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Colin McLarty [51]Colin Slator Mclarty [1]
  1. Numbers can be just what they have to.Colin McLarty - 1993 - Noûs 27 (4):487-498.
  2. 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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  3. 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.  20
    Elementary Categories, Elementary Toposes.Colin McLarty - 1991 - Oxford, England: 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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  5. 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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  6. 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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  7.  48
    ‘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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  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. Fluid Mechanics for Philosophers, or Which Solutions Do You Want for Navier-Stokes?Colin McLarty - 2023 - In Lydia Patton & Erik Curiel (eds.), Working Toward Solutions in Fluid Dynamics and Astrophysics: What the Equations Don’t Say. Springer Verlag. pp. 31-56.
    Of the seven $1,000,000 Clay Millennium Prize Problems in mathematics, just one would immediately appeal to Leonard Euler. That is “Existence and Smoothness of the Navier-Stokes Equation” (Fefferman 2000). Euler gave the basic equation in the 1750s. The work to this day shows Euler’s intuitive, vividly physical sense of mathematics.
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  10. 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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  11.  70
    Defining sets as sets of points of spaces.Colin McLarty - 1988 - Journal of Philosophical Logic 17 (1):75 - 90.
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  12.  57
    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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  13.  61
    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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  14.  18
    Structuralism in differential equations.Colin McLarty - 2024 - Synthese 203 (3):1-15.
    Structuralism in philosophy of mathematics has largely focused on arithmetic, algebra, and basic analysis. Some have doubted whether distinctively structural working methods have any impact in other fields such as differential equations. We show narrowly construed structuralism as offered by Benacerraf has no practical role in differential equations. But Dedekind’s approach to the continuum already did not fit that narrow sense, and little of mathematics today does. We draw on one calculus textbook, one celebrated analysis textbook, and a monograph on (...)
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  15.  55
    Failure of cartesian closedness in NF.Colin McLarty - 1992 - Journal of Symbolic Logic 57 (2):555-556.
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  16.  48
    The large structures of grothendieck founded on finite-order arithmetic.Colin Mclarty - 2020 - Review of Symbolic Logic 13 (2):296-325.
    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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  17.  44
    Elaine Landry.*Plato Was Not a Mathematical Platonist.Colin McLarty - 2023 - Philosophia Mathematica 31 (3):417-424.
    This book goes far beyond its title. Landry indeed surveys current definitions of “mathematical platonism” to show nothing like them applies to Socrates in Plat.
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  18. Philosophical Relevance of Category Theory.Colin McLarty - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford, England: Oxford University Press.
     
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  19. 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, England: Oxford University Press. pp. 187--208.
     
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  20.  91
    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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  21.  28
    Two Constructivist Aspects of Category Theory.Colin McLarty - 2006 - Philosophia Scientiae:95-114.
    Category theory has two unexpected links to constructivism: First, why is topos logic so close to intuitionistic logic? The paper argues that in part the resemblance is superficial, in part it is due to selective attention, and in part topos theory is objectively tied to the motives for later intuitionistic logic little related to Brouwer’s own stated motives. Second, why is so much of general category theory somehow constructive? The paper aims to synthesize three hypotheses on why it would be (...)
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  22.  22
    Two Constructivist Aspects of Category Theory.Colin McLarty - 2006 - Philosophia Scientiae:95-114.
    Category theory has two unexpected links to constructivism: First, why is topos logic so close to intuitionistic logic? The paper argues that in part the resemblance is superficial, in part it is due to selective attention, and in part topos theory is objectively tied to the motives for later intuitionistic logic little related to Brouwer’s own stated motives. Second, why is so much of general category theory somehow constructive? The paper aims to synthesize three hypotheses on why it would be (...)
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  23.  12
    Voir-Dire in the Case of Mathematical Progress.Colin McLarty - 2000 - In Emily Grosholz & Herbert Breger (eds.), The growth of mathematical knowledge. Boston: Kluwer Academic Publishers. pp. 269--280.
  24.  3
    Fermat’s Last Theorem.Colin McLarty - 2024 - In Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 2011-2033.
    For 300 years, Fermat’s Last Theorem seemed to be pure arithmetic little connected even to other problems in arithmetic. But the last decades of the twentieth century saw the discovery of very special cubic curves, and the rise of the huge theoretical Langlands Program. The Langlands perspective showed those curves are so special they cannot exist, and thus proved Fermat’s Last Theorem. With many great contributors, the proof ended in a deep and widely applicable geometric result relating nice curves in (...)
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  25.  46
    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.
  26.  37
    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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  27.  17
    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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  28.  13
    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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  29.  60
    Elementary axioms for canonical points of toposes.Colin McLarty - 1987 - Journal of Symbolic Logic 52 (1):202-204.
  30.  22
    Emmy Noether’s first great mathematics and the culmination of first-phase logicism, formalism, and intuitionism.Colin McLarty - 2011 - Archive for History of Exact Sciences 65 (1):99-117.
    Emmy Noether’s many articles around the time that Felix Klein and David Hilbert were arranging her invitation to Göttingen include a short but brilliant note on invariants of finite groups highlighting her creativity and perspicacity in algebra. Contrary to the idea that Noether abandoned Paul Gordan’s style of mathematics for Hilbert’s, this note shows her combining them in a way she continued throughout her mature abstract algebra.
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  31. in Arithmetic.Colin Mclarty - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford, England: Oxford University Press. pp. 370.
     
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  32.  21
    Mathematics as a love of wisdom: Saunders Mac Lane as philosopher.Colin McLarty - 2020 - Philosophical Problems in Science 69:17-32.
    This note describes Saunders Mac Lane as a philosopher, and indeed as a paragon naturalist philosopher. He approaches philosophy as a mathematician. But, more than that, he learned philosophy from David Hilbert’s lectures on it, and by discussing it with Hermann Weyl, as much as he did by studying it with the mathematically informed Göttingen Philosophy professor Moritz Geiger.
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  33.  25
    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.
    The creation of algebraic topology required “all the energy and the temperament of Emmy Noether” according to topologists Paul Alexandroff and Heinz Hopf. Alexandroff stressed Noether's radical pro-Russian politics, which her colleagues found in “poor taste”; yet he found “a bright trait of character.” She joined the Independent Social Democrats in 1919. They were tiny in Göttingen until that year when their vote soared as they called for a dictatorship of the proletariat. The Minister of the Army and many Göttingen (...)
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  34.  55
    Saunders Mac Lane (1909–2005): His mathematical life and philosophical works.Colin McLarty - 2005 - Philosophia Mathematica 13 (3):237-251.
  35.  42
    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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  36. 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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  37.  60
    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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  38.  28
    Book Review: Shaughan Lavine. Understanding the Infinite. [REVIEW]Colin McLarty - 1997 - Notre Dame Journal of Formal Logic 38 (2):314-324.
  39.  23
    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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  40.  14
    British Journal for the Philosophy of Science. [REVIEW]Colin McLarty - 2003 - Bulletin of Symbolic Logic 9 (1):43-44.
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  41.  9
    Book review: John Bell, "Toposes and Local Set Theories, an Introduction". [REVIEW]Colin McLarty - 1989 - Notre Dame Journal of Formal Logic 31:150-161.
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  42.  20
    La notion de nombre chez Dedekind, Cantor, Frege: Theories, conceptions, et philosophie by Jean-Pierre Belna. [REVIEW]Colin Mclarty - 1998 - Isis 89:145-146.
  43.  18
    La notion de nombre chez Dedekind, Cantor, Frege: Theories, conceptions, et philosophie. Jean-Pierre Belna. [REVIEW]Colin McLarty - 1998 - Isis 89 (1):145-146.
  44.  22
    La topologie et ses signes: Éléments pour une histoire sémiotique des mathématiques. [REVIEW]Colin Mclarty - 2002 - Isis 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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  45.  21
    Mathematics: Form and Function by Saunders MacLane. [REVIEW]Colin McLarty - 1987 - Journal of Philosophy 84 (1):33-37.
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  46.  19
    J. P. Mayberry, _The Foundations Of Mathematics In The Theory Of Sets. Encyclopedia Of Mathematics And Its Applications Ser._ , Vol. 82. Cambridge: Cambridge University Press (2000), xx+429 pp., index, cloth $80.00 (cloth). [REVIEW]Colin McLarty - 2002 - Philosophy of Science 69 (2):404-406.
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  47. Review of [Bell, 1988]. [REVIEW]Colin McLarty - 1990 - Notre Dame Journal of Formal Logic 31:151-161.
  48.  36
    Review of Charles S. Chihara, A Structural Account of Mathematics[REVIEW]Colin McLarty - 2004 - Notre Dame Philosophical Reviews 2004 (8).
  49.  31
    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.