV. I could customize the page and put it up for his review. This would be a “good faith” piece of work, but I would carefully record the time to do it, and bill for that time at the $40 per hour rate at some later time with his consent. I have basically run out of money, and cannot make the June mortgage payment.

2009 06 14

The BMAN telephoned at 8:30 PM to say he does not expect to have any work for me for two or three days.

But he did want to discuss – seriously – his web-site revision job, which today I told Henry and Carolyn is my Plan A for getting through the current Cooper Financial Crisis. He started by saying that the demands on him for refining the chart I made of all of his papers – with my preliminary assignment of scientific field to each one – would be enormous. I countered that first of all, that was a quick first draft, and that I could refine it a great deal on my own. Second, I insisted that the chart would be at most on the order of 15% of the job, that it is certainly not the most important part. He quickly said, “What is the most important part?” To which I replied that the objective is to make his scientific work available to search engines, and that using his Vita - Microsoft Word document – as a source of links to the pdfs of his papers is completely wrong. I explained that as such the Vita contents are not readily accessible to web search engines. (For example, at random I selected the paper title, “The Pareto-Lévy law and the distribution of income” from the Vita and did a Google search. It produced over 600 results, but the top 30 certainly did not link to his Vita or his web-site.)

He reiterated the story about how the web-site has a checkered history, and that he has given it too little attention. I went into sales mode and said the reasons why I must be the right person for this job: (1) I am a mathematician, (2) I have familiarized myself with his entire body of research for over a year, (3) I understand the objective of the web-site, (4) he and I work well together, (5) I have considerable experience at creating mathematics- oriented web-sites. He seems argumentative if not adversarial. Nevertheless, the conclusion was in my favor, because he asked me to prepare a 1 page proposal. This will have to be the greatest proposal I have ever written.

2009 06 15

My guess is that the primary intellectual-emotional tension tearing away at the aging BMAN is between the enmities of André Weil/Bourbaki style mathematicians, Herbert A. Simon and Paul Cootner style economists, the Elliot Wave “theorists,” and Jaffe- Quinn mathematicians, and the accolades bestowed upon him in the form of prizes, awards, honorary doctorates, and so forth. In other words, he has simultaneously contributed to a great many branches of science, and severely polarized some of them. He cares too much about what other people think of him.

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One more very important thing to know about the BMAN is that he freely edits the truth. That is to say, for example, he will meticulously edit the transcript of an interview with him. Or, he will modify a photograph to eliminate what he considers inessential. The choices he makes are governed by an ingrained aesthetic sense, together with the impulse to make himself appear as good as possible. These are conscious choices, and wherever it is obviously the ethically correct move, he publicly acknowledges such modifications have been made.

2009 06 18

On Tuesday, June 16 I emailed to the BMAN a 1 page proposal:

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On Wednesday, June 18 he called to say that he and Aliette were having a printer problem, so he couldn’t read the proposal. I offered some telephone advice on dealing with the printer. I haven’t heard from him since, and am rather anxious. Do I have a client for my web revision service, or not? This is very important, because I do not have a plan B.

The best would be if he writes me a check for $6,000 and I promise to implement everything in the proposal by, say, the Fall of 2009. I would perform the tasks

Page 125 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 56/66 step-by-step with full micro-management. Full hourly records would be maintained and reported monthly. If a grant from Yale or elsewhere is obtained to cover this contract, then the money would be returned to him. Presumably research & development grant funding would not taxable income to me, but he would have to inquire at Yale about that.

Although by telephone the other day I said 120 hours, that was before filling out details as in the 1 page proposal. The total time estimate for everything in the proposal is 150 hours, so that is where the $6,000 figure comes from. I would include maintenance for 2 years in the quote. This would cover repair of bugs discovered by visitors and addition of new material as sales of the memoirs mature.

Aha. I just received a telephone call from Maida Eickenberg at the IBM office informing me that Tuesday next they are tearing down the cubicle in which I have been working for the BMAN. I said I would come in tomorrow and move his belongings from the cubicle into his office, and take my belongings home.

She said we would call him to alert him about this, and sure enough, he called me a few minutes ago. He said he told her he would like me to continue to have a space to work there – okay, that is news – and that he expects to have work for me next week using material developed from his paid consultation with [Joel Segel]. I said there is a problem with me continuing with him next week because my dire financial situation calls for me to search for higher paying work elsewhere. I said that I did promise to finish the memoirs with him, but I could not have anticipated that it would take this long. He said we will go over the problem tomorrow at the office – and he also made it a point to say that he doesn’t want to do some of the things in the proposal. Let me guess: he will want Functionality (1), and only the Full Vita equivalent of (2), maybe a truncated chart of papers in (2); maybe a little of (4); only the first part of (6); memoirs part of (7); Graphics Design (1), light version of (4). This much is what I call 120 hours.

2009 06 19

The BMAN and I met at the IBM office. We discussed the latest letter from [Joel Segel] and the BMAN asked for my opinion about the items Segel suggested for omission. I agreed with some but not all, and said why for each. He said that he is still uncertain about how to add 4 chapters, on the Mandelbrot Set, on prices, and on mountains, plus a chapter on everything else, all to serve his “ulterior purpose,” which is to acquaint those in one field who don’t know what he accomplished in another field.

Then we discussed the proposal and I answered his questions, and showed him on-line evidence for flaws noted in the proposal. Bottom-line? He agreed to $5,000 for his choice of tasks, pretty much along the lines of my guess above. I should prepare a quote, as I said, on my letterhead as a consultant (see above). He will cut a personal check early next week – he does want to talk to Yale about

Page 126 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 57/66 getting reimbursed by the money ($45,000) that supposedly is in the bag from the Sloan Foundation. He says that there is “stupidity” going on with that whole process. I figure this buys me about 6 weeks. On the other hand, he is likely to be producing material for me to integrate into the book, so that will be – perhaps – back to a “normal” schedule. He was surprised to hear that we had not been working for about 5 weeks (May 18 – June 19, actually. I just checked.).

2009 07 11

The BMAN and I signed a contract for $5,000 to revise his web-site and he paid half up front, with the balance due “on approval.” Over the last couple of weeks I have completely revised the home page, the Table of Contents, and various additional parts – all ready for his “approval.”

On the other hand, he has not called me for a whole week – which is very unusual – so I have no idea where we stand on this project or on his book. I will call him on Monday July 13 if I don’t hear from him by noon time.

I do wish to record that he gave me two books that are continuing to be very useful for my book: the collected works of Gibbs on thermodynamics, and a book on free energy by Lewis-Randall. These books are rife with intuitions I need.

2009 07 21

The BMAN has not called until today. That is undoubtedly because of this email I sent to him yesterday:

To: [email protected] From: "Ellis D. Cooper" Subject: Web Site Revision Cc:

Dear Benoit,

I look forward to your approval of the direction I am taking with the revision of your web-site. With that I would expect to receive a check for the balance as per our agreement. Of course I will then proceed to implement all changes you require to match your expectations.

In the best of worlds I would complete the work we have agreed to in the month of August, except of course for adding material to be provided later on in coordination with your memoirs, and for other long-term site maintenance as discussed.

The "shadow" web-site can be examined at http://www.distancedrawing.com/mandelbrot . I estimate that about 20% of the whole job is done. A task not yet undertaken is to convert every WORD file linked by your Vitae into a PDF, and correct those links therein. However, the bulk of the remaining work is on presenting information about each of the approximately 400 PDF files. That is a job that will certainly require you to give me instruction in person before I proceed.

It will be a pleasure to see you again when we meet for continuing the work on your web-site and on your memoirs.

Sincerely with warm regards,

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Ellis

Ellis D. Cooper, Ph.D. 978-546-5228 (LAND) 978-853-4894 (CELL) [email protected]

He is dealing with several meetings and phone calls but said we will probably meet tomorrow. Fine.

2009 07 22

We met, it was a successful meeting: I know what to do for his web-site, he gave me a check. So, that email did the trick.

Regarding the memoirs, he said he has lost interest, but has written some new material, and is revising the front of the book. He is minimizing changes in existing text because he doesn’t want to give more trouble to the translators for the German publisher, who – he says – is not happy with his delays.

He asked about my time, and I explained that my book has hit a snag – I am in the midst of an intellectual struggle and am having a mental block to do with understanding thermodynamics because of a certain standard I set for myself. Hence, I said, these weeks I am not writing but studying and researching. So, I said, I have time to work on his web-site.When I said that my study has led me to the book by Herbert B. Callen on thermodynamics, he said, “I knew him. Sometime I will tell you the story.”

2009 08 21

I had to call him because IBM called to say they are dismantling the setup/office that we were using. He had also received that call, and we hope that next week we will meet to have him examine the work I am doing on the web-site, and discuss what he says is a few more weeks on the benoirs before he leaves for Germany. He is very excited about a news story to the effect that the Chancellor of Germany has been reading the (Mis)behavior book.

He also said that he would have me get involved with “Peter” who is or will be working on a follow-up to The Fractal Geometry of Nature, which he is always quick to point out is still selling, but he would like to juice the sales.

He also said he would like my help in sorting out material in the boxes of papers, etc., for long-term storage (Library of Congress? Stanford University?) of his intellectual remains.

Meanwhile, I am headed to personal financial collapse due to lack of adequate income. He is suggesting that he does not have the full level of work for me to which we have been accustomed. I have decided to thoroughly search for work as

Page 128 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\MeAndBenoit091106a.doc 59/66 a science writer, but not until I get a reply to an email I sent to World Scientific alerting them to this challenge of mine.

A reviewer of one of the BMAN’s selecta volumes says, “It is a psychogram of a conceited genius … Can be understood perhaps if you consider that success has been denied to him for a long time.”

2009 09 01

The BMAN never called last week. This is the email I sent to the BMAN today.

To: [email protected] From: "Ellis D. Cooper" Subject: Web-Site State Cc:

Dear Benoit,

The shadow web-site has reached a state for your review. http://www.distancedrawing.com/mandelbrot

The primary effort has been devoted to the Selecta book reviews and to your Curriculum Vitae. http://distancedrawing.com/mandelbrot/SelectaReviews.html http://distancedrawing.com/mandelbrot/Vitae/VitaVerdanaContents090726.htm

Photographs and audio interviews are included in the web-site. The attached journal -- for your records -- contains details of my work so far.

There remain a small number of subsidiary pages to revise, but I thought it wise to pause at this point and await your further instructions before proceeding.

I am investigating the idea of offering my services as a Science Writer via a new web-site. If you could write an endorsement for my new web-site I would certainly be most grateful (samples of my science writing would be provided as necessary).

Sincerely, Ellis

PS. Regarding your question about Wikipedia, perhaps the following article will help: http://www.nytimes.com/2009/08/25/technology/internet/25wikipedia.html?

2009 09 12

The BMAN finally called early this week (today is Saturday) and we reconvened on Thursday.

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He explained his “solution” to the problem of finishing the Benoirs – before he leaves for Europe early next month – which is to write another book. The idea is to accept an invitation to lecture at Princeton University and then publish the lectures. This would give him the opportunity – which I must observe is paramount now in his mind – to set the record straight and publicly that he was the man who knew long ago that there is something severely wrong with mathematical finance. I mentioned to him that I read Paul Krugman’s New York Times article, and that I was very appreciative of the information there, but also aghast that Mandelbrot was not mentioned at all. The BMAN said, well, maybe indirectly, and he also proceeded to give yet another of his character assassinations of Krugman. I told him that I sent an email to Krugman with my article, “Mathematics & The World-Wide Financial Crisis.”

I did give the BMAN a copy some weeks ago; there has been no acknowledgement of it. Nor did the BMAN acknowledge my request for an endorsement in the above email to him.

He also continues to repeat how The Fractal Geometry of Nature is still selling, and he has a little momentum with some other fellow (mentioned above somewhere) to create FGN2. I brought up LaTeX again in that context, saying that writing my book is making me considerably more expert at it.

Speaking of my book, I told him that it is not possible for me to give him the three or four days a week that he said he feels necessary to complete the benoirs, and also to have me resume doing his email chores. Evidently Aliette is balking about having to do them for him. I said that my book is my overwhelming priority this month, especially in view my daughter’s marriage next month. I said I could give him one full day a week, and take home work if he can organize it for that purpose. He was pleased when I said that if I deliver my book in December then come January I will be much more available to him.

I also showed him the work on the shadow web-site, and he immediately got engaged with aesthetic adjustments. For example, on his page for the Japan Prize he decided their logo – which is a red circle offset in a white circle all against a blue square background – is incorrect. He directed me to redesign it so as to include his name and the date of his award in white against the square extended rightward from the Japan Prize circles. No problem, I said.

We spoke of other things, such as my opinion of Obama’s speech on healthcare (I said he has tough going ahead), the BMAN’s opinion of physics and mathematics today (“dreadful”), and somehow we got to a point where he referred to Alexander Dumas’ “nigger.” I jumped a bit to hear the BMAN use that word, but he explained that when someone put his name on a piece of writing claiming it to be his own, he might have had a black man actually do the writing for him, “le negre.” All I could find on web relating to this is:

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2009 10 05 (Monday)

Six days to Nicole’s wedding.

The BMAN seemed to be receding in my rear view mirror, but, he did call last night. He is leaving on Thursday for what will probably be his last European trip. It seems that there is a big up-tick of interest in his work on finance, so he is being invited all over the place to give a talk. He needs me to make some changes in his powerpoint presentation, and asked what I think it will take to do this before he leaves.

It turned out there is maybe a half-hour of work. I had been rehearsing in my mind for some days how I would ask him for a loan to cover 5 months of mortgage payments. So, I just did it. I asked for the “big favor” of loaning me $10,000. (A zillion years ago I asked my boss, Tom Peterffy, for a $3000 loan at a time he was paying me $60,000 a year, and he replied quite brusquely, “I don’t have that kind of money.” This from a millionaire successful Wall Street trader. I did not ask him what “kind” he was referring to.) The BMAN wanted to know what I had in mind, so I explained. We know that the BMAN has not gotten around to doing the paperwork to get from the Sloan-Yale combination the $23,000 that was scheduled to reimburse him for money he paid me out-of- pocket. He had previously told me that according to the rules, Sloan-Yale would have to write the check to me, and the BMAN said that I would then cut him a check for $23,000. The new idea is that instead I would write him a check for $13,000 and owe him the balance. I suggested that over the next six months or so my book would come out. He said that royalties would probably not be immediate. So, I said – and this was part of my thoughts – that I would be available to work for him as much as he needed after delivery of my book manuscript. Therefore, I said, I would stop billing him for my help. The idea is that I would pay him back with time. When I made some noises about this being maybe a “surprise” and “shock” to ask of him, he replied that it is no surprise since I had been saying things for some weeks about my dire financial situation. And as for being “shocked,” he went into his charming mode about being an old man he is not susceptible to shocks any more. He also remarked about how flexible and accommodating I have always been.

We agreed to go over this when we meet on Wednesday to take care of burning a CD with his modified presentation.

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2009 10 10

Well, the BMAN called Tuesday evening to cancel the CD modification meeting. The usual explanation was about being tired from other activities. This cancellation put the kabosh on any momentum regarding my request for the favor of a loan. I have to persist until his return in some two or three weeks.

In any case, I have put on Carolyn's shoulders the burden of coming up with the October mortgage payment, due October 16. Before then, however, we are anticipating Nicole's marriage day tomorrow, so the excitement is corporeal.

I now have 154 pages of rather advanced material for my book, and a definite outline for completing the climactic chapter on muscle contraction. This involves MATLAB simulations, so I am on the learning curve for another programming language.

2009 10 27

This house I live in is a day away from not being able to make its October mortgage payment. This can way somewhat heavily on the creative mind.

I am deep enough into MATLAB to be creating a parallelizable timing machine interpreter with which to finish the penultimate chapter of my book. In other words, there is in the works a chapter on Timing Machinery, my greatest contribution to mankind. The climactic last chapter will have a few different models of muscle contraction, expressed in timing machines, and an explanation – somehow – of how the Theory of Substances integrates in the Power Balance Equation the relation between chemical hydrolysis of ATP, creation of entropy, and force to slide the actin filament. This will inevitably require a clear explanation of the connection from classical thermodynamics to statistical thermodynamics, since everybody seems to be committed in their explanations to use of Caby Tee and "detailed balance", whatever that means.

I attended the 50th reunion of my Stuyvesant High School graduating class. I was dressed in my primary self-image of a black "suit" and gold tie (it was not a suit, it was black slacks and a very dark navy blue jacket, probably Henry's). I bejeweled myself with my wedding ring (three intertwined rings of different alloys of gold) and my watch inscribed by Carolyn for 1980.

My expectations of being at least recognized by a few people were, of course, completely annihilated. Well, actually, I was a bit prepared for that because the final guest list on the reunion web-site had none save Bill Barth any familiar names: no Bob Burchess, no Steve Feinstein. But of those who signed by yearbook, Edward Mandelbaum was there – and he didn't recognize me or my name. He is a shriveled old psychoanalyst living across the stree from the old Stuyvesant building. His wife Bonnie is a heavyset very nice lady. He is totally boring.

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As I stood at the edge of the gathering after arriving and checking my backpack, I observed – somewhat to my chagrin – that my graduating class consists of a collection of well-groomed grey heads wearing suits. Bill Barth showed up, grey headed but with all his hair, a tad heavyset, but clearly a charming man. We hit it off, swapping stories and fooling around. He bought the first round – GlenFiddich at my suggestion – and I bought the second round. While bringing the second round to our table some woman had a siezure and the proceedings were highlighted by that for a while.

Okay, I had a couple of conversations with people I never knew in high school. Very nice. Reading the online publication of hundreds of pages of biographies I kind of already knew this is a collection of high achievers.

BUT, the main sentiment is this. Once again, but now as a fully conscious human being, I realize how truly alien I am. That is to say, for example, in the old days when I attended, say, an afternoon tea at the Columbia University mathematics department, I found myself in the midst of all these superior minds, with no one to talk to. The big difference at the reunion is that I found myself equally alien, except feeling superior to all of them. I asked a neighbor at my table, a Freudian psychoanalyst, whether anybody else at the reunion seriously considered themselves at the brink of the rest of their life. I meant, like me, whether anyone there felt on the verge of his or her greatest accomplishment, hungry for recognition, determined to complete the latest project at all costs. He said that everyone felt that they are at the beginning of the rest of their lives. Not what I meant, but I let that go.

Bottom line: I am an alien. With all of his "freakish gift" and "maverick" existence, the BMAN could not possibly in my humble opinion understand what I am.

2009 11 06

Breaking News Alert The New York Times Fri, November 06, 2009 -- 8:32 AM ET -----

U.S. Jobless Rate Hit 10.2% in October, Highest in 26 Years, as Employers Cut Payrolls by 190,000

Yesterday I was in considerable discomfort as I related to the BMAN at his apartment the severity of my personal financial crisis in the context of the severe decline in income from him since last May.

I had just completed solving yet another emergency need of his, to scan figures from a book to send to a person in England for inclusion in another book.

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Then we discussed continuing the work on his web-site, and pretty soon the challenges mushroomed beyond the agreement, including his change of mind about accepting graphic design that I thought was in good shape. (In engineering we would call this "feature creep," where sales would keep adding features after the engineers had already started their design and implementation with a previously agreed budget.)

I had mentally rehearsed how I would explain my idea of how he might lend me money from money that I believe he expects from Sloan-Yale. I also explained how in contrast to my brother I have a long history of borrowing – and paying back – and Professor Mandelbrot related how he himself never borrows, let alone lends. I explained my need and he explained his need for finishing the interrupted work on his memoirs, and his need to provide documents from his web-site because people have complained to him about missing or bad links, etc. He wanted to know whether I would be serving his needs by January 1, 2010, while I was trying to convey that without peace of mind about my financial situation and in the context of being determined to finish my book, I would be finding it very difficult to fulfill his needs. My thought basically has been that now is the time for him to perform an act of generosity on my behalf. Instead he explained how he provides some support to his sons who chose to work in hospitals rather than go into private practice, how one son needs help to finance a kitchen remodeling, and how a grandson evidently needs some kind of special support that costs more than usual. My heart sank, because this is basically saying he can't loan me money out of pocket. I indicated that I would pay him back with my time, but this did not go over very well. As I was leaving he said something about how perhaps I should send a bill to Yale, to which I replied that I just do not know the correct protocol for doing that, that as far Yale knows I am a name on his grant and no more, that I guess I would have to call them to find out how to do it properly, and of course as I expected, he vehemently said I should not communicate directly with Yale.

This was a very unpleasant experience, and I told Carolyn last evening on the train back to Rockport from North Station that before he left for Europe I estimated a 65% chance of getting a loan from him, but now I am down to 30%. Actually, I am a lot lower than that today. In fact I have started rearranging books and file folders as a symbolic gesture of putting Professor Mandelbrot very far back in my rear view mirror.

I am extremely disinclined to put any more of time on solving his problems, listening to his stories, working on his memoirs, or on his web-site. I feel that sure, I agreed to work for him at a certain rate of pay, and I signed a contract for a specific advance payment. He has some regrets about delaying his book and not taking the advice of some friends and relatives to leave it as is instead of trying to add chapters for his reasons as described earlier in this journal.

I committed to writing my book at a time when I had no reason to think that my income from him would take a sudden precipitous decline midway through the year, and I explained that to him yesterday. I said to him that I am on the verge of circling the wagons, and putting my self-interest completely ahead of his interests.

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To be completely frank: at $25 per hour I think he got an extremely generous deal from me, one-stop shopping for a very useful array of skills and professionalism. There is a large gap between his greatness as a scientist, and his generosity towards me.

Then again, I must admit the sensation of having painted myself into this corner, and my very long history of thinking that people would want to give me money or loan me money is a grave personal error that has finally come around to throttle me once and for all.

This has to do with how many times I went to Mommy for money, and she gave it to me if I explained carefully why I needed it.

It also has to do with my long history of not saving money when I had enough to save, but instead spending it on projects with the hope of great success, success that never materialized. The latest such project being the T_Rig business, where I thought for over a year that I was doing everything right: study of the market, product test, product manufacture, web-site development, and so on. I just sort of put my head down and charged ahead, optimistically.

I tell Carolyn the difference this time, with my book, is that well, a big-time publisher believes there is a market, so all I have to do is write the book and finally I will have a great success. Professor Mandelbrot points out that it can take quite a while before I see any money from sales of the book. I replied – and he accepted my reply, I think – that having published a book my chances of employment as a teacher, or obtaining contracts as a consultant, will be much greater.

I also do believe my book will be extremely interesting and useful to a wide audience of students and a certain category of researchers.

At this moment there is a not-quite-overwhelming amount of work to do on my book, I will most definitely need a time extension from the publisher, but I pretty much know exactly what has to be done to finish it. There are a couple of large understanding mountains to master, but the weather is clear and I can see the peaks. There is a lot of writing to do, also, but that for me is easy. Plus, of course, there are some tedious chores, like drawing a few more figures, and composing the index. But it is all doable.

Thus, as my financial condition completely deteriorates, with unpaid bills piling up, calls from creditors, a mortgage re-financing in jeopardy, very little operating cash and a rapidly dwindling amount of credit remaining on a card, I can't think of anything better to do than keep a very level head and write the book.

2009 11 06b Professor Mandelbrot called to continue the conversation interrupted by my departure yesterday to make a train.

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The bottom line is that basically all he wants from me is a few weeks at most to finish the memoirs, and there seems to be no real continuing work. He is hiring someone else to do the "son of fractals" book, someone who has the appropriate mathematical qualifications.

The only money that he will try to arrange for me to receive from Sloan-Yale is compensation for use of my software and hardware on his behalf, which will be on the order of $1,000, and I did leave him a bill for $238.

The rest of the conversation was basically about how in my circumstance it would not be fair for me to promise that in two or three weeks I could work on his memoirs. So we concluded by my offer to provide backup DVDs of his files, since the IBM computer is not accessible ever since it was moved around for the work they did on the offices there.

I could not advance any further with the idea of borrowing from him against future work to be done, because there is not enough to be done. He said that within a year he and his wife will be moving to Newton to a retirement community close to where his son lives, and very inconvenient for me to travel to.

After over two years of working together, and my very helpful, professional and great consideration of his needs, he did not even thank me.

So, I would say this ends "Me and Benoit." 2013 04 14 After me, Merry Morse at IBM Research took on the job to help him. By an automated alert from LinkedIn, her account asked me to link with her. So, I sent a note congratulating her on completing the mission. She replied that after he died she had worked with Aliette for a year to finish it off.

After a good deal of reflection I have concluded that (a) he was indeed generous to me, by giving me several excellent texts on thermodynamics, including the complete works of Gibbs; (b) I was not as perfect as I might have been, having made one very bad error when I sent the editor a complete document instead of only the second half. This gave rise to a problem for him because he only wanted them to see the corrections on the second half and not confuse them about the first half. I had done that basically because I wanted to preserve page numbering in WORD.

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Mathematics & The World-Wide Financial Crisis By Ellis D. Cooper, Ph.D., May, 2009

“Worldwide Worry” “Worldwide Worry” headlined The Boston Globe on Tuesday, October 7, 2008. Above the fold stories included “Ominous signs of pullback by businesses,” and “Markets tumble amid doubts on US stability.” The heat producing this meltdown had been turning up for a year or more. A large number of homeowners had to default on their mortgages; the major investment bank Bear Sterns built a house of cards to make money hand over fist, went belly up and was sold off; housing prices fell and the supply of credit dried up; the financial-services firm Lehman Brothers set the record for largest bankruptcy in US history; the banks in the entire country of Iceland melted away; and the largest automobile manufacturers in the US began to dissolve. Seemingly humble, Former Chairman of the Federal Reserve Bank said, “[W]e’re not smart enough as people. We just cannot see events that far in advance.” The “not smart enough” people were the mathematical economists on staff at the Federal Reserve Board. He implied that if the brilliant people hired by the government – product of the most prestigious business schools in the country – could not predict this calamity, then nobody could. What this leaves out is that several other people did foresee the possibility of the collapse based not just on readily available numerical data, but on a fundamental theoretical criticism of academic mathematical finance.

Alchemy and Free Money Limitless money and limitless time to play with it would be nice. Serious geniuses – including Isaac Newton – pursued the Alchemical goal of creating The Philosopher’s Stone, a special material with the power to convert any material object into gold and to confer immortality upon a human being. They failed. Second best would be a practical method for getting free money. Probability theory originated in the seventeenth century as a mathematical endeavor to solve gambling problems. As in so many other areas grounded in concrete problems – such as geometry – mathematical questions arose in probability theory that demanded more and more abstract levels of research. The work of a dozen or two geniuses yielded by 1900 a probabilistic mathematical machine for reasoning and calculating price and risk in financial markets. Gambling is all about getting money fast for free – with the risk of losing the wager. If there is no magic calculation that guarantees winning, at least one may hope for a risk formula. With that one could judge whether the reward would be worth the risk. Loaning money is all about gambling on getting paid back – with the reward of interest payment. Of course, a loan is not free from the borrower’s point of view, but at least it is fast money. Americans love borrowing. People increasingly “live on credit.” But a century ago except maybe in a bar or country store the idea of “Buy Now Pay Later” didn’t exist. Credit cards did not exist. The virtues of avoiding luxury, of frugality, and of avoiding debt governed many daily lives. However, mathematical, scientific and engineering innovations of the Industrial Revolution sparked huge demand for must-have commodities such as radios, phonographs, washing machines and refrigerators. By the

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1920’s and 30’s these wonderful but expensive things included automobiles. To compete with the Ford Motor Company “Layaway Plan” – basically an organized way to save up for a car – the idea of “Automobile Loan” was conceived, born and raised by the General Motors Assurance Corporation. In other words, General Motors – along with department stores – helped create “modern consumer capitalism.” After World War II the consumer credit business blossomed big-time and GMAC expanded into home mortgages. Banks gamble on their borrowers and big banks gamble big.

What is it Like to be a Mathematician? What has mathematics got to do with our worldwide financial convulsion? For sure, banks, businesses, investors, and traders perform arithmetic calculations – a lot of them, and fast. But there is more to mathematics than arithmetic, and there are mathematicians who are not even remotely as good at “doing the math” as a pit-bull trader. The problem with understanding the difference between mathematics and arithmetic is that nobody but mathematicians know what mathematics is. So, an analogy is needed. Just about everybody with any interest in sports knows about Michael Jordan. His grace, power, and competitiveness are legendary. Certainly he came to the game with extraordinary physical and mental gifts; we also know that he practiced extremely hard to refine his skills, making his performance appear almost effortless. But even if you were severely challenged to sink a free throw shot, you could watch him and feel vicariously what it’s like to fly and twist in space and dunk a seemingly impossible shot. Now, imagine a new kind of basketball played in outer space, with ever-changing teams and players, in a playing space the size of a space station, with multiple hoops, and machine shops where players are encouraged to invent and build inter-communicating machine helpers for playing defense, making multi-hoop trick shots, and so on. And assume that these helper machines can be passed from one generation of players to the next. The differences between this imaginary physical game and mathematics are that mathematics is a real “mental game,” that it is way more fun, and that – most importantly – over hundreds of generations it is cumulatively meaningful. Rigor cleans the window through which intuition shines. The mental gift of mathematical intuition is a light by which patterns are seen. But every light has some colors in it, and different colors bring out different patterns. In other words, mathematical intuition may err. But to reason and calculate correctly about whatever pattern is seen requires penetrating attention to details. Mathematicians seek truth – that is, truth about patterns, about patterns of patterns, searching ever more for profound underlying interconnecting truths. They strive to prove – not just state – the truths that they intuit. The statement of a truth in mathematics is called a theorem and for each theorem there is a logical proof. The chore of mathematical rigor is to make sure that there are no holes in the logic – that the arguments are airtight and streamlined. Well, ideally. What really happens is that there are claims and evidence. Ultimately, there must be agreement among the top experts about whether a proof – the logical evidence for a claim – suffices. Nevertheless, it is entirely possible that great logical care is lavished on erroneous intuition. When a lot of money is involved the results can be near catastrophic. Mathematics is not the same as mathematical science. If mathematics seeks profound truths underlying patterns, then a mathematical science seeks profound truths

Page 138 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\OpEd090525a.doc THIRD DRAFT 5,459 WORDS 3/11 underlying patterns that relate measurements. All measurements come down to counting. For example, time is measured by counting clock ticks, and money is measured by counting dollars. Of course, there are an unlimited number of quantities that can or might be counted. A mathematical scientist seeks the simplest patterns relating measurements to one another. That is science because a proposed pattern can be refuted by more measurements, and mathematical because studying patterns involving numbers – reasoning and calculating – is best expressed using mathematical machinery. For example, one of the best tricks in mathematics is to invent new kinds of numbers to expand the possibilities for reasoning and calculating. So, although counting cannot ever yield, say, the square-root-of-two, it is mysteriously useful to be able to reason about and calculate with measurements if square roots are allowed in reasoning and calculation. Making measurements is an engineering problem. Originating ideas about what patterns to seek requires intuition – and luck. The trick is to use the mathematical machinery to build a streamlined imaginary universe – often called a “model” – intended to mimic measurements in our messy real universe. The ideas come from recognizing analogies, making guesses, tinkering with the machinery, seeing the numbers it generates, comparing these numbers with measurements, and then guessing and tinkering some more. This activity is so much fun – and intellectually rewarding – that some people get deeply obsessed with it. We call them mathematicians and mathematical scientists. The super-star mental gymnasts of mathematics have contributed for thousands of years and from diverse cultures. Their understanding and inventiveness undergirds all of engineering, including electronics and computers, and increasingly all of science, including biology and medicine. The on-line Mathematics Genealogy Project presents a searchable “family tree” of mathematicians in which a “parent” is a supervisor – or among the supervisors – of the doctoral work of the “child.” For each individual the chart provides the year that the doctorate was granted together with the number of “children” and the total number of “descendants.” For example, Henri Poincaré achieved his doctorate in 1879, had 3 students, and they were his only descendants. Succinctly, (1879, 3/3). As a youth Poincaré was already called a “monster of mathematics” and went on to create entire fields of research, and among other prodigious achievements to anticipate key ideas leading to the relativity theory of Albert Einstein. In particular, Poincaré supervised the work of Louis Bachelier (1900, 0/0), who is universally acknowledged as the originator of mathematical finance. Subsequently, two of the students of Jacques Hadamard (1892, 6/1695) – Paul Lévy (1911, 4/377) and Szolem Mandelbrojt (1923, 10/928) – indirectly impinged on mathematical finance through Benoit Mandelbrot (1952, 8/12), who was a student of Lévy and a nephew of Mandelbrojt.

The Bachelier Financial Universe Of the previously obscure mathematician Louis Jean-Baptiste Alphonse Bachelier (1870-1946), it is known that his father was a businessman, a bureaucrat, and an amateur scientist. His mother was a banker’s daughter, and he had a grandfather in the “financial business.” In some sense, then, Louis Bachelier was born into the house of finance. His parents died when he was a teenager, he became the head of the family enterprise, and so he had to gain intimate acquaintance with the world of financial markets.

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At the Sorbonne, Bachelier was an undistinguished student. However, if a dumbed down definition of “genius” is originality that changes the world, his doctoral thesis Théorie de la Spéculation – accepted with praise by Poincaré – is a work of genius. It was the first high-level mathematical theory of finance, but attracted scant attention for half a century. A concept or idea or notion is interesting if there is much to say about it; it is important if there are numerous interesting ways to express it. Examples of concepts both extremely interesting and important are “computation” and “entropy.” Another such concept called “Brownian motion” entered the scientific zeitgeist in two different and seemingly unrelated ways just around the turn of the twentieth century. First, in physics – defined as the scientific study of motion – the acceptance of “molecular reality” underlying the illusory appearance of solidity in the everyday world sprung from the gradual physical and mathematical understanding of “Brownian motion.” Robert Brown (1773-1858) a botanist using a microscope observed that tiny particles within pollen grains suspended in water seemed to have a life of their own: they darted around like drunken sailors savaging a port. Previously, others had observed this motion and attributed it to life force, but he examined many, many substances and concluded that the movement was not a manifestation of life. In 1905 Albert Einstein (1879-1955), working on the theory of the physical existence of atoms, predicted the possibility of observing such motions – without knowing they had already been observed! Jean Perrin (1870-1942) knew about Brownian motion and about the prediction of Einstein, and after meticulous experiments – involving lots of counting – by 1909 published Brownian Movement and Molecular Reality. He concluded that the particles Brown observed were moving because they were being bombarded at random by invisible molecules of the water in which they were suspended. Perrin was awarded the 1926 Nobel Prize in Physics “for his work on the discontinuous structure of matter.” Scientists everywhere climbed on the molecular reality bandwagon and have been riding it ever since. Imagine a gigantic inflated balloon nudged by hundreds of ping-pong balls shot by playful little kids running every which way. Any impact moves the balloon a tiny bit in one direction or another at random. Of course, talking about “random” impacts introduces probability theory into any mathematics destined to provide a mathematical model of Brownian motion. It turns out that the most common probability concept of all, something called the “normal probability distribution” – high school students call its Cartesian graph the “bell curve” – is perfectly adapted to modeling Brownian motion. Norbert Wiener (1894-1964) originated the rigorous theory in a paper published in 1923 – fully aware of Einstein and Perrin – acknowledging a key conversation with Paul Lévy, Benoit Mandelbrot’s doctoral supervisor. The Wiener-Lévy process models Brownian motion and its cornerstone is the normal probability distribution. Second, the fundamental intuition of Bachelier finance is that price is like that giant balloon, impacted by “ping-pong” trading decisions governed by the Wiener-Lévy process, although of course in 1900 Bachelier could not call it that. The two most important implications of this model are that a price change at one time has virtually nothing to do with a change at a later time – price has no memory; and that price changes are overwhelmingly small. The giant balloon has inertia and does not move much per

Page 140 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\OpEd090525a.doc THIRD DRAFT 5,459 WORDS 5/11 ping-pong impact. The Bachelier intuition is that large price changes are extremely unlikely, that price changes continuously. The explosive history of the Bachelier financial universe is shot through with the Nobel Memorial Prize in Economic Science (briefly denoted by N and followed by the date of award). Around 1958, Bachelier’s largely ignored work was brought to the attention of Paul Samuelson (N, 1970) at the Massachusetts Institute of Technology (Norbert Wiener’s home base) by Leonard J. Savage, a student of economics under the supervision of Harry J. Markowitz (N,1990), who initiated “modern portfolio theory,” 1955-1956. William F. Sharpe (N, 1990), who also studied under the supervision of Markowitz, shared in the development of the “capital asset pricing model,” 1961-1966. Eugene F. Fama studied mathematical economics as a doctoral student of Benoit Mandelbrot, and originated the “efficient markets hypothesis,” 1964. Development of the “Black-Scholes option valuation formula,” 1973 was shared, among others, between Myron S. Scholes (N, 1997) who worked with Fama, and Robert C. Merton (N,1997), whose graduate work was supervised by Samuelson. The Bachelier financial universe is a Platonic realm of perfectly rational denizens who all have the same information about the financial objects – for example, “stocks” – on which they place bets. It is assumed, therefore, that the price of an object “reflects” the universally known information – this is the “efficient markets hypothesis.” These entities all wish to get the “greatest bang for the buck,” limited only by their tolerance for loss. Fear of loss is represented by a number called “risk” and is a statistical quantity – “variance” – calculated using a formula that assumes price changes conform to the “normal probability distribution.” Denizens can borrow as much money as they want at a guaranteed interest rate, so they can even borrow money for placing bets. They can also loan as much money as they wish at a guaranteed interest rate – by buying “bonds.” They can form sets of bets – “portfolios” – that combine stocks and bonds and all sorts of other financial objects. They guesstimate the expected value and the risk of the objects in the set, and calculate the combined expectation and risk of the whole set. By clever adjustment –“diversification” – of the quantities of objects in the set they minimize the risk for a given expectation – this is called “modern portfolio theory.” Given a guesstimate of the risk of the entire set of all possible financial objects – the “market risk” – they can also calculate the expectation of a financial object being considered for addition to the set – this is the “capital asset pricing model.” When someone has a great idea you can bet that others will try to do even better. In fact, that might even be the definition of “great idea.” In any case, human beings have invented many unique financial objects – with dependencies of the prices of some upon the prices of others. Those whose prices do not depend on the prices of others include basic objects such as bonds and stocks. Those whose prices do depend on others are called “derivatives,” such as an “option” whose price depends on the interest rates of bonds and the prices of stocks. In the Bachelier financial universe there is a calculation for the price of an option – “Black-Scholes option valuation formula” – that, again, assumes the normal probability distribution for stock price proportional changes. Companies use derivatives to reduce risk. Change in interest rates, currency exchange rates, supplies of raw materials, and chance of default on loans are some of the sources of corporate risk. Speculators use derivatives for hedging their bets.

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Calculating risk of a portfolio becomes ponderously complex as different sorts of derivatives are included. The Bachelier financial universe includes a calculation of portfolio risk – based of course on the normal probability distribution – using a more-or- less lengthy sequence of recent financial object prices. This is called the “value-at-risk calculation.” It comes up with a number that represents the probability of a portfolio not losing more than a specified number of dollars in value in the near future. In summary, the grand edifice of Bachelier Finance is erected on mathematical pillars called “efficient markets hypothesis,” “capital assets pricing model,” “Black- Scholes option valuation formula,” “value-at-risk calculation,” and “modern portfolio theory.” The pillars are set in the bedrock of the rigorous Wiener-Lévy process that models Brownian motion and hence captures Bachelier’s intuition that price changes continuously without memory.

The Mandelbrot Financial Universe The center of the Massachusetts Institute of Technology campus is Killian Court. Carved in large Roman letters on the friezes of the marble-clad buildings surrounding the court are the names of legendary scientific celebrities such as Aristotle, Newton, Franklin, Pasteur, Lavoisier, Faraday, Archimedes, da Vinci, Darwin, and Copernicus. Above each of these names are clusters of related names. In particular, at Building 2, Kepler, Newton and Poincaré appear together. As already recounted, Louis Bachelier was a student of Poincaré. His most basic tools for the study of price change were the differential and integral calculus – the mathematical study of change – independently originated by Isaac Newton (1643-1727), and the mathematical probability theory initiated in part by Pierre de Fermat (1601-1665) and Blaise Pascal (1623-1662). Reaching a generation further back in time, Benoit Mandelbrot (born 1924) aspired to emulate Johannes Kepler (1571-1630). After “playing” with a large number of alternative ideas, Kepler articulated laws of planetary motion, providing thus an impetus for the theory of universal gravitation originated by Isaac Newton. Mandelbrot is a Lithuanian whose family had immigrated to France. As a young man he skillfully survived German-occupied France and then he moved to the USA. His father had been an inventor and a not-too-successful businessman, and for a while his mother was a dental professional operating out of the family apartment in a low-class neighborhood of Paris. A grandfather was integral to an intellectual lineage of Judaism, and his uncle – Szolem Mandelbrojt, previously mentioned – was a world-class mathematician, and his nephew’s early mentor. Thus, it may be said that Mandelbrot was born into the house of mathematical science. The sort of mathematical super-stars Mandelbrot grew up knowing – and those he encountered along his extraordinary scientific career – would be like a young talented basketball player happening to play on teams with legendary stars such as Bill Russell, “Magic” Johnson, Larry Bird, and Michael Jordan. Add some time hanging out with super-stars of other sports, such as hockey player Wayne Gretsky and golfer Tiger Woods. In Mandelbrot’s life the names are Robert Oppenheimer (“the father of the atomic bomb”), John von Neumann (“the father of the computer”), Norbert Wiener (“the father of cybernetics”), and George A. Miller (an uncle of cognitive science). Add

Page 142 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\OpEd090525a.doc THIRD DRAFT 5,459 WORDS 7/11 anthropologist Margaret Mead, paleontologist Stephen Gould and psychologist Jean Piaget. A tree of a man, Benoit Mandelbrot has had a rough life, obsessed with his “Keplerian dream” to create a science of roughness. With a “freakish geometric gift,” prodigious curiosity, uncanny memory and utter mastery of probability theory, he is a ferociously competitive survivor and maverick scientist. He can be a completely charming companion and storyteller – and he can be dismissively argumentative. Thus, he has many, many friends – and he has had a small number of enemies. Nature is shot through with catastrophe, turbulence and roughness. Life is buoyed and lashed by luck. Mandelbrot’s singular achievement is that he originated the mathematical science for measurement and classification of roughness in nature – he calls it fractal geometry. He fulfilled his lifelong scientific ambition, “to feel the excitement of being the first to find a degree of order in some real, concrete and complex area … where everyone else saw a lawless mess.” Mandelbrot is to roughness as Kepler is to celestial physics. If Bachelier’s intuition was that price change is continuous, then Mandelbrot’s intuition is that, No! Nature is discontinuous – it is rough all over. A hard-core conflict between two intuitions, one half a century old emanating from a scientific genius, the other a century old insight of a mathematical genius; one flowering from an original geometry of turbulence and roughness, the other morphing into an elegant algebraic reasoning and calculating machinery; one advanced by a small band of revolutionaries, the other defended by an army of counter-revolutionary academic mathematicians and physicists – transmogrified into Wall street rocket scientists – frames the gripping story of how Mandelbrot’s David barely nudged the Bachelier Goliath. Until lately. The geometry studied by high school students is the geometry of nice flat objects such as lines and planes, triangles and circles. Likewise, the heart and soul of calculus is smoothness. Assuming that a mathematical variable is changing smoothly lubricates algebraic machinery for reasoning about – and calculating – the behavior of the variable. But the real world is neither flat nor smooth. For example, in engineering the behavior of an engine – be it hydraulic, pneumatic, acoustic or electrical – is mathematically analyzed in terms of its response to a shock. The resulting transient behavior settles down to a stable situation – until the next shock. For example, airplane wings must be designed to withstand the turbulence of air during flight. In biology one encounters the idea of “punctuated equilibrium” – long periods of stasis interrupted by rapid, catastrophic change – wherein the creation of new species is rare and rapid. This idea is contrasted with “gradualism,” the notion that evolution is generally smooth and continuous. And in physics the standard interpretation of quantum mechanics holds that an atomic system evolves quite smoothly – until it is measured. In that view, “measurement collapses the wave function,” a seriously discontinuous event. Bachelier published his thesis and five other books, all on probability theory, and altogether fifteen scientific papers, mostly on probability theory. Then, as already noted, his ideas evaporated into obscurity for decades. Mandelbrot is without any doubt a prolific scientific genius, having published hundreds of papers in a dozen different branches of science, and more than two dozen technical books in several languages. His vita alone is a fifty-page booklet that lists his publications, the television programs on him and his work, interviews with him for

Page 143 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\OpEd090525a.doc THIRD DRAFT 5,459 WORDS 8/11 dozens of periodicals, and honorary doctorates from institutions all over the world. Among the prizes listed are the Wolf Prize in Physics 1993: “by recognizing the widespread occurrence of fractals and developing mathematical tools for describing them, he has changed our view of nature”; and the Japan Prize 2003: “for the creation of universal concepts in complex systems - Chaos and Fractals.” Mandelbrot’s Hypothesis applied to price change is that – like most everything else in the real world – price change is discontinuous. This flatly contradicts the fundamental assumption of Bachelier finance – the one taught in business schools and trundled on Wall Streets worldwide – and the reaction of the mathematical finance community has been … largely dismissive. Until lately. If there is to be a Mandelbrot Financial Universe, it will extend the Bachelier Financial Universe as the Milky Way extends the Solar System. Mandelbrot has generalized the “mild” variability studied in Bachelier finance by providing mathematical machinery for simulating, reasoning and calculating about “wild” variability – abnormal probability distributions.

Cootner Assails the Mandelbrot Hypothesis By 1964 the idea that prices of financial objects such as stocks and commodities are best modeled by the continuous Wiener-Lévy process – or its discrete approximation called a “random walk” – was established enough to call for publishing a collection of academic papers on the origins, justification, refinement, testing, and re-examination of the idea. MIT Press –the publishing arm of the Massachusetts Institute of Technology – released The Random Character of Stock Market Prices, edited by Paul H. Cootner (1930-1978). It includes a translation of Bachelier’s doctoral thesis, a paper on Mandelbrot’s Hypothesis by Eugene Fama, a paper by Paul Samuelson on “rational pricing,” and a seminal work by Benoit Mandelbrot, The Variation of Certain Speculative Prices. As editor, Cootner certainly had every right to organize the book as he deemed useful, including the right to add comments of his own. Strangely, he chose to comment only on the one by Mandelbrot. He wrote that Mandelbrot “[w]ith determination and passion … has marshaled as an integral part of his argument evidence of a more complicated and much more disturbing view of the economic world than economists have hitherto endorsed.” He goes on to criticize Mandelbrot’s evidence, method, and conclusions. He concludes that if Mandelbrot is right, then “[a]lmost without exception, past econometric work is meaningless.” Mandelbrot was never given an opportunity of reply to Cootner’s excoriating diatribe, and could not withstand excommunication from the church of mathematical finance. Forty-five years later – in the midst of World-Wide Depression II – the Mandelbrot Hypothesis is being revived in articles appearing in the New York Times, Scientific American, and the Financial Times. In fact Mandelbrot has changed the English language twice – first with his word fractal, and now with the word (mis)behavior applied to markets and prices. His 2004 book with Richard L. Hudson, The (Mis)behavior of Markets, A Fractal View of Risk, Ruin, and Reward has been issued in paperback and recently reprinted with an up-to-date preface both in the US and UK.

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Taleb Assails Bachelier Finance Personal disclosure. Last year after a small reflection I concluded that luck – good and bad – had a very large influence on my life. This is not much of an Earth-shattering insight. In a bookstore one day I was riveted by Nassim Taleb’s 2004 book, Fooled by Randomness, The Hidden Role of Chance in Life and in the Markets. His 2007 book, The Black Swan, The Impact of the Highly Improbable, is dedicated to … Benoit Mandelbrot. Nassim Nicholas Taleb (born 1960) vehemently assails Bachelier Finance both as a very successful contrarian Wall Street trader and as an erudite mathematically trained philosopher. The philosophical question called “the problem of induction” is basically whether truth about the future can be induced from knowledge of the past. That question is even built into the prospectus of an investment fund, which is legally required to declare that “[t]he performance data shown represents past performance, which is not a guarantee of future results.” Taleb writes, “[b]efore the discovery of Australia, people in the Old World were convinced that all swans were white, an unassailable belief as it seemed completely confirmed by empirical evidence. The sighting of the first black swan … illustrates a severe limitation to our learning from observations or experience and the fragility of our knowledge.” Price discontinuities are like black swans. Taleb anti- religiously castigates the “value-at-risk calculations” derived from the comfort food for Wall Street cooked up by Bachelier.

Bachelier Finance is Flawed In the UK, Chairman Lord Turner of The Financial Services Authority – an independent non-government body funded by the firms it regulates – recently published The Turner Review of “the events that led to the financial crisis and to recommend reforms.” Regarding “value-at-risk calculations” it is written that, “[s]hort-term observation periods plus assumption of normal distribution can lead to large underestimation of probability of extreme loss events.” Once again, the point is very simple. Price change may roll along with mild jumps for a while, but there is no way to predict when it will hit a bump and make a wild leap, jarring the market and surprising everyone. The report declares that individual market participants, government regulators and academic researchers – the Bachelier rocket scientists – had “misplaced reliance” on sophisticated mathematics. The pristine ivory tower mathematics used to measure and manage risk engendered false confidence that other indicators of increasing risk – for example, rapid credit extension and balance sheet growth – could be ignored with impunity. Warren Buffett has said, “If you stand up in front of a business class and say a bird in the hand is worth two in the bush, you won’t get tenure … Higher mathematics may be dangerous and lead you down pathways that are better left untrod.”

A Post-Bachelier Financial Universe? Changing the mathematical culture in the world of finance is a formidable challenge. Perhaps a period of time which has witnessed solution of long-standing mathematical problems such as Fermat’s Last “Theorem” (1637; Theorem in 1994), the Four Color Conjecture (1852; Theorem in 1976), the Poincaré Conjecture (1904; Theorem in 2004), and the Weil Conjectures (1940’s; Theorems in 1974), may see

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creation of a new mathematics to supplant the useful but over-specialized Bachelier Finance. There is no formula for human fear, greed, and stupidity. A model of price behavior is no more a model of human behavior than a model of temperature behavior is a model of molecular behavior. Perhaps a mathematical science of financial behavior should take into account facts such as “markets are not perfectly efficient,” “people are not good guardians of their own self-interest,” “human beings evolved in a very uncertain world and therefore have inherent limitations and flaws in the way they think and act,” and “political systems necessarily bias perceptions of the powers-that-be in favor of maintaining their power.” A Post-Bachelier Financial Universe might do well to also model cooperative and feedback effects between the individuals and institutions acting in markets, and take account of the “butterfly effect,” especially because the speed of transactions is accelerating due to breakneck advances in electronic, optical, and quantum technology.

Bachelier Finance is So Twentieth Century Basketball teams have winning and losing streaks; individual players during games have runs of scoring shots and then go “cold” for a series of shot attempts. Clustering appears in price series too, but Bachelier’s theory is blind to these effects. Discontinuity and clustering are its bane. Mandelbrot offers new mathematical tools for studying price changes, but physicists flock to finance, attracted by increasing quantities of hard data – and by the high salaries on Wall Street. Thus, we have “econophysics.” The study of motion naturally includes the absence of motion – rest – which the physicists call “equilibrium.” A swinging pendulum eventually comes to rest because friction at the joint from which the bob hangs irreversibly converts motion energy into heat energy. Indeed, a fundamental law of physics says that any isolated physical system eventually comes to rest. Standard economic theory assumes there is an equilibrium price for each item traded. But no one has ever seen a price achieve equilibrium. Brownian motion and equilibrium are physicist concepts that have failed financial economics. If financial economists need new ideas, perhaps they ought to encourage their students to study more mathematics rather than so willingly allow physics students to invade their precincts. Mathematics is not shackled to physical reality, its ideas are free to roam a cosmos in which physics is nought but a single star. Mathematician Mohammad Bin Musa Al-Khowârizmi (790-840) gave to us in his name – transformed by translation into Latin from Arabic – the word algorithm which now means a method of calculation. His work Hisab al-jabr w’al-muqâ-balah was the first and enormously influential book on the “science of the reunion and the opposition” for solving equations. Its title originates the word algebra. Through the centuries algorithms and algebra have generated a universe of concretions including computer programs and abstractions including the algebra of algebra – which is called category theory. Its special study called “topos theory” is uniting theories of geometry and logic within mathematics; “higher-dimensional category theory” is actively studied by theoretical physicists; and category theory is unifying theories of parallel computation. Financial economists ought to be aware of this scientific work because it provides new options for inventing metaphors. The old-guard of financial economists probably will not be able to look away from their burnished ideas. Paul Samuelson is purported to have

Page 146 of 302 Unpublished.1998-2013 Emach c:\_edc\doc\OpEd090525a.doc THIRD DRAFT 5,459 WORDS 11/11 said, “Funeral by funeral, science makes progress.” Perhaps the next generation of finance students will boldly realize that mathematics is the ultimate tinker-toy of metaphor.

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Dear Henry,

I could not be more pleased that you are taking a course on Moral Philosophy (MP). As you well know MP has been of great interest to me for a long time, and we have had many great conversations about it.

In this letter I will direct your attention to two approaches to MP, both without religious intentions. First, I tell you about the “brights,” then I summarize my own approach to MP, with which you are already quite familiar, but I never wrote it up.

The bottom-line will be that both the “brights” and I are doomed to fail, if by success is meant widespread appreciation and adoption of one of our non-religious approaches to MP. I will also suggest an idea for a film that would succeed.

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The Brights

First, I know you are familiar with Daniel C. Dennett (“Consciousness Explained,” “Breaking the Spell, Religion as a Natural Phenomenon”) and Richard Dawkins (“The God Delusion”; inventor of memes). Both of these public intellectuals are prestigious members of an organization whose members call themselves – condescendingly – “brights.” From their web-site http://the-brights.net we get the definition:

A bright is a person who has a naturalistic worldview A bright's worldview is free of supernatural and mystical elements The ethics and actions of a bright are based on a naturalistic worldview

The slogan of the Bright’s is “illuminating and elevating the naturalistic worldview.” They put out an email newsletter to which I subscribe, and they are always proud to announce how much they are growing and how widespread is their influence. I don’t think they have more than 50,000 members world-wide. I think that works out to 0.0008 percent of the world population.

They have various projects, but the one on MP is of immediate interest to us. At the link http://the-brights.net/action/activities/organized/arenas/1/area_a/declaration.html they outline a framework for defining morality. I quote:

A: Morality is an evolved repertoire of cognitive and emotional mechanisms with distinct biological underpinnings, as modified by experience acquired throughout the human lifetime. (27 studies)

B: Morality is not the exclusive domain of Homo sapiens—there is significant cross- species evidence in the scientific literature that animals exhibit "pre-morality" or basic moral behaviors. (15 studies)

C: Morality is a "human universal" across all cultures worldwide, a part of human nature acquired during the course of human evolution. (18 studies)

D: Statement D: Young children and infants demonstrate some aspects of moral cognition and behavior which precede specific learning experiences and worldview development. (11 studies)

Now I quote from their web-site what they think is possible for advancing their framework for MP.

The idea of the “Reality about Morality” is to challenge the widespread presumption that ethical systems and morals are imparted to humankind by some form of divine being or power.

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We will build a straightforward foundation of scientific reality concerning the subject -- reality which is empirically substantiated.

Most of us have lived in a culturally religious milieu long enough to know that simply presenting scientific evidence won't necessarily change minds. Presenting a sound scientific explanation for the origins and presence in society of individual and group morals will be necessary but not be sufficient to quickly change social values. It is going to require all the dedication and resources the constituency can muster to confront one of society’s most deeply and broadly held beliefs.

However, as the Enlightenment did not happen overnight; neither will the scientific explanation concerning morals immediately easily take hold. However, in a sufficiently bright spotlight of truth, faith in an error has an opportunity to crumble. Individual Brights everywhere, armed with the “raw material” of scientific reality about morality in a suitable form, can work within their personal and local spheres of influence toward altering understandings.

At the Internet hub, The Brights’ Net, coordinated actions can take place to engender change within education, across media, and in the public at large. Brights will be collaborating and trying to subvert a false social belief that undermines brights everywhere. This is consistent with the broad civic aims of The Brights’ Net to improve the social and civic situation of persons whose worldview is free of supernatural (see the purpose statement at the home page). To the extent we are successful, we will also be advancing authentic public understanding of human nature on behalf of persons of all worldviews, not just brights. Such is sorely needed.

We are counting on many Brights to become involved at a point where project needs may match their expertise, interests and time, and on all who can to help financially support the overall enterprise. In a nutshell, here is what is initially planned for this adventure. No plans are cast in stone, of course. Refinement is expected!

Project Area A involves developing, in the form of a declaration, a scientifically defensible listing of statements regarding the naturalistic origins of morals. The declaration will be short, but it will cover the evolution of biological and social morals, starting from pre-human origins to the present.

Project Area B involves scientists in authenticating each statement in the declaration with citations and comments.

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Project Area C entails planning for the design and development of presentation and instructional materials for varied target audiences (through illustration and examples as necessary) on the final declaration statements.

Project Area D involves the development of volunteer mentors, and individuals schooled in the declaration statements and how to present and explain them to others.

Project Area E entails preparation for a public roll-out of the declaration and supporting instructional materials. It will lay the groundwork for presenting to the media our reclaiming of morality.

Project Area F is the culmination of all the work in Areas A through E. This is conducting the public roll-out of the Declaration.

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My Personal Opinion About Moral Philosophy

Second, my view on morality is parallel to the Bright’s view, but independently conceived and more formally elaborated.

AXIOM. The most basic distinction among items in the Universe is between objects and beings. A “being” is an object whose conception, development, and demise are governed by DNA. An item without DNA is an object. An object has no inclination to survive. DNA in beings inheres in them the genes for survival meticulously sculpted by three billion years of biological evolution.

AXIOM. Human beings are the beings of the species homo sapiens sapiens.

AXIOM. Consciousness inheres in beings alone and is associated with directing attention upon items of the Universe in the context of an ongoing model of the Universe implemented as a virtual machine over neural brain machinery. The purpose of consciousness is human survival. How that works in detail is a whole family of wide open scientific questions.

AXIOM. MP is an academic discipline. Its’ pursuit – and the practices of a moral life – are activities of conscious human beings.

AXIOM. The survival of a being depends on pursuit of various missions.

DEFINITION. Suffering is the feeling of interference with a mission.

THEOREM. Beings avoid suffering.

DEFINITION. Immorality is the unreasonable, deliberate infliction of involuntary suffering.

NOTE. Thus, for example, killing is immoral, because it not only interferes with, but ends, all missions of a being.

NOTE. The word “reasonable” is intended in the sense used in a court of law in reference to “reasonable doubt.” That is to say, an independent group of human beings with no stake in the outcome of a judgement consider the allegations to be unsupported by adequate evidence. “Reasonable” admits of degrees.

NOTE. The word “deliberate” refers to conscious, planned, thoughtful intention. “Deliberate” admits of degrees.

NOTE. “Infliction” means the use of force by one being upon another being. This includes operation of vocal muscles to create sound waves – sound waves that carry meanings, including hurtful ones.

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NOTE. “Involuntary” implies that the second being never intended the first being to inflict the suffering upon it.

This completes my MP. I would consider it a failure if it were not widely publicized, understood, and used to govern the amelioration of suffering in the Universe. Unfortunately, this is extraordinarily unlikely. The reason for that is that human nature – on the whole – responds not to logic but to emotion. Such a dry “mathematical theory” of morality will never appeal to emotions. Nor, of course would the “brights” rational MP succeed.

All the currently accepted religious MPs are grounded in gripping stories, ancient myths, and legendary sagas of struggling beings and Gods. Neither the “bright’s” scientific plans for advancing morality, nor my pithy moral theory have the slightest chance of enduring, wide influence unless somehow they are sheathed in stories that everybody can understand and repeat among themselves down through the ages.

In the old days these stories were word-of-mouth. Village storytellers learned the stories from their forerunners, elaborated on them to wide-eyed villagers gathered around them, who in turn elaborated on them, and so on. Some of these stories had such great emotional power that as meme complexes they grew to towering complexity, intellectual stature, and in the case of Christianity and Islam, political power. These Abrahamic religions put beliefs in God ahead of concern for amelioration of suffering. This inversion of priorities has been and still continues to be the underlying cause of a great deal of suffering.

Modern technology provides us with multiple means to tell long stories to large numbers of people – people who actually are willing to pay money to get the stories. Printed books, of course, allowed stories to be copied and handed around without any need for being at a place at a time with a person to get the story. Now we also have film and film theatres – and the internet and computers and cell phones, etc.

People disappear from themselves when they watch a film. The skillful film-maker molds emotions second by second. Rooms full of viewers can be brought to cheering or to tears within minutes. I claim that that kind of emotional power is what is needed to counter the millennia of religious MP.

Somebody has to create a film about a seriously flawed character who, nevertheless, transcends religious MP and embodies the non-religious MP according to the sorts of ideas summarized above. The movie poster says, “There is no redemption, no forgiveness. What is done is done, there is no going back. The arrow of time pierces everything.”

Love,

Your Dad.

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Qualitative Smooth Shape in the Plane

Ellis D. Cooper [email protected] March 3, 2012

Abstract

When you look at a shape in space, say a sculpture, from one position you only see one part of it. Once you see every part, you can imagine the whole sculpture. And, if you can imagine the whole sculpture, you can predict how it would look from any position. These inverse experiences depend less on precise measurements than on qualitative shape. What types of mathematics might make precise similar observations in the presumably easier case of qualitative shape in the plane?

Keywords ambient isotopy, anorthoscopic perception, Category Theory, Knot Theory, Morse Theory, phenomenology, qualitative shape

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Contents

1 Introduction 2 1.1 Phenomenology...... 2 1.2 Psychology of Visual Perception ...... 3 1.3 Optical Character Recognition ...... 3

2 Part I. Partial View of Opaque Shape from Near Viewpoint 4 2.1 Points,VectorsandLocatedVectors ...... 4 2.2 Curve,Circle,andShape ...... 6 2.3 Near Viewpoint and Partial View ...... 8 2.4 VisualEventandAbstractPartialView ...... 10 2.5 AspectGraph...... 12

3 Part II. Total View of Transparent Shape from Far Viewpoint 17 3.1 Orientation ...... 17 3.2 InﬂectaandExtrema...... 18 3.3 The Cell Complex Induced by a Far Viewpoint ...... 19 3.4 VerticallyOrderedGradedPoset ...... 20

4 Part III. Approximation and Experimentation 28 4.1 BezierShape ...... 28 4.2 Screenshots of Scultoro ...... 30

5 Part IV. An Imperfect Analogy: Knots and Shapes 32 5.1 Homotopy,Isotopy,andAmbientIsotopy ...... 34 5.2 Analogy Table with Imperfections ...... 38

6 Part V. Conclusion and Problems 39

Bibliography 41

Index 43

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1 Introduction 1.1 Phenomenology We may speak of the evidence that the identity is given only if a continuous transition from the one perception to the other is guaranteed in the unity of experience. The unity of the object demonstrates itself only in the unity of the synthesis continually joining the manifold perceptions, and this continuous synthesis must lie at the foundation in order for the logical synthesis, that of identification, to produce the evident givenness of the identity of the objects appearing in the various perceptions. The perceptions must be integrated as phases into the synthesis, and we see that precisely only if we carry out the synthesis. This important fact holds generally. In our case, it means that an identical and unchanged spatial body demonstrates itself as such only in a kinetic series of perceptions, which continually brings to appearance the various sides of that thing. The body must rotate or be displaced, or I must move, move my eyes, my Body, in order to see it from all around, and at the same time I must keep approaching it and receding. Or. finally. both I and the thing must move. That is how the state of affairs is expressed from the standpoint of the appearing thing [13].

I make no pretense of understanding the principle of “phenomenological reduction” introduced by Edmund Husserl for his general philosophy of phenomenology. I do know that his work includes mathematical diagrams per- taining to “the lawfulness of the modifications of appearances in the case of qualitative change” (e.g., [13], p.232). I have challenged myself to see what I can say on the topic of qualitative shape in terms of some basic modern mathematical technologies. By no means have I a mathematical theory of qualitative shape, with calculations and reasoning to prove theorems. I do have geometrical constructions and formal definitions of what seem to me to be relevant concepts. And I do make technical claims for which the evidence is provided by numerous diagrams. Thus, this document is less of a research paper and more of a preliminary research proposal. Due diligence leads me to conclude that there is a gap in the mathematical literature between the study of topological spaces that are considered equivalent if they are homeomorphic, and the study of rigid shapes entirely determined by the distances between pairs of distinct points. The hierarchy of geometries cod- ified in terms of the group of automorphisms of a space – as in the Erlanger Program of Felix Klein1 – includes, between topology with its floppy homeomor- phisms and Euclidean Geometry with its rigid isometries, many intermediaries such as projective geometry, algebraic geometry, and equiaffine geometry. I do

1Weisstein, Eric W. ”Erlanger Program.” From MathWorld – A Wolfram Web Resource. http://mathworld.wolfram.com/ErlangerProgram.html

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not find in that range a geometry of “qualitative shape,” which is a concept that remains to be defined. To be sure, the idea is not ignored by scientists and mathematicians. For decades Jan J. Koenderink has studied conceptually and mathematically the psychology and philosophy of perception, computer vision and ecological physics [15][16]. Also, “Differential topology, and specifically Morse Theory, provide a suitable setting for formalizing and solving several problems related to shape analysis” [4]. And there are “generative” theories of shape, such as [19] in which features of shapes “correspond to symmetry groups, and addition of features corresponds to group extensions.” However, to my knowledge only one study directly targets the fundamental question of defining “qualitative shape” and that is in the context of “Geographical Information Systems that simulate the mental processes of human beings” [7].

1.2 Psychology of Visual Perception A purpose of this proposal is to formalize some questions about integration of partial views of a shape to form a total perception of it. The phenomena of anorthoscopic perception also relate to integration of partial information, but that which is obtained by moving a viewing slit across the field of view. Our purpose was to investigate the contribution of pursuit eye movements to the integration of figures moving behind a slit (anorthoscopic perception). Outline shapes were moved across a simulated slit (without visible borders) on a CRT screen. Eye movements were monitored with a dual-Purkinje image eye-tracker. The slit width was adjusted so that there were spontaneous transitions between periods in which observers saw a horizontally moving shape and periods in which they saw only vertically moving contour segments. Subjects reported transitions between these alternative percepts. On half the trials the slit was retinally stabilised to eliminate pursuit- dependent retinal painting. During periods of figure perception, there was often low-amplitude spontaneous pursuit of the horizontal figure motion. Lower-amplitude pursuit was also sometimes observed when no figure was reported. Irrespective of figure perception, pursuit amplitudes were larger with stabilisation. However, the total percentage of time that a shape was seen, and the duration and frequency of episodes of figure perception, were not significantly different during stabilised and non-stabilised viewing. Our findings suggest that, although anorthoscopic figure percepts can elicit spontaneous pursuit under free-viewing conditions, this pursuit and the pattern it pro- duces do not contribute to the formation or maintenance of those percepts [10].

1.3 Optical Character Recognition My interest over thirty ﬁve years ago leading to this work centered on the problem of optical character recognition by computers. Nowadays this problem is

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largely solved, except perhaps for persistent diﬃculties with general handwriting. But for printed text in newspapers, books and journals one may use a scanner and commercially available OCR software to obtain a digital version of the material in a matter of minutes. Back in the day, however, pondering this problem I imagined a tiny spark of intelligence, a “pointhuman,” descending by parachute into the middle of a letter on a page, challenged to determine the identity of the letter. The assumptions are that the letter is a shape with empty interior and opaque boundary, and that (a) the pointhuman is mobile in any direction, (b) it can look in any one direction at a time and determine whether a far point of the boundary is occluded by a near portion of the boundary, and (c) it has unlimited calculating and reasoning ability, including unlimited memory. Of course, (d) the pointhuman must also have some sort of table of shapes against which to compare the information it gathers and integrates by exploring the interior of the letter [8]. One thing led to another, and I imagined a pointhuman landing, not inside a letter, but outside an island in the ocean of an alien planet, now challenged not to recognize but to cognize the shape of the island. That thought led to questions such as, what is the minimum number of exterior viewpoints from which the collected partial views of the island would be suﬃcient to claim knowledge of its entire shape? For a circular island that would be 3. And then I was thinking about similar explorers in three-dimensional space, and noting that a minimum of 4 viewpoints would be necessary to view the entire surface of a spherical planet. What would be the minimum number to see the entire surface of a doughnut-shaped world? In any case, the minimum number of comprehensive partial views should be an invariant of qualitative shape.

2 Part I. Partial View of Opaque Shape from Near Viewpoint 2.1 Points, Vectors and Located Vectors Definition 1. The plane is mathematically represented by the set R2. An element A = (x, y) of R2 is called a point. every point A = (x, y) corresponds −→ −−−→ −→ to a vector A = (x, y), and the space of vectors is denoted by R2, hence there −→ −→ is a 1-1 correspondence R2 −→ R2 : A → A between points and vectors. R2 −→ is an unstructured set of ordered pairs, whereas R2 is a 2-dimensional vector space defined by the usual operations of component-wise addition and scalar multiplication. −−−→ The translate of a point A = (u, v) along a vector B = (x, y) is the point −→ A + B := (u + x, v + y). See Fig.(1). −→ Translation of points along vectors is a map R2 × R2 −→+ R2 that satisfies the

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(u, v) (0, 0) −−−→ (x, y) −−−→ (u, v) + (x, y) (x, y)

Figure 1: A point, its vector, and a point translated by its vector. The vector of a point translates the origin (0, 0) to the point. following equations: −→ A = (0, 0) + A; (1) −→ −→ −→ −→ A + (B + C ) = (A + B ) + C. (2) −→ Furthermore, for any two points A, B there exists a unique vector C such that −→ B = A + C [30].

Deﬁnition 2. The unique vector that translates a point A to a point B is −→ −→ denoted by B − A ∈ R2. This means there exists a map R2 × R2 −→ R2 such that A + (B − A) = B. −→ −→ A located vector is a pair (A, C ), that is to say, an element of R2 × R2. Observation 1. Located vectors are the morphisms of a category [21][28] whose −→ −→ −→ objects are the points, where dom(A, C ) := A, cod(A, C ) := A+ C , the identity −→ morphism of A is (A, (0, 0)), and the composition of (A, C ) followed by (A + −→ −→ −→ −→ C, D) is (A, C + D). See Fig.(2).

−→ −→ Definition 3. For a located vector (A, C ) the ray Ray(A, C ) ⊂ R2 with −→ −→ vertex A in the direction of C is the set of points of the form A + sC where 0 ≤ s ∈ R. For any two points A and B there are intervals of points defined as follows. The closed interval [A, B] ⊂ R2 is the set of points of the form A + s(B − A) where 0 ≤ s ≤ 1; the half-open interval [A, B) is the set of points of the form A + s(B − A) where 0 ≤ s < 1; and (A, B], (A, B) are defined analogously, although the latter is ambiguously either an open interval, or just an ordered pair, so context must be consulted to clarify intention. See Fig.(3).

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−→ D −→ C

Figure 2: Addition of vectors is a composition of morphisms.

−→ D [D,E) Ray(A, C ) E

A −→ C

Figure 3: Intervals and a ray.

2.2 Curve, Circle, and Shape Deﬁnition 4. The (unit) circle S1 is the 1-dimensional submanifold of the plane R2 with underlying set { (x, y) | x2 + y2 = 1 }. If R2 is considered to be the underlying set of the complex plane C then also S1 = { eiθ | θ ∈ R }. A unit −→ −→ −→ vector −→u := eiθ ∈ S1 is called a direction, see Fig.(4). Every vector A has −→ −→ A magnitude A and direction, namely −→ . A

Deﬁnition 5. A shape is a smooth embedding S1 −→K R2 of the circle in the plane. “Smooth” means K is a diﬀerentiable map of manifolds, so there is a −→ −→ derivative tangent vector t = t (K, x) at each point of the image of K in the plane. “Embedding” means K is an injective map, i.e., distinct points of S1 map to distinct points of R2. It is convenient for the letter “K” to denote ambiguously the map and its image Im K = K(S1) ⊂ R2.

Observation 2. K smooth implies that at each point x of K there exists a

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0 + 1i eiθ θ −→ eiθ 1 + 0i

Figure 4: The “complex rotator.”

−→ unique tangent line L(K, x) consisting of all points of the form x + s t (K, x) for s ∈ R. See Fig.(5).

−→ t (K, x) x

L(y)

Figure 5: A tangent line and a unit tangent vector.

Definition 6. A curve (also called path) in the plane is a continuous map [0, r] −→α R2 defined on a closed interval [0, r] ⊂ R. See Fig.(6). Curve α is simple if it is an injective map, and closed if the end α(r) equals the start α α(0). The notation x −→ y signifies that path α starts at x and ends at y. Recall the Jordan Curve Theorem (first proved satisfactorily by Oswald Ve- blen) which asserts that for any simple closed curve the plane is partitioned into its bounded interior, the curve itself, and its unbounded exterior [3][24][25][31],

R2 = In K ∪ K ∪ Ex K .

Observation 3. Some might prefer to say α is a parametrization of its image α([0, r]), and that image is “the curve.” The advantage of deﬁning the curve to

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be the map is that the natural orientation of R (from 0 to 1) induces a direction along the curve [29]. Consequently, there exists a category with objects the points of R2 and morphisms the paths x −→α y. The constant path on [0, 0] with value x is the identity morphism of x, and composition is deﬁned by following one path and then the next [6].

x y

Figure 6: A curve.

2.3 Near Viewpoint and Partial View Deﬁnition 7. Let K be a shape. A point v ∈ Ex K is called a near viewpoint of K.A sight-interval from near viewpoint v to K is a closed interval [v, x] where x ∈ K and [v, x) ⊂ Ex K. Note that If [v, x] is a sight-interval and w ∈ [v, x] then [w, x] is also a sight-interval. See Fig.(7).

[v, x] x K

In K Ex K

Figure 7: A sight-interval.

Deﬁnition 8. The partial view of K from near viewpoint v is the subset of points x of K such that there exists a sight-interval from v to x. See Fig.(8).

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x x2 x4 K 3 x5 x6

Figure 8: Partial view. Little open circles at points x3, x5 represent occlusion of those points by x2, x4, respectively.

Definition 9. Let K be a shape and L a line in the plane. If L meets K and x ∈ L ∩ K then either L is tangent to K at x so L = L(K, x), or not, in which case L crosses K at x. Likewise, if v is a near viewpoint of K and −→ R = Ray(v, −→u ) meets K at x where −→u ∈ S1 is a direction, then either R is tangent to K or crosses K at x. Claim 1. For any shape K, near viewpoint v, and direction −→u , if R := Ray(v, −→u ) and n = #(R ∩ K), then either n = 0, or there exists a sequence of points (x1, . . . , xn) – the contact list – such that xi ∈ K for 1 ≤ i ≤ n, −→ [v, x1] is a sight-interval from v to K, and u is the direction of xi+1 − xi, for 1 ≤ i < n. Of course, R is either tangent to or crosses K at each contact point xi. Moreover, for every sight-interval [v, x] there exists a unique ray R with vertex v such that x is the first contact point of R with K. Thus, the partial view of K from v is the set of first contact points of rays with vertex v that meet K. See Fig.(9).

v x1 x2

x3 x4

Figure 9: Contact list.

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2.4 Visual Event and Abstract Partial View Definition 10. Let First(K, v, −→u ) be the partially defined function of −→u that is equal to the first contact point of Ray(v, −→u ) with K if it exists, and equal to ⊥ otherwise. The partial view of K from v is the set of values where First(K, v, −→u ) is defined. Claim 2. Let −→u := eiθ be a direction and −→u − := ei(θ−ǫ), −→u + := ei(θ+ǫ) be infinitely near directions, that is, ǫ > 0 is an infinitesimal.2 If Ray(v, −→u ) is tangent to K at First(K, v, −→u ), then exactly one of the following holds:

First(K, v, −→u +) = ⊥ and First(K, v, −→u −) ≈ First(K, v, −→u ) (3) First(K, v, −→u +) ≈ First(K, v, −→u ) and First(K, v, −→u −) = ⊥ (4) First(K, v, −→u +) ≈/ First(K, v, −→u ) and First(K, v, −→u −) ≈ First(K, v, −→u ) (5) First(K, v, −→u +) ≈ First(K, v, −→u ) and First(K, v, −→u −) ≈/ First(K, v, −→u ) (6) First(K, v, −→u +) ≈ First(K, v, −→u ) and First(K, v, −→u −) ≈ First(K, v, −→u ) (7) where ≈ is the relation “infinitely near,” which means “equal standard parts,” and the ≈/ ≈/ case is impossible since K is continuous. See Fig.(10). Definition 11. In case Equ.(3) the partial view drops to infinity from the right; in case Equ.(4) it drops to infinity from the left; in case Equ.(5) it drops back from the right; in case Equ.(6) it drops back from the left; and, last, in case Equ.(7) the partial view sees an inflection. Claim 3. Let K be a shape, v a near viewpoint, and −→u a direction. Let R := Ray(v, −→u ). Then

A ∞ First(K, v, −→u ) = ⊥ so R ∩ K = ∅; otherwise, let x := First(K, v, −→u ) ∈ K. Then either B =: X R crosses K at x; or R is tangent to K at x, and either C =: S x is an inflection of K; or D =: C K drops to infinity from the right at x; or E =: C K drops to infinity from the left at x; or F =: C K drops back from the right at x; or G =: C K drops back from the left at x.

Observation 4. Consequent to this claim, there exists a function associated to K and v assigning to each direction −→u one of the values A, B, C, D, E, F, G. As a mnemonic aid, a special symbol is associated with each of these letters. Claim 4. Values ∞, X, S, C, ,C C, and C occur at most in a ﬁnite number of directions, and on an open interval separating two of those directions, the value is constantly one of ∞ or X.

2Mikhail G. Katz and David Tall, “Tension between Intuitive Inﬁnitesimals and Formal Mathematical Analysis,” http://arxiv.org/ftp/arxiv/papers/1110/1110.5747.pdf

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−→u +−→u −→u − (C)( )C

(C)( )C

(Z)(S)

Figure 10: Contact classiﬁcation.

Observation 5. A direction with value ∞ just means that there is no sight- interval from the near viewpoint to the shape. Value X means there is a sight- interval along a ray that immediately crosses the shape. Otherwise, either there is a sight-interval along a ray tangent to the shape at an inﬂection, S, or there is a sight-interval along a “horizon” of the shape, Cor ,C or a sight-interval tangent to the shape at an extremum that occludes a point of the shape further away, Cor .C See Fig.(8).

Definition 12. The values S, C, ,C C, or C are called visual events, and the values ∞, Xare called visual non-events. Observation 6. As a direction at angle θ rotates from θ = 0 counter-clockwise, there exists a periodic sequence of visual events separated by constant ∞ or Xvalues along open intervals. There may or may not be a visual event at θ = 0. If there is, then there is a final open interval as θ approaches 2π before the sequence repeats. If there is not, then there is an open interval between the last visual event before θ = 2π and the first visual event after θ = 0. Either way, there exists a unique sequence of symbols ∞, X, S, C, ,C C, or inC one period of the sequence as θ varies from 0 to 2π. Any occurrence of ∞ signifies an interval of directions in which no point of the shape is in sight. However, distinct occurrences of Xrepresent intervals in which distinct portions of the partial view are visible. Definition 13. The finite sequence of visual events and non-events associated −−−→ with directions at v rotating counter-clockwise from 1 + i0 is called the abstract partial view of K at v. Two near viewpoints v, w for which there exists a path v −→α w such that the abstract partial view is invariant up to a cyclic permutation are abstractly equivalent near viewpoints, for which the notation is v =∼ w. Any two near viewpoints of a circle are abstractly equivalent, with abstract partial view C ∞.

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2.5 Aspect Graph Claim 5. Abstract equivalence of near viewpoints is an equivalence relation on Ex K, and there exists a ﬁnite number of abstract equivalence classes of near viewpoints. An equivalence class is called an aspect of the shape. Each aspect is a (possibly unbounded) connected region of the plane. The boundary of an aspect may or may not contain a subset of the points of the shape. The boundary of an aspect is composed of sets of points – its components – of one or more of the following kinds:

a continuous portion of the shape itself; (8) a ray contained in the exterior of the shape; (9) a tangent ray with vertex on the shape, and not meeting its interior; (10) an interval [x, y), x, y ∈ K, and [x, y) ∩ In K = ∅; (11) an interval [x, z], x ∈ K on the shape, y ∈ Ex K, and [x, y] ∩ In K = ∅. (12)

Deﬁnition 14. Two aspects that share a common boundary component are called contiguous. Claim 6. Two contiguous aspects share exactly one boundary component. The shared boundary component of two contiguous aspects is a subset of exactly one of them.

C C C Z

C C

Figure 11: Abstract partial view.

Referring to Fig.(12), examples of abstract partial views of an opaque smooth

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x1 v8 v6 x v x −→ 3 7 2 u2 v y1 2 v5 x6 x4 x v 5−→ 3 u1 v4 x7

Figure 12: An opaque shape, near viewpoints, and aspectual boundary components. shape K are as follows:

v1 ∞ C XC (13)

v2 XC ∞ C (14)

v3 ∞ C XC (15)

v4 ∞ C XC (16)

v5 ∞ C X1 C X2CX3C (17)

v6 X1C ∞ C X2C (18)

v7 XC ∞ C (19)

v8 X C (20) Likewise, examples of boundary components of aspects are as follows:

[x1, x2] closed interval between points of K, once tangent to K (21)

[x3, x4] likewise (22) −→ Ray(x5, u1) ray tangent to K (23)

[x6, y1] closed interval from point of K toanexternalpoint (24) −→ Ray(y1, u2) ray entirely external to K (25)

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Definition 15. The directed graph whose vertices are the aspects of a shape, and in which there exists an edge from one aspect to another provided they are contiguous and the second contains their shared boundary component, is called the aspect graph of the shape [26][27]. Observation 7. A line L ⊂ R2 may or may not meet a shape K. If L meets K then L may or may not be tangent to K. If K has a “flattened” portion then there exists a line L tangent to K at an infinite number of points. See Fig.(13).

Figure 13: A shape that is not rounded.

Definition 16. For line L and shape K let Tn(L, K) denote the number of points at which L is tangent to K, and define Tn(L, K) = −∞ if L∩K = ∅, and Tn(L, K) = +∞ if L is tangent to K at an infinite number of points. Otherwise, Tn(L, K) is a non-negative integer, and Tn(L, K) = 0 means L ∩ K = ∅ but L is not tangent to K at any point. Say K is a rounded shape if Tn(L, K) < +∞ for every line L. Henceforth, assume all shapes are rounded. Claim 7. If Tn(L, K) = 0 then #(L ∩ K) is a positive even integer. Evidence 1. K is bounded so there exist oppositely directed rays contained in L that do not meet K. At every point of L ∩ K there exists a crossing between Ex K and In K. But the extreme end points of L ∩ K have Ex K on opposite sides. 2 The circle is a “trivial” rounded smooth shape. First in complexity beyond the circle is the bean, which serves throughout this proposal as the generic example of a rounded smooth shape. [C]onsider an “oval with a dent”. The dent will involve a bitangent, two inflection points, and a vertex (of the opposite curvature to the major part). The configuration is not arbitrary, either, for

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instance, the pair of inflections must lie between the points of tan- gency of the bitangent. Such “feature clusters” are certain to be important in practice, yet they don’t figure in the literature. One way to get a handle on this is to view the dent as a result of some operation on an oval. When you “put the dent in” you must pass through a stage where the oval develops a flat point, and the features are seen to ”originate” from an explosion of the flat point into inflection points, etc. Since the features were “once” contained in a single point, they really belong together! Similar analyses can be done in surface theory. The authors consider any surface as the end result of the deformation of an initially simple surface such as an ovoid. As the surface ”develops,” feature clusters are created, and by noting the history one may build a hierarchical order of features. It is like noting the changes put by a sculptor on an originally shapeless blob of clay that made it into a simile of a human face [16]. Observation 8. The bean shape also appears in diverse contexts such as Morse Theory [5] and computer vision.3

∞ C X1 C X2 C

XC ∞ C

X1 CX2 C ∞ C

∞ C XC

Figure 14: The bean and its abstract partial views.

3Daniel Keren, “Advanced Topics in Computer Vision: 3D vision, Chapter 23, “Aspect Graphs,” http://cs.haifa.ac.il/~dkeren/acv/index.html

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∞ C X C X C 8 1 2 pp8 NNN ppp NNN ppp NNN ppp NN& XC C C XC ∞ ∞8 NNN pp8 NNN ppp NNN ppp NN& ppp X1CX2C∞ C

Figure 15: The abstract opaque shape of the bean.

Observation 9. Every shape has an abstract opaque shape speciﬁed by an algebraic structure, namely a directed graph with dots labeled by abstract partial views. For example, the abstract opaque shape of the bean is given by the diagram in Fig.(15).

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3 Part II. Total View of Transparent Shape from Far Viewpoint 3.1 Orientation Deﬁnition 17. Given a line L in the plane, an orienter of the line L is a direction −→w anchored at a point x of L, and a second orienter −→w ′ at x′ has the same orientation if −→w ′ = −→w . Hence, any line L has exactly two orientations. An oriented line is an ordered pair (L, −→w ) where L is a line and −→w is the direction of an orienter of L. An orienter of the plane R2 is an ordered pair (−→u , −→v ) of orthogonal vectors anchored at the same point. Two plane orienters have the same orientation if one is a translate and rotation of the other. Hence, the plane has exactly two orientations. The equation eiπ + 1 = 0, with “rotator function” θ → eiθ of the complex plane C suggests the “natural orienter” (1 + 0i, 0 + 1i) = (1, eiπ/2).

−→u −→′ −→v w y −→ x′ w ′ L y x −→′ u −→v ′

Figure 16: Orientation of the line and of the plane.

Henceforth, assume that the plane has the natural orientation. Deﬁnition 18. Given an oriented line (L, −→w ) the natural orienter of the plane may be translated and rotated until 1 + 0i coincides with −→w . In that case, 0 + 1i ends up lying in exactly one of the two half-planes into which L splits R2. Call that one the positive half-plane of (L, −→w ), and other its negative half-plane.

Observation 10. Relative to the natural orientation, each oriented line (L, −→w ) induces a partition of the plane, R2 = H+(L, −→w ) ∪ L ∪ H−(L, −→w ) where H+(L, −→w ) (respectively, H−(L, −→w )) is the positive (respectively, negative) half- plane associated with (L, −→w ). K −→ For a shape S1 −→ R2, the derivative unit tangent vector t (x) at x = K(eiθ) is an orienter for the tangent line L(x), hence at each point x of a shape there

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−→ exists a natural positive half-plane H+(x) := H+(L(x), t (x))) and negative −→ half-plane H−(x) := H−(L(x), t (x))).

3.2 Inﬂecta and Extrema Claim 8. Let K be a rounded shape in the naturally oriented plane. For any angle θ ∈ R and positive inﬁnitesimal ǫ > 0, if x = K(θ), b := K({ eiβ | θ − ǫ < θ }) (“before”), and a := K({ eiα | θ < α < θ + ǫ }) (“after”), then exactly one of the following holds:

a, b ⊂ H+(x); (26) a ⊂ H+(x) and b ⊂ H−(x); (27) a ⊂ H−(x) and b ⊂ H+(x); (28) a, b ⊂ H−(x). (29)

Definition 19. The point x ∈ K is a positive inflection of K if a ⊂ H+(x) and b ⊂ H−(x), and it is a negative inflection if a ⊂ H−(x) and b ⊂ H+(x). Otherwise, x is an extremum of K. Claim 9. (1) For any rounded shape K, either a, b ⊂ H+(x) for all extrema x, or a, b ⊂ H−(x) for all extrema x. (2) The inflections of K alternate from positive to negative and back.

Deﬁnition 20. For every shape S1 −→K R2, eiθ → K(eiθ) there exists its reverse shape K deﬁned by S1 −→K R2, eiθ → K(e−iθ). Observation 11. The orientations of tangent lines of K are the reverses of the orientations of the tangent lines of K, and so H+(x) and H−(x) are inter- changed at all θ.

φ Observation 12. Every shape induces a smooth map S1 × S1 −→ R given by φ(τ, θ) := Re(eiθ K(τ)), where τ is the parameter along K, and θ is the angle of a direction. In other words, viewing a shape at direction angle θ is equivalent to rotating the shape through angle θ and viewing it along the imaginary axis. Then φ measures the height along the real axis of a point with parameter τ. The result is essentially a smooth real-valued function on a 2-dimensional surface – “the torus” S1 × S1 – for each 1-dimensional shape, which is possibly of interest to Morse theorists [5]. It seems that these functions are unique for each shape. But there is the question, exactly which functions on the torus arise in this way from shapes? Deﬁnition 21. The set of all lines parallel to a given line in the plane is called a parallel pencil [22]. A far viewpoint −→v is the set of all oriented lines parallel to a given line with the same direction. A line belonging to a far viewpoint is called a sight-line, and a sight-line paired with one of its two directions is called a sight-ray.

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Observation 13. A parallel pencil corresponds to a “point at infinity.” That is, at infinity the lines in a parallel pencil “converge” to the same point in both directions. The set of all such points constitute “the line at infinity,” which together with the lines in the plane constitute the real projective plane [11], and one might be tempted to adopt the terminology and theory of the real projective plane in this work on shape. However, that would be misleading, because the direction of view can make a difference (by duality – see Claim (15) and Observation (19)) even if along the same sight-lines. For a rounded shape K, and for each sight-line L of a far viewpoint, Tn(L, K) is −∞ if the sight-line does not meet K, 0 if L meets K but is nowhere tangent to K, or is a positive integer that is the number of times L is tangent to K. Definition 22. The set of sight-rays of the far viewpoint determined by direction −→u such that 0 < Tn(L, K) < ∞ is called the view of K in direction −→u . Observation 14. The contact list of a sight-ray in a view consists of a point or points other than the first one which are not “directly visible.” In this sense the shape is considered to be “transparent” to a view. Claim 10. No matter the shape or view, since the shape is bounded there always exists at least two sight-lines in the view.

3.3 The Cell Complex Induced by a Far Viewpoint Observation 15. The sight-lines of a view divide the shape into connected subsets of the plane. Indeed, there exists a 2-dimensional cell complex [12] whose 1-dimensional skeleton is comprised of segments of sight-lines and portions of the shape itself. As the direction of the view rotates, the cell complex changes piecewise continuously. That is to say, 2-cells may change in shape without change in boundary 1-cells, they may suddenly pop into being, and they may just as suddenly wink out of existence. The same 2-cell may be the result of wink-outs in more than one way. Deﬁnition 23. A ﬁnite connected 1-dimensional cell complex is called a cycle- like complex if every 0-cell joins exactly two 1-cells, and is a chain-like complex if two 0-cells meet exactly one 1-cell each, and all other 0-cells join exactly two 1-cells.4 Claim 11. Every view of a shape K subdivides it into a cycle-like complex and determines a dimension-preserving projection of C onto a chain-like complex. See Fi.g(17).

4“-like” must be appended to the terms “cycle” and “chain” because these words have speciﬁc very well-known meanings in algebraic topology, and there must be no confusion introduced by my appropriation of them, which is based merely on visual appearance and my predilection for short suggestive words.

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Figure 17: The cycle-to-chain projection of a view of a shape.

Definition 24. For shape K, a far viewpoint −→u is a regular viewpoint if no sight-line of the view of −→u meets K at an inflection nor at more than one tangent point. Otherwise, −→u is a critical viewpoint. Claim 12. For every shape K and far viewpoint −→u there exists a 2-dimensional −→ cell complex CK ( u ) with underlying set In K ∪ K such that −→ −→ the 0-cells of CK ( u ) are the intersections of sight-lines of u with K; (30) −→ the 1-cells of CK ( u ) are either segments of K, or intervals of intersections of sight-lines of −→u with In K ∪ K; (31) −→ K itself is a cycle-like complex, and is the boundary of CK ( u ). (32) Claim 13. For every shape K there exists a regular viewpoint. There exists at most a finite number of critical viewpoints. Any non-zero infinitesimal variation of a critical viewpoint is a regular viewpoint. Let M be the maximum number of 2-cells among the associated cell complexes of the regular viewpoints. For an arbitrary far viewpoint −→u , let i be the number of inflection points amongst the −→ sight-lines of u , and let nk be the number of k-tangents amongst the sight-lines −→ −→ of u . Then nk = 0 for sufficiently large k, and the number M2( u ) of 2-cells in the cell complex associated with −→u is −→ M2( u ) = M − 2i − X(nk − 1) . (33) k≥2

Claim 14. (1) If [θ1, θ2] is an interval of regular view angles, then CK (θ1) and CK (θ2) are isomorphic cell complexes. (2) If 0 < ε ∈ R, and θ is the only critical view angle in the range from θ − ε to θ + ε, and both endpoints θ ± ε are regular view angles, then CK θ is a deformation retract of CK (θ + ε) or vice versa, and likewise it is a deformation retract of CK (θ − ε), or vice versa.

3.4 Vertically Ordered Graded Poset Deﬁnition 25. A grading of a ﬁnite non-empty poset (P, <) is an order- preserving surjection of (P, <) upon a totally ordered poset. A structure (P, <

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, g) such that g is a grading of (P, <) is called a graded poset.

g Observation 16. If P −→ X is a grading then it is just as well that X = [n] := { 1, 2, . . . , n } for n := #X, where the ordering of X is 1 < 2 < < n. With that understanding, g(x) is a natural number called the grade of x by g. If the longest chain in (P, <) is of length n, then the Hasse diagram of (P, <) automatically determines a grading where the least elements of (P, <) have grade 1, and the greatest elements have grade n. Every ﬁnite poset has a trivial grading, namely any function with codomain a singleton, but in general there are possibly several more interesting gradings.

g Definition 26. The fiber over i, 1 ≤ i ≤ n of a grading P −→ [n] is the inverse image g−1(i). A vertically ordered graded poset is a structure (P, <, g, ∨) where ∨ is a total ordering of each fiber such that if x ∨ y in the fiber over i, and x < u, y < v for u, v in the fiber over i + 1, then u ∨ v:

< < if x / u then x / u

∨ ∨ ∨ y / v y / v < <

In the sequel, “Vertically ordered graded poset” is shortened to VOG poset, and the whole structure is represented by (P, <, g, ∨), where (P, <) is the “horizontal” poset, and for each i, 1 ≤ i ≤ n the vertically ordered ﬁber at i is (g−1(i), ∨). Every VOG poset (P, <, g, ∨) has a dual VOG poset (P ∗, <∗ , g∗, ∨∗) in which (P ∗, <∗) is the reverse poset of (P, <), g∗(x) := n − g(x), and ∨∗ is the reverse ordering of ∨ on each ﬁber. Claim 15. VOG posets are the objects of a category VOG in which the morphisms are order-preserving morphisms of horizontal posets which commute with the grading and preserve the vertical ordering. Moreover, dualization is a (co- variant) functor ∗ VOG −→ VOG .

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E J G D H

K F A I

B L C

(34)

D/ e E/ j J/ g (35) B BB G BB! BBB F/ H/ / k N K I h / NN'N 7ppp NN ppp L i 8ppp A C C pp / b / c / l p

∆A ∆A⊗1A ⋄A⊗⋄A⊗1A 2D ⊗1⊗b A −−→ A ⊗ A −−−−−→ A ⊗ A ⊗ A −−−−−−−→ I ⊗ I ⊗ A −−−−−−→ (36)

e⊗2F ⊗c j⊗k⊗l g⊗i D ⊗ I ⊗ B −−−−−→ E ⊗ F ⊗ C −−−−→ J ⊗ K ⊗ L −−→ G ⊗ I −→h H (37)

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R G′′ ′′ E H′′ D′′ P S N T V Q F ′′ M

U A′′ B′′ W

C′′

(38)

′′ ′′ D ′′ E R ′′ / e / r / g ′′ (39) NNNG N' NN p ′′ V o NN P H p ′′ o7 o NN' ′′ p7 p F T o NN D pp / / v oo n / ′ p / t / / n QQ M Q { QQ( Q {= {{ Q ′′ ′′ ′′ {{ A B C U W {{ / b′′ / c′′ / u / w / q

∆ ′′ ∆ ′′ ⊗1 ′′ ⋄ ′′ ⊗⋄ ′′ ⊗1 ′′ A′′ −−A−→ A′′ ⊗ A′′ −−−−−−−→A A A′′ ⊗ A′′ ⊗ A′′ −−−−−−−−−−→A A A I ⊗ I ⊗ A′′ (40) ′′ ′′ ′′ ⋄ ′′ ⊗1I⊗b e ⊗⋄ ′′ ⊗c −−−−−−−−→D D′′ ⊗ I ⊗ B′′ −−−−−−−−→F E′′ ⊗ F ′′ ⊗ C′′ ′′ q ⊗v⊗w ⊗ ⊗ p⊗q ′ −−−−−−→ R ⊗ T ⊗ U −−−−→r t u G′′ ⊗ V ⊗ W −−→ P ⊗ Q −→n N −→n H′′ ⊗ M (41)

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′ ′ E D G′

H′

F ′ A′

I′ B′ C′

(42)

′ ′ D ′ E ′ / e / g ′ (43) NN G NN& ′ NN H h′ / ′ I yy ′

∆ ′ ∆ ′ ⊗1 ′ ⋄ ′ ⊗⋄ ′ ⊗1 ′ A′ −−−→A A′ ⊗ A′ −−A−−−−→A A′ ⊗ A′ ⊗ A′ −−−−−−−−−→A A A I ⊗ I ⊗ A′ (44) ′ ′ ′ ′ ′ ′ ⋄D′ ⊗1I⊗b e ⊗2F ′ ⊗c g ⊗i −−−−−−−→ D′ ⊗ I ⊗ B′ −−−−−−−→ E′ ⊗ F ′ ⊗ C′ −−−→ G′ ⊗ I′ −→h H′

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Observation 17. Diagrams (34,38,42) exhibit VOG posets associated with three views of a smooth shape. The corresponding diagrams (35,39,43) are “string diagram” representations constructed by a self-evident algorithm where 2-cells become “wires” and 1-cells become “boxes.” The diagrams (36,40,44) are the formal algebraic equivalents of the string diagrams. Observation 18. These representations of VOG posets by diagrams of arrows and “tensor products” are inspired by graphical notations encountered in the context of Higher Category Theory. The “Copy”, “Erase”, “Merge”, and “Ini- tial” diagrams associated with biproduct categories in Peter Selinger’s survey of graphical languages for monoidal categories seem to ﬁt.5 One may speculate that the (ﬁnite) set of all 2-cells of all cell complexes determined by all far viewpoints of a rounded smooth shape may be the objects of a biproduct category generated by VOG poset morphisms.

5 http://www.mscs.dal.ca/~selinger/papers/graphical.pdf

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moving retraction of 2-cell to 0-cell bitangent sight-ray vertically VOG poset morphism ordered graded x b” poset b b’ a’ a a”

view sight-ray y

c” z” z’ z

c u

c’

e’

e e” view rotation d’ d” v d

moving homeomorphism of 2-cell

moving retraction of 2-cell to 0-cell inflection sightray

u x b”

a b

b’ c’

c w c” z w’

moving retraction of 2-cell to 1-cell

Figure 18: For a bean shape, bitangent and inﬂectional critical views in the middle, with neighboring regular views to the left and the right.

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Fig.(18) exhibits algebraic and topological structure associated with the qualitative shape of a bean. Top center is the shape viewed along a sight- ray meeting the shape at two tangent points a, e. Parallel sight-rays are tangent at c, z, and intersect the shape at b, d. These rays subdivide the interior of the shape together with the shape itself into a 2-dimensional cell complex comprised of 0-cells a,b,c,d,e,z, 1-cells ab,bc,cd,de,ez,za, and 2-cells abz,bcdz,dez. This view direction is critical, since slight rotation of the view direction clockwise or counterclockwise dramatically alters the cell complex determined by tangent sight-rays. Counterclockwise, the cell complex on the left consists of 0- cells a′, x, b′, c′, d′, e′, z′, y, 1-cells a′x, x′b′, b′c′, c′d′, d′e′, e′z′, z′y, ya′, and 2-cells a′xy, xb′z′y, b′c′d′z′, d′e′z′. Clockwise, the cell complex on the right is comprised of 0-cells a′′, b′′, c′′, d′′, v, e′′, u, z′′, 1-cells a′′b′′, b′′c′′, c′′d′′, d′′v, ve′′, e′′u, uz′′, z′′a′′, and 2-cells a′′b′′z′′, a′′b′′c′′c′′z′′, d′′vuz′′, ve′′u. Similar cell complexes are associated with the view along an inflection sight-ray, bottom center. The 2-cells of each cell complex in Fig.(18) are the elements of a VOG poset, indicated by red arrows. At lower right is the trivial 1-element VOG poset, which represents the fact that the view direction from afar does not see any difference between the transparent bean shape and a circle. Then again, at lower left the view direction from afar upon the transparent bean shape yields a 4 element VOG poset. The VOG posets associated with regular views infinitesimally clockwise or counterclockwise from a critical view are related by VOG poset morphisms as indicated by green arrows. Moreover, as view direction varies continuously away from a critical view direction, cells change continuously. 0-cells move along the shape itself, hence 1-cells change within the interior of the shape, and consequently 2-cells change via a continuous homeomorphism as well. Indeed, cells may come into existence or disappear, but systematically. That is to say, for example, a 2-cell may retract continuously to a moving 1-cell or even to a moving 0-cell. Such transformations are indicated by purple arrows. A purple double arrrowhead signifies moving retraction onto a lower-dimensional cell, and a two-headed arrow is a moving homeomorphism.

Deﬁnition 27. A smooth 2-dimensional variable cell-decomposition of a space X is a structure (X0,X1,X2) such that Xi is a ﬁnite set of smooth maps Di × [0, r] −→a X, and if

i ( ,t) Xi(t) := { D −−→ X | a ∈ Xi }, t ∈ [0, r], i = 0, 1, 2 (45) (46) then (X0(t),X1(t),X2(t)) is a cell-decomposition of X. Claim 16. If a space X has a 2-dimensional cell-decomposition, then there exists a category C(X) whose objects are cell-decompositions of X, and a morphism from (U0,U1,U2) to (V0,V1,V2) is a smooth 2-dimensional variable cell- decomposition (X0,X1,X2) such that Xi(0) = Ui and Xi(r) = Vi for i = 0, 1, 2.

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Composition of variable cell-decompositions with intervals [0, r] and [0, s] is a variable cell-decomposition with interval [0, r +s] by concatenation of maps, and the identity morphism of (U0,U1,U2) consists of constant maps (cf. [6], p.78). Observation 19. Given a rounded smooth shape K let X := In K ∪ K. Each view direction from afar upon K considered to be transparent yields a cell- decomposition of X. As the view direction rotates from 0 to 2π, there exists a ﬁnite circular diagram in C(X) in which the dots correspond to critical views of K, and the arrows are smooth 2-dimensional variable cell-decompositions. As suggested by Fig.(18), arrows may alternate in direction. Furthermore, for any view direction, the VOG poset in the exact opposite direction is the dual VOG poset. Deﬁnition 28. For any rounded smooth shape K the circular diagram in C(X) is the abstract transparent shape of K.

4 Part III. Approximation and Experimentation 4.1 Bezier Shape The commercially available Adobe Illustrator application program has a mod- estly named functionality called the “Pen Tool.” It is used to place and join a sequence of “anchor points.” The first fact about an anchor point is that it may be selected and moved to an arbitrary position on the “artboard.” The amazing fact about an anchor point is that it is an inflection point on the curve joining the anchor points in their sequence. This is amazing because these inflection points have “handles” which may be used to rotate, shrink, or expand the size of the inflection. An infinite variety of open or (not necessarily simple) closed curves may be constructed using the Pen Tool, see Fig.(19). Indeed, one may claim that given any smooth curve and a tolerance not only on the position of the points of the curve, but on the direction of the derivative at each point of the curve, there exists a curve constructed by the Pen Tool which matches the given curve within those tolerances. Complete mathematical programming code for so-called Bezier curves is freely available.6 The implication is that a finite amount of data may approximate arbitrarily closely any given smooth curve. There is a disturbing question of whether it makes sense to ask for the group of transformations which preserve qualitative shape of Bezier shapes.

6http://processingjs.nihongoresources.com/bezierinfo/

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Figure 19: Closed curve constructed with the Pen Tool in Adobe Illustrator. Some anchor points and handles are shown. (All the smooth shapes in this proposal were created with Adobe Illustrator.)

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4.2 Screenshots of Scultoro A first theorem in elementary Morse Theory is about a smooth real-valued map on a smooth shape (a.k.a., “smooth compact manifold”). If the extrema of the map are “non-degenerate,” that is, not like inflecta, then the shape may be re- constructed as a cell complex step-by-step from knowledge of the critical values of the map [5]. Observation 20. Given a far viewpoint upon a transparent shape, suppose the sight-ray through the center-of-gravity of the shape is singled out. With respect to the standard orientation of the plane there exists a directed line orthogonal to that sight-ray also through the center-of-gravity. These orthogonal rays constitute a rectangular coordinate system, with sight-ray as abscissa and its orthogonal ray as ordinate. Since the shape is actually a smooth map on the circle to the plane, each image of a point on the circle determines a height from the chosen sight-ray along the orthogonal. Hence, for each far viewpoint there exists a smooth real-valued map on the shape. Since far viewpoints correspond also to points on the circle, we have defined for the given shape a smooth map on S1 × S1 (cf. Observation (12)). One thing to do with this map, which I claim is determined by the shape, and up to the choices made in its construction determines the shape, is to apply Morse Theory to it. Another thing to do with it is to graph it. Fig.(20) is a screenshot of my Mathematica program called “Scultoro” which calculates and displays the map in this case for a circle shape. One axis represents view direction from 0 to 2π, a second axis represents the parameter of the shape, and the third axis is the calculated height. As one would expect, the graph is a sinusoidal wave.

Figure 20: The built-in circle shape invoked by the CRCL button.

“Scultoro” has a virtual button which enables a blue dot to behave like a hammer, so that if the real mouse button is pressed while over the dot, then the shape nearby becomes dented, as though struck with a hammer. The strength of the blow is greater as the blue dot is moved closer to the shape. Fig.(21)

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shows how the circle becomes a bean when it is banged by the hammer, and the resulting crinkle in the wave is exhibited on the right. A more complex

Figure 21: Toggling the TRAK button to BANG enables the blue hammer to bang the circle into a bean shape. dented shape and its wave is exhibited in Fig.(22). The TRAK button converts

Figure 22: The built-in dented shape banged from a circle is invoked by the DNTD button. the mouse from a hammer banger to a free-hand drawing tool, as illustrated by Fig.(23). Comparison of constructions based on a far viewpoint of the transparent dented shape, and on a near viewpoint of the opaque dented shape, are enabled by the DRWG button of “Scultoro,” as shown in Fig.(24).

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Figure 23: Toggle to the TRAK button for free-hand shape drawing.

5 Part IV. An Imperfect Analogy: Knots and Shapes

Aside. At the Joint Meeting of the AMS and MAA in Boston, January, 2012, I had a conversation with magister Barry Mazur after his talk on mathematical plausibility, which emphasized examples from history, including multiple references to Euler. With a seemingly false analogy between Knot Theory and Shape Theory in the back of my mind, after his talk I waited my turn and said to him, “You spoke of analogy guiding mathematical inquiry. But what if the analogy turns out to be a false analogy?” Mazur: “Then there is something much deeper going on. For example Kronecker’s liebster Jugendtraum.” Pause. Me: “Did you say Euler’s ... what?” Mazur – looking down his nose at me even though I am taller than he – said, “No, Kronecker. His Jugendtraum. Don’t you know?” 7 My personal liebster Altertraum (“dearest old-age dream”) is to relate abstract algebraic structure of far views of a transparent shape to abstract algebraic structure of near views of the opaque shape, and thus better to understand qualitative shape.

Fortunately, Barry Mazur distributed copies of “rough notes” for his talk, “Why is it plausible?”.

But reasoning by analogy is, I think, the keystone: it is present in much (perhaps all) daily mathematical thought, and is also of-

7Triggered by an analogy articulated by David Mumford, Barry Mazur developed new ideas relating Knot Theory (speciﬁcally, the Alexander polynomial of a knot) to Number Theory. See http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf.

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Figure 24: At the left are sight-lines of an inﬁnitely far viewpoint, and a cycle- like complex calculated for the freehand transparent shape. At the right is the partial view of the opaque shape from a near viewpoint.

ten the inspiration behind some of the major long-range projects in mathematics. And, Andr´eWeil was right when he said: “nothing also gives more pleasure to the researcher.”

I add that the role of analogy in physics is equally important [23]. Mazur mentions, among the grand analogies, the Langlands Program in Number The- ory, and that “[t]he analogy between number fields and function-fields of one variable over finite fields is a more elementary, and older example.” In a footnote, he writes “as with all great analogies, its imperfections sparkle, raising questions that may lead to future theories, far deeper than the ones we currently are at home with.” To suggest the inestimable importance of Knot Theory in mathematics and physics I paraphrase the already brief history of Knot Theory recounted by Michael Atiyah in [1]. Physicist Lord Kelvin in 1867 published the idea that atoms are actually “knotted vortex tubes of ether,” based on the observations that such ethereal knots would be stable, would come in many varieties – the chemical elements – and would vibrate in ways to account for experimentally measured “spectral lines.” In modern times, transmutation of elements would correspond to cutting and recombination of knots. The idea was taken seriously for a while, giving rise to long lists – “knot tables” – of knots represented by diagrams. But the idea was soon discarded by physicists and the study of knots became merely “an esoteric branch of pure mathematics,” with some efforts devoted to proving abstract conjectures about the classification of knots. Early on it was realized that a knot, defined as a specific kind of subset of three-dimensional

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space, might be deformed – as though made of flexible wire – yet retain its essential“qualitative” nature even after deformation, so long as no portion of it is forced to pass through another portion during the deformation. In other words, it is not a knot per se that is the proper object of study, but its equivalence class “up-to-deformation.” In 1928 James W. Alexander published the first algebraic structure associated with each knot class, and in 1984 Vaughan F. R. Jones constructed a polynomial for each knot class. Nothing is more algebraical than a polynomial, and several polynomials have subsequently been invented for each knot class [14][20]. Generally, the mathematical objective is to define an algebraic structure for each knot class which uniquely identifies the knot class: every knot class should correspond to exactly one distinct structure. Such a structure should not vary even if a knot class representative is deformed in space, so is called an invariant of the knot. The Alexander algebraic structure, the Jones polynomial, and their children, are called knot invariants. It is a separate question to find means of accurately representing knots, which are intrinsically subsets of 3- dimensional space, in terms of notations, called knot presentations, on the 2-dimensional plane of writing paper.8 The surprise is that knots have re-appeared big-time in theoretical physics. So much so, that already by 1989 there appeared a one thousand page collection of papers on “new developments in the theory of knots” published not as a work of mathematics, but in an advanced series on mathematical physics [17]. Indeed, Knot Theory – with related concepts, such as links, braids, and tangles – is entwined in a truly grand analogy between Category Theory, physics, topology, and logic [2]. Aside. Perhaps I have “knot envy” – analogous to the erstwhile “physics envy” of social sciences – since I recently asked myself whether knots in space are intrinsically richer mathematical structures than smooth shapes in the plane, and that therefore I should not harbor hopes that studying such shapes should give rise to a particularly interesting theory. I console myself with the rejoinder that mathematical riches arise from the invention of questions, and from the difficulty of answering the questions, and from how surprising, clever, or far- reaching might be the answers. Simply, do the questions yield to breathtaking feats of calculation and reasoning?

5.1 Homotopy, Isotopy, and Ambient Isotopy A central idea in algebraic topology is “continuous reshaping.” In this con- f text a continuous map X −→ Y consists of an abstract shape X which via f parametrizes the actual shape f(X) inside the ambient space Y . For example in this work, the circle S1 is the abstract shape, R2 is the ambient space, and a smooth shape is a smooth embedding S1 −→K R2. In the case of knots, S1 is

8Speciﬁcation of the requirements for knot presentations are exactly stated, and several historically important examples are reviewed, in “Knots as processes: a new kind of invariant,” L. G. Meredith and David F. Snyder, http://128.84.158.119/abs/1009.2107.

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also the abstract shape, but a knot is a continuous map S1 −→K R3. A three- dimensional smooth shape would be a smooth embedding S2 −→K R3, where the abstract shape is a 2-dimensional sphere, and so on. An abstract shape realized in two possibly diﬀerent actual shapes is modeled by a pair of maps,

f ! X Y (47) = g Aside. The idea of “simple” things like the unit circle serving as a template or prototype for shapes in an ambient space permeates algebraic topology. For example, we have simple things like points, segments, triangles, and higher dimensional triangles as abstract shapes mapping into spaces, and we have algebraic tools such as simplicial homology calculated from these maps to study properties of the ambient space. F. W. Lawvere discusses this idea in [18] with the language “X-shaped figure in Y ,” and he writes, “An ancient principle of mathematics holds that a figure is a locus of a varying element.” (See Observa- tion (3).) On the other side, physics is shot through with maps from a space to a relatively “simpler” structure such as the real or complex numbers. Namely, measurements assign numbers to points of space, where space may, for example, represent states of a physical system. A modern elaboration of this idea is provided in the context of a proposal to “radically revise” the theory of quantum gravity: Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics uses the topos of sets. Other theories involve a different topos. For the types of theory discussed in this proposal, a key goal is to represent any physical quantity A with an arrow Aφ :Σφ → Rφ where Σφ and Rφ are two special objects (the ‘state-object’ and ‘quantity-value object’) in the appropriate topos, τφ [9]. Back down to Earth, Morse Theory begins with study of real-valued maps defined on a shape, and in this proposal each shape determines such a map, as discussed in Observation (20) about Scultoro. The technical word for “continuous reshaping” is homotopy. Definition 29. A homotopy from f to g in diagram (47) is a new diagram of continuous maps of the form

( 1X 0 ) X f (48) tt AA tz t H A X × I / Y Jd > JJ }} ( 1X 1 ) J } g X

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and this is abbreviated to the diagram

f X H Y . (49) B g

If there exists at least one homotopy between coterminous maps f, g then they are called homotopic maps. All the maps between two spaces are partitioned into sets of mutually homotopic maps. In other words, “homotopic to” is an equivalence relation on coterminous maps [6].

In a contractible space any two shapes are homotopic [12]. In particular, R2 or R3 are contractible. In them a homotopy can shrink f(X) to a point and then expand the point to g(X). In this way a homotopy can continuously change any shape into any shape. So, homotopy is not a good candidate for “continuous re-shaping” while preserving “qualitative shape.” A more restrictive condition on a homotopy is that at all times t ∈ I the image of X in Y should be homeomorphic to X.

Definition 30. A homotopy (49) is an isotopy if H ◦ ( 1X t ) is a homeomorphism at all times t ∈ I. Maps between two spaces fall into distinct mutually isotopic sets, but these sets are more exclusive than the homotopy classes of maps. This is going in the right direction, but it has been observed that any knot is isotopic to the circle. For, given a knot in space, pull a part of it outward to the shape of a very large circular arc, and then continuously shrink the rest of the knot down towards a single point. “[I]t is easy to see that every sufficiently smooth simple, closed curve can be deformed continuously into a circle without ever acquiring a multiple point, even though the curve may be one which ought to be regarded as knotted. In figure 1, for example, we have a series of self- explanatory diagrams showing a trefoil knot in the process of being deformed into a circle. It will be noticed that the curve acquires no multiple point even at the moment it becomes a circle.”9 An even more restrictive idea is not that one shape is deformed to another inside space, but that the whole space is deformed and thereby re-shapes one shape to another.

9J. W. Alenxander, “Some problems in topology”, http://www.mathunion.org/ICM/ ICM1932.1/Main/icm1932.1.0249.0257.ocr.pdf .

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Deﬁnition 31. An ambient isotopy of f to g in (47) is a diagram pair

1Y X H YX (50) B ~ @@ f ~~ @@g L ~~ @@ ~~ @ Y / Y L where now H is an isotopy of the identity map of Y to the spatial homeomorphism L, and g = L ◦ f. Ambient isotopy is also an equivalence relation on coterminous maps. Also, given an ambient isotopy (50), there exists an isotopy H ◦ ( g 1I ) between f and g. The point of recapitulating these technical deﬁnitions is merely that Knot Theory has in ambient isotopy a mature deﬁnition of what it means for two knots to be qualitatively the same.

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5.2 Analogy Table with Imperfections Knot Shape Analogy (1) A knot is a map that embeds A (2D smooth) shape is a the circle into the space. smooth map that embeds the circle into the plane. (2) A knot diagram is the image of The dimension preserving a regular knot projection of the cellular map induced by a space upon the plane, regular far viewpoint of a annotated with an ordered pair transparent shape totally of points of the circle to orders the 1-cells that project indicate which is “over” and upon the same 1-cell on the which is “under” in the line. diagram. (3) At a crossing of a knot diagram Over a 1-cell on the line there the inverse image has exactly 2 is an even number of 1-cells in points. the inverse image. (4) If a sufficiently small circle If a sufficiently small interval surrounds each crossing, then surrounds an interior point of a each circle intersects the knot 1-cell in the projection, then its diagram in 2 × 2 points. inverse image consists of n × 2 intervals. (5) Thesetof2 × 2 small segments The set of n × 2 small segments at intersections are “wired” to corresponding to the choice of form the whole knot. 1 interior point in each 1-cell of the projection are connected to form the whole shape. (6) For every knot projection there For every far viewpoint there exists an infinitely nearby exists an infinitely nearby regular knot projection ([20], regular viewpoint. p.22). (7) Every knot projection infinitely Every far viewpoint infinitely near to a regular knot near to a regular far viewpoint projection is a regular knot is a regular far viewpoint. projection.

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Knot Shape Imperfection (8) A Reidemeister move of a knot Certain moves of the 1-cells in diagram is a “local sliding” the projection corresponding to that is equivalent to a local a far viewpoint also correspond rotational distortion of the to a change in the far space containing the knot viewpoint, but not to a change above the plane. in the shape. (9) Cutting a knot at two distinct Cutting across a shape along a points yields two endpoints a, b non-tangential sight-line yields of one connected part, and two two (not necessarily connected) endpoints c, d of the other. parts with multiple endpoints Joining a to b and c to d yields at specific locations in the two new knots. Conversely, plane together with specific given two knots, a new knot is tangent directions at each composed from them by endpoint. These endpoints cutting one to yield endpoints may be paired off and a, b and cutting the other to connected smoothly by yield endpoints c, d, then end-caps in at least one way joining a to c and b to d. ultimately to yield two new smooth shapes. Conversely, two “suitable” figures in the plane with matching data involving locations and tangent directions could be re-arranged to join up and compose a new smooth shape (cf. composition of 1-dimensional tangles in ([2], p.9)). (10) Although there exist algebraic Knot Theory has a plethora of constructions associated with a combinatorial-algebraic smooth shape, there does not invariants or near-invariants of exist an independent definition knots-up-to-ambient-isotopy. of qualitative equivalence of shape.

6 Part V. Conclusion and Problems

Abstract algebraic structures are associated with near views of opaque shapes and with far views of transparent shapes. There exist references to relevant literature on phenomenology, psychology, computer vision, mathematics, and mathematical physics. Formal deﬁnitions of concepts relating to shapes in the plane are articulated, and several claims are made, with drawings and diagrams oﬀered in evidence. One problem is to convert these claims and that evidence into theorems and

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proofs. Another is to answer questions such as how to determine the minimum number of near views that are adequate to see the entirety of a plane shape. Observation (18) alludes to a biproduct category. If there exists a biproduct category in which certain diagrams of chains of morphisms correspond to VOG posets, then a problem is to translate the circle of VOG poset morphisms corresponding to the far views of a transparent shape into a circle of morphisms of such chains. If there exists a category of aspect graphs corresponding to the near views of opaque shapes, and if there exists a category of circular diagrams of VOG posets corresponding to the far views of transparent shapes, then there is the problem of constructing a pair of adjoint functors between these categories. This adjoint pair might answer the question implicit in the Abstract, which is, “How to formalize the inverse experiences going from mental model to partial views, and back?”. As knots are included among links, which are multiple knots in space, so shapes in the plane may collect into systems of shapes, like the British Isles. Also, a shape may have more than one hole, so that its boundary consists of more than one connected 1-dimensional complex. Then again, shapes may be nested, as islands in lakes on islands in oceans. All these cases require generalizations of constructions in this proposal based on near and far views of single opaque and transparent shapes. The most important problem is to solve the analogy, “qualitative shape is to shape as ambient isotopy is to knot” for the unknown, qualitative shape. Another way of putting this, is to observe that I have constructed candidates (aspect graphs, VOG posets) for invariants, but there remains the question, what is it that varies? Perhaps a way forward is to create the category of Bezier shapes, in which the objects are smooth shapes in the plane constructed by joining some finite number of anchor points using the pen tool. The morphisms are finite sequences of operations of adding or deleting anchor points, or adjustments of handles at inflecta. Maybe this Bezier category is to shapes in the plane as the category of cell complexes is to the category of topological spaces, and is the ground in which to grow the concept of qualitative shape, then to pick the low hanging fruit. Beyond the scope of this proposal is the project of generalizing from shapes in the plane to shapes in space like sculptures, shapes which if sliced by planes can sometimes lead back to smooth shapes in the plane.

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References

[1] Michael Atiyah, The Geometry and Physics of Knots, Cambridge Univer- sity Press, 1990. [2] John C. Baez and Mike Stay, Physics, Topology, Logic and Computation: A Rosetta Stone, March 2009. [3] Nestor Bertoglio and Rolando Chuaqui, An elementary geometric nonstandard proof of the Jordan curve theorem, Geometriae Dedicata 51 (1994), no. 1, 15–27.

[4] S. Biasotti, L. De Floriani, B. Falcidieno, P. Frosini, D. Giorgi, C. Landi, L. Papaleo, and M. Spagnuolo, Describing shapes by geometrical-topological properties of real functions, ACM Computing Surveys 40 (2008), no. 4. [5] Raoul Bott, Morse theory indomitable, Publications Mathématiques de L’H.E.Sˆ 68 (1988), 99–114, Fig. (3) exhibits critical points of a function defined on the circle in terms of a bean diagram. [6] Ronald Brown, Topology and Groupoids, www.booksurge.com, 2006. [7] Eliseo Clementini and Paolino Di Felice, A global framework for qualitative shape description, GeoInformatica 1 (1997), 11–27. [8] Ellis D. Cooper, Point-humans: optical perception by people and computers, Interface Age 2 (1977), no. 12. [9] Andreas Döring and Chris Isham, ’What is a Thing?’: Topos theory in the foundations of physics, (2008), arXiv:0803.0417v1 [quant-ph] 4 March 2008. [10] Rieger J. W. Fendrich and H-J Heinze, The contribution of eye movements to anorthoscopic percepts under free-viewing conditions, Perception, vol. 33, 27th European Conference on Visual Perception, 2004, Abstract Supple- ment. [11] F.R.S. H. S. M. Coxeter, The Real Projective Plane, 2 ed., Cambridge At The University Press, 1961. [12] Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002. [13] Edmund Husserl, Thing and Space: Lectures of 1907, Kluwer Academic Publishers, 2010, Translated by Richard Rojcewicz. [14] Louis H. Kauffman, On Knots, Princeton University Press, 1987. [15] J. J. Koenderink and A. J. van Doorn, The shape of smooth objects and the way contours end, Perception 11 (1982), 129–137.

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[16] Jan J. Koenderink, Two- and Three-Dimensional Patterns of the Face, The Mathematical Intelligencer 22 (2000), no. 3, Review of book authored by Peter W. Halliman, Gaile G. Gordon, A. L Yuille, Peter Giblin, and David Mumford published by A. K. Peters, 1999. [17] Toshitake Kohno (ed.), New Developments in the Theory of Knots, Ad- vanced Series in Mathematical Physics, vol. 11, World Scientific, 1989. [18] F. William Lawvere and Stephen Schanuel, Conceptual Mathematics, Cam- bridge University Press, 1997. [19] Michael Leyton, A Generative Theory of Shape, Springer-Verlag, 2001. [20] Charles Livingston, Knot Theory, The Mathematical Association of Amer- ica, 1993. [21] Saunders Mac Lane, Categories for the Working Mathematician, Springer- Verlag, New York, 1971. [22] George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag, 1975. [23] Ciara Aisling Muldoon, Shall I Compare Thee to a Pressure Wave? Vi- sualisation, Analogy, Insight and Communication in Physics, Doctor of Philosophy, University of Bath, May 2006. [24] Louis Narens, A nonstandard proof of the Jordan curve theorem, Pacific Journal of Mathematics 36 (1971), no. 1, 219–229. [25] Lawrence Neffstout, Two discrete forms of the Jordan curve theorem, Amer- ican Mathematical Monthly 95 (1988), no. 4, 332–336. [26] N. Ahuja C. Dyer A. Pentland R. Jain O. Faugeras, J. Mundy and K. Ikeuchi, Why Aspect Graphs are Not (Yet) Practical for Computer Vi- sion, Workshop panel report, University of South Florida, Department of Computer Science and Electrical Engineering, 1992. [27] Robert D. Schiffenbaur, A Survey of Aspect Graphs, Technical Report TR- CIS-2001-01, Polytechnic University, Department of Computer and Infor- mation Science, 2001. [28] Giandomenico Sica (ed.), What is Category Theory?, Polimetrica, Monza, Italy, 2006. [29] Hassler Whitney, On regular closed curves in the plane, Compositio Mathe- matica 4 (1937), 276–284, http://archive.numdam.org/ARCHIVE/CM/CM_ 1937__4_/CM_1937__4__276_0/CM_1937__4__276_0.pdf. [30] , Geometric Iintegration Theory, Princeton University Press, 1957, Appendix I. Section 10, Affine spaces.

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[31] Freek Wiedijk, Formal proof - getting started, Notices of the American Mathematical Society 55 (2008), no. 11, 1408–1414, The Jordan Curve Theorem was formally proved twice by independent researchers in 2005.

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Index liebster Altertraum, 32 dimension preserving cellular map, 38 drops adjoint pair, 40 back, 10 ambient isotopy, 37 to infinity, 10 analogy, 32, 34, 40 false, 32 embedding, 6 anorthoscopic perception, 0, 3 Erlanger Program, 2 Euclidean Geometry, 2 Bezier extremum, 18 category, 40 curve, 28 fiber, 21 shape, 28, 40 graded poset, 21 category grading, 20 aspect graph, 40 group of transformations, 28 Bezier, 40 biproduct, 25, 40 Hasse diagram, 21 cell-decomposition, 27 homeomorphism, 2 circular diagram, 40 homotopy, 35 located vector, 5 horizontal poset, 21 path, 8 vertically ordered graded poset, 21 inflection Category Theory, 0 positive (negative), 18 cell complex, 20 invariant, 34 complex isotopy, 0, 36 1-dimensional, 40 cell, 19, 20, 25, 27, 30, 40 Jordan Curve Theorem, 7 chain-like, 19 knot cycle-like, 19 class, 34 number, 35 classification, 33 plane, 6, 17 invariant, 34 computer vision, 3 presentation, 34 contact table, 33 first point, 10 Knot Theory, 0 list, 9, 19 continuous synthesis, 2 Langlands Program, 33 cross, 9 line at infinity, 19 curve, 7 links, 40 closed, 7 simple, 7 mental cyclic permutation, 11 model, 40 process, 3 deformation retract, 20 Morse Theory, 0 derivative tangent vector, 6

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optical character recognition, 3 generative theory, 3 oriented line, 17 partial view of, 3 positive (negative) half-plane, 17 qualitative, 0, 2, 32 orienter qualitative invariant, 4 of a line, 17 reverse, 18 of the plane, 17 rigid, 2 natural, 17 rounded, 14 transparent, 19 parallel pencil, 18 unit tangent vector, 17 parametrization, 7 sight- partial view, 4, 8–11, 40 interval, 8, 9, 11 abstract, 11, 12, 16 line, 18, 20, 39 the bean, 15 ray, 18, 19, 27, 30 integration, 3 perception, 2 Theory phenomenology, 0, 2 Category, 34 plane, 4 Higher Category, 25 point, 4 Knot, 32–34, 37, 39 vector, 4 Morse, 3, 15, 18, 30, 35 point Number, 32, 33 interval, 5 Shape, 32 translate by a vector, 4 topological space, 2 point at inﬁnity, 19 total perception, 3 pointhuman, 4 poset unit circle, 6 VOG, 25 unity of experience, 2 psychology of perception, 3 variable cell-decomposition, 27 qualitative change, 2 vector qualitative shape, 0 located, 5 quantum gravity, 35 vertically ordered graded (VOG) poset, 21 ray, 5 view, 19 direction, 5 viewpoint vertex, 5 abstractly equivalent near, 11 real projective plane, 19 exterior, 4 rigid isometry, 2 far, 18, 20, 25, 30, 38 near, 8–11 shape, 6, 11 regular, 38 abstract opaque, 16 regular (critical), 20 abstract transparent, 28 visual aspect, 12 event, 11 component, 12 perception, 3 contiguous, 12 aspect graph, 14 bean, 14

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Page 202 of 302 This PDF is for review purposes ONLY. Unpublished.1998-2013 This PDF is for review purposes ONLY.

Virtuality, Reality and Phenomenological Spirituality - ID: 43466

Type of Funding Request: Online Funding Inquiry Status of this Funding Request: Draft/Saved

Instructions:

This PDF document is a read-only version of your funding request as it currently stands. You may print or save this document for your records. However, you should not attempt to edit this document. Any edits made to this PDF will not be saved in the Templeton Portal. If there are further edits that you would like to make to your funding request, please return to the request inside the Templeton Portal, select the appropriate section(s) of the form, make your edits, and save your work. After saving your work in the portal you will be able to generate a new PDF version of the funding request that includes your recent edits.

When you are satisfied with your description of the proposed project, you may go to the Templeton Portal to submit your request. Please note that all requests must be submitted through the Templeton Portal and the Foundation will not accept this PDF as a valid application for funding.

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Funding Request Info

Request Type

Request Type: Core Funding Area (JTF's full range of grantmaking areas)

Core Area: Philosophy and Theology

What prompted you to request funding from the John Templeton Foundation?: The Foundation’s website

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Project

Project title: Virtuality, Reality and Phenomenological Spirituality

Executive Summary:

People ought to get mental models of the world with widest possible temporal and spatial scope. Mental models are structures in minds. Mentality sees the world as interacting towers of virtuality grounded in physical reality. Core questions are: What is the mathematical essence of a virtuality grounded in a reality? What is a mental model of the phenomenology of subjective time? What conscious-thought experiments could lead to spirituality? The main activities of this project are:

Surveying a ground on which to build mathematical foundations, erecting upon that ground a theory of towers of virtuality over a reality such as physics, axiomatizing a theory of substances for a physical reality, axiomatizing a theory of qualitative shape of physical objects in space, explicating subjective time relative to objective time, systematizing conscious-thought experiments, and investigating phenomenological spirituality by applying the developed formalism to conscious-thought experiments. Outputs include a book manuscript and articles for peer review. This project argues that humanity is immersed in virtuality grounded in a reality. Properly understood, virtuality includes a vast fund of mental models of itself at all levels from the ground up.

Project Description:

Activities: Reveal the ubiquity of virtuality and the paucity of its acknowledgement; Precisify the concept of the microlect of a mental model and its variation during discourse; Delineate the microlect of virtuality organized in towers of systems; Elaborate a ground for foundations of mathematics with a distinction between Discourse and Diagram; Erect a foundation of mathematics with structures constituted by data satisfying conditions; Specialize to modal transition systems representing mental models and their simulations; Formalize interaction of towers of virtuality grounded in physical reality; Expand the basal level of physical reality in terms of a theory of substances; Investigate the apical level of a human tower of virtuality with conscious-thought experiments; Construct a mathematical model of subjective time relative to objective time; Explore phenomenological spirituality at the apical level of human towers of virtuality. The ubiquity of virtuality shall be documented by examples from everyday life (driving a car, using a computer), scholarly disciplines (psychology, neuroscience), and engineering (computers, the internet). Human-human and human-computer interaction are behaviors of coupled pinnacles of towers of virtuality. Is there a flexible mathematical structure to found a theory of virtuality? What adaptation of discourse analysis is required to formalize the concept “microlect,” i.e., the limited natural sublanguage for communicating a domain of discourse? What is the microlect of virtuality? What structure represents the relation of mental models to their microlects? What alternative mental models of reality ground towers of virtuality? Can any mental model be represented by a mathematical structure? The proposed mathematics treasures the dots and arrows notation of category theory, but converts it to a ground for all foundations of mathematics. Namely, structure as data plus conditions. The core structures – basal, medial, and apical levels – for building towers of virtuality are generalized differential equations called modal transition systems. E. Husserl introduced a geometrical metaphor for subjective time, based on primal impressions, retentions, and protentions. The open problem of how to interpret this geometry is solved by appeal to a basic conscious-thought experiment involving counting foci of attention. The flexible relationship between objective and subjective time is represented by certain partial orderings on the objectively time-varying space of these foci. Apprehension of spatial shape by nonconscious integration of partial views is the normal visual experience of physical objects. There is no general theory of qualitative shape to model this process of integration. Modal transition systems, as in Kripke semantics of modal logic, suffice to formalize the concept of qualitative shape. Blending subjective time and qualitative shape is a foundation for building arbitrary mental models within medial and apical levels of towers of virtuality. Basal levels of towers of virtuality may be supernatural, or grounded in one or the other physical theory. A specific mental model of the general mental state includes nonconscious and conscious states, both with perceptual foci. Conscious states include nonverbal and verbal conscious thoughts. As category theorists emphasize the importance of morphisms between mathematical structures, this research emphasizes that transitions between mental states also need formalization. This model supports experiments in meditation suggested by G. I. Gurdjieff, C. Tart, F. Varela, and A. B. Wallace, leading to the possibility of phenomenological spirituality intimated by E. Husserl and especially E. Fink. Funds are required by Ellis D. Cooper for living expenses plus travel and per diem costs of consultation with experts in mathematics, linguistics, phenomenology, and spirituality.

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Project Description Attachment:

Strategic Promise:

Crucial to education are mental models of widening temporal and spatial scope are. Missing is a rigorous mental model of mental models grounded in lived experience. Virtuality is encountered in entertainment (science fiction“virtual reality” themes), in philosophy (“brain in a vat” thought experiments) and in computer science (virtual machines, e.g., spreadsheets). It is plausible that a theory of virtuality would offer a unifying framework for observation and theory in philosophy, phenomenology, and applications to practical questions about human-human communication and human-computer interaction. Such a theory could clarify important questions about mental models, microlects, and levels of understanding organized in interacting towers of virtuality. Finally, the open problem in “phenomenology of the consciousness of internal time” of elucidating “primal impressions, retentions,” and especially “protentions” may be solved by a new mental model of subjective time.

Capacity for Success:

As independent scholar I am not subject to an institutional agenda that might constrain free inquiry. Likewise, without any particular research group affiliation, my inter-disciplinary investigations are at liberty to gather and consider viewpoints from widely disparate if not dissenting resources. As a mathematician trained in the habits of semantic precision and notational simplicity, I may deploy the modern mathematical technologies of Category Theory and Philosophical Logic for the most recondite domains of discourse. Humility recognizes the limitations of any language encountering the Ultimately Ineffable. On the other hand, high precision delimitation and delivery of mental models is a worthy goal. Preliminary investigations have already reticulated a scaffolding in preparation of the proposed book and technical papers.

Expected Outputs:

The main output is a book manuscript. A mathematics paper on the ground and a foundation for mathematics with new details on mathematical structure will be based on the book. Another paper will be on modal transition systems, which are generalized differential equations augmented by statements associated with states and transitions. A chapter on the theory of physical substances will undergird another paper. Levels of towers of virtuality are modal transition systems, which, as simulators operating outside of objective time, are models of imagination. Behavior and interaction of towers of virtuality will occupy another chapter. The theory will be applied to modeling subjective time – including retentions and protentions and foci of attention – in relation to objective time. Such higher levels of towers of virtuality reflect lived experience of conscious thought. The book is expected to close with a chapter on the possibilities of phenomenological spirituality.

Expected Outcomes:

Relating appearance to reality is always a philosophical problem, so interpreting appearance as a structure called virtuality could be welcomed by philosophers and especially phenomenologists. Expressing this interpretation mathematically opens new lines of inquiry. The tinkertoy of metaphor called mathematics offers philosophers high-precision general-purpose tools for building and refining mental models. The proposed emphasis on mental models could change how educators in general view their objectives. Foundational mathematicians will have a baggage-free streamlined diagram logic for calculating and reasoning, and physics educators will have an axiomatic theory of substances as ground level for their towers of virtuality. Phenomenologists will have a formal theory of qualitative shape in space, and of foci of attention in subjective time. Seekers of Ultimate Reality will have a reference manual of new conscious-thought experiments.

Enduring Impact:

Business models, political models, physical models, theological models are mental models. The project hopes to develop a formal theory of mental models and their microlects, which ultimately are the stuff of all education. One hopes the theory is applied beneficially by educators in formulating their curricula over the long-term. The proposed ground for foundations of mathematics, which sharply

Page 206 of 302 Date generated: 10/14/2012 4 / 7 This PDF is for review purposes ONLY. Unpublished.1998-2013 This PDF is for review purposes ONLY. distinguishes the natural language verbal expression of conscious mathematical thought from rigorous artificial language for calculating and reasoning, is used by foundational mathematicians seeking to minimize their presuppositions. Another hope is to accumulate in one place conscious-thought experiments that may propel seekers towards phenomenological spirituality. This kind of spirituality is intimated in the under-valued work of E. Husserl and especially E. Fink, and deserves wider appreciation among those concerned with the advancement of human conscious thought.

Currency: US dollars

Request Amount: 180000

Total Project Amount: 180000

Additional Funding from Other Sources:

Not applicable.

Proposed Project Start Date: 07/04/2013

Proposed Project End Date: 07/09/2015

Relation to Sir John Templeton's Donor Intent:

Education is acquiring, using, and adapting physical and mental skills of mental models with widening temporal and spatial scope. Alongside Materialism and Humanism is the mental model, “Infinite Intellect created the universe. Humanity must progress towards spirituality. Visions and scriptures energize progress. Science of finite intellects deploying experiment and theory may accelerate spirituality.” Sir John Templeton's Philanthropic Vision is that financial support for advancing scientific spirituality may convince skeptics of spirituality and Infinite Intellect. Conscious thought is a prerequisite for intellect and spirituality. This project advances a framework for reasoning about mental models. Conscious-thought experiments suggested by E. Husserl, G. I. Gurdjieff, C. Tart, and F. Varela, among others, may culminate in practices for experiencing phenomenological spirituality, with the option but not the obligation of postulating an Infinite Intellect.

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Personnel

Primary Contact Information Prefix: Dr.

First Name/Given Name: Ellis

Middle Names: Dexter

Last Name, Family Name, or Surname: Cooper

Suffix: Ph.D.

Title: Adjunct Instructor of Mathematics

E-mail: [email protected]

Phone Number: 978-853-4894

Project Leader

Project Leader Status: The Primary Contact is the Project Leader

Project Leader’s CV/Resume Upload: StandardResume120928a.pdf

Project Co-Leader

Project Co-Leader Status: There is no Project Co-Leader for this request

Personnel Relationship to JTF: No

Additional Request Contact Name(s):

Not applicable

History with the Foundation:

Not applicable

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Organization

I am seeking funding as an individual

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The U.S. Electoral College For Liberal Arts Mathematics Students

Dr. Cooper November 6, 2012

1 Introduction

The textbook [Angel et al.(2013)Angel, Abbott and Runde] for MAT140-L, “Lib- eral Arts Mathematics” at North Shore Community College, Fall 2012, includes Chapter 15, “Voting and Apportionment.” There is a sidebar on page 927 entitled, “The “Flawed” Electoral College System,” with a sketch of the U.S. Elec- toral College system and a brief chronology of this controversial Constitutional electoral requirement. I never fully understood how the Electoral College works, and this sidebar leaves me unfulﬁlled. In view of the upcoming Presidential Elec- tion, I decided to build a mathematical model of the U.S. Electoral College Sys- tem using only mathematical technology in [Angel et al.(2013)Angel, Abbott and Runde] about ﬁnite sets (Chapter 2, especially pp. 43-46, pp.58-59), their functions (Chapter 6, especially p.362),1 and logic (Chapter 3, p.105). It seems to me this topic is an excellent opportunity for contextual learning.2

2 Finite Sets

The formal model of the U.S. Electoral System in this report is a simpliﬁcation. For example, it includes only two political parties, called Ob and Ro, the concept of “faithless elector” is ignored, and the number of voters is not accurate. The distinctions between rules used by states to select electors is ignored. No information about the timing of events is included, but it is intended that this report be available for discussion prior to November 6, 2012.

There are a few ﬁnite sets involved in this model of the U.S. Electoral System. Notations and basic facts about them are given in Fig.(1). Page 45 in Chapter 2 of [Angel et al.(2013)Angel, Abbott and Runde] introduces “set-builder notation” with the example “{ x x ∈ N and 2 ≤ x < 8 },” which represents the set of natural numbers not less than 2 and less than 8. In

1The arrow notation for functions between sets is deployed in the hope of added conceptual clarity. 2“Reality-based, outside-of-the-classroom experience.”

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k ∈ Stat set of states n(Stat) = 50 x ∈ Votr set of voters n(Votr) ≅ 90, 000, 000 h ∈ Hous House of Representatives n(Hous) = 435 a ∈ Snte Senate n(Snte) = n(Stat) ∗ 2 = 100 Cong = Snte ∪ Hous ∅ = Snte ∩ Hous y ∈ Cndt set of candidates for President Cndt = { Ob, Ro } p ∈ Prty set of political parties Prty = { Dem, Rep } e ∈ Eltr set of electors n(Eltr) = 538

Figure 1: Notations for variables of the sets are shown at the left. Interpretations of the sets and information about the size or relationships between sets are also provided. The number of electors 538 is the number 100 of senators plus the number 435 of members of the House of Representatives, plus 3 electors from the District of Columbia in Washington. this report the same information is represented by { x ∈ N 2 ≤ x < 8 }. This variant of set-builder notation speciﬁes an ambient set to the left of the symbol, and a condition to its right that determines a subset of the ambient set. The text uses the notation n(A) for the cardinality of the ﬁnite set A. See page 46.

Prior to the National Election, in each state k ∈ Stat there is a process for selecting for each Presidential candidate a “slate” of potential electors, and the union of all the slates for each candidate are denoted by SlatOb and SlatRo, and of course, SlatOb ∩ SlatRo = ∅, where ∅ is the empty set introduced in [Angel et al.(2013)Angel, Abbott and Runde] on page 47. The number of potential electors for each state equals the total number of senators (2) plus the number of representatives from the state in the House of Representatives. There- fore, the cardinalities n(SlatOb) = n(SlatRo) = n(Eltr) = 538.

3 Finite Maps

In everyday language the word “map” usually refers to a drawing-to-scale on paper of some geographical territory. Naturally, if the territory covers some number of square miles and the piece of paper is measured in square inches, then there must be considerable loss of detail from territory to its map. For example, individual people and automobiles are not shown. On the other hand, boundaries of political subdivisions such as states in a map of the U.S. are often important to exhibit. See Fig.(2). The basic idea of a map is that for every

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Figure 2: A map of the U.S., plus the number of (potential) electors for each state. point of the territory there is exactly one corresponding point of the map – and due to scaling, many points of the territory may be compressed, so to speak, into one point of the map. There are usually points of the paper that do not correspond to any points of the territory. So, one may say there is a mapping of the territory into but not necessarily onto the map.

Among finite sets the word “map” is used metaphorically to refer to the idea of associating with each element of a set exactly one corresponding element of a (not necessarily different) set. For example, with each state k ∈ Stat there is exactly one positive whole number of electors associated to that state. This assignment is also shown in Fig.(2). If N denotes the set of natural numbers defined as N = { 1, 2, 3, 4,... } in [Angel et al.(2013)Angel, Abbott and Runde] on page 43, then this Figure is one way to show a mapping from Stat to N. Another way to show this mapping would be a table with two columns and 50 rows. Each row could name a state in the left column, and the number of electors for it in the right column. It is extremely convenient to have names for mappings, and even more so to have an abstract notation. The mapping assigning elector count to states could be named, say, E, in which case the abstract notation for the assignment is

E Stat → N (1)

In this notation the arrow has the “territory” so to speak at the tail end, and (a set containing) its map at the right end, and the name of the mapping placed

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over the arrow. Mathematicians tend to simplify the story of this notation, and just say that E is a map from Stat to N.4 Each potential elector for either candidate comes from exactly one state, so there is a diagram of maps

SlatOb (2) II II fOb II II I$ Stat u: uu uu uu uu fRo SlatRo so for each state the set { n ∈ SlatOb fOb(n) = k } is the slate of electors for Ob from state k ∈ Stat, and likewise { n ∈ SlatRo fRo(n) = k } is the slate of potential electors for Ro from k.

The popular vote v is a map that assigns to each voter exactly one candidate for the Presidency, so

v Votr → Cndt (3)

Hence, for example, if, according to v, voter x ∈ Votr votes for candidate Ro, v(x) = Ro. Since every voter is registered to vote in exactly one state, there is also a diagram of maps

Cndt (4) u: uu vuu uu uu Votr II II II w II I$ Stat so that the set of voters for candidate p ∈ Cndt from state k is

{ x ∈ Votr v(x) = p ∧ w(x) = k } (5) where the symbol ∧ for logical conjunction of statements is introduced on page 97 in the text. The choice of which slate of electors from each state that actually goes to Washington to vote for the President is determined by the outcome of the s popular vote in each state. Let Eltr → Stat be the result of the popular votes in the states. The “winner-take-all” process deployed in almost all states says that whatever candidate wins the majority of the popular votes in a state, that candidate’s slate of potential electors becomes the actual slate of electors that goes to Washington for the electoral vote count. This is exactly the moment in

4This story tends to confuse “mapping” with “map,” but one gets used to that.

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the U.S. Electoral System at which the outcome is not necessarily proportional to the popular vote, since the slate of a candidate usually pledges all of its votes to its candidate.5

⎧ ⎪{ n ∈ SlatOb fOb(n) = k } if n({ x ∈ Votr v(x) = Ob ∧ w(x) = k }) > ⎪ ⎪ n({ x ∈ Votr v(x) = Ro ∧ w(x) = k }) { e ∈ Eltr s(e) = k } = ⎨ ⎪ ⎪{ n ∈ SlatRo fRo(n) = k } if n({ x ∈ Votr v(x) = Ro ∧ w(x) = k }) > ⎪ ⎩⎪ n({ x ∈ Votr v(x) = Ob ∧ w(x) = k }) (6) where the case = of a tie is handled in real-life but not here. Assuming that all ev electors are faithful, the electoral vote is a map Eltr → Cndt deﬁned by

⎧ ⎪Ob if e ∈ { n ∈ SlatOb fOb(n) = k } for some k ∈ Stat ev(e) = ⎨ (7) ⎪Ro if e n Slat f n k for some k Stat ⎩⎪ ∈ { ∈ Ro Ro( ) = } ∈ and the winner of the electoral vote is deﬁned by ⎧ ⎪Ob if n({ e ∈ Eltr ev(e) = Ob }) > n({ e ∈ Eltr ev(e) = Ro }) President = ⎨ ⎪Ro if n({ e ∈ Eltr ev(e) = Ro }) > n({ e ∈ Eltr ev(e) = Ob }) ⎩⎪ (8)

4 Example of Losing Popular Vote and Winning Electoral Vote

5Those electors in a candidate’s slate from a given state who do not cast their electoral vote for their own candidate are the “faithless electors.”

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5 Conclusion

The formal model of the U.S. Electoral College System detailed here represents my mental model, which is the result of on-line research. Therefore, it may be in error if only because it depends on my interpretation of texts of varying depth and clarity, even the best of which do not meet the rigorous standards routinely expected by mathematicians.

6 On-line Search Results 6.1 How does the electoral college work? The candidate who receives the most votes receives all the electoral votes in that state. The candidate with a majority of the electoral votes is elected to oﬃce. ... Presidential elections are held on the Tuesday after the ﬁrst Monday in November every four years. Af- ter the results have been tallied, the Electoral College meets on the Monday after the second Wednesday in December to cast their votes. In most elections, the electors will vote for the candidates that the people chose through the popular vote. However, this is not required by law. The Founding Fathers felt that there would potentially be times when the people would not know which candidate would be the best president, and as a result the electors can choose to vote against the popular vote.6 Today this is rare, but it is possible.

Some states have made laws that require the electors to vote according to the will of the people. Other states have separate groups of electors for each party, and these electors are bound to vote for the their party’s candidate. When the votes are tallied, these states send the appropriate electors to the December vote, thus ensuring that the choice the people made is followed. However, not all states have these provisions, so it is entirely possible for an elector to vote for a candidate that did not win the popular vote in her state.7

6.2 Losing the Popular Vote and Winning Under this system, each state is assigned a speciﬁc number of votes that is proportional to its population, so that each state’s power is representative of its population. So, while winning the popular vote may not ensure a candidate’s victory, a candidate must gain popular support of a particular state to win the votes in that state. The goal

6The so-called faithless electors. 7http://wiki.answers.com/Q/How_does_the_electoral_college_work

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of any candidate is to put together the right combination of states that will give him or her 270 electoral votes.8

6.3 Who really elects the President of the United States? When you vote for a presidential candidate you are really voting to instruct the electors from your state to cast their votes for the same candidate. For example, if you vote for the Republican candidate, you are really voting for an elector who will be “pledged” to vote for the Republican candidate. The candidate who wins the popular vote in a state wins all the pledged votes of the state’s electors.9

6.4 The Electoral College Each state determines how its electors are chosen by state law and the process varies from state to state. In states with primary elections, each presidential candidate usually designates a slate of electors who then appear on the November ballot. The voters are then actually voting for a slate of electors pledged to one candidate or another. In caucus states, the electors may be chosen at the state caucus. Electors are actual human beings, with houses, children, jobs, and very occasionally, their own opinions. In most states, the slate of electors that gets the most votes wins and gets to travel to the state capital in December to vote for president and vice president.

Many criticisms have been leveled at this 18th Century system. First, why have electoral votes at all? Why not just elect the president by popular vote? The reason this system has never changed is simple: politics. States with many buffalo and few people, like Wyoming, benefit from it and are not keen on changing it. Since every state gets at least three electors, low-population states have proportionally far more political power than they would have in a direct election system. The number of voters per elector is about four times smaller in the three-elector states than in the most-populous states. The fact that nearly all the low-population states are heavily Republican adds to the difficulties of changing the system. Direct election of the president would eliminate the current bias in favor of the Republi- cans.

A second criticism of the electoral college is its winner-take-all character. If Florida’s 25 electoral votes had been split 13 for George Bush and 12 for Al Gore, then Al Gore would now be president.

8http://people.howstuffworks.com/question4721.htm 9http://usgovinfo.about.com/od/thepoliticalsystem/a/electcollege.htm

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There is nothing in the constitution mandating winner-take-all. The manner for choosing electors is regulated by state law. In fact, two states, Maine and Nebraska, do not use winner-take-all. In those states, the winner of each congressional district gets one elector and the winner of the state as a whole gets an additional two. Any state that wanted to adopt this system need only pass a state law to do so. No constitutional amendment is required.

The third objection to the electoral college is the so-called faithless elector problem. As mentioned above, electors are actual human beings, with all the properties that go along with that. In particular, when they meet in December in their respective state capi- tals, they sometimes do not vote for the candidate they are oﬃcially pledged to. In 1948, for example, Truman elector Preston Parks of Tennessee, voted for Strom Thurmond who was running on the pro-segregation Dixiecrat ticket. In 1960, Nixon elector Henry D. Irwin of Oklahoma voted for Virginia Senator Harry F. Byrd, then an ardent segregationist. In 2000, one of Gore’s D.C.’s electors, Bar- bara Lett-Simmons, cast a blank ballot in protest of the District’s lack of congressional representation. None of these faithless electors changed the outcome of their respective elections, but in a close election in the future, it might be possible for a candidate to bribe enough electors to swing the election. At the very least, state law could make voting the wrong way a felony with life in prison as the punishment.10

6.5 Slate (elections) The United States presidential elections use an electoral college to determine the winner and the electors are chosen by popular vote in each state. Voters choose a slate of electors who support one of the candidates, although this may not be obvious to the voter at the time.11

6.6 Electoral College (United States) A result of the present functionality of the Electoral College is that the national popular vote bears no legal or factual signiﬁcance on determining the outcome of the election. Since the national popular vote is irrelevant, both voters and candidates are assumed to base their campaign strategies around the existence of the Electoral College; any close race has candidates campaigning to maximize

10http://www.electoral-vote.com/evp2012/Info/electoral-college.html 11http://en.wikipedia.org/wiki/Slate (elections)

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electoral votes by capturing coveted swing states, not to maximize national popular vote totals. ... Most states use a winner-take-all system, in which the candidate with the most votes in that state receives all of the state’s electoral votes.12

6.7 Who are the Electors? The process for selecting Electors varies throughout the United States. Generally, the political parties nominate Electors at their State party conventions or by a vote of the partys central committee in each State. Each candidate will have their own unique slate of potential Electors as a result of this part of the selection process.

On Election Day, the voters in each State choose the Electors by casting votes for the presidential candidate of their choice. The Elec- tors names may or may not appear on the ballot below the name of the candidates running for President, depending on the procedure in each State. The winning candidate in each Stateexcept in Nebraska and Maine, which have proportional distribution of the Electorsis awarded all of the States Electors. In Nebraska and Maine, the state winner receives two Electors and the winner of each congressional district receives one Elector. This system permits the Electors from Nebraska and Maine to be awarded to more than one candidate.

There is no Constitutional provision or Federal law that requires Electors to vote according to the results of the popular vote in their States. Some States, however, require Electors to cast their votes according to the popular vote. These pledges fall into two categories- Electors bound by State law and those bound by pledges to political parties.

The U.S. Supreme Court has held that the Constitution does not require that Electors be completely free to act as they choose and therefore, political parties may extract pledges from electors to vote for the parties nominees. Some State laws provide that so-called “faithless Electors”; may be subject to fines or may be disqualified for casting an invalid vote and be replaced by a substitute elector. The Supreme Court has not specifically ruled on the question of whether pledges and penalties for failure to vote as pledged may be enforced under the Constitution. No Elector has ever been prose- cuted for failing to vote as pledged.

12http://en.wikipedia.org/wiki/United States Electoral College

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Today, it is rare for Electors to disregard the popular vote by casting their electoral vote for someone other than their partys candidate. Electors generally hold a leadership position in their party or were chosen to recognize years of loyal service to the party. Throughout our history as a nation, more than 99 percent of Electors have voted as pledged.13

13http://www.archives.gov/federal-register/electoral-college/electors.html#restrictions

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References

[Angel et al.(2013)Angel, Abbott and Runde] Angel, A. R., Abbott, C. D. and Runde, D. C. (2013). A Survey of Mathematics with Applications, 9th edn. (Pearson).

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0.1. MEMO TO THE GODS 1

0.1 Memo To The Gods

TO: All Gods DATE: Eternity FROM: Ultimate Superior Being RE: The Facts on Earth

This is to inform all Gods not of Earth, and forcefully to remind all Gods of Earth, about the facts on Earth.

1. Massive trade in arms is a fact of life, even between nations committed to destruction of their trading partners, and serving diﬀerent Gods.

2. There is an enormous number of ways human beings interfere with or irre- vocably interrupt the lives of other beings, including: abortion for personal reasons; terrorism in the names of diﬀerent Gods; etc.

3. Most human beings are incapable of cognizing any but the simplest logical and empirical data.

4. There is no reliable evidence that any two human beings selected at random, whether they be neighbors or distant strangers, would trust each other.

5. Each of you insists that you are the only God in my universe of Gods. This dogged insistence of yours has produced more misery among human beings than any other factor in their already diﬃcult existence.

6. Most human beings will take the easy path over the hard; the short term gain over the long; the simple idea over the complex; and the old way over the new.

7. Human beings are almost universally possessed with the capacity to detect patterns in any data.

8. Most human beings are easily “hypnotized” by a certain minority of either courageous, or imaginative, or persuasive, or demagogic persons.

9. Few human beings can recount connected events from their first couple of Earth years of existence. Self-consciousness, and a belief in one of you, develops gradually and is normally dominated by the external influence of parental figures.

10. In any suﬃciently large collection of human beings there must always exist at least one schism which cuts across any emotional or intellectual bound-

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aries.

11. Your message ought to be that belief in you is to serve merely as a temporary scaﬀolding around the building of Morality. The building of Morality is the highest aim of a primate, or any other sentient matter.

12. Good luck deserves no reward; bad luck deserves no punishment.

13. Human beings in general lack

1. imagination 2. patience 3. generosity 4. worldview education 5. capacity for delayed gratiﬁcation 6. ability to deny media-induced delusions 7. multi-track contingency prioritization.

0.2 Memo To The Humans

TO: All Humans DATE: Eternity FROM: Ultimate Superior Being RE: Rules For Public Discussion Let Public Blabber stand for any politician, pundit, entertainer, journalist, or even layperson, with a public forum.

1. A Public Blabber who announces a prediction must also provide publicly available references to prior public predictions, and the actual outcomes. In other words, they must automatically display their “track record,” plus explanations of their failures to predict correctly.

2. If a Public Blabber receives a public counter-argument from another Pub- lic Blabber, then they must publish a rejoinder, recursively.

3. If a Public Blabber deploys a number as premise of an argument, then they must provide a publicly available reference for the means of measurement, and for the resulting magnitude.

4. If a Public Blabber expresses a political, scientiﬁc, or religious belief, then they must be willing to provide details about the grounds for their belief.

5. If a Public Blabber claims to represent some set of individuals (or corporate entities, etc.), then they must be able to provide evidence that they have been accepted by that set as their representative.

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0.2. MEMO TO THE HUMANS 3

6. If a Public Blabber is asked a question, then an answer must be directly, grammatically and meaningfully coordinated with the question.

One more thing. American Culture must re-focus far away from entertainment, sports, toys, and games, towards focus on concentrated study. These other things are rewards for successful study, not ends in themselves. The purpose of study is to improve mental models of the world: how it works, and how to work it. Study is hard work, so work hard.

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A Category of Natural Numbers

Ellis D. Cooper Rockport, Massachusetts February 4, 2013

0.1 Abstract A category is described whose objects are the natural numbers and whose arrows represent addition or multiplication. The Division Equation & Inequality translate into a (non-commutative) diagram, and the irrationality of 2 is deduced using diagram deduction rules.

The fruit may be hanging low enough for young students in high school and developmental mathematics students in college to pick and enjoy.

0.2 Introduction The set N of natural numbers 0, 1, 2, 3,... supports a structure including ad- h / dition, multiplication, and comparison. Deﬁne a 1-graph N ∶ A / N with t natural numbers as dots, and two kinds of arrow as follows. For every equation b a + b = c there exists an arrow a Ð→ c, and for every equation a × b = c there b 9 7 exists an arrow a Ô⇒ c. For examples, 13 Ð→ 22 and 6 Ô⇒ 42. In general, such ● arrows are true statements. If an arrow label is ●, as in a Ð→ b, that means there n ● exists a natural number n such that a Ð→ b. Therefore, a Ð→ b is equivalent to ● the comparison a ≤ b, and a Ô⇒ b is equivalent to a∣b, that is, a divides b evenly. n ● n ● For every natural number n there exist arrows 0 Ð→ n, 0 Ð→ n, 1 Ô⇒ n, and 1 Ô⇒ n.

The 1-graph N structure is contained in a category structure determined by the arithmetic of natural numbers. That is to say, arrows may be composed according to the following deﬁnitions:

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? y ? ;C y ??? ;C y ? ? y ??? a ??b a ???b a ??b a ???b ? ???? ? ???? ? # ? # (1) x / z x +3 z x / z x / z a+b a×b x×(b−1)+a×b x×(a−1)+b

0 Each natural number n has an identity arrow represented either by n Ð→ n or 1 by n Ô⇒ n, so there exists a commutative diagram

0 ' n N 3; n (2)

The symbol N now denotes the underlying 1-graph, the category, and the notion of equality in the category.

Division For natural numbers d (divisor) and D (dividend) there exist unique natural numbers Q (quotient) and r (remainder) such that D = Q ∗ d + r and 0 ≤ r < d. This statement may be translated into a single (non-commutative) diagram o DO v r (3) vv vv vv1 {vv − + D d wr 1 ww ww w● ww w{ w d Q +3 Q × d but no meaning is lost by expunging the variables, ● o ● O (4) 1 ● ● ● ● +3 ●

2 is Irrational ● Assume p is a natural number satisfying 2 Ð→ p, which is to say, 2 ≤ p. The claim that p is a prime is expressed by the diagram deduction rule

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⎡ ⎤ ⎢ ● 1 ⎥ ⎢ a Ô⇒ p 0 Ð→ a ⎥ ⎢ ⎥ (5) ⎢ a ⎥ ⎣ 0 Ð→ p ⎦ which is a grounded1 rendering of the statement “if a divides p then a = 1 or a = p.” Suppose that it has been deduced that ⎡ ● ● ⎤ ⎢ Ô⇒ × Ô⇒ × ⎥ ⎢ a b a c b d ⎥ ⎢ ● ⎥ (6) ⎣ c Ô⇒ d ⎦ and also suppose that it has been deduced that if p is prime then ⎡ ● ● ⎤ ⎢ Ô⇒ × Ô⇒ ⎥ ⎢ p a b p a ⎥ ⎢ ● ⎥ (7) ⎣ p Ô⇒ b ⎦ which is conventionally stated as, “if p ∣ a × b then p ∣ a or p ∣ b.”

Assume now that q q p p 1 +3 ● +3 ● 2 +3 ●ks ● ks 1 (8) = p2 which is a possible translation of 2 q2 . Assume also that r 4< p (9) r●rrrrr rrrrr 2 LLLLL LLLLLL ● L "* q which, following C. S. Peirce [3], represents a negation of the assertion that 2 p divides both p and p. This is implied by assuming that the fraction q is in reduced form, but that more comprehensive assumption is not called for below.

● ● ● Then 2 Ô⇒ p2, so 2 Ô⇒ p by (7), and therefore, 4 Ô⇒ p2 by (6). Therefore, ● ● ● 4 Ô⇒ 2q2, so 2 Ô⇒ q2, hence 2 Ô⇒ q. But then, r 4< p (10) r●rrrrr rrrrr 2 LLLLL LLLLLL ● L "* q contradicting (9). Therefore, (8) cannot hold, and 2 is irrational.

1This term alludes to forthcoming work entitled, “A Ground for Foundations of Mathe- matics,” wherein is provided a full explanation of Structure Logic with general mathematical structures deﬁned in terms of labelled dots and arrows subject to diagram deduction rules. Premonitions of this ground-breaking work appear in Ch. 2 of [1].

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Comments 1. The category of natural numbers may be augmented by an additional arrow opposite in direction for each addition arrow. Then conditions may be imposed to quotient this enlarged category to make each addition arrow into an isomorphism. The result is a category of integers. Likewise, (non 0) multiplication arrows may be inverted, thus forming a category of rational numbers. These constructions are reminiscent of “calculus of fractions” as in [2].

2. The objects and morphisms of a category of rational numbers as above may be decorated with “units of measurement,” so that for example

5280 feet x mile ÔÔÔÔ⇒mile y feet

may be incorporated in commutative diagrams that automatically represent dimensionally correct physical calculations. b 3. The (wide) subcategory of just the multiplication arrows a Ô⇒ c has ﬁnite products and ﬁnite coproducts. These are traditionally called “greatest common divisor” and “least common multiple.” 4. This category of natural numbers is a strict symmetric monoidal category in two ways, namely, objects can be added with unit 0, and multiplied with unit 1. ● 5. The statement “a ≤ b” becomes the arrow a Ð→ b, but the statement “a < b” 1 ● is rendered by a Ð→ a Ð→ b. 6. Commutativity of addition and multiplication, of course, are rendered by commutative diagrams

● a / ● ● a +3 ● b N b b N b ● / ● ● +3 ● (11) a a

x 7. Suppose to each hom set N (a, b) is adjoined a new arrow, a Ð→ b, where x n n−1 is a symbol for variable. The brutally literal rendering of anx +an−1x + th ⋯ + a1x + a0 – the general n -degree polynomial with coeﬃcients in N –

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is the diagram (12)

● (12) an ● ● x an−1 ● ● x x ● ● ● a1 ● ● ● x x x a 0 ●/ ● ●/ ● ● ●/ ● 0 / ●

I thank Florian Lengyel for pointing out that the simpler but equivalent translation into a diagram given by (13)

a x a − x a x a 0 Ð→n ● Ô⇒ ● ÐÐ→n 1 ● Ô⇒ ● Ð→ ●⋯⋯● Ð→1 ● Ô⇒ ● Ð→0 ● (13)

corresponds to Horner’s Method for evaluating a polynomial, trading 2n−1 multiplications in (12) for n of them in (13), which is the idea behind the recursive computation called “synthetic division” [4].

8. The quadratic equation ax2 + bx + c = 0 becomes the “circular” diagram

o7 0 OO (14) ooo OOO ooco OaOO ooo OO' ● [c ● @@@@ ~~~ @@@x@ x~~~~ @@ {Ó~~~ ● o ● b

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References

[1] Ellis D. Cooper, Mathematical Mechanics: From Particle to Muscle, World Scientiﬁc Series, on Nonlinear Science Series A, World Scientiﬁc, Singapore, 2011. [2] P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, Springer-Verlag, 1967. [3] Eric Hammer, Peircean Graphs for Propositional Logic, Studies in Logic and Computation, vol. Logical Reasoning with Diagrams, Oxford University Press, 1996.

[4] F. B. Hildebrand, Introduction to Numerical Analysis, 2 ed., McGraw-Hill, Inc., 1974.

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Ground and Foundation For Mobile Diagram-Oriented Functional Programming

Ellis D. Cooper Rockport, Massachusetts March 15, 2013

ABSTRACT: Chasing along paths of arrows in diagrams for category-theoretic reasoning and calculation arrives at results without “side eﬀects.” Since arrows may represent functions, and category theory oﬀers multiple constructions of arrows from others, it seems natural to envision a functional programming environment based on diagram chasing. A computer application program for calculating and reasoning about mathematical structures represented by diagrams of labeled dots and arrows requires a mathematical foundation. A possible way forward based on a small number of axioms and conventions is suggested. The goal is an application program running on a tablet with a stylus for writing mathematical expressions and drawing structure diagrams. The program should recognize options for transforming diagrams in conformity with “structure logic” deduction rules, so the mathematician or programmer could choose an option to witness the transformation. Any move shall be revocable, so that alternatives may be explored. Drawing tools should allow the mathematician or programmer to re-arrange, delete, or add diagrams from a library, or at will, dot by dot and arrow by arrow. The usual, expected New, Save, Save As, etc. functionality shall be supported. Little of that design and none of that implementation is discussed in this paper, which concentrates on a mathematical ground and foundation.

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Contents

1 Introduction 3

2 Ground: Rules 3 2.1 The Surface ...... 3 2.2 Axiom PTH1: First Path Rule ...... 4 2.3 Axiom PTH2: Second Path Rule ...... 5 2.4 Axiom MEQU: Map Equality ...... 5 2.5 Axiom NEQU: Map InEquality ...... 6 2.6 Axiom FUNC: Function Rule ...... 6 2.7 Binding and Ligature ...... 7

3 Foundation: Category S of Sets and Maps 8 3.1 Axiom SETC: S is a Category ...... 10 3.2 Axiom PBPO: S has Pullbacks and Pushouts ...... 12 3.3 Axiom PRCP: S has Products and Coproducts ...... 13 3.4 Axiom MTUN: S has Distinct Empty and Unit Sets ...... 14 3.5 Axiom TRSP: S has Product and PowerTransposes ...... 14 3.6 Axiom BTST: S has a Boolean Truth Structure ...... 16 3.7 Axiom AXCH: S has Arbitrary Choice ...... 18 3.8 Axiom NNST: S has a Natural Numbers Structure ...... 18

4 Startup Theorems 19

5 Calculation 21 5.1 Central Processing Unit and Flowchart ...... 21 5.2 Theorem: Category S is “Turing Complete” ...... 24

6 Conclusion 26

Bibliography 27

Index 27

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1 Introduction Philosophical versus Mathematical Foundations; Law- vere versus Kreisel

2 Ground: Rules 2.1 The Surface The mathematician considers the tablet to be a window upon the outer surface of a 2-sphere. The radius of the sphere grows automatically to accomodate expressions and diagrams drawn upon its outer surface. Gestures for zooming in or out, or for panning latitudinally or longitudinally facilitate navigation across the surface. Moves are physical operations with symbols, expressions, and diagrams on the surface. The symbols ●, /// and ///\ have special signiﬁcance, to be elaborated shortly.

Cosmetic moves slide, twist, rotate, scale, or reﬂect diagrams. Likewise, copying and pating diagrams does not inﬂuence mathematical content.

The fundamental notion is neither deduction (as in logic) nor membership (as in set theory), but formation and transformation of structure. Structure is a more complicated idea than these alternatives, but no less intuitively accessible, provided one keeps in mind a metaphor centered on building, construction, and architecture. A structure is constituted by data and conditions satisﬁed by the data. A structure is represented by a diagram on the surface. Cosmetic moves applied to a diagram do not inﬂuence mathematical content, but may very well facilitate understanding, and even suggest more substantive moves. The data for a structure are connected supports. The supports of a structure are represented in diagrams by (labeled) dots,1 and the connectors are represented by (labeled) arrows between dots. Conditions are represented by structure logic deduction rules that show the “before” and “after” of diagram moves.2

A structure support may or may not have members. Choice of a member x in a support labeled A is a structure with data

● x / A and no conditions. Of course, intuition for membership comes from experience with the notion of set membership, as in x ∈ A, but there is considerable baggage carried by the latter notation, namely, the entire history of naive and axiomatic

1But typically, the label for a dot actually replaces the dot. 2Structure logic deduction rules are just a diagrammatic version of “natural deduction” or “rewrite rules.”

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set theory. But supports are not necessarily sets, nor are members of supports necessarily elements of sets.

Unlike some alternative foundations of mathematics, here there is no presumption of the circle of ideas around the notion of “ﬁrst(or higher)-order theory.” In fact, one intention here is to explain a modern version of “theory” and “model” in terms of structure, which is taken as a primitive. Ground rules are intended at least to replace the widespread reliance on “naive set theory” [3] to deﬁne formal systems.3 At best, a ground for foundations ought to provide a replacement for the “set-theoretic assumptions” adopted as a context for distinguishing “internal” versus “external” mathematical development ([4], p.235).

2.2 Axiom PTH1: First Path Rule Two structure logic rules (1)-(2) govern coterminous paths on the surface.

ddddd1 O/ OO ddddd1 /OOO dddddddd OO dddddddd OO dddddddd OOO dddddddd OOO d? d /' / ?dd /' / / /// / I // I 4 / // / hhhh / / hhhh / /K ;hhhh ' // ww ww ww Q ww QQQ 1-graph G 1-graph G QQQ QQQ QQQ QQQ /// QQ QQ(V 7 VVVV VVVV VVV* /

Ground Equality Rule I

(1)

In words, rule (1) says that if two coterminous paths that surround an arbitrary

1-graph are declared equal by the primitive commutativity symbol /// appearing in the region of the surface exterior to the region bounded by the paths, then either path may be pushed through the enclosed 1-graph – or around the backside of the surface – so that the coterminous paths bound no 1-graph in the view window. Moreover, the commutativity symbol must be moved inside the region bounded by the coterminous paths.4//

3To start defining a formal language, model theory assumes undefined terms such as “finite set of symbols,” “finite sequence of symbols,” and “countable set of symbols.” 4 This is “symmetry” of ///.

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Cosmetic moves include duplication and translation across the surface of diagram, and of paths in particular. Individual dots in paths may be moved while retaining the arrow or arrows connecting them to other dots. Any two distinct dots with the same label may be merged to a single dot. Any dot may be duplicated and translated, provided identical copies of its connecting arrows are created as well. If moves duplicating and translating an open path are followed by merging the new termini with their corresponding termini in the original path, then any void region bounded by the resulting closed diagram 5 should be planted with ///.

2.3 Axiom PTH2: Second Path Rule

o7 O/ O o7 O/ O ooo OOO ooo OOO ooo OOO ooo OOO o? '// o? /' / /// / / / / G ? / / // G // // // // ? // ?J J / /// J /O J JJ OO JJ JJ OOO JJ JJ OO' JJ JJ JJ JJ JJ J$O 7 JO$ 7 OOO OOO OO /// OO OO' / OO' /

Ground Equality Rule II (2) The second ground equality rule (2) says that if two pairs of equal coterminous paths share a path, then the modiﬁed diagram obtain by deleting the shared path consists of a pair of equal coterminous paths.6

2.4 Axiom MEQU: Map Equality

u u f f f " /// " a / a / " ● X >AY ● X >AY X /// >Y /// (3) g g g v v

Map Equality

5 This is the “reﬂexivity” of ///. 6 This is “transitivity” of ///.

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The Map Equality ground rule (3) says that if coterminous connectors in a structure have the property that their compositions with any choice are equal 7 at the ground level (///), then they themselves are equal.

2.5 Axiom NEQU: Map InEquality

u f f " " \ a / \ X /// >Y ● X /// >AY (4) g g

v ///\ Map InEquality

Conversely, the Map InEquality ground rule (2.5) says that if two maps are not equal, there is a reason for that, namely, existence of a choice in their shared support at which they are not equal.8

2.6 Axiom FUNC: Function Rule The surface has diagrams and moves. Diagrams represent structures, and moves are modifications of diagrams according to structure logic rules. Thus, diagrams and moves are reminiscent of formations and transformations as in, say, [1]. Again, these come down to dots and arrows. At the ground level, an identified- by-name sequence of moves leading from membership in a set to a uniquely defined membership in a set (whether the same or another one) justifies creation of a map from the first set to the second set with the same name as the sequence of moves. A map created in this way from a sequence of moves is called a function. On the surface, therefore, a rule diagram indicated schematically as in (5)

a / F (a) / ● X ● ● ● ● Y (5) a: Uniquely

F creates a map X Ð→ Y . Therefore, there exists a second-order ground rule, as in (6)

a / F (a) / ● X ● ● ● ● Y F X / Y a: Uniquely (6) F: Uniquely

f g 7In “naive set theory” this principle could be expressed by “if X Ð→ Y and X Ð→ Y and f(a) = g(a) for all a in X, then f = g.” f g 8“If X Ð→ Y and X Ð→ Y and f ≠ g, then there exists a in X such that f(a) ≠ g(a).”

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2.7 Binding and Ligature The ﬁrst-order formula (7) deﬁnes continuity of a map f at a point a of an f open set U and U Ð→ R, in a context where R is an element of the set N of normed linear spaces.

f (∀R)(R ∈ N / (∀f)(∀a)((U Ð→ R ∧ U ∈ O(R) ∧ a ∈ U) / (f ∈ C(U, R, a) ↔ (7) (∀ε(ε > 0 / (∃δ)((∀x)((x ∈ U ∧ ∣x − a∣ < δ) / ∣f(x) − f(a)∣ < ε))))))

In technical English, the formula says that if R is a normed linear space, for f any f and a such that U Ð→ R with U in the set O(R) of open sets in R and a in U, then f is a continuous map from U to R at a if and only if for any ε if ε is positive then there exists δ such that for all x such that x is in U and its normed distance from a is less than δ, then the normed distance of f(x) from f(a) is less than ε.9 Note that (7) is not a sentence, since U is not bound by any quantiﬁer, although all other variables are. The universal convention is that the choice of symbols for variables that are bound is entirely arbitrary, provided that changing from one choice to another does not inadvertently introduce a double bind. For example, substitution of δ for x in (∀x) and elsewhere in the scope of the quantiﬁer ∀ in (∀x) would also be within the scope of the (∃δ), so δ would be bound both by ∀ and ∃, a double bind. This entire convoluted discussion would be moot, however, if choices of symbols for variables could be entirely avoided to begin with.

A way to do that is to exhibit binding with ligatures, as in Fig.(2.7). This convention eliminates bound variables, but, of course, the price to pay is that the formula is no longer a linear expression. Indeed, it corresponds to a certain structured 1-graph diagrammed on the surface.

Observe, also, that not all symbols appearing in the diagram are eliminated. For one thing, U is a free variable, which, if one so chose, could be bound by a universal quantifier at the appropriate position in the formula, and five ligatures created to eliminate all occurrences of U. In the context of mathematical logic, we are bound to the arbitrary conventions that the symbols ∀ and ∃ stand for universal and existential quantification. So, these two symbols might also be eliminated, with ligatures that stretch from their occurrences in the formula back to declaration in the context of mathematical logic. Likewise, the symbols / , ↔, and ∧ are introduced in that context. Furthermore, the symbol “N” for “normed linear space” and the symbol “C” for continuous are introduced in the context of mathematical analysis, along with 0 as an overloaded symbol for the additive unit in any normed linear space. Indeed, even the absolute value

9For the sake of brevity, to ﬁt Fig.(2.7) into the page width, the formula omits to exhbiit the sets to which ε and x belong.

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symbol ∣ ... ∣, the comparison symbol <, the grouping symbols “(” and “)” are arbitrary conventions introduced in the context of mathematics. Bottom line? Every mathematical symbol is bound in some suﬃciently comprehensive context.

Figure 1: Elimination of arbitrary choice of symbols for bound variables using ligatures.

3 Foundation: Category S of Sets and Maps

One mathematical foundation for mathematics is a category of sets and maps. Deﬁnition 1. A 1-graph is a structure with data

h # A < D (8)

t and no conditions. Generally, if A has members, a member of A is called a connector – graph- ically represented in a diagram by an arrow – and if D has members, a member α of D is called a support. If ● Ð→ A is a connector, then t(α) and h(α) are its starting and ending terminals, respectively. The compound statement,

alpha x ● ÐÐÐ→ ● Ð→ D ∧ ●yD ∧ t(α) = x ∧ h(α) = y is represented by the concise diagram

x α / y (9) and the labeled dot representing the starting terminal of the connector α is called its tail, while the other dot is called its head. The data for mathematical structures are usually composed of supports and connectors, and these may be represented by diagrams of arrows with heads and tails. Deﬁnition 2. A category is a structure C with data including a 1-graph,

f h " # / o T c < A i < D (10)

l t

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and conditions

f T / A T c / A T c / A

l h f t l h /// /// /// (11) A / D A / D A / D t t h

y y G O // G O // // // // // /// /// // /// /// // }> x @@ / }> x @@ / }} @ / }} @ / }} @@ // }} @@ // } @@/ } @@/ (12) }} }} /// w / z w / z ///

Associativity in S

y ? x ?? ? >> ?? >> iy ix ?α? α >> /// ? /// > ? > x / y / x / y / α x α y α x α y

Left Identity Right Identity (13)

Diagrams are compact, readable representations of compound logical state- α ments. For a category C with connectors A and supports D, the diagram x Ð→ y α represents the (external) logical statement ● Ð→ A ∧ t(α) = x ∧ h(α) = y. More, the diagram

? y ?? (14) ??β α ?? C ?? / x γ z

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corresponds to a compound statement

∆ α β γ ● Ð→ T ∧ ● Ð→ A ∧ ● Ð→ A ∧ ● Ð→ A ∧ (15) f(∆) = α ∧ l(∆) = β ∧ c(∆) = γ ∧ t(α) = t(γ) ∧ h(α) = t(β) ∧ h(γ) = h(β)

Typically, one just writes

γ = β ○ α (16) where the discourse context informs the reader about the supports of the connectors α, β and γ. That information may be re-supplied with the “linear notation,”

x = y ○ x γz βz αy (17)

Still, the category in which the composition (17) is evaluated must be learned x from the discourse context. Note, also, that the notation, for example, αy in (17), is both a connector, and a statement about the supports of that connector. All told, the diagram (14) is the most perspicuous of the above alternative representations of the basic structure in category theory.

3.1 Axiom SETC: S is a Category Deﬁnition 3. A pullback is a structure with data

> A A (18) ~~ AA ~~ AAf ~~ A P @ > X @@ }} @@ }} @ }} g B and two conditions, namely

> A A (19) ~~ AA ~~ AAf ~~ A P @ /// > X @@ }} @@ }} @ }} g B

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and 7 7 /// ooo~> A AA /// ooo~> A AA ooo ~~ AAf ooo ~~ AAf ooo ~~ AA ooo ~~ AA oo ~ oo /// ~ o o / Z PPP P @ /// > X Z PPP P @ /// > X PP @@ }} PP/// @@ }} PPP @ } PPP @ } PP@@ }} PP@@ }} (20) PP' } g PP ' } g B B

Uniquely The abbreviation for a pullback is

> A A (21) ~~ AA ~~ AAf ~~ A P @ PBK > X @@ }} @@ }} @ }} g B

The special case of a pullback where A = B and f = g is called an equalizer and its abbreviation is

? A @ (22) ~~ @@ e~~ @@f ~~ @ EQU E @@ ~>X @@ ~~ @@ ~ e ~~ f A

Deﬁnition 4. A pushout is deﬁned exactly the same data and conditions as a pullback, except that all arrows in (18)-(20) are reversed head for tail. Hence, the abbreviation for a pushout is

> A @ (23) }} @@ f }} @@ }} @ X A PSH > P AA ~~ AA ~~ g A ~~ B

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The special case where A = B and f = g is called a coequalizer and its abbreviation is

? A ? (24) ?? f ?? ? X ? CQU ? Q ?? ?? f ? A

3.2 Axiom PBPO: S has Pullbacks and Pushouts

A @ > A A @@ ~~ AA @@f ~~ AAf @@ ~~ A @ P @ PBK > X >X @@ }} ~~ @@ }} ~~ @ }} g (25) ~~ ~~ g B B

Pullbacks

> A > A @ ~~ }} @@ f ~~ f }} @@ ~~ }} @ ~~ X A PSH > P X @ AA ~~ @@ AA ~~ @@ g A ~~ (26) @@ g @ B B

Pushouts

Deﬁnition 5. A product is a structure with data

A _@@ (27) @@ @@πA @@ P ~~ ~~ ~~ ~~ πB B

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and condition

~> A _@@ ~> A _@@ ~ @@ ~ @@ u ~~ @πA u ~~ @πA ~~ @@ ~~ @@ ~~ @ ~~ /// @ / X @@ ~P X @@ ~P @@ ~ @@ ~ @ ~~ @ /// ~~ @@ ~~ @@ ~~ (28) v @ ~~ πB v @ ~~ πB B B

Uniquely The abbreviation for a product structure is

A Fb F (29) FF FπFA FF × Ax B xx xx xx |xx πB B Deﬁnition 6. A coproduct is deﬁned exactly the same data and conditions as a product, except that all arrows in (27)-(29) are reversed head for tail, and × is replaced by +. Hence, the abbreviation for a coproduct is A (30) xx µA xx xx |xx A + BFb FF FF FF µB FF B

3.3 Axiom PRCP: S has Products and Coproducts

A A Fb F FF FπFA FF A × B xx (31) xx xx xx πB B |x B Product

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A A xx µA xx xx |xx A + BFb FF (32) FF FF µB FF B B Coproduct

3.4 Axiom MTUN: S has Distinct Empty and Unit Sets ∅ There exists an initial set ● Ð→ D, called the empty set, and a terminal set 1 ● Ð→ D, called the unit set. Formally, there are structure logic rules

∅ ι τ x x / x / x / 1 / ● D ∅ x ● D x 1 (33) Uniquely Uniquely satisfying the condition / 1 i1 ~? 1 /// ~~ α ~β α ~~ /// / ∅ i∅ ∅ (34)

Never

In other words, ∅ ≇ 1.

3.5 Axiom TRSP: S has Product and Power Transposes There exists a Right Transpose

f / f / A ×O B C BO CA × p q /// r q /// rp × / / X X Y c Z Y Z (35) c

Right Transpose: Uniquely

The map p × q in the left frame of (35) is deﬁned by p × q ∶= (p ○ πX , q ○ πY ). However, in the right frame the map rp is created by the rule (35) to satisfy the

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condition speciﬁed in the right frame. The uniqueness in this rule implies that r s p p p p for C Ð→ Z Ð→ W , it is true that (s ○ r) = s ○ r , and also that (1C ) = 1CA .

A special case of (35) is obtained by substitutions leading to

1A×B / 1A×B / A ×O B A × B BO (A × B)A 1 ×1 /// f /// 1 A B 1B f A / A × B C / A f B C (36) f

Left Transpose: Uniquely

1A In linear notation, this case means f = f ○ 1A×B. In other words, the right f transpose of an arbitrary map A × B Ð→ C is expressible in terms of just f and the special map 1A×B. For technical reasons that shall become clear, this map A ∶= is called co-evaluation and denoted by ηB 1A×B.

Right transpose has an inverse operation, Left Transpose, as in (37).

g / ̃g / BO CA A ×O B C × q /// rp p q /// r / Y / ZX X × Y Z (37) d d̃

Left Transpose: Uniquely

As above, there exists a special case:

1 1̃ CA / CA / COA CA A ×O CA C g × /// 1CA 1A g /// 1C / B / CA A × B C (38) g ̃g

Left Transpose: Uniquely ̃ and again there is a special equation ̃g = 1CA ○ (1A × g), and this leads as above εA A ∶= ̃ × A Ð→C to deﬁning εC 1CA , so that A C C. This map is called evaluation, and is the internal version of the idea of evaluating a map at a point. It is used later to build a Turing machine in a category of sets and maps. Axiom TRSP is conventionally expressed by saying S is a “cartesian closed category,” a concept unpacked in terms of relatively advanced concepts such as “bifunctor,” “natural transformation,” and “adjoint functor.” Reliance on advanced concepts is inadmissible in a foundational enterprise. Fortunately, these advanced notions are entirely captured in Axiom TRSP.

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3.6 Axiom BTST: S has a Boolean Truth Structure m Deﬁnition 7. For map x Ð→ y of category C, m is called a monic map (or, a monomorphism) if the bilateral structure logic rule

? x ?? f ??m ?? f # z /// y ?? ? /// : x ?? ?? g / m / g m x y (39) x

Monic applies to m.

e Deﬁnition 8. For map x Ð→ y of category C, e is called an epic map (or, an epimorphism) if the bilateral structure logic rule

? y ? ??f e ?? ? f C $ x ?? ? z y C : z ?? ?? g e / / e g x y (40) y

Epic applies to m. ⊺ applies to e. There exists a truth structure with data 1 Ð→ Ω and condition

? A @@ @@⊺ @@ x / m / y x PBK ? Ω ?? ~~ ?? ~~ m ? ~~ χm (41) y

Uniquely

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In particular, there exists a structure

? 1 >> (42) > ⊺ >> > ∅ > PBK ? Ω >> >> ∅= ∅ > χ∅ ι1 τ1 1

and, by deﬁnition, ∶= χ∅. By Axiom PRCP there exists a unique structure

{ 1 ?? (43) {{ ? µ1{{ ??⊺ {{ S ?? {} [⊺ ] + , / 1 1aCC ? Ω CC S CC µ CC 2 C 1

and, it is declared, ﬁrst, the induced map [⊺, ] in (43) is an isomorphism, and second, that ⊺ and are the only maps from 1 to Ω. There exists also a unique structure

? 1 @@ (44) @ ⊺ @@ @ PBK 1 >> ? Ω >> >> χ Ω

and, by deﬁnition, ¬ ∶= χ.

Axiom BTST is conventionally expressed by saying S has a “subobject classiﬁer,” which is a concept unpacked in terms of external notions, such as “equivalence relation.”

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3.7 Axiom AXCH: S has Arbitrary Choice

? x ?? ??f ?? /// ?? f / / x y y / y iy (45)

3.8 Axiom NNST: S has a Natural Numbers Structure 0 s There exists a structure with data 1 Ð→ N Ð→ N and condition

N s / N x< 0 xx xx xx b n G f f 1 / x / x 1 GG GG (46) GG b # / x n x Uniquely

A member of the set N is called a natural number, 0 is called zero, and s is the successor function.

x y Deﬁnition 9. For category C with choices of dots ● Ð→ D and ● Ð→ D, notation x ≅ y signiﬁes that x is isomorphic to y, which is expressed by the bilateral structure logic rule / x 1x ? x C α β α C x ≅ y 1 / y y y (47)

Isomorphism

Either α of β is called an isomorphism.

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4 Startup Theorems

Theorem 1. A category satisfying Axiom NNST has a predecessor map, that is to say

s / N N _> >> >>0 ( N ) g > 1 ,g /// /// > > N × h / o a0 A A 1 N × A / A o 1 h a0 (48)

Uniquely

Proof. ([5], pp.158-159). Consequently, there exists a unique structure

s / N N _> (49) >> >>0 ( N ) p > 1 ,p /// /// > > N × N / N o 1 π1 0

so that p(0) = 0 and p(s(n)) = n, which justiﬁes calling p the predecessor function. This (together with the successor function s) is used to model the move-to-the-left (while s is used to model the move-to-the-right) operation of the tape head in a Turing machine, see below. Deﬁnition 10. For category C, an image factorization with image I of a f structure A Ð→ B is a structure

? I (50) ? ? e ?? C ? A / B f

such that for any structure

? J (51) ?? e ?? C ? A / B f

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there exists a structure

JF O (52) 1 1 11 O 1 CC 1 11 = I !B 1 || B 11 e || BB 1 || C BB11 || B! A / B f

The image factorization is an epic-image factorization if the map e is epic, as in

? ? I ? (53) e ?? C ? A / B f

Theorem 2. For a category C satisfying Axiom BTST, every monic map is an equalizer map. Theorem 3. For a category C satisfying Axiom TRST and Axiom PBPO, every map has an epic-image factorization. Theorem 4. Any two pullbacks of the same structure are isomorphic. More precisely, there exists a structure logic rule

> A A ? A @ ~~ AA @@ ~~ AAf @@f ~~ A @ P PBK X PBK @@ }> Q ? ~?X @@ }} ?? ~ ≅ @@ }} ?? ~~ P Q } g ? ~~ g (54) B B

Pullback Isomorphism

1 Theorem 5. There exists exactly one structure 1 Ð→ 1, namely 1 Ð→1 1. Like- wise, there exists exactly one structure ∅ Ð→ ∅. Also, there exists exactly one ∅ Ð→ ∅ = ∅ structure 1, namely, ι1 τ1 . α Theorem 6. There does not exist a map 1 Ð→ ∅.

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α Consequently, there does not exist a structure x Ð→ ∅ for any set x. Therefore, ∅ ι the initial map ∅ Ð→x x is a monic map for any set x. Theorem 7. If a set is non-void then it has a member. More precisely, if A ≇ ∅, then there exists a choice ● Ð→ A. Theorem 8. Every identity map is both epic and monic.

Theorem 9. If a category C has epic-image factorizations, then any structure

A / / / B is an isomorphism.

Theorem 10. The monic map of an epic-image factorization of an epic map is an identity.

Theorem 11. ‘ The epic map of the epic-image factorization of a monic map is an isomorphism.

5 Calculation

An abstract concept of calculation may boil down to changes of state resulting from performance of operations. That would be little more than a labeled 1- graph, where dots represent states and labels on arrows represent operations. A slightly more concrete concept permits operations to be organized in terms of tests on the state, leading to the idea of a “ﬂowchart” to represent an algorithm. Thus, “hardware” consists of primitive “instructions” that change “memory state,” and “software” is a “ﬂowchart.”

A much more concrete concept involves storage of the ﬂowchart in the memory, eventually leading to the very concrete notions of “von Neumann Architec- ture” and “Harvard Architecture,” which are very much foundations for modern computer hardware. This report stops short of such concrete ideas, and instead formalizes two related abstractions.

5.1 Central Processing Unit and Flowchart Deﬁnition 11. A central processing unit (CPU) is a structure

γ U ∶ Σ × I Ð→ Σ (55)

A member of Σ is called a state, and a member of I is called an instruction. For state x and instruction i, γ(x, i) is “the state entered by invocation of” i in state x.

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A ﬂowchart for C is a structure

< DO (56) xx πD xx xx t xx /// D × Σ σ / A h / D

p I

A member of D is called a test, a member of A is called a flow arrow. Each flow arrow α invokes an instruction, p(α), and its tail t(α) and head h(α) are tests. The flow arrow σ(δ, x) is determined by test δ and state x. Ordinarily, a flowchart is a diagram composed of boxes and diamonds connected by arrows. Boxes contain executable instructions that change memory state (including controls and indicators). Diamonds typically have one incoming arrow and two outgoing arrows, where control flow passes from the incoming arrow to the choice of outgoing arrow depending on a test of memory state specified in the diamond. This convention implies that the diamond condition partitions memory states into two disjoint and mutually exhaustive parts, and they correspond to the two outgoing arrows.

Usually, a flowchart also has a START circle with no incoming arrows and exactly one outgoing arrow; also, there is a STOP circle, with one or more incoming arrows, and no outgoing arrow. A path from START to STOP along arrows through diamonds and boxes represents one possible flow of instruction execution. Different flows result from different initial states.

More generally, one may have a “multi-faceted” diamond which partitions memory states into some number of disjoint mutually exclusive parts, each of which corresponds to exactly one outgoing arrow. That is the role of σ, where members of D are like multi-faceted diamonds. There may be a test with exactly one outgoing arrow and no incoming arrow, so that test could represent a START circle. Similarly, a test may have one or more incoming arrows, and only a single outgoing arrow back to itself, so that test could represent a STOP circle. For test δ and state x, σ(δ, x) is also called the “transition” undergone from test δ in state x, and h(σ(δ, x)) is the next test, while p(σ(δ, x)) is the instruction invoked by the transition. The equation t((σ(δ, x)) = δ expresses the statement that the tail of any outgoing arrow is δ.

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Structure logic deduces from (55)-(56) new structures

6 hP mmm Σ PPP (57) mmm PPP πΣmm PπPΣ mmm PPP mmm PP × q / × D ΣFF Σz I FF zz FF zz σ FF zz π F" z I / z| A p I where q ∶= (πΣ, pσ), and

h / u: A D cFF (58) uu FF σ uu FπD uu FF uu F χ / D × ΣI D × Σ II x II xx II xx q I xx π I$ |xx Σ × / Σ I γ Σ with χ ∶= (hσ, γ(πΣ, pσ)). The map q assigns, to a test and state, that same state, but coupled to the instruction associated with the outgoing arrow selected by testing that state. The map χ determines the next test and the new state based on the results of q and γ. Therefore, by Axiom NNST, there exists a unique structure

9 N s / N ss (59) 0ss ss ss K f f 1 KK /// /// KK KK b % × / × D Σ χ D Σ

The “dynamical system” χ with “initial condition” b induces a (possibly inﬁnite) structure

χb χ2b 1 b / D × Σ / D × Σ / ⋯ (60)

This is the “trace” that results from execution of the ﬂowchart algorithm, in response to the speciﬁc initial state and test.

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5.2 Theorem: Category S is “Turing Complete” There are many references for definition of “Turing machine,” including, of course, Alan Turing’s original paper. The following definition is based on [2] and [6]. One slight deviation from these equivalent alternative definitions is that here the case where the tape head does not move, and therefore there is no change of state or symbol (see (62)), is included just for the sake of making the rules into a totally defined function.

Deﬁnition 12. A Turing machine is a structure with data

S × C α / S (61)

β S × C / C

φ S × C / M

A member of S is called a a state of the tape head. A member of C is m called a symbol, and a member of M is called a move. Every move 1 Ð→ M is s 1N either the successor map N Ð→ N, the identity map N Ð→ N, or the predecessor p map N Ð→ N. Map α speciﬁes the next state for a state and symbol; likewise, β determines the replacing symbol for a state and symbol. The move of the tape head – one cell to the right by s, one cell to the left by p, and no move by 1N – is determined by state and symbol via φ. These data conform to the condition o / / / 1 @ P E E S × C (62) @ E y uu @@ EE yy uu @ EE yy uu @@ PBK E /// y EQU u 1N E" y| y uz u 1S×C M o S × C / S × C φ (α,β)

In (62), P is the set of state-symbol pairs to which φ assigns 1N – no move. E is the set of state-symbol pairs that are invariant under (α, β): the state and symbol remain the same. The condition expressed by P / / E is that state-symbol pairs for which there is no movement also do not change state or symbol. The “global state” of a Turing machine is a member of

T ∶= S × N × CN

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|> S B` B (63) || BB α̃|| BπB1 || BB | ̃ T φ / NTo π2 AA } AA }} AA }} A } π β̃ ~} 3 CN

(x,h,t) The components of a member 1 ÐÐÐÐ→ T in (63) are a state x, tape head position t h, and (transposing 1 Ð→ CN by Axiom TRSP) a tape, t. Thus, for example, h t the compositon 1 Ð→ N Ð→ C determines the symbol t(h) at cell h of tape t. The “dynamical system” map (̃ ̃ ̃) T ÐÐÐÐ→α,φ,β T will determine the next state of the tape head, the next head position, and the contents of the tape, once α,̃ φ̃, and β̃ are deﬁned.

k5 Qh kkk S QQQ (64) kkk QQQ π1 kk Qπ1 kkk QQQ kkk QQ kkk Q T k δ / × FF Sy C FF yy FF yy FF yy π23 # |yy π2 N × N / C N C evC

First, α̃ ∶= α ○ δ deﬁnes α̃ as the map which calculates the symbol in the current cell of the tape, and via δ in (64) yields the state-symbol pair at that cell. Then, α gives the next state.

i4 N jUU iiii UUUU (65) iiii UUU π2 ii UUUπ1 iiii UUUU iiii UUUU iiii U T i ε / N × NN DD w DD ww DD ww DD ww δ D! {ww π2 × / / NN S C φ M

N Second, the next head position is deﬁned by φ̃ ∶= evN ○ ε, where ε in (65) uses φ to determine the move based on the state-symbol pair resolved by δ in (64).

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N ( ) 1 ×ev / ′ / x,h,t / S C/ 1 1 1 h 1 1 T S × CG }} GG h } GGβ (x,h,t) ⊺ ⊺ }} /// G h /// /// } GG }~ } G# / / / F (x,h,t) / T N Ω N Ω Ω N `A ; C χh χh ¬ A x AA xx AA xx A /// xx ` xx t h′ / / N

(1) (2) (3) (4)

Third, and ﬁnally, Ground Rule FUNC is applied as follows to determine the tape obtained by replacing the symbol in the current cell. Transition from (1) to (2) is accomplished by projection map from Axiom PRCP (product) to isolate h, and then applying Axiom BTST to yield the characteristic map χh. In (2) to ′ (3), Axiom BTST gets from characteristic map ¬ ○ χh the complement h in N. F (x,h,t) Then (4) uses Axiom PRCP (coproduct) to deﬁne the new tape N ÐÐÐÐÐ→ C using the replacement symbol supplied by β at the current cell, and retention of all symbols elsewhere.

6 Conclusion

There exists a constructive foundation for mathematics, and therefore functional programming, built upon a ground consisting of structure logic deduction rules that replace “naive set theory.” A small number of axioms for a category of sets and maps, entirely motivated by some deep work of F. William Lawvere and colleagues, is put in structural form, and elementary theorems are derived.

A very general concept of computation with central processing unit (“hardware”) and ﬂowchart (“software”) is articulated in structural terms, and the more speciﬁc concept of Turing machine is is shown to exist. Thus, the system is demonstrably “Turing complete.” Ongoing research intends to show that it is “mathematically complete,” in the sense of ZF(C) axiomatic set theory, as well as in the sense of type theory, and category theory of every order.

With a complete, constructive ground and foundation of mathematics at hand, it would be merely an exercise in computer programming to implement a mobile application program for a tablet equipped with stylus to serve as a portable structure-oriented proof and programming assistant. Indeed, proofs and programs are the same thing.

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References

[1] Rudolf Carnap, The Logical Syntax of Language, Littleﬁeld, Adams & Co., 1959, Translated by Amethe Smeaton (Countess von Zeppelin).

[2] Daniel I. A. Cohen, Introduction to Computer Theory, 2 ed., John Wiley & Sons, Inc., 1991.

[3] Paul R. Halmos, Naive Set Theory, Springer-Verlag, New York, 1960. [4] Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic, a First Introduction to Topos Theory, Springer-Verlag, 1992. [5] F. William Lawvere and Robert Rosebrugh, Sets for Mathematics, Cam- bridge University Press, 2003. [6] Stephen Wolfram, A New Kind of Science, Wolfram Media, Inc., 1995.

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The Human Condition Ellis D. Cooper, May 22, 2013

I will eventually have a philosophy. Preparation for my philosophy will consist in learning the philosophies of others, and observations of the life around me. Naturally, mine will absorb some ideas, and reject others. I think I can set the limit on human knowledge, and thereby answer all the questions of philosophy (Age 17, February 15, 1960).

Key Terms: awareness, conscious thought, evil, hidden mind, human condition, mental model, metaphor, microlect, self, suﬀering, virtuality.

Now at Age 71, I will ask you, at the end, whether I have successfully conveyed to you my “mental model” of the human condition.1

Structure & Mathematics Constructing a building usually entails a foundation in the ground upon which supports are erected and between which connecting beams are attached. A sequence of transformations concludes with a ﬁnished structure. In use, as people and objects come into the building, move about in it, and leave, the ﬁxed structure endures changes in internal state as time goes on. Distinct from structural artifacts such as buildings, human beings discern natural structures in their World. Mathematics is highly structured reasoning about structure wherever it is discerned and interesting. Mathematicians discover and invent intricate, deep, and clever ideas about structure. Mathematics is the tinkertoy of metaphor,2 and rigor cleans the window through which intuition shines.3 If geometry is the mathematical study of symmetry, then algebra is the geometry of notation.4

Mental Model & Microlect A mental model is a structure in the mind that simulates – that is, imagines – more or less how the World works, including “how to work” the World.5 All animals have mental models. Mental models are malleable and mostly incom- municable. Human beings have some reportable mental models,6 and adult human beings have a reportable mental model of birth, the biological necessity of other lives to end for One to persist, and inevitable death. The human mental model diﬀers from animal mental models because it situates One in the World. This mental model is the Self.7

Each reportable mental model is conveyed with a specialized part of natural language. I call it the natural microlect of the mental model. A microlect is

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the language of topical discourse. Coherent discourse glides among microlects, thus reﬂecting turns from model to model in mind. Mathematical discourse is conducted in specialized artiﬁcial microlects called “formal languages.”8 The neologism “microlect” belongs to the microlect of my mental model of the human condition.9

Education & Imagination Education is the gradual increase in scope of mental models both in time and space. Education acquires, uses, and adapts mental models with widening temporal and spatial scope.10 The primary function of any educational institution is to further the education of its students. There are useless mental models, and there are eﬃcacious mental models. Two mental models may conﬂict.11 There are delightful mental models, and there are inimical mental models.

Any narrative is expressed in the microlect of some mental model. Reading ﬁctional or other narratives is a most highly eﬃcient means of education, since it brings to mind another’s mind. Dreams and waking imaginations are operations of mental models within the mind. These operations simulate conditions beyond the mind.12

Intellect & Spirituality Alongside Materialism and Humanism is the Monotheism mental model, “God created the universe. Humanity must progress towards spirituality. Visions and scriptures energize progress. Science of ﬁnite intellects deploying experiment and theory may accelerate spirituality.” Whatever, conscious thought is a prerequisite for intellect and spirituality. Conscious-thought experiments13 may culminate in practices for experiencing spirituality, with the option but not the obligation to postulate some awesome, wonderful God.

Suffering & Radicalization Evil is the unreasonable, deliberate infliction of involuntary suffering.14 Actions of human beings ought to be coordinated somehow to minimize the Totality of Suffering. This is the Principle of Least Suffering. Monotheism15 considers Adoration of God more important than relieving people from suffering. This infinitely pernicious inversion of priorities has always had execrable both intended and unintended consequences.

Science, engineering, and technology evolve gradually by toiling corporate- funded researchers and exuberant home tinkerers. Occasionally the work is

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punctuated by revolutionary discoveries and inventions.16 Happily, this amplifies human capacity to ameliorate suffering, but sadly, also to inflict it. Personal, group, and community factors; social, political, economic, and ideological factors; all may contribute to spiraling human radicalization. Unless homeland security agents identify a pattern of radicalization behavior, deploy counter- radicalization strategies, and if necessary, interfere with susceptibility to cat- alytic events or perceptions, the radicalized individual accelerates towards support of action, including identification of opportunities, capabilities, and targets. Unless interdicted, such radicalized, mobilized individuals could inflict massive suffering upon others, and themselves. Therefore, the human condition is all the more fragile as science, engineering, and technology provide ever more capabilities for radicalization, mobilization, and infliction of suffering.17,18

Science & Religion Philosophers are dumbstruck by the ineﬀable. Science is the eyesight of philosophy in a dark mansion.19 It seeks the simplest mental models of the widest scope based at best on repeatable and at least on reportable evidence. Evidence is expressed always in the microlect of some tentative model.20 But, religious faith is belief without evidence. As a chain is only as strong as its weakest link; as a highway is only as safe as its worst driver; so a religion is only as moral as its extremists. Government funds, extracted from widely diverse citizens, must never be used for religious prosyletizing. In fact, religious prosyletization in general, and evangelization in particular, ought to be voluntarily eschewed if not vigorously proscribed.

Hidden Mind & Conscious Thought Consciousness21 is not experienced, but conscious thought is. Conscious thought is nothing if not focus of attention changing over objective time.22 Simple chemistry and physics experiments performed, say, on a kitchen table with materials readily available in the home, can convince children that science is fun, and extremely interesting. The “kitchen table” conscious thought experiment is quietly to count successive foci of attention, regardless of intent, or absence of intent. Anybody can perform this experiment, and with practice it can be pretty interesting, especially when it comes to indescribable, ﬂitting thoughts. “Hidden mind” is the support system for conscious thought.23 It is throughout life a continuously running mental model – called hidden mind – that rapidly recognizes patterns and reacts accordingly. It is the invisible director of attention in conscious thought. Conscious thoughts bubble up from hidden mind.24 Hidden mind is the abode of intuition, and conscious thought is the home of rigorous deliberation.25

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Temporality & Awareness The microlect of Edmund Husserl corresponding to his mental model of temporal awareness includes the terms (1) primal-impression for the immediate sensation – whether attended to or not, (2) retention for the memory of the primal-impression – possibly mutable as objective time runs forward, and (3) protention for anticipated but not yet having occurred primal-impressions. At one instant of objective time – that is, the time of ordinary physics measured by clocks – the mind (including both hidden mind and conscious thought) is a temporal structure of retentions and protentions.26,27

The continuously running mental model updates and anticipates. This is awareness.

Retentions need not be ordered in conformity with the objective times at which their instigating primal-impressions occurred, nor need every primal- impression promote retentions that persist beyond some further time. Proten- tions need not ever “come true” in terms of actually occurring future primal- impressions. Protentions are imaginary futures, and as such are concocted by operation of mental models in hidden mind, and also maybe by deliberate conscious thought. In a diagram, this structure of temporal awareness – subjective time – is drawn at right angles to the horizontal line that is conventionally drawn to represent objective time. So, subjective time is “orthogonal” to objective time, an ever-changing structure rolling forward along the track of objective time.28

Meaning & Understanding Meaning leads to understanding; it is both a process and a product. The process, meaning, is the production of means, and a mean is a step in the sense of “intuitive leap.” Concatenation of intuitive leaps is how a proof works in informal mathematics, and how we come to understand one another in conversation.29 In mathematics the leaps tend to be extremely tiny, so that any perseverant person can check every step of the proof. The mathematician who writes, at some point in a proof, “Obviously,... ” is trusting the reader’s agility to take the leap without hand-holding. The Principle of Least Thought corresponds to the idea of ﬁnding the shortest possible proof: understanding is best achieved with the minimum number of largest possible steps. People diﬀer in “mental agility” to make these leaps, but for each person there is an optimum size, possibly increasing with training and practice.

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Existence & Virtuality Stranger. I suggest that anything has existence, that is so constituted as to possess any sort of power either to aﬀect anything else or to be aﬀected, in however small a degree, by the most insignif- icant agent, though it be only once. I am proposing as a mark to distinguish existence, that it is nothing but power.30

The virtual is contingent existence. It is that which exists and persists only if something else exists and persists, and in terms of which it is eventually understood.31

Virtuality & Reality The mathematical study of computation, like the mathematical study of reasoning,32 ﬁnds abstract limitations. But concrete computation is limited by engineering, which is based on physics. The repeated discovery, that from simple computational rules emerge complex structures, is balanced by the existence of simple phenomena grounded ultimately in complex physical reality.33 The spectrum of medial levels between basal reality and apical virtuality is called, in my microlect, a tower of virtuality. It must be observed that “emergence” of observed phenomena is entirely governed by hidden mind, bubbling up from which is articulation by fragmentary inner speech34 in conscious thought, and then possibly verbally reported. Identifying “reality” as the basal level of a tower of virtuality is always as provisional as science itself. Science ﬁction literature and cinema, and computer application programs, exhibit virtual un-reality. These are manifestations of imaginary realities, worlds not consistent with the current laws of physics, nor with ordinary expectations of behavior.

Luck & Love There are angelic humans among us. These are individuals, triggered by some experience, and with purity of motive, called to devote themselves to uplift- ing others.35 There is more luck – both good and bad – in life than generally acknowledged. The magnitude of any kind of luck follows a power law. That is to say, daily life is strewn with many small occasions of luck.36 Occasions of extraordinary good luck, or bad luck, are rare. Each person has their own power, which may vary over time. Life is as exquisitely tuned to the World as is possible at this time, and keeps getting better at it. Greatly varied versions of the structure of the deoxyribonucleic acid molecule all record rules for survival in confrontation with mostly impartial natural events.37 If one decides to try for a fully lived life, it is a good idea to know and treasure another’s life. Hence, usually, marriage – or any equivalent of a lifetime together – which may begin with or end with love, or both. Giving beats getting, and forgetting beats

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forgiving.

So, did I?

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Notes

1The psychologist Philip N. Johnson-Laird’s focus of attention on mental models is exceptional. In [12] he credits Kenneth Craik for sketching in 1943 a theory: “If the organism carries a “small-scale model” of external reality and of its own possible actions within its head, it is able to try out various alternatives, conclude which is the best of them, react to future situations before they arise, utilize the knowledge of past events in dealing with the present and the future, and in every way to react in a much fuller, safer, and more competent manner to the emergencies which face it.” For me, mental models are the twirling nuts and bolts of mind, which at large scale are called “belief systems” or “world hypotheses” [25]. 2The point of this metaphor is that there are aspects of nature, such as quantum mechanics, for which there are no adequate metaphors. Therefore, only mathematics has provided artificial structure to reflect natural structure. 3Here the point of this metaphor is that often, not always, an effort to formalize one’s thoughts does help to clarify them. But, magisterial mathematician Jacques Hadamard has been quoted, “Logic merely sanctions the conquests of the intuition” [13]. Also, gratuitous formalization, if not pretentious, is merely a hindrance to thinking clearly. 4Mathematics is nothing without its focus of attention on interplay between geometry and algebra. For example, there has existed over more than a century a global enterprise devoted to the idea that a geometric structure is determined by its group of symmetries, where a group is perhaps the quintessential algebraic structure in mathematics (Look up “Erlangen Program.”). A possibly new observation is that even the dance of symbols used by mathematicians to articulate their ideas have choreographed symmetries. For example, the so-called “adjoint” pattern (F a, b) ⇔ (a, Gb) is so ubiquitous throughout geometry, algebra, and logic, that it has risen to the status of a foundational idea in mathematics [19][21][18]. 5Computer scientists explore how machines might automate “formal models of a user’s knowledge about a given system,” as in [3], where the system is a semi-automatic transmission of a large vehicle, and the authors demonstrate “one possible, non-deterministic, mental model for the vehicle transmission system.” in the mathematical microlect of labelled transition systems. 6Reports are delivered with natural language. 7“Writing in the Principles of Psychology almost a century ago, William James suggested that the key to the consciousness is self-reference: “The universal conscious fact is not ’feelings’ exist and ’thoughts exist’ but ’I think’ and ’I feel’”. In other words, in order for ongoing experience, thought, and action to become conscious, a link must be made between its mental representation and some mental representation of the self as agent or experiencer – as well, perhaps, as some representation of the environment in which these events take place” [16]. 8Currently, the most popular ground for (several alternative) foundations of mathematics is an artificial microlect consisting of symbols admitting substitution by other symbols (the “variables”), symbols that do not admit substitution by other symbols (“constants”), and symbols that represent compounded substitutions yielding new symbols (“functions” and “relations”). 9 Everyday discourse swerves from topic to topic according to what the people are interested in from moment to moment. Sometimes the topic changes suddenly, for example listening to the news on the radio leads from breaking stories in disparate parts of the world, to local news, almost without hesitation. However, in a given context key terms are used consistently. Otherwise, people do not know what what they are talking about. Unless they are singing songs or speaking in poetry, the grammar of the ambient natural language they speak is invariant while the set of key terms changes from context to context. Discourse analysis is the scientific study of sequences of sentences as they occur in natural settings, and goes beyond linguistics towards the social and psychological aspects of communication among human beings [32]. A number of standard terms refer to subsets of natural language distinguished by variations in grammar or vocabulary. Synonyms: dialect, vernacular, jargon, cant, argot, lingo, patois

These nouns denote forms of language that vary from the standard. Dialect usually applies to the vocabulary, grammar, and pronunciation characteristic of

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specific geographic localities or social classes. The vernacular is the informal everyday language spoken by a people. Jargon is specialized language understood only by a particular group, as one sharing an occupation or interest. Cant now usually refers to the specialized vocabulary of a group or trade and is often marked by the use of stock phrases. Argot applies especially to the language of the underworld. Lingo is often applied to language that is unfamiliar or difficult to understand. Patois is sometimes used as a synonym for jargon or cant, but it can also refer to a regional dialect that has no literary tradition.1 To my ear “jargon” bears a tinge of pejoration. Therefore, I prefer to use the word “microlect” for a specialized language conforming to standard grammar but with its own set of key terms. Two microlects may share common terms – the microlect within a basketball team shares the names of its players with the microlect of the fans of the team, but the players have agents and managers and contractual language not normally known by fans. In mathematics a collision of microlects may explode in a shower of ideas. For example, a particular number might show up in two seemingly unrelated mathematical microlects. The mathematicians pounce, they want to know why that number shows up in those seemingly distinct areas of research. Such questions may lead to new research. Microlects are associated with “owner’s manuals,” “user guides,” or “instruction books” for tools, appliances, instruments, or software. For example, the owner’s manual for a wet/dry vacuum appliance uses the words “filter,” “filter cage,” “rubber gasket,” “powerhead assembly,” “vacuum port,” “locking tabs,” “release button,” “wet nozzle,” and “hose.” A “visual quickstart guide” for the Macintosh personal computer iMovie application program uses the words “footage,” “project file,” “playback control,” “import button,” “clip,” “titles list,” Over Black option,” and “Export Movie.” Every one of these microlects is associated with a mental model of the tool, appliance, instrument, or software. We learn the microlect and the model together.

10Babies know little about the past, nor about the far away. An adult probably knows something about the history of the world, and something about unvisited remote places. 11Cf. “cognitive dissonance.” 12Phenomenology of reading is beautifully discussed in [28]. The phenomenology and relationships between simulation, metaphor, and imagination are detailed in [2][7] and [29]. 13As suggested by E. Husserl, G. I. Gurdjieff, C. Tart, and F. Varela, among others, not to mention Buddha before all. 14Moral psychologist Jonathan Haidt refers to suffering indirectly in Moral Foundations Theory http://www.moralfoundations.org/. The five statistically salient foundations are 1. care/harm, 2. fairness/cheating, 3. liberty/oppression, 4. loyalty/betrayal, 5. authority/subversion As I see it, harm, cheating, oppression, betrayal and subversion are just possible ways to inflict suffering. Connection of suffering to mental models is suggested by Haidt’s critic, Sam Harris, who writes, “Haidt draws comfort from the fact that even biblical literalists occasionally yield to common sense and ignore their holy books. Of course they do: their holy books are not only bursting with ancient ignorance - they are actually self-contradictory. Is Haidt suggesting that there are no real religious fundamentalists out there at all, or that their numbers are negligible? According to a recent poll, thirty-six percent of British Muslims (ages 16-24) think apostates should be put to death for their unbelief. Just how much exculpatory sociology is Haidt inclined to do in this area so as to get Islam entirely off the hook? When is a belief system not only false, but so encouraging of falsity and needless suffering as to be worthy, not merely of our understanding, but of our contempt?”2

1The American Heritage Dictionary of the English Language, Fourth Edition copyright 2000 by Houghton Miﬄin Company. Updated in 2009. Published by Houghton Miﬄin Com- pany. All rights reserved. 2http://www.edge.org/discourse/moral_religion.html#haidt

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15Judaism, Christianity, and Islam. 16This point is stimulated by the notion of “punctuated equilibrium” introduced into biology in [5]. 17The microlect deployed in this paragraph is from [15]. Almost always the “subject terms” or “key words” of a technical paper are listed beneath the Abstract. These are the core microlect of the work. The index of a technical book lists in detail the microlect of the work. A Glossary of Terms is especially valuable in reports of original research. 18Early exposure to a captivating viral counter-narrative might tend to neutralize radicalization. Radicalizing web-sites might be hacked to cleverly insinuate a benign counter-narrative. 19That being said, the greatest scientists attend closely to philosophy – Einstein already as a teenager was quite familiar with the writings of Immanuel Kant. 20Physics is the scientific study of motion, which is one sort of change. In general, the infinitesimal calculus is the mathematical study of change. 21Research on consciousness by top-level scientists proliferates. For example, see [4]. Philo- sophical theories of consciousness are advancing as well. For example, [17] summarizes many ideas. It leads off with the sentence, “Phenomenal consciousness is the property of mental states, events, and processes have when, and only when, there is something it is like for their subject to undergo them, or be in them.” This formulation or variations of it recur throughout modern philosophical literature. My infuriating problem with it is in the words “something” (what?) and “like” (what?). Fortunately, philosopher Greg Janzen calms me down [11]. In detail, he re-formulates Thomas Nagel’s mystifying “definition” in successive ways: W LF An organism has conscious mental states if and only if there is something that it is like to be that organism - something it is like for the organism.

W LF2 A subject S is in a conscious mental state M at t if and only if there is something it is like for S to be in M at t.

W LF3 Necessarily, a subject S is in a conscious mental state M at t if and only if there is something it is like for S to be in M at t.

W LF4 Necessarily, a subject S is in a conscious mental state at t if and only if S undergoes a type of psychological internality at t.

W LF5 Necessarily, a subject S is in a conscious mental state at t if and only if S is, in some suitable way, aware of being in M at t. Setting aside my bridling at the pretentious use of mathematical symbols, for me the key to these re-formulations is that the concluding equivalent W LF5 = W LF has no words “something” and “like,” and instead uses “awareness,” and not just naked awareness, but self -awareness. Please see [30]. 22Greater emphasis on attention is not possible. Scientiﬁc and philosophical analysis relating attention to consciousness and memory is represented by [23][6] and especially [35]. 23Poetry transiently springs the latch of the gate to hidden mind, and then it ampliﬁes the unspeakable. A metaphor is a portal between universes of discourse. 24“Intuitions are the judgements, solutions, and ideas that pop into consciousness without our being aware of the mental processes that led to them. When you suddenly know the answer to a problem you’ve been mulling, or when yo know that you like someone but can’t telll why, your knowledge is intuitive. Moral intuitions are a subclass of intuitions, in which feelings of approval or disapproval pop into awareness as we see or hear about something someone did, or as we consider choices for ourselves” [8]. 25 Hidden mind (also known as “unconscious thought,” or “phenomenal awareness,” or “System 1”) among other things, coordinates the organism with the immediately present physical environment, but conscious thought (also known as “System 2”) among other things, integrates the human being into its social and cultural environment over wide temporal and spatial horizons. The value of conscious thought is its evolutionary adaptive contribution to survival of human beings in groups. Survival is enhanced by abilities to simulate the future based partly on counterfactual simulations of the past, and to infer and anticipate the conscious thoughts of other human beings [1].

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Psychologist Daniel Kahneman writes, “System 1 operates automatically and quickly, with little or no eﬀort and no sense of voluntary control. System 2 allocates attention to the eﬀortful mental activities that demand it, including complex computations. The operations of System 2 are often associated with the subjective experience of agency, choice, and concentration. ... The automatic operations of System 1 generate surprisingly complex patterns of ideas, but only the slower System 2 can construct thoughts in an orderly series of steps” [14]. 26A genius marshalling a geometric “Diagram of Time” as a metaphor for subjective time, Edmund Husserl introduced the microlect centered on “primal-impression,” “retention,” and “protention” [9]. A more modern version of this geometrization of subjectivity is provided by Izchak Miller in [22]. In these diagrams a vertical line represents memory and anticipation. My elaboration of this idea expands the vertical line into an objectively time-varying structure that intertwines partially ordered sets representing relative focus of attention, relative duration, and temporal ordering. This work also responded to the philosophical analyis of temporal properties by Ian Phillips in [26]. 27

Technically, protention will be given by a temporal mirror image, as it extends retention forwards into time. Protention is, above all, the tropism inherent to action performed by any life form. This point is at the center of our approach: we call retention and protention these particular aspects of memory and of anticipation that are speciﬁc to all life forms – a sort of present which is extended in both directions. Thus we do not limit our analysis to the phenomenological use of these words, inasmuch it limits their meaning to situations that can be examined through conscious activities. We believe that this extension to pre-conscious activities remains compatible with (and helps to understand) its classical usage, particularly such as described by Van Gelder and Varela who develop the concepts of intentionality, retention and of protention, introduced and discussed in length by Husserl in his analysis of human consciousness [20]. 28William James wrote, “ the portion of time which we directly intuit has a breadth of several seconds, a rearward and a forward end, and may be called the specious present” [10]. The “specious present” is discussed from the Husserlian phenomenological perspective, for example, in [33] and [27]. 29For example, if I say something and you say, “I don’t quite follow you,” that means I have taken a leap too large. 30Adapted from Plato, “The Sophist,” 247E. “Plato’s Theory of Knowledge, the Theaete- tus and the Sophist of Plato,” Translated with a Running Commentary 1951, by Francis MacDonald Cornford, with special regard to “Nature and Historical Experience, Essays in Naturalism and in the Theory of History,” John Herman Randall, Jr., p.130, on “The Nature of Metaphysics.” 31Latin vir means a man, male person. So, we have English virtue meaning manly strength or courage, and of course, virile meaning capable of functioning as a male in copulation. But, more importantly, we have virtual meaning – according to a dictionary and not in the sense of the microlect in this paper – having the power of acting without the agency of material part. 32Namely, malthematical logic. 33Some are hypnotized by emergence of complexity from simplicity (Stephen Wolfram); others rejoice in the contraction of complexity to simplicity (Steve Jobs). 34The fascinating paper [24] includes a Glossary of the microlect for discourse on inner speech. An important early work is [31], and, even earlier, there is [34]. 35The best teachers and caregivers. 36Whether it involves bumping into the corner of some furniture, or jumping the curb while driving, or getting a refund just at the right time. 37Or worse: “Earth is an insigniﬁcant speck in a vast and overwhelmingly hostile universe,” wrote prestigious mathematical physicist Bryce DeWitt, in “God’s Rays,” Physics Today, January, 2005.

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References

[1] Roy F. Baumeister and E. J. Masicampo, Conscious thought is for facil- itating social and culteral interactions: How mental simulations serve the animal-culture interface, Psychological Review 117 (2010), no. 3, 945–971. [2] Edward S. Casey, Towards a phenomenology of imagination, Journal of the Biritsh Society for Phenomenology 5 (1974). [3] SébastienCombéfisand Charles Pecheur, A bisimulation-based approach to the analysis of human-computer interaction, Symposium on Engineering Interactive Computing Systems (Pittsburgh, PA), vol. (EICS 2009), ACM SIGCHI, July 2009. [4] Gerald M. Edelman, Naturalizing consciousness: a theoretical framework, Proceedings of the National Academy of Sciences 100 (2003), no. 9, 5520– 5524.

[5] N. Eldredge and S. J. Gould, Punctuated equilibria: an alternative to phyletic gradualism, Models in Paleobiology (Schopf, ed.), TJM Freeman, Cooper & Co., 1972, pp. 82–115. [6] Daryl Fougnie, The Relationship between Attention and Working Memory, New Research on Short-Term Memory (Noah B. Johansen, ed.), Nova Sci- ence Publishers, Inc., 2008.

[7] Alvin Goldman and Kelby Mason, Simulation, Philosophy of Psychology and Cognitive Science (Paul Thagard, ed.), Elsevier B.V., 2007.

[8] Jonathan Haidt and Craig Joseph, Intuitive ethics: how innately prepared intuitions generate culturally variable virtues, Daedalus (2004), 55–66.

[9] Edmund Husserl, On the Phenomenology of the Consciousness of Internal Time (1893-1917), Kluwer Academic Publishers, 1991. [10] William James, The Principles of Psychology, Dover Publications, Inc., 1918, Authorized Edition, in two volumes. [11] Greg Janzen, In Defense of the What-It-Is-Likeness of Experience, The Southern Journal of Philosophy 49 (2011), no. 3, 271–293. [12] Philip N. Johnson-Laird, The history of mental models, Psychology of Rea- soning: Theoretical and Historical Perspectives (K. Manktelow and M. C. Chung, eds.), Psychology Press, 2004, pp. 179–212.

[13] Jean-Pierre Kahane, Jacques Handamard, The Mathematical Intelligencer 13 (1991), no. 1, 23–29.

[14] Daniel Kahneman, Thinking, Fast and Slow, Farrar, Strauss and Giroux, 2011.

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[15] Karen D. Keys-Turner, The Violent Islamic Radicalization Process: A Framework for Understanding, Masters thesis, Naval Postgraduate School, Monterey, California, December 2011, Approved for public release; distribution is unlimited.

[16] John F. Kihlstrom, The cognitive unconscious, Science 237 (1987), 1441– 1452.

[17] Uriah Kriegel, Theories of Consciousness, Philosophy Compass 1 (2006), no. 1, 58–64.

[18] Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic, a First Introduction to Topos Theory, Springer-Verlag, 1992.

[19] F. William Lawvere, Adjointness in foundations, Dialectica 23 (1969), 281– 296.

[20] Giuseppe Longo and Ma¨elMont´evil, Protention and retention in biological systems, Theory in Biosciences 130 (2011), no. 2, 107–117.

[21] Saunders Mac Lane, Categories for the Working Mathematician, Springer- Verlag, New York, 1971.

[22] Izchak Miller, Husserl’s account of our temporal awareness, Husserl, Inten- tionality, and Cognitive Science (Hubert L. Dreyfus, ed.), The MIT Press, 1982, in collaboration with Harrison Hall, pp. 125–146. [23] Christopher Mole, Attention and consciousness, Journal of Consciousness Studies 15 (2008), no. 4, 86–104. [24] Alain Morin, Inner speech, Encyclopedia of Human Behavior, Elsevier, 2 ed., 2012, p. 436443. [25] Stephen C. Pepper, World Hypotheses, a Study in Evidence, University of California Press, 1966.

[26] Ian Phillips, Perceiving temporal properties, European Journal of Philoso- phy 18 (2008), 176–202.

[27] Susan Pockett, How long is “now”? phenomenology and the specious present, Phenomenology and the Cognitive Sciences 2 (2003), 55–68.

[28] Georges Poulet, Phenomenology of Reading, New Literary History, vol. 1, The Johns Hopkins Press, 1969, New and Old History, pp. 53–68.

[29] Jr. Raymond W. Gibbs and Teenie Matlock, Metaphor, imagination, and simulation, The Cambridge Handbook of Metaphor and Thought (Jr. Ray- mond W. Gibbs, ed.), Cambridge University Press, 2008, pp. 161–176. [30] Philippe Rochat, Five levels of self-awareness as they unfold early in life, The Southern Journal of Philosophy 12 (2011), 271–293.

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[31] A. N. Sokolov, Inner Speech and Thought, A Plenum/Rosetta Edition, 1975, Translation by George T. Onischenko, Translation edited by Donald B. Lindsley.

[32] Teun A. van Dijk, Semantic Discourse Analysis, Handbook of Discourse Analysis (Deborah Schiﬀrin, Deborah Tannen, and Heidi Hamilton, eds.), Academic Press, 1985.

[33] Francisco J. Varela, The Specious Present: A Neuorophenomenology of Time Consciousness, Naturalizing Phenomenology: Issues in Contempo- rary Phenomenology and Cognitive Science (Jean Petitot, Francisco J. Varela, Berhanrd Pachoud, and Jean-Michel Roy, eds.), Stanford Univer- sity Press, 2000, pp. 266–329. [34] L. S. Vygotsky, Thought and Language, The M.I.T. Press, 1962, Edited and translated by Eugenia Hanfmann and Gertrude Vakar. [35] Sebastian Watzl, Attention as structuring of the stream of consciousness, Attention: Philosophical and Psychological Essays (C. Mole, D. Smithies, and W. Wu, eds.), Oxford University Press, 2010.

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The Design of Cognocity.org

Ellis D. Cooper, Ph.D. June 15, 2013

1 Mission Statement

Everybody knows something worth knowing, and everybody needs to know something. Cognocity is a Knowledge Auction Market for matching Seekers with Knowers. Mobile technology for communicating by text, audio, and video empowers people with knowledge, and others with need to know, to meet and convey understanding, regardless of physical location. Cognocity is not for bricks-and-mortar or online classroom use. In fact, it is a separate web-wide virtual classroom.

2 Cognocity Forms

1. Registration Form 2. Seeker Login Form 3. Knower Login Form 4. Meeting Conclusion Form 5. Knowledge Request Form

3 New Thoughts

1. The payment rules are designed for fair compensation, even though Meet- ings are held oﬀ-site. Seekers do not recover their deposits, and Knowers are not paid their due, unless both Report back to Cognocity and agree on the Start Time and End Time of their Meeting, to the nearest 5 minute time point.

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2. The goal of a Knower for a Meeting is to convey Knowledge efficiently, to collect the Start Fee accordingly (plus the Time Fee), and to get a high Rating. 3. The goal of a Seeker for a Meeting is to acquire Knowledge efficiently at the lowest cost, and to achieve a high Rating. 4. A possible game-like aspect of Cognocity could be that each month Cog- nocity rewards highest Rating Cognocenters with a Cogno-cent Prize. 5. Cogno-centi who sign up for alerts will receive email announcements of bids or offers on topics of personal interest. 6. Replace piggy picture at Cognocity.org home page EDC: Perhaps a photograph of EDC home office, emphasis on books. EDC: ooVoo appears to offer desktop and mobile options. Maybe it is wise to finish this design, and then run it by ooVoo to con- firm feasibility. Perhaps users are responsible for avoiding abuse or any inappropriate behavior, plus they may report the like to cognocity. 7. How to popup example problems from topic selection page EDC: Testgen can export HTML with an images sub-directory. Maybe it is wise to finish this design, and then run it by Pearson to check re licensing rights. 8. bootstrap.js for Cognocity.org site adaptibility to mobile devices 9. FWL: learn HTML5, CSS3, JavaScript 10. research bid/ask trading screens EDC: It is not just a “knowledge market,” it is a Knowledge Auction Market. 11. registration/login/forgotpassword “openid.net” EDC: Verio Control Panel has http://cognocity.org/ControlPanel/#website/password_rules.xsl 12. Cognocity.org CHAT (a) continuous audio, (b) text chat, (c) two-way video, (d) archiving (cf. Google Hangout) 13. drill down from “Utilities and Files” in Control Panel to source files for web-site 14. how to “shadow” the VERIO web-site locally, especially with regard to the database 15. seeker-table gets education-level 16. knower-table gets credentials 17. every user gets 10,000 free cogno-cents (can purchase blocks of 100 cogno- cents for real dollars, later). Thus, an offer to provide knowledge for 2 cogno-cents per second translates into 2×3600 = 7200 cogno-cents per hour, or 72 dollars per hour. This would be an offer from a very highly qualified knower. It is in the interest of the knower to turn around knowledge at a high rate, since the start-fee is a fixed income per meeting. It is in the interest of the seeker to acquire knowledge at a high rate so as to reduce the total after-start-fee up-front cost. Users of Cognocity.org are called cogno-centi (singular, cogno-cente). 18. 1-5 seeker-rating and 1-5 knower-rating per meeting in meeting-table

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19. Cognicity archives all meetings; meetings are owned by knowers; knowers may market (portfolios of) meetings with the permission of corresponding seeker to whom royalties to seekers, and commission to Cognicity. 20. “Problem-Solving Market” 21. NOO! Knowledge Auction Market. 22. If meetings are external to Cognocity.org (e.g., via Skype, Google Hang- out, ooVoo, etc.), then there must be an incentive for seekers and knowers to come back to Cognocity.org to report on the type of conclusion of the meeting. So, (a) knowers do not get paid if they don not come back and report; (b) seeker has to put up a refundable deposit, which is refunded only if they come back and report. 23. Cognocity.org provides short demo videos of alternative technologies for meetings. In particular, it must be shown (a) how rearview camera of smartphone can be aimed at a pad, and (b) how a small upright whiteboard could be in front of a webcam attached to a PC or built-in to a laptop.

4 Glossary of Cognocity Terms

Seeker An individual who has registered at Cognocity.org for the purpose of acquiring knowledge from Knowers. Knower An individual who has registered at Cognocity.org for the purpose of providing Knowledge to Seekers. Topic The name of a speciﬁc Skill or limited unit of Knowledge that can be taught by a Knower to a Seeker during a single Meeting. Meeting An audio, video, text chat continuous connection between a Knower and a Seeker that lasts for no more than 15 minutes for the purpose of conveying Knowledge of a Topic.

Cogno-cent The unit of payment by a Seeker to a Knower for Knowledge on a Topic.

Knowledge A basic Skill or small Mental-Model and its Microlect. Skill

Mental-Model Microlect

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5 Cognocity Database Design 5.1 Rules 1. Not for use in any classroom, ever. 2. At any time, there exists at most one login for a given user email address. 3. If the same email address is registered more than once, then that user is registered exactly twice, once as seeker and once as knower, and with distinct user-id’s. 4. Only a seeker can select a knower oﬀer. 5. Only a knower can select a seeker bid. 6. cnclsn-type is one of the following: 1. both SEEKER and KNOWER satisﬁed, hence full payment made to KNOWER. 2. SEEKER ended before TRIAL PERIOD, hence START FEE refunded to SEEKER. 3. KNOWER ended before TRIAL PERIOD, hence START FEE refunded to SEEKER. 4. SEEKER ended after TRIAL PERIOD, hence full payment to KNOWER. 5. KNOWER ended after TRIAL PERIOD, hence KNOWER retains only START FEE.

5.2 user-table

user-table user-id role (S/K) active (Y/N) email-addr password reg-date reg-time balance

5.3 site-visit-table site-visit-table user-id visit-id login-date login-time logout-date logout-time

5.4 page-visit-table page-visit-table visit-id page-id start-time end-time

5.5 dead-table dead-table user-id deactivate-date deactivate-time lifetime-count

5.6 topic-table topic-table topic-id topic-name topic-description

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5.7 seeker-table

seeker-table user-id topic-name bid-rate start-date start-time preferred-duration

5.8 knower-table knower-table user-id topic-name start-fee ask-rate trial-period

5.9 meeting-table

meeting-table seeker-id knower-id topic-name start-date start-time end-date end-time cnclsn-type (1-5)

6 Algorithms 6.1 SELECTION

Figure 1: After registering, a SEEKER may login to select offerings of one or more KNOWERS. If there are no scheduling conflicts among the selections, then COGNOCITY alerts the selected KNOWERS that their offers have been selected, and COGNOCITY confirms to the SEEKER that those alerts have been sent. SEEKERS and KNOWERS use COGNOCITY CHAT to negotiate schedules for MEETINGS.

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6.2 MEETING 6.3 BID FORM Modal dialog for a registered seeker to add the record of knowledge sought. 1. TOPIC 2. BID RATE 3. START TIME DATE 4. PREFERRED DURATION

6.4 ASK FORM Modal dialog for a registered knower to add the record of knowledge oﬀered. 1. TOPIC 2. PAY RATE 3. START FEE 4. TRIAL PERIOD

7 PAGE DESIGN 7.1 HOME PAGE 1. COGNOCITY logo at upper left. 2. COGNOCITY mission statement at upper right. 3. Registration/Login/ForgotPassword buttons. 4. For registered users only, a text-only CHAT box:

1. For a seeker to discuss with any knower what topic to meet about. 2. For a seeker and a knower to arrange a meeting. 3. simple display of TOPICS with rollover popup of existing SEEKERS and KNOWERS per topic.

7.2 TOPIC PAGE 1. TOPIC-NAME at upper left 2. SEEKER columns:

1. SEEKER ID 2. BID RATE 3. START TIME DATE 4. PREFERRED DURATION 5. KNOWER columns: 1. TOPIC 2. PAY RATE 3. START FEE

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4. TRIAL PERIOD

Each side may be sorted on any column.

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Figure 2: If a MEETING is to begin, COGNOCITY turns MEETING LIGHT to green on the screens of the two participants. Continuous audio (rectangular box) is enabled. The participants alternately draw diagrams on tablets, or type text, or take snapshots of pads, and then press SEND. At MEETING conclusion, participants press DONE to inform COGNICITY of meeting completion. GOGNICITY turns MEETING LIGHTs to RED.

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130525 Android SDK downloaded, installed without incident. I am following the online tutorial. 1. The Android Virtual Device Manager is a choice under the Window menu. 2. My emulator fails to launch due to some memory allocation limitation. Try a different device besides “Galaxy Nexus”? 3. OK, VirtualCognocityB launches – take care to set a low value (e.g., 500 MB) for virtual RAM. 4. In fact, more than one emulator may be launched. 5. Following the nose, open src∥org.CognocityA∥MainActivity.java, and PRESS Run, SEE “Hello world!” in emulator. 6. I do not know where the “Hello world!” text is sourced... 7. I went to Package Explorer∥res∥values∥strings.xml and created a new string name and value, then edited activity main.xml to use the new string, and it works in the emulator. 8. Actually, I had edited, not created, my new string. I cannot make the Add button behave... 9. It is easier to just edit the strings.xml file. 10. I have created a horizontal layout with a text entry field and a “Send” button. 11. I do not get a swift reaction to sending code to the emulator. This is an impediment. It has gotten there a couple of times, but not exactly reliably. One of my emulators just showed me a whole bunch of apps, like camera, clock, etc. But CTL-F11 does not seem to load my tutorial code. 12. I have the terrifying impulse to go buy a Samsung Galaxy Note unit, a major expense at this point. 13. Also, I have not setup command line operation, so I cannot try that for the moment. 14. Before leaping off a cliff in that direction, a google search on “android virtual device slow” turns up a rowboat load of advice that has to be checked out... 15. Upping the heap from 64 to 512 makes no difference. 16. One story is to use “Android X86” instead of the ggogle emulator. 17. There is a HAXM story, and some other alternative emulator stories. Some people say, just use a real phone. 18. Exit Eclipse and fire everything up cold. 19. The emulator device just says “charging 50%” and doesn’t do anything. 20. The story about using Android X86 virtual machine is a rather elaborate exercise, involving a new partition, etc. 21. Purchase of a hard device could be rationalized by pointing out that only a hard device has a stylus. 22. A GalaxyNexus emulator shows several buttons that “do” things, but my app never seems to get to it.

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23. It did get there earlier today, the first time pretty OK, the second it did, but not so great, now not at all. 24. At Windows∥Android SDK Manager∥Packages I installed “Intel x86 Em- ulator Accelerator (HAXM). 25. Now I get a Console message, “Adb connection Error:An existing connection was forcibly closed by the remote host.” This is not going well. 26. This kind of experience with the App Inventor emulator does not bode well for Android emulators in my life. 27. On stackoverflow I found the suggestion to go to Run∥Run Configurations and set Launch to org.CognocityA.MainActitivity. This yields nice looking Console messages:

------Android Launch! adb is running normally. Performing org.CognocityA.MainActivity activity launch Automatic Target Mode: using existing emulator ’emulator-5554’ running compatible AVD’VirtualCognocity’ Uploading CognocityA.apk onto device ’emulator-5554’ Installing CognocityA.apk... Success! Starting activity org.CognocityA.MainActivity on device emulator-5554 ActivityManager: Starting: Intent { act=android.intent.action.MAIN cat=[android.intent.category.LAUNCHER] cmp=org.CognocityA/.MainActivity }

28. but nevertheless, nothing shows up in the emulator. It just shows date, time, “Charging 50%”, and a lock in a circle. 29. In SDK Manager I installed the documentation for the Android SDK. 30. OK, that’s it for day 1.

130528 The task is to replace the little piggy image at upper-right of index.html at Cognocity.org . 1. I photographed my office and uploaded TheHomeOffice.jpg to VERIO using (the equivalent of the Easy Site) multimedia image button. However, I do not yet know where that resides on the server. 2. On the other hand, I did call technical support and got informed that their templates cannot be edited except for very simple changes using Easy Site, and that to replace the little piggy I would need to get into the HTML. No problem. 3. I have indeed located the little piggy as kv 7567.jpg at Utilities and Files∥All Files∥www∥htdocs ∥templates∥Delphi∥img∥2B559. This little piggy actually has parts of the nice framing in the template, so the game is to get into, say, Photoshop, and just replace the little piggy by the home office.

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4. It is not clear how to download the little piggy, and as I said, it is not clear where an upload goes. On the other hand, copies and moves in VERIO server are easy. 5. In the VERIO HTML editor it is trivial to locate where the little piggy is referenced – just look for ‘.jpg” since it is the only image on the home page. 6. I used Easy Site to stick TheHomeOffice.jpg as a new paragraph, and then examine HTML to see where it is coming from. Kind of interesting, it is from src=”./mediac/450 0/media/< longhexstring >.jpg” .... . 7. Ah, no problema. This mediac is a sub-directory of htdocs. 8. But, heh heh, every since I attempted to align my home office as ”center” instead of ”left”, I am unable to get index.htm into the editor, instead I see “An error has occurred. Could not parse the VSAP Response: Empty String at /usr/local/cp/lib/ControlPanel/MetaProc.pm line 186.” 9. Technical support is at: 800-438-8374 x1 x1. “Patrick”. CASE NUMBER: INC419825. He said he does not know how to get past that error, for now, and that I should just right click the home page to save the source and edit locally and then upload it. 10. OK, no problema, I aligned the home office to the right using WinEdt over here. 11. So now I want to get a hold of the little piggy... 12. Here’s how you do it: in VERIO under htdocs I had obtained a copy of the piggy, so click on that to bring it up in a browser, then right click and click View Page Info. You get a dialog with a Media button at the top, so click that, and select the piggy Image, then Save As. Done. Now I have the beast in Photoshop Elements. 13. Things almost worked to replace the piggy by my PiggyHome.jpg, but there is a scaling problem...0 x 0 ? 14. It was merely a typo. All is well, I have replace the piggy by my home office.

130531 Focus is now entirely on creating a “Topics Page.”

The 56 topics from the last page of the BHCC MAT097 syllabus is a terrific set of 56 skills to master for “Foundations of Algebra” at the level of developmental mathematics. So, that page was converted to an HTML document (Save As in WORD) and uploaded to Cognocity.org , no problema. What I want is that (1) when a registered site visitor does mouseover on one of those topics, a box at a fixed size and location displays an example of a problem that requires mastery of that skill. The effect should be that if the visitor slides the mouse down the list, she sees like a movie of examples in a fixed frame at the right. (Mind you, Florian, this is NOT the same effect as a tooltip.) Also, (2) each topic is a clickable link to a “Knowledge Auction Page” for that topic, with the

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bids of seekers and oﬀers of knowers on that topic.

The game is to see just how close I can get to this on my own, without spending money, or depending on anybody else. OK, I might spend $20.

130602 OK, FrontPage 2003 is installed, functioning, and perfectly adequate for my simple needs. The desired functionality of a topics page has been implemented, uploaded to Cognocity.org , and behaves properly.

I am thinking that the entire design for Cognocity.org could be published as a “Frequently Asked Questions” page, so that I can just send a link to various folks. For example, perhaps even importantly, to Kiruba and people at BHCC, maybe even to people at ENDC. Before I die I want to make a breakthrough positive impact on mathematical science education.

130603 1. skill.psd for PhotoShop is a template for 2.5” x 6” skill box, with upper-left number in font Regular Arial 18. Text in box is Regular Arial 12. 2. Text to list skills is Bold Arial 14. 3. At Cognocity.org Control Panel there was a small problem preventing editing directly from on-line ﬁle – an error message came up (using the Properties selection). But work continued. 4. However, yesterday a vanilla install of Drupal (which promises upload functionality) lead to complete site crash and inoperability. 5. Therefore, a completely new fresh start is demanded at Verio, and I would use FileZilla for uploading: HOST = Cognocity.org , LOGIN = cogno3/S55k-n20w, PORT = 21. 6. Technical Support at Verio: 800-438-8374. #422878. They are investigating (I told the edit failure and Drupal story). 7. Once past these issues, I should complete the topics pages for all 56 MAT097 skills before exposing this work to the authorities at BHCC (and ENDC?). 8. “One of perhaps the most compelling things about Silicon Valley is that it is a place where you can fail, and if you do, you can raise money and try again, said Mark Leslie, a retired entrepreneur and lecturer at the Stanford Graduate School of Business. Its a miraculous place; the streets are lined with gold here.”

I have implemented skills 1-45 and have 11 to go to ﬁnish the MAT097 Syllabus course. It would be most useful to have the web-site functioning for tomorrow’s ﬁrst class. So, a call to Verio is in order...

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I am told it is a “known server issue,” and not a problem caused by a mistake of mine. I cannot be switched to another server. I have to sit tight and wait for an email. I did say that I have to be up and running by tonight. I am now (4:43PM) in a chat with Maria Boton:

Customer Chat Chat Transcript Please wait while we find an agent to assist you... All agents are currently busy. Please stand by. An agent will be with you in a moment. Thank you for your patience. You have been connected to Maria Boton. Maria Boton: Thank you for contacting Web Hosting Customer Care. Maria Boton: Hi Ellis, how are you doing today? ELLIS COOPER: Your Customer Care Notice says "Kindly check your htaccess file, because when we tried to rename it, your website showed up. It could be the reason why its says internal server error." I do not know what to do with this information. Maria Boton: I understand you contacted us about the notice you have received regarding .htaccess. Am I correct? ELLIS COOPER: Yes. Maria Boton: I apologize for any inconvenience you might have experienced with this issue. Maria Boton: For verification purposes, am I chatting with Ellis Cooper? ELLIS COOPER: Yes, this is Ellis Cooper. Maria Boton: Thank you for that confirmation. Maria Boton: Please give me a moment while I pull up the account. ELLIS COOPER: Thank you. Maria Boton: Thank you for patiently waiting, Ellis. ELLIS COOPER: No problema. Maria Boton: For security purposes, kindly provide me the answer to the account‘s security question. Maria Boton: dogs name? ELLIS COOPER: madison Maria Boton: That is correct. Maria Boton: I am now checking the previous ticket you have with us and it is INC000000422878. ELLIS COOPER: That is correct. Maria Boton: It appears that you are having issue with your website. ELLIS COOPER: The site has been delivering an internal server error since last night. Maria Boton: Yes, Ellis. ELLIS COOPER: I was told by telephone that there is a "known server issue". Maria Boton: Kindly hold on while I am checking that said issue for you. ELLIS COOPER: Fine. Maria Boton: Thank you for your patience, Ellis. Maria Boton: I have checked the server and it is currently working fine. Maria Boton: Please be advised that we have tried to rename the .htaccess and successfully accessed your website. ELLIS COOPER: Internal Server Error The server encountered an internal error or misconfiguration and was unable to complete your request. Please contact the server administrator, [email protected] and inform them of the time the error occurred, and anything you might have done that may have caused the error. More information about this error may be available in the server error log. Apache/1.3.42 Server at cognocity.org Port 80 Maria Boton: For more information about .htaccess, please see the link below:

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The agent is sending you to http://supportcenter.verio.com/KB/questions.php?questionid=308. ELLIS COOPER: Is that it for help from you? Does that mean it is not a "known server issue," and that it is something I am supposed to figure out on my own in .htaccess? Maria Boton: My apology, Ellis. However, I have successfully accessed the server and unable to find any error. ELLIS COOPER: As shown above, when I enter my domain name cognocity.org in a browser I get "internal server error." It does not help me if people cannot get to my web-site and that you "have successfully accessed the server." Maria Boton: Yes, Ellis. I have duplicated that issue. I am now checking if the error is due to coding or permission. ELLIS COOPER: Thank you. Maria Boton: I appreciate your patience, Ellis. Kindly hold on. ELLIS COOPER: I’m sitting tight. Maria Boton: Thank you for patiently waiting, Ellis. Maria Boton: Please be informed that I have run mv .htaccess .htaccess.bak to see if your website will work. Maria Boton: We would like to inform you that this is a coding issue. We regret to inform you that this is out of our scope. Maria Boton: Please be advised to check your website codes or contact your webmaster. ELLIS COOPER: I wrote the index.htm code, and I will check it over carefully. Maria Boton: You may also contact our Professional Website Design to assist you further. Please see the link below: The agent is sending you to http://www.verio.com/professional-website-services/website-design-packages/. ELLIS COOPER: I must say that it has been confusing to be told by telephone that issue is a "known server issue," and then be told that the problem is my code -- which, by the way, works just fine when run in a browser independently of verio. Maria Boton: I apologize for any inconvenience you might have experienced with this issue. ELLIS COOPER: Thank you for your support. Maria Boton: You’re welcome, Ellis. I just want to remind that I have rename the .htaccess to .htaccess.bak. ELLIS COOPER: Where exactly in my Control Panel file system is the .htaccess file located? ELLIS COOPER: What is the exact path? Maria Boton: You can find the .htaccess under Utilities and Files. Please see the link below: The agent is sending you to http://198.63.33.133/ControlPanel/index.xsl#files/fileman.xsl?view_path=%2Fwww%2Fhtdocs&. Maria Boton: This can be found in www/htdocs. ELLIS COOPER: Nope, it is not there. I have extras, flash, images, img, include, includes, jpg, mediac, misc, modules, mysqladmin, profiles, scripts, sites, templates, themes, and my index.htm, but no ".htaccess". Maria Boton: Please click All Files under Utilities and Files. Maria Boton: Click www folder under Directory Contents to see the htdocs folder. ELLIS COOPER: Nope, it is not there. I have extras, flash, images, img, include, includes, jpg, mediac, misc, modules, mysqladmin, profiles, scripts, sites, templates, themes, and my index.htm, but no ".htaccess". Maria Boton: I am now trying to check that in my end. Please be advised that there are files that is automatically invisible. ELLIS COOPER: OK, I will use my invisible keyboard to edit an invisible file, and then people can visit my invisible website. Maria Boton: If you are going to shell in or access the server, you will see that in www/htdocs. You may also use the FileZilla to see the path. Maria Boton: Please be informed that I have successfully see the .htaccess via FileZilla. ELLIS COOPER: OK, I do see .htaccess using FileZilla. Maria Boton: Sounds good. ELLIS COOPER: Actually, I see .htacces.bak. Maria Boton: Yes, Ellis. That is correct because I have rename .htaccess to .htaccess.bak to view your website. ELLIS COOPER: Please advise. Maria Boton: Please clear cache and cookies before accessing your website. Maria Boton: You may also use another browser to access the site. ELLIS COOPER: OK, we are done. Thank you.

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Maria Boton: You’re welcome, Ellis. Maria Boton: Is there anything else I can help you with? ELLIS COOPER: No. Bye. Maria Boton: For your reference, the ticket number for this transaction is INC000000422878. OK, so I cleared cache and cookies, and at least the index.htm page came up. The .jpgs don’t work, but that can be ﬁxed. The nice thing is that FileZilla uploaded a boatload of .jpgs without a problem. So, the development can continue.

130604 cognocity.org/FAQ130602g.htm has all 56 “Foundations of Algebra” skills mouseovered.

130607 This Cognocity.org project has to be considered an experiment, with all the attendant doubts and hopes.

The highest priority is to create the registration/login functionality on the web-site. Evidently, this requires server-side scripting, e.g., with PHP (and MySQL), in which I have no experience. However, I have found at least one complete tutorial with code, and also I have a textbook with plenty of detail.

FWL says he might push for something called “oauth 2” and that he would consult with some CS expert professor who he knows.

I have entered a chat with Verio:

Please wait while we find an agent to assist you... You have been connected to Christel. Christel: Thank you for contacting Verio Web Hosting Customer Care. Christel: Hi Ellis, how are you doing today? ELLIS COOPER: Hi, Christel. I am interested in deploying server-side PHP scripts for my web-site. I will be experimenting with code, and need to be sure I know the exact folder to place .PHP files. Please advise.

Christel: I understand that you are contacting us because you want to know the path/folder you would place your PHP files. ELLIS COOPER: Yes.

Christel: I’m sorry for that inconvenience. Let me help you with your concern. Christel: But before we proceed, we must first pass with our authentication process. ELLIS COOPER: OK. Christel: Am I chatting with Ellis Cooper?

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ELLIS COOPER: Yes. Christel: Thank you, Ellis. Christel: I can see that your account is not currently listed with a Security Question and Answer; therefore, I will need to assist you in updating this information. Christel: May I have the username and first 4 characters of your account Backroom password? ELLIS COOPER: Actually, the security question is dog’s name, and the answer is madison. Christel: Thank you for letting me know, Ellis. Please hold on. ELLIS COOPER: username = cogno3, xxxxn20w ELLIS COOPER: Oops, password = S55kyyyyy Christel: Thank you for creating and answering the security question, Ellis. Christel: Now, let’s proceed again to your concern. ELLIS COOPER: Great. Christel: May I know if you are using an FTP client is uploading those PHP files? ELLIS COOPER: Yes, FileZilla. Also, I have experience with the Website Management Upload button. Christel: I see, first in Filezilla the proper path to place the PHP files are on /www/htdocs. Christel: With regard to Website Management Upload it is on your Control Panel, right? ELLIS COOPER: Yes, and I have already placed .HTM files in ../htdocs, no problem. So, it seems nothing could be simpler, the .PHP files go into the same folder as the .HTM files. Nice. Christel: That’s great! I also upload a test file. Actually it’s your PHP information : http://cognocity.org/phpinfo.php ELLIS COOPER: I see that phpinfo.php file. What would be the simplest way to test it right now? Should I write an .HTM file with a call to phpinfo.php? I can do that... Christel: That’s so smart of you, Eliis. Sure you can do that also or just a siple HTML or PHP file. ELLIS COOPER: OHH, you already told me when you wrote http://cognocity.org/phpinfo.php. That works already by itself. Christel: Yes it is. By the way with regard to Website Management upload button, unfortunately you can only used that for viewing of files or downloading. We recommend you to use an FTP client just like what are you are using right now. Christel: Is there anything else I can help you with? ELLIS COOPER: Well, thank you, Christel. That is already most helpful. But, please, let me just ask, this phpinfo.php file has a LOT of information. Is there any particular part that would be important for me to focus on, if my intention is to work with MySQL and forms and stuff like that? Christel: Yes those are all the information for your PHP/MySQL configuration. You say which are the Important part, well mostly just the version you are currently using that suites on the code you are prefered to use. Also the extension,support enabled. ELLIS COOPER: OK, very good. Thank you so much for your support. Christel: It was my pleasure, Eliis. Christel: For your reference, kindly take note of this case number, INC000000426450. Christel: If I have answered your questions/resolved your concerns, I will close this ticket. Wed love to hear your feedback on the support you received from us today. Well send you a survey through email. Your response will surely help us improve our service. Christel: We are happy to know that we were able to help you about your concern. I will end this session now. Have a great day! ELLIS COOPER: Done, and bye.

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130608 Starting my eduation in phpMyAdmin: 1. CREATE DATABASE ‘cognocity-db‘; 2. DROP DATABASE ‘user-id‘; 3. CREATE TABLE ‘cognocity-db‘.‘cognocenter-table‘ ( ‘user-id‘ VARCHAR( 16 ) NOT NULL , ‘role‘ TEXT NOT NULL , ‘active‘ TEXT NOT NULL , ‘email-address‘ VARCHAR( 64 ) NOT NULL , ‘password‘ VARCHAR( 16 ) NOT NULL , ‘registration-datetime‘ DATETIME NOT NULL , ‘education-level‘ TEXT NOT NULL , ‘cognocent-balance‘ INT NOT NULL , ) ENGINE = MYISAM 4. CREATE TABLE ‘cognocity-db‘.‘site-visit-table‘ ( ‘user-id‘ VARCHAR( 16 ) NOT NULL , ‘visit-id‘ INT NOT NULL , ‘login-datetime‘ DATETIME NOT NULL , ‘logout-datetime‘ DATETIME NOT NULL , ) ENGINE = MYISAM OK, I get the idea. Maybe it will be wise to purchase the book on ph- pMyAdmin. Also, it seems like there are some interesting databases already built-in to the web-site, such as information schema, cogno3 drp, etc., with lots of tables and stuﬀ. It looks like there is a preference for underscore as word separator, not ‘-” which I have been using. Perhaps that is to avoid confusion with subtraction.

130610 I use collation = utf8 general ci when asked.

I wanted to eliminate all use of “-” in naming, and had to rename “cognocenter- table”. But the MySQL interpreter would not accept occurrence of “-”, so the trick was to enclose “cognocenter-table” in backticks “‘”.

I have created 5 table in cognocity database, namely bid table, cognocenter table, meeting table, oﬀer table, and topic table. Currently, I am adding test records into the cognocenter table, e.g., with INSERT INTO cognocenter_table VALUES ( ’cbcdear’,’K’,1,’imright’,’hotdaddy.com’,’whynot’,130610081010,8,540 ); Let me think through what forms are required for Cognocity.org .

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1. Registration and Login Form

2. Seeker Bid Form

3. Knower Oﬀer Form

4. Topic Request Form

5. Meeting Conclusion Form

6. Seeker Meeting Start Form

7. Knower Meeting Start Form

Chapter 27 of Welling-Thomson shows in detail, with code on the CD (much anticipated delivery on Wednesday), shows how to do user registration and login. In the meantime, here is an outline of their ﬁles, and what Cognocity might need to do:

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bookmarks.sql SQL statements to create the PHPbookmark database. I do not need this.

login.php Front page with login form for the system. Cognocity front page. Possibly high-end video of “actual” Meet- ing. register form.php Form for users to register in the system. Adapt to Cognocity.

register new.php Script to process new registrations. Adapt to Cognocity. forgot form.php Form for users to ﬁll out if they’ve forgotten their passwords. Adapt to Cognocity. forgot passwd.php Script to reset forgotten passwords. Adapt to Cognocity. member.php A user’s main page, with a view of all his current bookmarks. All bids of a Seeker, or all oﬀers of a Knower; also show current cogno-cents balance; also show all pending Meetings and their Topics.

add bm form.php Form for adding bookmarks. Seeker Bid Form, or Knower Oﬀer Form.

add bms.php Script to actually add new bookmarks to the database. Adapt to Cognocity for adding Cognocenters, adding Topics, and adding Meetings.

delete bms.php Script to delete selected bookmarks from a user’s list. Cancellation of Meetings, deletion of Cognocenters. recommend.php Script to suggest recommendations to a user, based on users with similar interests. Not currently need.

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change passwd form.php Form for members to ﬁll out if they want to change their passwords. Adapt to Cognocity. change passwd form.php Script to change a user’s password in the database. Adapt to Cognocity.

logout.php Script to log a user out of the application. Adapt to Cognocity. bookmark fns.php A collection of includes for the application. Adapt to Cognocity.

data valid fns.php Functions to validate user-input data. Adapt to Cognocity. db fns.php Functions to connect to the database. Adapt to Cognocity.

user auth fns.php Functions for user authentication. Adapt to Cognocity. url fns.php Functions for adding and deleting bookmarks and for making recommendations. Not currently needed. output fns.php Functions that format output as HTML. Adapt to Cognocity. bookmark.gif Logo for PHPbookmark. Cognocity logo to be done.

130612 While waiting for today’s delivery of Welling-Thomson , I shall install Apache, PHP, and MySQL for local development and testing independently of VERIO. I have cleaned up EMACH4, and will rely on http://www.ultraedit.com/support/tutorials_power_tips/uestudio/local_php_mysql_dev_environment.html for instructions.

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1. APACHE: http://httpd.apache.org/download.cgi leads to http://apache.mesi.com.ar//httpd/binaries/win32/README.html which says “Install the Windows XP Service Pack 3.” All done, I have Win- dows XP Professional SP3 installed. However, there are warnings about possible problems, as in

Most Firewall programs, Web Spam ﬁlters and other TCP/IP driver-based products (including spyware!) do not correctly implement the entire WinSock API. The shortcuts taken by the developers of such products cause Apache to fail. If you insist on leaving such programs installed, and have problems with your Apache installation, consider the suggestion below. 2. Anyhow, proceed to http://www.carfab.com/apachesoftware//httpd/binaries/win32/. I will go for

httpd-2.2.22-win32-x86-openssl-0.9.8t.msi 30-Jan-2012 14:06 6.1M MSI Installer Package

basically because it is the latest dated .msi. Good luck. I created restore point called “Pending APACHE installation.” 3. OK, I see the Apache HTTP Server 2.2 - Installation Wizard, and will try to follow instructions. 4. I am supposed to see “It Works!” if I enter “localhost” in my browser. Instead, I see

This site is running TeamViewer.

Free Port 80 for other applications in advanced settings.

I guess I should turn oﬀ TeamViewer, and try again. 5. OK, now I get “Unable to connect.” I should say, my toolbar apache icon upon mouseover says, “Running none of 1 apache services.” So, I told the little thingy to START. AHA! Now I do see “It works!” Move on to PHP. 6. Go to http://windows.php.net/download/#php-5.4 and download php- 5.4.16-nts-Win32-VC9-x86.zip. Extraction was performed into

C:\Documents and Settings\Administrator\My Documents\Downloads\php-5.4.16-nts-Win32-VC9-x86

7. OK, there is a sub-directory and plenty of ﬁles, but where is the installation .msi? I only have php-cgi.exe, php-win.exe, and php.exe. 8. I peek into the .zip for php-5.2.17-nts-Win32-VC6-x86.zip, and see, again in the install.txt ﬁle that it says

Normal Install

Run the MSI installer and follow the instructions provided by the installation wizard. You will be prompted to select the Web Server you wish to configure first, along with any configuration details needed.

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So, where is this vaunted “MSI installer”? 9. Hmm, PHP 5.3.26 does have a download called “Installer,” but no later version has that. Try it, it IS php-5.3.26-nts-Win32-VC9-x86.msi. Try it. 10. The instructions do not match exactly the wizard, but what the heck, try a phpinfo.php test file anyway to see what happens. 11. I copied the phpinfo.php file into the htdocs sub-directory of APACHE, but I do not get the info, I just see the contents of the phpinfo.php file. I do not think PHP is running. Instructions say to look at error log, all it has is

[Wed Jun 12 10:42:54 2013] [notice] Apache/2.2.22 (Win32) configured -- resuming normal operations [Wed Jun 12 10:42:54 2013] [notice] Server built: Jan 28 2012 11:16:39 [Wed Jun 12 10:42:54 2013] [notice] Parent: Created child process 1792 [Wed Jun 12 10:42:54 2013] [notice] Child 1792: Child process is running [Wed Jun 12 10:42:54 2013] [notice] Child 1792: Acquired the start mutex. [Wed Jun 12 10:42:54 2013] [notice] Child 1792: Starting 64 worker threads. [Wed Jun 12 10:42:54 2013] [notice] Child 1792: Starting thread to listen on port 80. [Wed Jun 12 11:18:36 2013] [notice] Parent: Received restart signal -- Restarting the server. [Wed Jun 12 11:18:36 2013] [notice] Child 1792: Exit event signaled. Child process is ending. [Wed Jun 12 11:18:36 2013] [notice] Apache/2.2.22 (Win32) configured -- resuming normal operations [Wed Jun 12 11:18:36 2013] [notice] Server built: Jan 28 2012 11:16:39 [Wed Jun 12 11:18:36 2013] [notice] Parent: Created child process 2764 [Wed Jun 12 11:18:37 2013] [notice] Child 2764: Child process is running [Wed Jun 12 11:18:37 2013] [notice] Child 1792: Released the start mutex [Wed Jun 12 11:18:37 2013] [notice] Child 2764: Acquired the start mutex. [Wed Jun 12 11:18:37 2013] [notice] Child 2764: Starting 64 worker threads. [Wed Jun 12 11:18:38 2013] [notice] Child 1792: All worker threads have exited. [Wed Jun 12 11:18:38 2013] [notice] Child 1792: Child process is exiting [Wed Jun 12 11:18:38 2013] [notice] Child 2764: Starting thread to listen on port 80.

12. I think the problem has to do with the missing APACHE detail in Web Server Setup of the PHP installation. I used the .msi of PHP 5.3.26 to REMOVE it, and I intend to re-install it. 13. In Web Server Setup my only choices are “IIS FastCGI,” “Other CGI” (which is what I used last time), and “Do not setup a web server.” This time I will use the latter, and then see how to set up web server for PHP later...Right, so now the question is, how to attach PHP and APACHE outside of the wizardry. 14. Googling quickly turned up a youtube “How to install PHP 5.3.0 for Win- dows Apache 2 Server” which might be edifying. 15. The ﬁrst thing he says is to use the VC6 Thread Safe download. I had used the VC9 Non Thread Safe installer...Well, what he does is perfectly understandable, so I will un-install what I did, and do what he did, which is at http://www.youtube.com/watch?v=5erNUqAKwJE. 16. Uninstall done.

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17. My best choice is php-5.2.17-Win32-VC6-x86.msi. (There is no installer for VC6 version 5.3.x.) 18. The really big move he makes is to create a php directory under the APACHE directory. This makes the Web Server Setup show the servers that I never saw. 19. The next big thing he does is to set the Apache Conﬁguration Directory to conf. 20. I enable just the MySQL extension. Install. 21. Restart APACHE. Oops, the requested operation has failed. Now what? 22. The answer is to edit the httpd.conf ﬁle at the very end to be

#BEGIN PHP INSTALLER EDITS - REMOVE ONLY ON UNINSTALL PHPIniDir "C:\Program Files\Apache Software Foundation\Apache2.2\php" LoadModule php5_module "C:\Program Files\Apache Software Foundation\Apache2.2\php\php5apache2_2.dll" #END PHP INSTALLER EDITS - REMOVE ONLY ON UNINSTALL 23. OK, now the phpinfo.php ﬁle works like a charm. On to MySQL. 24. Go to http://dev.mysql.com/downloads/installer/5.6.html. 25. Download mysql-5.6.12-win32.zip. This is not an .msi, so what gives? 26. What I want is a MySQL Installer... 27. OK, I found this: mysql-installer-web-community-5.6.12.0.msi. 28. I seem to have completed a fairly elaborate installation, including “SQL Workbench.” Anyhow, try to follow the instructions for command line action. 29. Click on Command Line Client brings it up, then it ﬂashes out of existence. BUT, who needs the command line client when I have the MySQL WorkBench, a fabulous IDE for MySQL? At this point, I would say I am all set for local development with APACHE, PHP, and MySQL. Done. Oracle MySQL = /20r1cl5My Window Service Name= MySQL56 Configuration Type = Development Machine Port Number = 3306 MySQL Root Password= 5LL10sD5xt5r MySQL User Password= ellis/ytilautriV

130614 Welling-Thomson arrived, I have backed up the CD, and the Chapter 27 files copied to htdocs for APACHE. The login and registration .PHP files operate fine. Some time should be devoted to creating a “bookmarks” database and getting a feel for the Chapter 27 system, before attempting to modify/adapt it for Cognocity.org .

There are interesting possibilities for quick installation of the APACHE- MySQL-PHP etc., say, on VAIO2, at “WampServer” and “xampp”. Not to

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mention, Welling-Thomson has a whole set of installations for the book. I am deﬁnitely not clear on the workﬂow using the AMP triumvirate.

1. Create the database using bookmarks.sql. HOW? Welling-Thomson use the command line

mysql -u root -p < boomarks.sql

but I would be using MySQL Workbench. So, I chose to open bookmarks.sql in the editor, and did use the MySQL Root Password= 5LL10sD5xt5r and saved it in the vault. 2. Maybe I will edit the ﬁle a tad, but mainly I want to see what happens when it is executed...I edited the last line to

to bm_user@localhost identified by ’t5stw20rd’;

NO! You can’t do that. Leave it at ’password’. Then the code runs, and supposedly a database has been created. But where? And what is it called? And what is in it? And how do you add to it? How do you print it out? 3. Well, OK, I will run login.php. It does seem to work up to the point of running member.php, but nothing shows up. My login is ellis/dexter, by the way. 4. I created a new member with [email protected]/henry/silly3p128, but p.582 of Welling-Thomson says I should see “Registration successful” from register new.php, and I see a blank for this .PHP ﬁle. No error message, either. 5. Just for fun, how about a simple echo version of it, say “register newTEST.php”.

works just ﬁne. OK, add a little more...

also works just ﬁne. WinEdt seems to be able to check matching of parentheses, so I want to check register new.php carefully...Alright, no problema.

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6. Ah, the real problem that keeps things from showing up is the line session start(). What is that? Is it a PHP thingy? 7. Um, I have to read Chapter 23, “Using Session Control in PHP.” 8. By the way, it is always fun to examine php-errors.log in C:/Windows/temp. 9. I shall try out page1.php, page2.php, page3.php of Chapter 23. 10. OK, Welling-Thomson description of events pp.514-516 is correct, the value of ’sess var’ is indeed preserved and then evaporated. 11. Just for fun, I deleted all cookies from my browser. 12. Watching cookies page by page is fun, and I see that page1.php creates a session cookie, and it persists until the browser is closed. Then it is gone. 13. Login as henry gets me to member.php, which has a session start() line, and nothing shows up. My guess is the reason is that there is nothing in the database for henry. I am still in the dark as to where are the tables of databases in this project. Gee, maybe I should look in the MySQL directory structure. 14. Interesting. MySQL directory has no my.ini conﬁgured; the default is just my-default.ini :

# For advice on how to change settings please see # http://dev.mysql.com/doc/refman/5.6/en/server-configuration-defaults.html # *** DO NOT EDIT THIS FILE. It’s a template which will be copied to the # *** default location during install, and will be replaced if you # *** upgrade to a newer version of MySQL.

[mysqld]

# Remove leading # and set to the amount of RAM for the most important data # cache in MySQL. Start at 70% of total RAM for dedicated server, else 10%. # innodb_buffer_pool_size = 128M

# Remove leading # to turn on a very important data integrity option: logging # changes to the binary log between backups. # log_bin

# These are commonly set, remove the # and set as required. # basedir = ..... # datadir = ..... # port = ..... # server_id = .....

# Remove leading # to set options mainly useful for reporting servers. # The server defaults are faster for transactions and fast SELECTs. # Adjust sizes as needed, experiment to find the optimal values. # join_buffer_size = 128M # sort_buffer_size = 2M

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# read_rnd_buffer_size = 2M

sql_mode=NO_ENGINE_SUBSTITUTION,STRICT_TRANS_TABLES

According to http://www.mkyong.com/mysql/where-does-mysql-stored-the-data-in-my-harddisk/ I am interested in getting configured at datadir. It seems I had better run the MySQL Configuration Wizard carefully (Welling-Thomson p.900). But, when I revisit the MySQL Installer, it skips over the Configuration choice, and wants me to go right to MySQL Workbench. Maybe there is something to do there... 15. In MySQL Workbench at Manage Service Instances I do find Configuration File is at

C:\Documents and Settings\All Users\Application Data\MySQL\MySQL Server 5.6\my.ini

so that ought to be interesting... 16. And, the answer is:

datadir="C:/Documents and Settings/All Users/Application Data/MySQL/MySQL Server 5.6/data\"

Let’s go there and see who is home...Yup, we go the bookmark.frm, and other files for the database, including, aha, user.frm. I wonder what is in it? 17. There is no question that the SQL Editor in MySQL Workbench DOES see (an empty) bookmarks database. OK, so what keeps the PHP files in Chapter 27 from allowing addition of bookmarks to, say, the henry account? 18. Even after creating a completely new account with my email and lsd- cooper/t5rr10bl5, the register new.php file shows nothing. And, that has got to be because the user table is NOT getting new information. Its table is empty. (I exported the .CSV using MWB). 19. I am not confident that my PHP installation is really working with MySQL, or something along those lines. 20. Hmm, maybe the smartest move is to follow exactly the AMP installation from Welling-Thomson Appendix, and do it on VAIO2. Or, more fun, would be one of these “WampServer” and “xampp” things. Maybe, try one of those on VAIO1 just to see how it goes...

130614b My initial impression of WampServer is quite positive. 1. Download and installation to VAIO1 over a hard line is trivial. 2. The index.php file works fines. 3. There is a MySQL command line window, MySQL Console, that works fine.

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4. The register new.php file appears to work, up to a point, and then detailed warning messages with pointer to a specific line in user aut functions.php shows up. The problem well may be that I never created the databases – I have to run bookmarks.sql. 5. there is easy access to functionalities for AMP at the button down at lower right. 6. A little problem is that I do not recall defining a password... 7. No password is needed. START the console, leave password blank, CLICK Enter, SEE mysql prompt. 8. EXECUTE

source c:\wamp\www\bookmarks.sql

9. In browser, EXECUTE localhost/login.php, CREATE account < email >/ellis/dextercooper, SEE registration completed, and end up as a registered member and ready to add bookmarks. All good, all working, no problema. I added some bookmarks, and logged out. That was a totally satisfying experience, and it means I can move on to Cog- nocity.org development. But I shall need to uninstall the AMP currently on EMACH4, and install WampServer because it is soooo friendly.

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