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Colin McLarty [46]Colin Slator Mclarty [1]
  1. What Does It Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory.Colin McLarty - 2010 - Bulletin of Symbolic Logic 16 (3):359-377.
    This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.
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  2. Emmy Noether's “Set Theoretic” Topology: From Dedekind to the First Functors.Colin McLarty - 2006 - In José Ferreirós Domínguez & Jeremy Gray (eds.), The Architecture of Modern Mathematics: Essays in History and Philosophy. Oxford University Press. pp. 187--208.
     
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  3.  64
    What Does It Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory.Colin McLarty - 2010 - Bulletin of Symbolic Logic 16 (3):359-377.
    This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.
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  4.  16
    The Large Structures of Grothendieck Founded on Finite Order Arithmetic.Colin Mclarty - forthcoming - Review of Symbolic Logic:1-30.
    The large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in practice, yet published foundations for them go beyond ZFC in logical strength. We reduce the gap by founding all the theorems of Grothendieck’s SGA, plus derived categories, at the level of Finite-Order Arithmetic, far below ZFC. This is the weakest possible foundation for the large-structure tools because one elementary topos of sets with infinity is already this strong.
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  5.  65
    ‘Mathematical Platonism’ Versus Gathering the Dead: What Socrates Teaches Glaucon &Dagger.Colin McLarty - 2005 - Philosophia Mathematica 13 (2):115-134.
    Glaucon in Plato's _Republic_ fails to grasp intermediates. He confuses pursuing a goal with achieving it, and so he adopts ‘mathematical platonism’. He says mathematical objects are eternal. Socrates urges a seriously debatable, and seriously defensible, alternative centered on the destruction of hypotheses. He offers his version of geometry and astronomy as refuting the charge that he impiously ‘ponders things up in the sky and investigates things under the earth and makes the weaker argument the stronger’. We relate his account (...)
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  6. The Uses and Abuses of the History of Topos Theory.Colin Mclarty - 1990 - British Journal for the Philosophy of Science 41 (3):351-375.
    The view that toposes originated as generalized set theory is a figment of set theoretically educated common sense. This false history obstructs understanding of category theory and especially of categorical foundations for mathematics. Problems in geometry, topology, and related algebra led to categories and toposes. Elementary toposes arose when Lawvere's interest in the foundations of physics and Tierney's in the foundations of topology led both to study Grothendieck's foundations for algebraic geometry. I end with remarks on a categorical view of (...)
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  7.  98
    Numbers Can Be Just What They Have To.Colin McLarty - 1993 - Noûs 27 (4):487-498.
  8. Learning From Questions on Categorical Foundations.Colin McLarty - 2005 - Philosophia Mathematica 13 (1):44-60.
    We can learn from questions as well as from their answers. This paper urges some things to learn from questions about categorical foundations for mathematics raised by Geoffrey Hellman and from ones he invokes from Solomon Feferman.
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  9.  83
    Axiomatizing a Category of Categories.Colin McLarty - 1991 - Journal of Symbolic Logic 56 (4):1243-1260.
    Elementary axioms describe a category of categories. Theorems of category theory follow, including some on adjunctions and triples. A new result is that associativity of composition in categories follows from cartesian closedness of the category of categories. The axioms plus an axiom of infinity are consistent iff the axioms for a well-pointed topos with separation axiom and natural numbers are. The theory is not finitely axiomatizable. Each axiom is independent of the others. Further independence and definability results are proved. Relations (...)
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  10.  76
    The Last Mathematician From Hilbert's Göttingen: Saunders Mac Lane as Philosopher of Mathematics.Colin McLarty - 2007 - British Journal for the Philosophy of Science 58 (1):77-112.
    While Saunders Mac Lane studied for his D.Phil in Göttingen, he heard David Hilbert's weekly lectures on philosophy, talked philosophy with Hermann Weyl, and studied it with Moritz Geiger. Their philosophies and Emmy Noether's algebra all influenced his conception of category theory, which has become the working structure theory of mathematics. His practice has constantly affirmed that a proper large-scale organization for mathematics is the most efficient path to valuable specific results—while he sees that the question of which results are (...)
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  11.  15
    Elementary Categories, Elementary Toposes.Colin Mclarty - 1997 - Studia Logica 59 (1):143-146.
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  12. Elementary Categories, Elementary Toposes.Colin McLarty - 1995 - Oxford University Press.
    Now available in paperback, this acclaimed book introduces categories and elementary toposes in a manner requiring little mathematical background. It defines the key concepts and gives complete elementary proofs of theorems, including the fundamental theorem of toposes and the sheafification theorem. It ends with topos theoretic descriptions of sets, of basic differential geometry, and of recursive analysis.
     
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  13.  75
    Poincaré: Mathematics & Logic & Intuition.Colin Mclarty - 1997 - Philosophia Mathematica 5 (2):97-115.
    often insisted existence in mathematics means logical consistency, and formal logic is the sole guarantor of rigor. The paper joins this to his view of intuition and his own mathematics. It looks at predicativity and the infinite, Poincaré's early endorsement of the axiom of choice, and Cantor's set theory versus Zermelo's axioms. Poincaré discussed constructivism sympathetically only once, a few months before his death, and conspicuously avoided committing himself. We end with Poincaré on Couturat, Russell, and Hilbert.
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  14.  25
    Foundations as Truths Which Organize Mathematics.Colin Mclarty - 2013 - Review of Symbolic Logic 6 (1):76-86.
    The article looks briefly at Fefermans own foundations. Among many different senses of foundations, the one that mathematics needs in practice is a recognized body of truths adequate to organize definitions and proofs. Finding concise principles of this kind has been a huge achievement by mathematicians and logicians. We put ZFC and categorical foundations both into this context.
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  15.  36
    Defining Sets as Sets of Points of Spaces.Colin McLarty - 1988 - Journal of Philosophical Logic 17 (1):75 - 90.
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  16.  35
    Anti-Foundation and Self-Reference.Colin McLarty - 1993 - Journal of Philosophical Logic 22 (1):19 - 28.
    This note argues against Barwise and Etchemendy's claim that their semantics for self-reference requires use of Aczel's anti-foundational set theory, AFA, semantics for self-reference requires use of Aczel's anti-foundational set theory, AFA, ones irrelevant to the task at hand" (The Liar, p. 35). Switching from ZF to AFA neither adds nor precludes any isomorphism types of sets. So it makes no difference to ordinary mathematics. I argue against the author's claim that a certain kind of 'naturalness' nevertheless makes AFA preferable (...)
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  17. Philosophical Relevance of Category Theory.Colin McLarty - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press.
     
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  18. Review of S. Duffy, Virtual Mathematics: The Logic of Difference (Clinamen, 2006). [REVIEW]Colin McLarty - 2008 - Australasian Journal of Philosophy 86 (2):332-336.
    This book is important for philosophy of mathematics and for the study of French philosophy. French philosophers are more concerned than most Anglo-American with mathematical practice outside of foundations. This contradicts the fashionable claim that French intellectuals get science all wrong and we return below to a germane example from Sokal and Bricmont [1999]. The emphasis on practice goes back to mid-20th century French historians of science including those Kuhn cites as sources for his orientation in philosophy of science [Kuhn (...)
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  19.  19
    Failure of Cartesian Closedness in NF.Colin McLarty - 1992 - Journal of Symbolic Logic 57 (2):555-556.
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  20.  2
    Voir-Dire in the Case of Mathematical Progress.Colin McLarty - 2000 - In Emily Grosholz & Herbert Breger (eds.), The Growth of Mathematical Knowledge. Kluwer Academic Publishers. pp. 269--280.
  21.  27
    Review of Real Numbers, Generalizations of the Reals, & Theories of Continua by Philip Ehrlich. [REVIEW]Colin McLarty - 1999 - Philosophy of Science 66 (3):500-501.
  22.  32
    Saunders Mac Lane (1909–2005): His Mathematical Life and Philosophical Works.Colin McLarty - 2005 - Philosophia Mathematica 13 (3):237-251.
  23.  11
    Two Constructivist Aspects of Category Theory.Colin McLarty - 2006 - Philosophia Scientiae:95-114.
  24.  54
    Category Theory in Real Time.Colin Mclarty - 1994 - Philosophia Mathematica 2 (1):36-44.
    The article surveys some past and present debates within mathematics over the meaning of category theory. It argues that such conceptual analyses, applied to a field still under active development, must be in large part either predictions of, or calls for, certain programs of further work.
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  25.  3
    Two Constructivist Aspects of Category Theory.Colin McLarty - 2006 - Philosophia Scientae:95-114.
  26.  28
    2000-2001 Spring Meeting of the Association for Symbolic Logic.Michael Detlefsen, Erich Reck, Colin McLarty, Rohit Parikh, Larry Moss, Scott Weinstein, Gabriel Uzquiano, Grigori Mints & Richard Zach - 2001 - Bulletin of Symbolic Logic 7 (3):413-419.
  27.  28
    Elementary Axioms for Canonical Points of Toposes.Colin McLarty - 1987 - Journal of Symbolic Logic 52 (1):202-204.
  28.  24
    Palmer House Hilton Hotel, Chicago, Illinois April 23–24, 2004.Warren Goldfarb, Erich Reck, Jeremy Avigad, Andrew Arana, Geoffrey Hellman, Colin McLarty, Dana Scott & Michael Kremer - 2004 - Bulletin of Symbolic Logic 10 (3).
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  29.  37
    Book Review: John Bell. Introduction to Toposes and Local Set Theory. [REVIEW]Colin McLarty - 1989 - Notre Dame Journal of Formal Logic 31 (1):150-161.
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  30.  23
    Review of Charles S. Chihara, A Structural Account of Mathematics[REVIEW]Colin McLarty - 2004 - Notre Dame Philosophical Reviews 2004 (8).
  31.  22
    Saunders Mac Lane. Saunders Mac Lane: A Mathematical Autobiography.Colin McLarty - 2007 - Philosophia Mathematica 15 (3):400-404.
    We are used to seeing foundations linked to the mainstream mathematics of the late nineteenth century: the arithmetization of analysis, non-Euclidean geometry, and the rise of abstract structures in algebra. And a growing number of case studies bring a more philosophy-of-science viewpoint to the latest mathematics, as in [Carter, 2005; Corfield, 2006; Krieger, 2003; Leng, 2002]. Mac Lane's autobiography is a valuable bridge between these, recounting his experience of how the mid- and late-twentieth-century mainstream grew especially through Hilbert's school.An autobiography (...)
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  32.  10
    Mathematics: Form and Function by Saunders MacLane. [REVIEW]Colin McLarty - 1987 - Journal of Philosophy 84 (1):33-37.
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  33.  7
    British Journal for the Philosophy of Science.Colin McLarty - 2003 - Bulletin of Symbolic Logic 9 (1):43-44.
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  34.  7
    La Notion de Nombre Chez Dedekind, Cantor, Frege: Theories, Conceptions, Et Philosophie. Jean-Pierre Belna.Colin McLarty - 1998 - Isis 89 (1):145-146.
  35.  10
    Poor Taste as a Bright Character Trait: Emmy Noether and the Independent Social Democratic Party.Colin McLarty - 2005 - Science in Context 18 (3):429-450.
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  36.  9
    La Topologie Et Ses Signes: Éléments Pour Une Histoire Sémiotique des Mathématiques. [REVIEW]Colin Mclarty - 2002 - Isis: A Journal of the History of Science 93:328-328.
    Topology uses simple geometric and algebraic ideas, but its huge success and vast ramifications make it a tough nut for historians of twentieth‐century mathematics. Two books have addressed it well: Dieudonné chronicles about one thousand key definitions and theorems, and essays in James focus on forty central themes. Both assume considerable mathematics, but neither offers a historical synthesis of the simplest core ideas. Now, Alain Herreman uses semiotics to watch these leading ideas develop through the founding works of Henri Poincaré, (...)
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  37.  6
    Alain Herreman. La topologie et ses signes: Éléments pour une histoire sémiotique des mathématiques. 348 pp., figs., tables, index. Paris/Montreal: L'Harmattan, 2000. [REVIEW]Colin McLarty - 2002 - Isis 93 (2):328-328.
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  38.  8
    La Notion de Nombre Chez Dedekind, Cantor, Frege: Theories, Conceptions, Et Philosophie by Jean-Pierre Belna. [REVIEW]Colin Mclarty - 1998 - Isis: A Journal of the History of Science 89:145-146.
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  39.  7
    Mayberry, J. P.: The Foundations Of Mathematics In The Theory Of Sets. Encyclopedia Of Mathematics And Its Applications Ser., Vol. 82. [REVIEW]Colin Mclarty - 2002 - Philosophy of Science 69 (2):404-406.
  40.  8
    Raymond J. Nelson 1917-1997.Chin-Tai Kim & Colin McLarty - 1997 - Proceedings and Addresses of the American Philosophical Association 71 (2):125 - 126.
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  41.  1
    Como Grothendieck Simplificou a Geometria Algébrica.Colin McLarty & Norman R. Madarasz - 2016 - Veritas – Revista de Filosofia da Pucrs 61 (2):276-294.
    Alexandre Grothendieck foi um dos maiores matemáticos do século 20 e um dos mais atípicos. Nascido na Alemanha a um pai anarquista de origem russa, sua infância foi marcada pela militância política dos seus pais, assim passando por revoluções, guerras e sobrevivência. Descoberto por sua precocidade matemática por Henri Cartan, Grothendieck fez seu doutorado sob orientação de Laurent Schwartz e Jean Dieudonné. As principais contribuições dele são na área da topologia e na geometria algébrica, assim como na teoria das categorias. (...)
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  42.  8
    Book Review: Shaughan Lavine. Understanding the Infinite. [REVIEW]Colin McLarty - 1997 - Notre Dame Journal of Formal Logic 38 (2):314-324.
  43. Book Review: John Bell, "Toposes and Local Set Theories, an Introduction". [REVIEW]Colin Mclarty - 1990 - Notre Dame Journal of Formal Logic 31:150-161.
     
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  44. In Arithmetic.Colin Mclarty - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press. pp. 370.
     
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  45. Review of [Bell, 1988]. [REVIEW]Colin McLarty - 1990 - Notre Dame Journal of Formal Logic 31:151-161.