The Project Gutenberg Ebook #43006: Space–Time–Matter
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Computer Oral History Collection, 1969-1973, 1977
Computer Oral History Collection, 1969-1973, 1977 INTERVIEWEES: John Todd & Olga Taussky Todd INTERVIEWER: Henry S. Tropp DATE OF INTERVIEW: July 12, 1973 Tropp: This is a discussion with Doctor and Mrs. Todd in their apartment at the University of Michigan on July 2nd, l973. This question that I asked you earlier, Mrs. Todd, about your early meetings with Von Neumann, I think are just worth recording for when you first met him and when you first saw him. Olga Tauskky Todd: Well, I first met him and saw him at that time. I actually met him at that location, he was lecturing in the apartment of Menger to a private little set. Tropp: This was Karl Menger's apartment in Vienna? Olga Tauskky Todd: In Vienna, and he was on his honeymoon. And he lectured--I've forgotten what it was about, I am ashamed to say. It would come back, you know. It would come back, but I cannot recall it at this moment. It had nothing to do with game theory. I don't know, something in.... John Todd: She has a very good memory. It will come back. Tropp: Right. Approximately when was this? Before l930? Olga Tauskky Todd: For additional information, contact the Archives Center at 202.633.3270 or [email protected] Computer Oral History Collection, 1969-1973, 1977 No. I think it may have been in 1932 or something like that. Tropp: In '32. Then you said you saw him again at Goettingen, after the-- Olga Tauskky Todd: I saw him at Goettingen. -
How Should the College Teach Analytic Geometry?*
1906.] THE TEACHING OF ANALYTIC GEOMETRY. 493 HOW SHOULD THE COLLEGE TEACH ANALYTIC GEOMETRY?* BY PROFESSOR HENRY S. WHITE. IN most American colleges, analytic geometry is an elective study. This fact, and the underlying causes of this fact, explain the lack of uniformity in content or method of teaching this subject in different institutions. Of many widely diver gent types of courses offered, a few are likely to survive, each because it is best fitted for some one purpose. It is here my purpose to advocate for the college of liberal arts a course that shall draw more largely from projective geometry than do most of our recent college text-books. This involves a restriction of the tendency, now quite prevalent, to devote much attention at the outset to miscellaneous graphs of purely statistical nature or of physical significance. As to the subject matter used in a first semester, one may formulate the question : Shall we teach in one semester a few facts about a wide variety of curves, or a wide variety of propositions about conies ? Of these alternatives the latter is to be preferred, for the educational value of a subject is found less in its extension than in its intension ; less in the multiplicity of its parts than in their unification through a few fundamental or climactic princi ples. Some teachers advocate the study of a wider variety of curves, either for the sake of correlation wTith physical sci ences, or in order to emphasize what they term the analytic method. As to the first, there is a very evident danger of dis sipating the student's energy ; while to the second it may be replied that there is no one general method which can be taught, but many particular and ingenious methods for special problems, whose resemblances may be understood only when the problems have been mastered. -
Greece Part 3
Greece Chapters 6 and 7: Archimedes and Apollonius SOME ANCIENT GREEK DISTINCTIONS Arithmetic Versus Logistic • Arithmetic referred to what we now call number theory –the study of properties of whole numbers, divisibility, primality, and such characteristics as perfect, amicable, abundant, and so forth. This use of the word lives on in the term higher arithmetic. • Logistic referred to what we now call arithmetic, that is, computation with whole numbers. Number Versus Magnitude • Numbers are discrete, cannot be broken down indefinitely because you eventually came to a “1.” In this sense, any two numbers are commensurable because they could both be measured with a 1, if nothing bigger worked. • Magnitudes are continuous, and can be broken down indefinitely. You can always bisect a line segment, for example. Thus two magnitudes didn’t necessarily have to be commensurable (although of course they could be.) Analysis Versus Synthesis • Synthesis refers to putting parts together to obtain a whole. • It is also used to describe the process of reasoning from the general to the particular, as in putting together axioms and theorems to prove a particular proposition. • Proofs of the kind Euclid wrote are referred to as synthetic. Analysis Versus Synthesis • Analysis refers to taking things apart to see how they work, or so you can understand them. • It is also used to describe reasoning from the particular to the general, as in studying a particular problem to come up with a solution. • This is one general meaning of analysis: a way of solving problems, of finding the answers. Analysis Versus Synthesis • A second meaning for analysis is specific to logic and theorem proving: beginning with what you wish to prove, and reasoning from that point in hopes you can arrive at the hypotheses, and then reversing the logical steps. -
Council for Innovative Research Peer Review Research Publishing System
ISSN 2347-3487 Einstein's gravitation is Einstein-Grossmann's equations Alfonso Leon Guillen Gomez Independent scientific researcher, Bogota, Colombia E-mail: [email protected] Abstract While the philosophers of science discuss the General Relativity, the mathematical physicists do not question it. Therefore, there is a conflict. From the theoretical point view “the question of precisely what Einstein discovered remains unanswered, for we have no consensus over the exact nature of the theory's foundations. Is this the theory that extends the relativity of motion from inertial motion to accelerated motion, as Einstein contended? Or is it just a theory that treats gravitation geometrically in the spacetime setting?”. “The voices of dissent proclaim that Einstein was mistaken over the fundamental ideas of his own theory and that their basic principles are simply incompatible with this theory. Many newer texts make no mention of the principles Einstein listed as fundamental to his theory; they appear as neither axiom nor theorem. At best, they are recalled as ideas of purely historical importance in the theory's formation. The very name General Relativity is now routinely condemned as a misnomer and its use often zealously avoided in favour of, say, Einstein's theory of gravitation What has complicated an easy resolution of the debate are the alterations of Einstein's own position on the foundations of his theory”, (Norton, 1993) [1]. Of other hand from the mathematical point view the “General Relativity had been formulated as a messy set of partial differential equations in a single coordinate system. People were so pleased when they found a solution that they didn't care that it probably had no physical significance” (Hawking and Penrose, 1996) [2]. -
Arxiv:1601.07125V1 [Math.HO]
CHALLENGES TO SOME PHILOSOPHICAL CLAIMS ABOUT MATHEMATICS ELIAHU LEVY Abstract. In this note some philosophical thoughts and observations about mathematics are ex- pressed, arranged as challenges to some common claims. For many of the “claims” and ideas in the “challenges” see the sources listed in the references. .1. Claim. The Antinomies in Set Theory, such as the Russell Paradox, just show that people did not have a right concept about sets. Having the right concept, we get rid of any contradictions. Challenge. It seems that this cannot be honestly said, when often in “axiomatic” set theory the same reasoning that leads to the Antinomies (say to the Russell Paradox) is used to prove theorems – one does not get to the contradiction, but halts before the “catastrophe” to get a theorem. As if the reasoning that led to the Antinomies was not “illegitimate”, a result of misunderstanding, but we really have a contradiction (antinomy) which we, somewhat artificially, “cut”, by way of the axioms, to save our consistency. One may say that the phenomena described in the famous G¨odel’s Incompleteness Theorem are a reflection of the Antinomies and the resulting inevitability of an axiomatics not entirely parallel to intuition. Indeed, G¨odel’s theorem forces us to be presented with a statement (say, the consistency of Arithmetics or of Set Theory) which we know we cannot prove, while intuition puts a “proof” on the tip of our tongue, so to speak (that’s how we “know” that the statement is true!), but which our axiomatics, forced to deviate from intuition to be consistent, cannot recognize. -
K-Theory and Algebraic Geometry
http://dx.doi.org/10.1090/pspum/058.2 Recent Titles in This Series 58 Bill Jacob and Alex Rosenberg, editors, ^-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara) 57 Michael C. Cranston and Mark A. Pinsky, editors, Stochastic analysis (Cornell University, Ithaca) 56 William J. Haboush and Brian J. Parshall, editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991) 55 Uwe Jannsen, Steven L. Kleiman, and Jean-Pierre Serre, editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R. O. Wells, Jr., editor, The mathematical heritage of Hermann Weyl (Duke University, Durham, May 1987) 47 Paul Fong, editor, The Areata conference on representations of finite groups (Humboldt State University, Areata, California, July 1986) 46 Spencer J. Bloch, editor, Algebraic geometry - Bowdoin 1985 (Bowdoin College, Brunswick, Maine, July 1985) 45 Felix E. -
On the Constructibility of the Axes of an Ellipsoid
ON THE CONSTRUCTIBILITY OF THE AXES OF AN ELLIPSOID. THE CONSTRUCTION OF CHASLES IN PRACTICE AKOS´ G.HORVATH´ AND ISTVAN´ PROK Dedicated to the memory of the teaching of Descriptive Geometry in BME Faculty of Mechanical Engineering Abstract. In this paper we discuss Chasles’s construction on ellipsoid to draw the semi-axes from a complete system of conjugate diameters. We prove that there is such situation when the construction is not planar (the needed points cannot be constructed with compasses and ruler) and give some others in which the construction is planar. 1. Introduction Who interested in conics know the construction of Rytz, namely a construction which determines the axes of an ellipse from a pair of conjugate diameters. Contrary to this very few mathematicians know that in the first half of the nineteens century Chasles (see in [1]) gave a construction in space to the analogous problem. The only reference which we found in English is also in an old book written by Salmon (see in [6]) on the analytic geometry of the three-dimensional space. In [2] the author extracted this analytic method to prove some results on the quadric of n-dimensional space. However, in practice this construction cannot be done using compasses and ruler only as we will see in present paper (Statement 4). On the other hand those steps of the construction which are planar constructions we can draw by descriptive geometry (see the figures in this article). 2. Basic properties The proof of the statements of this section can be found in [2]. -
11.-HYPERBOLA-THEORY.Pdf
12. HYPERBOLA 1. INTRODUCTION A hyperbola is the locus of a point which moves in the plane in such a way that Z the ratio of its distance from a fixed point in the same plane to its distance X’ P from a fixed line is always constant which is always greater than unity. M The fixed point is called the focus, the fixed line is called the directrix. The constant ratio is generally denoted by e and is known as the eccentricity of the Directrix hyperbola. A hyperbola can also be defined as the locus of a point such that S (focus) the absolute value of the difference of the distances from the two fixed points Z’ (foci) is constant. If S is the focus, ZZ′ is the directrix and P is any point on the hyperbola as show in figure. Figure 12.1 SP Then by definition, we have = e (e > 1). PM Note: The general equation of a conic can be taken as ax22+ 2hxy + by + 2gx + 2fy += c 0 This equation represents a hyperbola if it is non-degenerate (i.e. eq. cannot be written into two linear factors) ahg ∆ ≠ 0, h2 > ab. Where ∆=hb f gfc MASTERJEE CONCEPTS 1. The general equation ax22+ 2hxy + by + 2gx + 2fy += c 0 can be written in matrix form as ahgx ah x x y + 2gx + 2fy += c 0 and xy1hb f y = 0 hb y gfc1 Degeneracy condition depends on the determinant of the 3x3 matrix and the type of conic depends on the determinant of the 2x2 matrix. -
Relativistic Quantum Mechanics 1
Relativistic Quantum Mechanics 1 The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. The emphasis 1.1 SpecialRelativity 1 is given to those elements of the formalism which can be carried on 1.2 One-particle states 7 to Relativistic Quantum Fields (RQF), which underpins the theoretical 1.3 The Klein–Gordon equation 9 framework of high energy particle physics. We begin with a brief summary of special relativity, concentrating on 1.4 The Diracequation 14 4-vectors and spinors. One-particle states and their Lorentz transforma- 1.5 Gaugesymmetry 30 tions follow, leading to the Klein–Gordon and the Dirac equations for Chaptersummary 36 probability amplitudes; i.e. Relativistic Quantum Mechanics (RQM). Readers who want to get to RQM quickly, without studying its foun- dation in special relativity can skip the first sections and start reading from the section 1.3. Intrinsic problems of RQM are discussed and a region of applicability of RQM is defined. Free particle wave functions are constructed and particle interactions are described using their probability currents. A gauge symmetry is introduced to derive a particle interaction with a classical gauge field. 1.1 Special Relativity Einstein’s special relativity is a necessary and fundamental part of any Albert Einstein 1879 - 1955 formalism of particle physics. We begin with its brief summary. For a full account, refer to specialized books, for example (1) or (2). The- ory oriented students with good mathematical background might want to consult books on groups and their representations, for example (3), followed by introductory books on RQM/RQF, for example (4). -
Quantum Mechanics Digital Physics
Quantum Mechanics_digital physics In physics and cosmology, digital physics is a collection of theoretical perspectives based on the premise that the universe is, at heart, describable byinformation, and is therefore computable. Therefore, according to this theory, the universe can be conceived of as either the output of a deterministic or probabilistic computer program, a vast, digital computation device, or mathematically isomorphic to such a device. Digital physics is grounded in one or more of the following hypotheses; listed in order of decreasing strength. The universe, or reality: is essentially informational (although not every informational ontology needs to be digital) is essentially computable (the pancomputationalist position) can be described digitally is in essence digital is itself a computer (pancomputationalism) is the output of a simulated reality exercise History Every computer must be compatible with the principles of information theory,statistical thermodynamics, and quantum mechanics. A fundamental link among these fields was proposed by Edwin Jaynes in two seminal 1957 papers.[1]Moreover, Jaynes elaborated an interpretation of probability theory as generalized Aristotelian logic, a view very convenient for linking fundamental physics withdigital computers, because these are designed to implement the operations ofclassical logic and, equivalently, of Boolean algebra.[2] The hypothesis that the universe is a digital computer was pioneered by Konrad Zuse in his book Rechnender Raum (translated into English as Calculating Space). The term digital physics was first employed by Edward Fredkin, who later came to prefer the term digital philosophy.[3] Others who have modeled the universe as a giant computer include Stephen Wolfram,[4] Juergen Schmidhuber,[5] and Nobel laureate Gerard 't Hooft.[6] These authors hold that the apparentlyprobabilistic nature of quantum physics is not necessarily incompatible with the notion of computability. -
Attoclock Ultrafast Timing of Inversive Chord-Contact Dynamics
Attoclock Ultrafast Timing of Inversive Chord-Contact Dynamics - A Spectroscopic FFT-Glimpse onto Harmonic Analysis of Gravitational Waves - Walter J. Schempp Lehrstuhl f¨urMathematik I Universit¨atSiegen 57068 Siegen, Germany [email protected] According to Bohr's model of the hydrogen atom, it takes about 150 as for an electron in the ground state to orbit around the atomic core. Therefore, the attosecond (1 as = 10−18s) is the typical time scale for Bloch wave packet dynamics operating at the interface of quantum field theory and relativity theory. Attoclock spectroscopy is mathematically dominated by the spectral dual pair of spin groups with the commuting action of the stabilized symplectic spinor quantization (Mp(2; R); PSO(1; 3; R)) In the context of the Mp(2; R) × PSO(1; 3; R) module structure of the complex Schwartz space 2 2 ∼ S(R ⊕ R ) = S(C ⊕ C) of quantum field theory, the metaplectic Lie group Mp(2; R) is the ∼ two-sheeted covering group Sp(2f ; R) of the symplectic group Sp(2; R) = SL(2; R), where the ∼ special linear group SL(2; R) is a simple Lie group, and the two-cycle rotation group SO(2f ; R) = U(1e ; C) is a maximal compact subgroup of Mp(2; R). The two-fold hyperbolically, respectively ∼ elliptically ruled conformal Lie group SU(1; 1; C) is conjugate to SL(2; R) = Spin(1; 2; R) in the ∼ ∼ complexification SL(2; C) = Spin(1; 3; R); the projective Lorentz-M¨obiusgroup PSO(1; 3; R) = ∼ PSL(2; C) = SL(2; C)=(Z2:id) represents a semi-simple Lie group which is isomorphic to the "+ four-dimensional proper orthochroneous Lorentz group L4 of relativity theory. -
A Survey of the Development of Geometry up to 1870
A Survey of the Development of Geometry up to 1870∗ Eldar Straume Department of mathematical sciences Norwegian University of Science and Technology (NTNU) N-9471 Trondheim, Norway September 4, 2014 Abstract This is an expository treatise on the development of the classical ge- ometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of non-Euclidean geometry, about 40 years earlier, had just been demonstrated to be a ”true” geometry on the same footing as Euclidean geometry. These were radically new ideas, but henceforth the importance of the topic became gradually realized. As a consequence, the conventional attitude to the basic geometric questions, including the possible geometric structure of the physical space, was challenged, and foundational problems became an important issue during the following decades. Such a basic understanding of the status of geometry around 1870 enables one to study the geometric works of Sophus Lie and Felix Klein at the beginning of their career in the appropriate historical perspective. arXiv:1409.1140v1 [math.HO] 3 Sep 2014 Contents 1 Euclideangeometry,thesourceofallgeometries 3 1.1 Earlygeometryandtheroleoftherealnumbers . 4 1.1.1 Geometric algebra, constructivism, and the real numbers 7 1.1.2 Thedownfalloftheancientgeometry . 8 ∗This monograph was written up in 2008-2009, as a preparation to the further study of the early geometrical works of Sophus Lie and Felix Klein at the beginning of their career around 1870. The author apologizes for possible historiographic shortcomings, errors, and perhaps lack of updated information on certain topics from the history of mathematics.