DOCSLIB.ORG

The Algebraic Interpretation of Quantifiers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Algebraic Interpretation of Quantifiers Andrzej Mostowski and Foundational Studies 289 A. Ehrenfeucht, V.W. Marek and M. Srebrny (Eds.) IOS Press, 2008 ©2008TheauthorsandIOSPress.Allrightsreserved. The Algebraic Interpretation of Quantifiers: Intuitionistic and Classical Dana S. SCOTT Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic (Emeritus) Computer Science Department, Carnegie Mellon University Pittsburgh, Pennsylvania, USA [email protected] Keywords. Algebraic logic, completeness, Boolean and Heyting algebras “Die Mathematiker sind eine Art Franzosen: redet man zu ihnen, so übersetzen sie es in ihre Sprache und dann ist es alsobald ganz etwas Anderes.” J.W. von Goethe 1. Introduction In their 1963 book, The Mathematics of Metamathematics [Rasiowa and Sikorski, 1963], the authors present an approach to completeness theorems of various logics using alge- braic methods. Perhaps they put the quotation from Goethe over their preface to suggest that mathematicians can be forgiven if they introduce new interpretations of old ideas. The idea of an algebra of logic can of course be traced back to Boole, but it was revived and generalized by Stone and Tarski in the 1930s; however, the most di- rect influence on the work of Rasiowa and Sikorski came from their well-known col- league, Andrzej Mostowski, after WW II. Mostowski’s interpretation of quantification [Mostowski, 1948] can as well be given for intuitionistic as classical logic. The present expository paper will briefly review the history and content of these ideas and then raise the question of why there was at the time no generalization made to higher-order logic and set theory. Entirely new light on this kind of algebraic semantics has more recently been thrown by the development of topos theory in category theory. Some reasons for pursuing this generalization will also be discussed in Section 12. In order to dispel any confusion that might arise about the title of this paper, the author would like to point out that there are other meanings to “algebraic” as relating to the semantics of Logic. Perhaps the first interpretation of quantifiers in an algebraic style goes back to consideration of early work in Descriptive Set Theory. Quantifier manipula- tion was explained in this regard in the well known book on Topology by K. Kuratowski (also quoting Tarski). In 1946 C.J. Everett and S. Ulam published their paper on Pro- 290 D.S. Scott / The Algebraic Interpretation of Quantifiers: Intuitionistic and Classical jective Algebras. A. Tarski, building on his famous paper on Truth in formalized lan- guages and on his studies of Relation Algebras (along with J.C.C. McKinsey, L. Henkin and many collaborators) developed an elaborate equational theory of Cylindric Algebras (duly quoting Everett and Ulam). In a related but different style, P.R. Halmos and collab- orators studied Polyadic Algebras. A generalization of Relation Algebras was proposed by P.C. Bernays and that research has been continued over many years by W. Craig. Most recently, F.W. Lawvere unified many of these ideas – along with the lattice-theoretic se- mantics explained in the present paper – by showing how quantification can be construed in suitable categories by using the idea of adjoint functors. None of these works are cited in our bibliography here, because the literature is too vast to be explained in such a short paper. 2. Preliminaries, Terminology, Notation The following standard terminology and notation will be used throughout the paper. Acompletelattice(cLa)isapartiallyorderedset(poset) A, where every subset ! !" has a least upper bound (lub) under !.(Duallywecansay,everysubsethasagreatest lower bound (glb).) A broader notion of a lattice requires only every finite subset to have alubandaglb.Alubisalsocalledasup;andaglbaninf. Theorem 2.1. If a poset has all lubs, it has all glbs; and conversely. The proof is well known: the greatest lower bound of a family is the least upper bound of all the lower bounds of the family. Theorem 2.2. Using the following notation, lubs and glbs can be uniquely characterized as follows: (LUB) y X z[ x X. x z = y z] ! ⇐⇒ ∀ ∀ ∈ ! ⇒ ! (GLB) X y z[ x X. z x = z y] ! ⇐⇒ ∀ ∀ ∈ ! ⇒ ! (TOP) 1=!A = ∅ (BOT) 0=" A = ! " ∅ Theorem 2.3. Lattices" (not! necessarily complete) are characterized by these axioms: • ! is a partial order • x z & y z x y z ! ! ⇐⇒ ∨ ! • z x & z y z x y ! ! ⇐⇒ ! ∧ • 0 ! x ! 1 Examples of complete lattices abound in set theory. Here is an example of a direct consequence of Theorem 2.1. Theorem 2.4. The powerset (A) is always a cLa; hence, if a family (A) is P C⊆P closed under all intersections (unions), then , is also a cLa. !C ⊆" We need to remember that in the cLa (A) the intersection of the empty family is P taken to be A. D.S. Scott / The Algebraic Interpretation of Quantifiers: Intuitionistic and Classical 291 3. Closures on a Power Set (A) P Closure operations C : (A) (A) give interesting examples of complete lattices. There are several kinds ofP closures→P axiomatized as follows. General: • X C(X)=C(C(X)) • X ⊆ Y = C(X) C(Y ) ⊆ ⇒ ⊆ Topological: • X C(X)=C(C(X)) • C(⊆)= • C(∅X Y∅)=C(X) C(Y ) ∪ ∪ Algebraic: • X C(X)=C(C(X)) • C(⊆ X)= C(X) for directed (A) X X W⊆P #∈W #∈W For any closure operation we say that a set X A is closed provided C(X) X. ⊆ ⊆ Theorem 3.1. The family of closed sets of a general closure operation is closed under arbitrary intersections and so forms under inclusion a complete lattice. Moreover, an isomorph of any complete lattice can be found in this way. For the easy proof see [Rasiowa and Sikorski, 1963]. The complete lattices formed from topological or algebraic closure operations have additional properties. We will dis- cuss the topological case further below. 4. Implication and Distribution Definition 4.1. Alatticeoperation is called an implication if and only if it satisfies this axiom: → • x y z x y z ∧ ! ⇐⇒ ! → Note. The axiom uniquely determines an implication on a lattice if it exists. There is an equational axiomatization where we can define the partial ordering in several ways. • x ! y x y = y ⇐⇒ x ∨ y = x ⇐⇒ x ∧ y =1 ⇐⇒ → See [Rasiowa and Sikorski, 1963] for details. A lattice with implication is called a Heyt- ing algebra.(Haisusedforshort.) Theorem 4.1. All Ha’s are distributive lattices. 292 D.S. Scott / The Algebraic Interpretation of Quantifiers: Intuitionistic and Classical Proof. From basic lattice properties we have x y (x y) (x z).Theny ∧ ! ∧ ∨ ∧ ! x ((x y) (x z)).Similarlywehavex z (x y) (x z).Thusz → ∧ ∨ ∧ ∧ ! ∧ ∨ ∧ ! x ((x y) (x z)).Itfollowsthaty z x ((x y) (x z)).Butthen → ∧ ∨ ∧ ∨ ! → ∧ ∨ ∧ x (y z) ! (x y) (x z).ThisishalfoftheDistributiveLaw.Theotherhalfholds in∧ all lattices.∨ ∧ ∨ ∧ Definition 4.2. AHeytingalgebrawhichhasallglbsandlubs(= complete lattice) is called a complete Heyting algebra (or cHa, for short). Theorem 4.2. AcompletelatticeisacHaif,andonlyif,itsatisfies: x y = (x y ). ∧ i ∧ i i I i I $∈ $∈ The proof of the above equation is a direct generalization of the proof of Theorem 4.1. For the converse we can use the following definition of implication: y z = x x y z . → { | ∧ ! } $ The results about distributivity of Heyting algebras were proved independently by Stone [Stone, 1937] and Tarski [Tarski, 1938]. 5. Boolean Algebras The difference between classical and intuitionistic logic lies in the fact that the former satisfies the law of the Excluded Middle. In algebraic terms we have the following. Definition 5.1. AHa(resp.cHa)isaBooleanalgebra(orBa)(resp.completeBoolean algebra or cBa) iff it satisfies x (x 0) = 1. ∨ → The element x 0 is called the negation of x and is denoted by x. → ¬ Perhaps we should pause to note this passage from [Rasiowa and Sikorski, 1963] on pp. 8–9: The inclusion of two chapters on intuitionism is not an indication of the authors’ positive atti- tude towards intuitionistic ideas. Intuitionism, like other non-classical logics, has no practical application in mathematics. Nevertheless many authors devote their works to intuitionistic logic. On the other hand, the mathematical mechanism of intuitionistic logic is interesting: it is amazing that vaguely defined philosophical ideas concerning the notion of existence in mathematics have lead to the creation of formalized logical systems which, from the math- ematical point of view, proved to be equivalent to the theory of lattices of open subsets of topological spaces. Finally, the formalization of intuitionistic logic achieved by Heyting and adopted in this book is not in agreement with the philosophical views of the founder of in- tuitionism, Brouwer, who opposed formalism in mathematics. Since in treating intuitionistic logic we have limited ourselves to problems which are directly connected with general alge- braic, lattice-theoretical and topological methods employed in the book, we have not included the latest results of Beth and Kreisel concerning other notions of satisfiability which we have adopted. D.S. Scott / The Algebraic Interpretation of Quantifiers: Intuitionistic and Classical 293 As a matter of fact the topological modeling of intuitionistic formal rules is not really to be regarded as “amazing”. For, in view of Theorem 4.2, we conclude at once this theorem and corollary. Theorem 5.1. Every sublattice of a cBa (also cHa) closed under and is a cHa. ∧ Corollary 5.1. The lattice of open subsets of a topological space forms a! cHa. There is also a reverse connection between Heyting algebras and Boolean algebras. An element x of a Heyting algebra is called stable (sometimes regular) if it satisfies x = x. ¬¬ Theorem 5.2. The stable elements of a Ha (cHa) form a Ba (cBa). Aproofisgivenin[RasiowaandSikorski,1963],pp.134–135.Anotherconnection is an embedding theorem. Theorem 5.3. Every cHa can be embedded in a cBa so as to preserve and .
Load more

Recommended publications
  • Creating New Concepts in Mathematics: Freedom and Limitations. the Case of Category Theory
    Creating new concepts in mathematics: freedom and limitations. The case of Category Theory Zbigniew Semadeni Institute of Mathematics, University of Warsaw Abstract In the paper we discuss the problem of limitations of freedom in mathematics and search for criteria which would differentiate the new concepts stemming from the historical ones from the new concepts that have opened unexpected ways of thinking and reasoning. We also investigate the emergence of category theory (CT) and its origins. In particular we explore the origins of the term functor and present the strong evidence that Eilenberg and Carnap could have learned the term from Kotarbinski´ and Tarski. Keywords categories, functors, Eilenberg-Mac Lane Program, mathematical cognitive transgressions, phylogeny, platonism. CC-BY-NC-ND 4.0 • 1. Introduction he celebrated dictum of Georg Cantor that “The essence of math- Tematics lies precisely in its freedom” expressed the idea that in mathematics one can freely introduce new notions (which may, how- Philosophical Problems in Science (Zagadnienia FilozoficzneNo w Nauce) 69 (2020), pp. 33–65 34 Zbigniew Semadeni ever, be abandoned if found unfruitful or inconvenient).1 This way Cantor declared his opposition to claims of Leopold Kronecker who objected to the free introduction of new notions (particularly those related to the infinite). Some years earlier Richard Dedekind stated that—by forming, in his theory, a cut for an irrational number—we create a new number. For him this was an example of a constructed notion which was a free creation of the human mind (Dedekind, 1872, § 4). In 1910 Jan Łukasiewicz distinguished constructive notions from empirical reconstructive ones.
    [Show full text]
  • What Is Abstract Algebraic Logic?
    What is Abstract Algebraic Logic? Umberto Rivieccio University of Genoa Department of Philosophy Via Balbi 4 - Genoa, Italy email [email protected] Abstract Our aim is to answer the question of the title, explaining which kind of problems form the nucleus of the branch of algebraic logic that has come to be known as Abstract Algebraic Logic. We give a concise account of the birth and evolution of the field, that led from the discovery of the connections between certain logical systems and certain algebraic structures to a general theory of algebraization of logics. We consider first the so-called algebraizable logics, which may be regarded as a paradigmatic case in Abstract Algebraic Logic; then we shift our attention to some wider classes of logics and to the more general framework in which they are studied. Finally, we review some other topics in Abstract Algebraic Logic that, though equally interesting and promising for future research, could not be treated here in detail. 1 Introduction Algebraic logic can be described in very general terms as the study of the relations between algebra and logic. One of the main reasons that motivate such a study is the possibility to treat logical problems with algebraic methods. This means that, in order to solve a logical problem, we may try to translate it into algebraic terms, find the solution using algebraic techniques, and then translate back the result into logic. This strategy proved to be fruitful, since algebra is an old and well{established field of mathematics, that offers powerful tools to treat problems that would possibly be much harder to solve using the logical machinery alone.
    [Show full text]
  • Explanations
    © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. CHAPTER 1 Explanations 1.1 SALLIES Language is an instrument of Logic, but not an indispensable instrument. Boole 1847a, 118 We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic; the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two. De Morgan 1868a,71 That which is provable, ought not to be believed in science without proof. Dedekind 1888a, preface If I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems whilst the root drives into the depthswx . Frege 1893a, xiii Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians. Russell 1903a, 451 1.2 SCOPE AND LIMITS OF THE BOOK 1.2.1 An outline history. The story told here from §3 onwards is re- garded as well known. It begins with the emergence of set theory in the 1870s under the inspiration of Georg Cantor, and the contemporary development of mathematical logic by Gottlob Frege andŽ. especially Giuseppe Peano. A cumulation of these and some related movements was achieved in the 1900s with the philosophy of mathematics proposed by Alfred North Whitehead and Bertrand Russell.
    [Show full text]
  • Ralph Martin Kaufmann Publications 1. Kaufmann, Ralph
    Ralph Martin Kaufmann Department of Mathematics, Purdue University 150 N. University Street, West Lafayette, IN 47907{2067 Tel.: (765) 494-1205 Fax: (765) 494-0548 Publications 1. Kaufmann, Ralph M., Khlebnikov, Sergei, and Wehefritz-Kaufmann, Birgit \Local models and global constraints for degeneracies and band crossings" J. of Geometry and Physics 158 (2020) 103892. 2. Galvez-Carillo, Imma, Kaufmann, Ralph M., and Tonks, Andrew. \Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects" Comm. in Numb. Th. and Physics (CNTP), vol 14,1 (2020), 1-90. 3. Galvez-Carillo, Imma, Kaufmann, Ralph M., and Tonks, Andrew. \Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation" Comm. in Numb. Th. and Physics (CNTP), vol 14,1 (2020), 91-169. 4. Kaufmann, Ralph M. \Lectures on Feynman categories", 2016 MATRIX annals, 375{438, MATRIX Book Ser., 1, Springer, Cham, 2018. 5. Kaufmann, Ralph M. and Kaufmann-Wehfritz, B. Theoretical Properties of Materials Formed as Wire Network Graphs from Triply Periodic CMC Surfaces, Especially the Gyroid in: \The Role of Topology in Materials", Eds: Gupta, S. and Saxena, A., Springer series in Solid State Sciences. Springer, 2018 6. Kaufmann, Ralph and Lucas, Jason. \Decorated Feynman categories". J. of Noncommutative Geometry, 1 (2017), no 4 1437-1464 7. Berger, C. and Kaufmann R. M. \Comprehensive Factorization Systems". Special Issue in honor of Professors Peter J. Freyd and F.William Lawvere on the occasion of their 80th birthdays, Tbilisi Mathematical Journal 10 (2017), no. 3,. 255-277 8. Kaufmann, Ralph M.
    [Show full text]
  • An Interview with F. William Lawvere
    An Interview with F. William Lawvere You have written a paper, published for the first time in 1986, entitled \Taking categories seriously"1. Why should we take categories seriously ? In all those areas where category theory is actively used the categorical concept of adjoint functor has come to play a key role. Such a universal instrument for guiding the learning, development, and use of advanced mathematics does not fail to have its indications also in areas of school and college mathematics, in the most basic relationships of space and quantity and the calculations based on those relationships. By saying \take categories seriously", I meant that one should seek, cultivate, and teach helpful examples of an elementary nature. The relation between teaching and research is partly embodied in simple general concepts that can guide the elaboration of examples in both. No- tions and constructions, such as the spectral analysis of dynamical systems, have important aspects that can be understood and pursued without the complications of limiting the models to specific classical categories. The application of some simple general concepts from category theory can lead from a clarification of basic con- structions on dynamical systems to a F. William Lawvere (Braga, March 2007) construction of the real number system with its structure as a closed cate- gory; applied to that particular closed category, the general enriched category theory leads inexorably to embedding the- orems and to notions of Cauchy completeness, rotation, convex hull, radius, and 1Revista Colombiana de Matematicas 20 (1986) 147-178. Reprinted in Repr. Theory Appl. Categ. 8 (2005) 1-24 (electronic).
    [Show full text]
  • GRASSMANN's DIALECTICS and CATEGORY THEORY in Several
    F. WILLIAM LAWVERE GRASSMANN'S DIALECTICS AND CATEGORY THEORY PROGRAMMATIC OUTLINE In several key connections in his foundations of geometrical algebra, Grassmann makes significant use of the dialectical philosophy of 150 years ago. Now, after fifty years of development of category theory as a means for making explicit some nontrivial general arguments in geometry, it is possible to recover some of Grassmann's insights and to express these in ways comprehensible to the modem geometer. For example, the category J/. of affine-linear spaces and maps (a monument to Grassmann) has a canonical adjoint functor to the category of (anti)commutative graded algebras, which as in Grassmann's detailed description yields a sixteen-dimensional algebra when applied to a three­ dimensional affine space (unlike the eight-dimensional exterior algebra of a three-dimensional vector space). The natural algebraic structure of these algebras includes a boundary operator d which satisfies the (signed) Leibniz rule; for example, if A, B are points of the affine space then the product AB is the axial vector from A to B which the boundary degrades to the corresponding translation vector: d(AB) = B-A (since dA = dB = I for points). Grassmann philosophically motivated a notion of a "simple law of change," but his editors in the 1890' s found this notion incoherent and decided he must have meant mere translations. However, translations are insufficient for the foundational task of deciding when two formal products are geometrically equal axial vectors. But if "Iaw of change" is understood as an action of the additive monoid of time, "simple" turns out to mean that the action is internal to the category J/.
    [Show full text]
  • How Peircean Was the “'Fregean' Revolution” in Logic?
    HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? Irving H. Anellis Peirce Edition, Institute for American Thought Indiana University – Purdue University at Indianapolis Indianapolis, IN, USA [email protected] Abstract. The historiography of logic conceives of a Fregean revolution in which modern mathematical logic (also called symbolic logic) has replaced Aristotelian logic. The preeminent expositors of this conception are Jean van Heijenoort (1912–1986) and Don- ald Angus Gillies. The innovations and characteristics that comprise mathematical logic and distinguish it from Aristotelian logic, according to this conception, created ex nihlo by Gottlob Frege (1848–1925) in his Begriffsschrift of 1879, and with Bertrand Rus- sell (1872–1970) as its chief This position likewise understands the algebraic logic of Augustus De Morgan (1806–1871), George Boole (1815–1864), Charles Sanders Peirce (1838–1914), and Ernst Schröder (1841–1902) as belonging to the Aristotelian tradi- tion. The “Booleans” are understood, from this vantage point, to merely have rewritten Aristotelian syllogistic in algebraic guise. The most detailed listing and elaboration of Frege’s innovations, and the characteristics that distinguish mathematical logic from Aristotelian logic, were set forth by van Heijenoort. I consider each of the elements of van Heijenoort’s list and note the extent to which Peirce had also developed each of these aspects of logic. I also consider the extent to which Peirce and Frege were aware of, and may have influenced, one another’s logical writings. AMS (MOS) 2010 subject classifications: Primary: 03-03, 03A05, 03C05, 03C10, 03G27, 01A55; secondary: 03B05, 03B10, 03E30, 08A20; Key words and phrases: Peirce, abstract algebraic logic; propositional logic; first-order logic; quantifier elimina- tion, equational classes, relational systems §0.
    [Show full text]
  • Abstract Algebraic Logic an Introductory Chapter
    Abstract Algebraic Logic An Introductory Chapter Josep Maria Font 1 Introduction . 1 2 Some preliminary issues . 5 3 Bare algebraizability . 10 3.1 The equational consequence relative to a class of algebras . 11 3.2 Translating formulas into equations . 13 3.3 Translating equations into formulas . 15 3.4 Putting it all together . 15 4 The origins of algebraizability: the Lindenbaum-Tarski process . 18 4.1 The process for classical logic . 18 4.2 The process for algebraizable logics . 20 4.3 The universal Lindenbaum-Tarski process: matrix semantics . 23 4.4 Algebraizability and matrix semantics: definability . 27 4.5 Implicative logics . 29 5 Modes of algebraizability, and non-algebraizability . 32 6 Beyond algebraizability . 37 6.1 The Leibniz hierarchy . 37 6.2 The general definition of the algebraic counterpart of a logic . 45 6.3 The Frege hierarchy . 49 7 Exploiting algebraizability: bridge theorems and transfer theorems . 55 8 Algebraizability at a more abstract level . 65 References . 69 1 Introduction This chapter has been conceived as a brief introduction to (my personal view of) abstract algebraic logic. It is organized around the central notion of algebraizabil- ity, with particular emphasis on its connections with the techniques of traditional algebraic logic and especially with the so-called Lindenbaum-Tarski process. It Josep Maria Font Departament of Mathematics and Computer Engineering, Universitat de Barcelona (UB), Gran Via de les Corts Catalanes 585, E-08007 Barcelona, Spain. e-mail: [email protected] 1 2 Josep Maria Font goes beyond algebraizability to offer a general overview of several classifications of (sentential) logics—basically, the Leibniz hierarchy and the Frege hierarchy— which have emerged in recent decades, and to show how the classification of a logic into any of them provides some knowledge regarding its algebraic or its metalogical properties.
    [Show full text]
  • CONCEPTUAL MATHEMATICS, a Review by Scott W. Williams
    CONCEPTUAL MATHEMATICS, a review by Scott W. Williams Not long ago, I spoke with a professor at strong HBCU department. Her Ph.D. was nearly twenty years ago, but I shocked her with the following statement, "Most of our beginning graduate students [even those in Applied Mathematics] are entering with the basic knowledge and language of Category Theory. These days one might find Chemists, Computer Scientists, Engineers, Linguists and Physicists expressing concepts and asking questions in the language of Category Theory because it slices across the artificial boundaries dividing algebra, arithmetic, calculus, geometry, logic, topology. If you have students you wish to introduce to the subject, I suggest a delightfully elementary book called Conceptual Mathematics by F. William Lawvere and Stephen H. Schanuel" [Cambridge University Press 1997, $35 on Amazon.com) (ISBN: 0521478170 | ISBN- 13:9780521478175)]. From the introduction: "Our goal in this book is to explore the consequences of a new and fundamental insight about the nature of mathematics which has led to better methods for understanding and usual mathematical concepts. While the insight and methods are simple ... they will require some effort to master, but you will be rewarded with a clarity of understanding that will be helpful in unraveling the mathematical aspect of any subject matter." Who are the authors? Lawvere is one of the greatest visionaries of mathematics in the last half of the twentieth century. He characteristically digs down beneath the foundations of a concept in order to simplify its understanding. Though Schanuel has published research in diverse areas of Algebra, Topology, and Number Theory, he is known as a great teacher.
    [Show full text]
  • Toposes of Laws of Motion
    Toposes of Laws of Motion F. William Lawvere Transcript from Video, Montreal September 27, 1997 Individuals do not set the course of events; it is the social force. Thirty-five or forty years ago it caused us to congregate in centers like Columbia University or Berkeley, or Chicago, or Montreal, or Sydney, or Zurich because we heard that the pursuit of knowledge was going on there. It was a time when people in many places had come to realize that category theory had a role to play in the pursuit of mathematical knowledge. That is basically why we know each other and why many of us are more or less the same age. But it’s also important to point out that we are still here and still finding striking new results in spite of all the pessimistic things we heard, even 35 or 40 years ago, that there was no future in abstract generalities. We continue to be surprised to find striking new and powerful general results as well as to find very interesting particular examples. We have had to fight against the myth of the mainstream which says, for example, that there are cycles during which at one time everybody is working on general concepts, and at another time anybody of consequence is doing only particular examples, whereas in fact serious mathematicians have always been doing both. 1. Infinitesimally Generated Toposes In fact, it is the relation between the General and the Particular about which I wish to speak. I read somewhere recently that the basic program of infinitesimal calculus, continuum mechanics, and differential geometry is that all the world can be reconstructed from the infinitely small.
    [Show full text]
  • Colin S. Mclarty Truman P. Handy Professor of Intellectual Philosophy, and of Mathematics Case Western Reserve University
    Colin S. McLarty Truman P. Handy Professor of Intellectual Philosophy, and of Mathematics Case Western Reserve University 211 Clark Hall Phone: (w) 216 368-2632 (h) 440 286-7431 Cleveland OH 44106 Fax: 216 368-0814 Email: [email protected] Homepage: cwru.edu/artsci/phil/mclarty.html Education Ph.D. Philosophy, Case Western Reserve University, 1980. Dissertation: Things and Things in Themselves: The Logic of Reference in Leibniz, Lambert, and Kant. Advisors: Raymond J. Nelson and Chin-Tai Kim. M.A. Philosophy, Case Western Reserve University, 1975. B.S. Mathematics, Case Institute of Technology, 1972. Area of Specialization: Logic; history and philosophy of mathematics. Areas of Competence: Philosophy of Science; History of Philosophy esp. Plato, Kant; Contemporary French Philosophy. Employment Shanxi University, China. Visiting Lecturer, Philosophy of Science and Technology, 2011. University of Notre Dame,Visiting Associate Professor, Philosophy, Fall 2002. Harvard University, Visiting Scholar, Mathematics 1995–1996. Case Western Reserve University, Philosophy 1987–. Chair of Department 1996–2010. Cleveland State University, Lecturer, Mathematics 1984–1986. Cleveland Art Institute, Lecturer, Philosophy 1984–1986. Current Research My Summer School in Denmark on a structural foundation for mathematics, expanded my Beijing (2007) talk and is now a book forthcoming with OUP. I pursue the perspective one mathematician expressed say- ing “philosophers should know our objects only have the properties we say they do.” Those are structural properties and in practice rely on category theory. My recent articles show how this ontology suits current research. And I have a series of articles on Emmy Noether (1882–1935, generally considered the greatest of woman mathematicians) and Saunders Mac Lane (1909–2005, greatly influenced by Noether) showing how deep the attitude goes in 20th century mathematics.
    [Show full text]
  • Algebraic Logic, Where Does It Stand Today? Part 2: Algebraic Logic; Counting Varieties and Models, Vaught’S Conjecture and New Physics
    1 What happened since ’ Algebraic Logic, where does it stand today? Part 2: Algebraic logic; Counting varieties and models, Vaught’s conjecture and new physics Tarek Sayed Ahmed Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt. Dedicated to my mentors Andreka´ and Nemeti´ No Institute Given Summary. * In the article ’Algebraic logic, where does it stand today?’ published in the Bulletin of Symbolic logic in 2005 logic, we gave a fairly comprehensive review of Tarskian algebraic logic reaching the boundaries of current research back then. In this paper we take it from there, giving the reader more than a glimpse of the recent develoments till the present day. In the process, we take a magical tour in algebraic logic, starting from classical results on neat embeddings due to Henkin, Monk and Tarski, all the way to recent results in algebraic logic using so–called rainbow constructions. Highlighting the connections with graph theory, model theory, and finite combinatorics, this article aspires to present topics of broad interest in a way that is hopefully accessible to a large audience. Other topics dealt with include the inter- action of algebraic and modal logic, the so–called (central still active) finitizability problem, Godels’s¨ incompleteness Theorem in guarded fragments, counting the number of subvarieties of RCAw which is reminiscent of Shelah’s stability theory and the interaction of algebraic logic and descriptive set theory as means to approach Vaught’s conjecture in model theory. Rainbow constructions conjuncted with variations on Andreka’s´ methods of splitting (atoms) are used to solve problems adressing classes of cylindric–like algebras consisting of algebras having a neat embedding property.
    [Show full text]