Results for 'Foundation of mathematics'

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  1.  46
    Non-Monotonic Set Theory as a Pragmatic Foundation of Mathematics.Peter Verdée - 2013 - Foundations of Science 18 (4):655-680.
    In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also (...)
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  2.  13
    Foundation of Mathematics Between Theory and Practice.Giorgio Venturi - 2014 - Philosophia Scientiae 18 (1):45-80.
    Je me propose dans cet article de traiter de la théorie des ensembles, non seulement comme fondement des mathématiques au sens traditionnel, mais aussi comme fondement de la pratique mathématique. De ce point de vue, je marque une distinction entre un fondement ensembliste standard, d'une nature ontologique, grâce auquel tout objet mathématique peut trouver un succédané ensembliste, et un fondement pratique, qui vise à expliquer les phénomènes mathématiques, en donnant des conditions nécessaires et suffisantes pour prouver les propositions mathématiques. Je (...)
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  3.  68
    Thomists and Thomas Aquinas on the Foundation of Mathematics.Armand Maurer - 1993 - Review of Metaphysics 47 (1):43 - 61.
    SOME MODERN THOMISTS claiming to follow the lead of Thomas Aquinas, hold that the objects of the types of mathematics known in the thirteenth century, such as the arithmetic of whole numbers and Euclidean geometry, are real entities. In scholastic terms they are not beings of reason but real beings. In his once-popular scholastic manual, Elementa Philosophiae Aristotelico-Thomisticae, Joseph Gredt maintains that, according to Aristotle and Thomas Aquinas, the object of mathematics is real quantity, either discrete quantity in (...)
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  4. What is Required of a Foundation for Mathematics?John Mayberry - 1994 - Philosophia Mathematica 2 (1):16-35.
    The business of mathematics is definition and proof, and its foundations comprise the principles which govern them. Modern mathematics is founded upon set theory. In particular, both the axiomatic method and mathematical logic belong, by their very natures, to the theory of sets. Accordingly, foundational set theory is not, and cannot logically be, an axiomatic theory. Failure to grasp this point leads obly to confusion. The idea of a set is that of an extensional plurality, limited and definite (...)
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  5.  51
    Quotient Completion for the Foundation of Constructive Mathematics.Maria Emilia Maietti & Giuseppe Rosolini - 2013 - Logica Universalis 7 (3):371-402.
    We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere hyperdoctrine for which we describe a notion of quotient completion. That notion includes the exact completion on a category with weak finite limits as an instance as well as examples from type theory that fall apart from this.
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  6. Reductions of Mathematics: Foundation or Horizon?Felix Mühlhölzer - 2019 - In Gabriele M. Mras, Paul Weingartner & Bernhard Ritter (eds.), Philosophy of Logic and Mathematics. Berlin, Boston: De Gruyter. pp. 327-341.
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  7. Reductions of Mathematics: Foundation or Horizon?Felix Mühlhölzer - 2019 - In Gabriele M. Mras, Paul Weingartner & Bernhard Ritter (eds.), Philosophy of Logic and Mathematics: Proceedings of the 41st International Ludwig Wittgenstein Symposium. De Gruyter. pp. 327-342.
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  8. Strict Finitism an Examination of Ludwig Wittgenstein's Remarks on the Foundation of Mathematics.Charles F. Kielkopf - 1970 - Mouton.
     
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  9.  36
    A Neglected Thomistic Text on the Foundation of Mathematics.Armand Maurer - 1959 - Mediaeval Studies 21 (1):185-192.
  10. A Neglected Thomistic Text on the Foundation of Mathematics. --.Armand A. Maurer & Thomas - 1959 - Pontifical Institute of Mediaeval Studies.
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  11. The Foundation of Mathematics.Pieter Jacobus van Heerden - 1968 - Wassenaar, Uitg. Wistik.
  12.  37
    Discussion on the Foundation of Mathematics.John W. Dawson - 1984 - History and Philosophy of Logic 5 (1):111-129.
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  13.  5
    Aron Gurwitsch's Ordinal Foundation of Mathematics and the Problem of Formalizing Ideational Abstraction.Gilbert T. Null & Roger A. Simons - 1981 - Journal of the British Society for Phenomenology 12 (2):164-174.
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  14.  14
    Leonard Greenberg. The ‘is’ of Identity in Definitions. ETC.: A Review of General Semantics, Vol. 1 , Pp. 109–111. - Charles Morris. Science and Discourse. Synthese , Vol. 5 , Pp. 296–308. - Brugt H. Kazemier. Remarks on Logical Positivism. Synthese , Vol. 5 , Pp. 327–332. - Arnold Reymond. Congrès de Berne de I'unité Et de la Méthode Dans les Sciences. Synthese , Vol. 5 , Pp. 475–485. - Anonymous. The Relativistic Standpoint with Regard to the Foundation of Mathematics. Synthese , Vol. 5 , Pp. 519–521. - Jean-Louis Destouches. Logique El Réalité. Synthese , Vol. 6 , Pp. 300–304. - F. Denk. Sprache, Modell Und Exaktheit. Synthese , Vol. 5 , Pp. 487–494. - P. H. Esser. Inaugural Address. English with French Abstract. Synthese , Vol. 7 , Pp. 16–22. - Karl Dürr. Logislik Als Forschungsmethode. Synthese , Vol. 5 , Pp. 27–31. - Louis van Haecht. Les Aspects Psychologique Et Logique de I'analyse du Langage. Synthese , Vol. 5 , Pp. 100–108. [REVIEW]Alonzo Church - 1950 - Journal of Symbolic Logic 15 (3):236-236.
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  15.  9
    Special Section: Computability Theory and the Foundation of Mathematics.Chi Tat Chong & Stephen G. Simpson - 2017 - Annals of the Japan Association for Philosophy of Science 25:23-24.
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  16.  11
    Review: Zyoiti Suetuna, On the Intuition Which Forms the Foundation of Mathematics[REVIEW]M. Kondô - 1952 - Journal of Symbolic Logic 17 (1):63-63.
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  17.  11
    Review: Alonzo Church, The Present Situation in the Foundation of Mathematics[REVIEW]Saunders MacLane - 1940 - Journal of Symbolic Logic 5 (2):78-78.
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  18.  10
    The Trend of Logic and Foundation of Mathematics in Japan in 1991 to 1996.Yuzuru Kakuda, Kanji Namba & Nobuyoshi Motohashi - 1997 - Annals of the Japan Association for Philosophy of Science 9 (2):95-110.
  19.  10
    Pierre Joray (Ed.), Contemporary Perspectives on Logicism and the Foundation of Mathematics. Switzerland: Centre de Recherches Semiologiques Universite de Neuchaˆtel, 2007. VI Þ 208 Pp. Issn 1420-8520, No. 18. [REVIEW]Gregory Landini - 2008 - History and Philosophy of Logic 29 (4):383.
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  20.  16
    New Programs and Open Problems in the Foundation of Mathematics.G. Longo & P. Scott - 2003 - Bulletin of Symbolic Logic 9 (2):129-130.
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  21.  6
    Fraenkel Abraham A.. The Recent Controversies About the Foundation of Mathematics. Scripta Mathematica, Vol. 13 Nos. 1–2 , Pp. 17–36. [REVIEW]A. R. Turquette - 1948 - Journal of Symbolic Logic 13 (1):56-56.
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  22.  7
    Review: L. P. Gokieli, On the Problem of the Axiomatization of Logic; L. P. Gokieli, The Mathematical Manuscripts of Karl Marx and Problems of the Foundation of Mathematics[REVIEW]George L. Kline - 1950 - Journal of Symbolic Logic 14 (4):243-244.
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  23.  4
    Review: Kiyoshi Ito, Set Theory as Foundation of Mathematics[REVIEW]S. Kuroda - 1956 - Journal of Symbolic Logic 21 (1):95-95.
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  24.  3
    Church Alonzo. The Present Situation in the Foundation of Mathematics. Philosophie Mathematique, by Gonseth F., Actualités Scientifiques Et Induatrielles 837, Hermann Et Cie, Paris 1939, Pp. 67–72. [REVIEW]Saunders MacLane - 1940 - Journal of Symbolic Logic 5 (2):78-78.
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  25.  4
    Review: Abraham A. Fraenkel, The Recent Controversies About the Foundation of Mathematics[REVIEW]A. R. Turquette - 1948 - Journal of Symbolic Logic 13 (1):56-56.
  26.  2
    The Recent Controversies About the Foundation of Mathematics.Abraham A. Fraenkel - 1948 - Journal of Symbolic Logic 13 (1):56-56.
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  27.  37
    The Ideological Background of the Foundation of Mathematics.G. Mannoury - 1955 - Synthese 10 (1):315 - 317.
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  28.  8
    Foundation of Mathematics Between Theory and Practice.Giorgio Venturi - 2014 - Philosophia Scientae 18:4580.
  29.  7
    Beyond Wittgenstein's Remarks on the Foundation of Mathematics: Explication of Piaget's Suggestion of a Biological Foundation.Hans-Otto Carmesin - 1992 - Science & Education 1 (2):205-215.
  30. The Relativistic Standpoint with Regard to the Foundation of Mathematics.G. Mannoury - 1947 - Synthese 5 (11-12):519-521.
     
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  31.  49
    The Sensible Foundation for Mathematics: A Defense of Kant's View.Mark Risjord - 1990 - Studies in History and Philosophy of Science Part A 21 (1):123-143.
  32.  32
    Kurt Gödel. Diskussion Zur Grundlegung der Mathematik . A Reprint of 4184. Collected Works, Volume I, Publications 1929– 1936, by Kurt Gödel, Edited by Solomon Feferman, John W. DawsonJr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort, Clarendon Press, Oxford University Press, New York and Oxford1986, Pp. 200, 202. - Kurt Gödel. Discussion on Providing a Foundation for Mathematics . English Translation by John Dawson of the Preceding. Collected Works, Volume I, Publications 1929– 1936, by Kurt Gödel, Edited by Solomon Feferman, John W. DawsonJr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort, Clarendon Press, Oxford University Press, New York and Oxford1986, Pp. 201, 203. , Pp. 125-126.) - Kurt Gödel. Nachtrag. A Reprint of 4185. Collected Works, Volume I, Publications 1929– 1936, by Kurt Gödel, Edited by Solomon Feferman, John W. DawsonJr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoor. [REVIEW]Martin Davis - 1990 - Journal of Symbolic Logic 55 (1):343-343.
  33.  23
    F. William Lawvere. The Category of Categories as a Foundation for Mathematics. Proceedings of the Conference on Categorical Algebra, La Jolla 1965, Edited by S. Eilenberg, D. K. Harrison, S. MacLane, and H. Röhrl, Springer-Verlag New York Inc., New York 1966, Pp. 1–20. [REVIEW]Calvin C. Elgot - 1974 - Journal of Symbolic Logic 39 (2):341.
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  34.  18
    Foreword. Bibliography of Polish Mathematics 1944–1954, Translated Reprint From the Roczniki Polskiego Towarzystwa Matematycznego, Seria II, Wiadomości Matematyczne, Published for the Department of Commerce and the National Science Foundation, Washington, D.C., on the Order of Centralny Instytut Informacji Naukowo-Technicznej I Ekonomicznej, by Państwowe Wydawnictwo Naukowe, Warsaw 1963 , Pp. 1–2. - A. Mostowski and J. Łoś. I. Foundations of Mathematics, Theory of Sets and Mathematical Logic. Bibliography of Polish Mathematics 1944–1954, Translated Reprint From the Roczniki Polskiego Towarzystwa Matematycznego, Seria II, Wiadomości Matematyczne, Published for the Department of Commerce and the National Science Foundation, Washington, D.C., on the Order of Centralny Instytut Informacji Naukowo-Technicznej I Ekonomicznej, by Państwowe Wydawnictwo Naukowe, Warsaw 1963 , Pp. 4–17. - S. Drobot and S. Straszewicz. XI. History, Teaching, Popularization and Organization of Mathematics. Bibliog. [REVIEW]Alonzo Church - 1966 - Journal of Symbolic Logic 31 (3):517-517.
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  35.  15
    Aubert Daigneault. Introduction. Studies in Algebraic Logic, Edited by Aubert Daigneault, Studies in Mathematics, Vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, Pp. 1–5. - William Craig. Unification and Abstraction in Algebraic Logic. Studies in Algebraic Logic, Edited by Aubert Daigneault, Studies in Mathematics, Vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, Pp. 6–57. - J. Donald Monk. Connections Between Combinatorial Theory and Algebraic Logic. Studies in Algebraic Logic, Edited by Aubert Daigneault, Studies in Mathematics, Vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, Pp. 58–91. - Helena Rasiowa. Post Algebras as a Semantic Foundation of M-Valued Logics. Studies in Algebraic Logic, Edited by Aubert Daigneault, Studies in Mathematics, Vol. 9, The Mathematical Association of America, [Washington, D.C.], 1974, Pp. 92–142. - Gonzalo E. Reyes. From Sheaves to Logic. Studies in Algebraic Logic, Edited B. [REVIEW]Anne Preller - 1978 - Journal of Symbolic Logic 43 (1):145-147.
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  36.  9
    Review: F. William Lawvere, S. Eilenberg, D. K. Harrison, S. MacLane, H. Rohrl, The Category of Categories as a Foundation for Mathematics[REVIEW]Calvin C. Elgot - 1974 - Journal of Symbolic Logic 39 (2):341-341.
  37.  16
    Gender Differences in Pupil Attitudes to the National Curriculum Foundation Subjects of English, Mathematics, Science and Technology in Key Stage 3 in South Wales.Dave Hendley, John Parkinson, Andrew Stables & Howard Tanner - 1995 - Educational Studies 21 (1):85-97.
    The attitudes of pupils in South Wales to mathematics, English, science and technology were tested using a Likert‐type attitude scale. Pupils were selected from National Curriculum Key Stage 3, specifically Year 9 . Schools were selected by their position in the 1992 National League tables produced by the Welsh Office, the schools being placed into one of four bands. The number of schools involved was 34 and the number of pupils 4263. This represents 15.3% of the total population of (...)
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  38.  64
    Husserl's Epistemology of Mathematics and the Foundation of Platonism in Mathematics.Guillermo E. Rosado Handdock - 1987 - Husserl Studies 4 (2):81-102.
  39.  36
    Depth and ClarityFelix Mühlhölzer. Braucht Die Mathematik Eine Grundlegung? Eine Kommentar des Teils III von Wittgensteins Bemerkungen Über Die Grundlagen der Mathematik [Does Mathematics Need a Foundation? A Commentary on Part III of Wittgenstein's Remarks on the Foundations of Mathematics]. Frankfurt: Vittorio Klostermann, 2010. ISBN: 978-3-465-03667-8. Pp. Xiv + 602. [REVIEW]Juliet Floyd - 2015 - Philosophia Mathematica 23 (2):255-276.
  40. Leibnizian Mathematics and Physics-(2e Partie) Divine Immutability as the Foundation of Nature Laws in Descartes and the Arguments Involved in Leibnizs Criticism.Laurence Devillairs - 2001 - Revue d'Histoire des Sciences 54 (3):303-324.
  41.  67
    Twin Paradox and the Logical Foundation of Relativity Theory.Judit X. Madarász, István Németi & Gergely Székely - 2006 - Foundations of Physics 36 (5):681-714.
    We study the foundation of space-time theory in the framework of first-order logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for space-time theory (or relativity). First we recall a simple and streamlined FOL-axiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove the usual (...)
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  42.  28
    Wigner’s Puzzle on Applicability of Mathematics: On What Table to Assemble It?Cătălin Bărboianu - 2019 - Axiomathes 1:1-30.
    Attempts at solving what has been labeled as Eugene Wigner’s puzzle of applicability of mathematics are still far from arriving at an acceptable solution. The accounts developed to explain the “miracle” of applied mathematics vary in nature, foundation, and solution, from denying the existence of a genuine problem to designing structural theories based on mathematical formalism. Despite this variation, all investigations treated the problem in a unitary way with respect to the target, pointing to one or two (...)
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  43.  14
    Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts.Stefania Centrone, Deborah Kant & Deniz Sarikaya (eds.) - 2019 - Springer Verlag.
    This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. -/- The first two sections focus on the two most prominent candidate theories for a foundation of mathematics. Readers may (...)
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  44. The Importance of Developing a Foundation for Naive Category Theory.Marcoen J. T. F. Cabbolet - 2015 - Thought: A Journal of Philosophy 4 (4):237-242.
    Recently Feferman has outlined a program for the development of a foundation for naive category theory. While Ernst has shown that the resulting axiomatic system is still inconsistent, the purpose of this note is to show that nevertheless some foundation has to be developed before naive category theory can replace axiomatic set theory as a foundational theory for mathematics. It is argued that in naive category theory currently a ‘cookbook recipe’ is used for constructing categories, and it (...)
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  45. From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s.Paolo Mancosu (ed.) - 1998 - Oxford University Press.
    From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important (...)
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  46.  63
    Deleuze and the History of Mathematics: In Defense of the New.Simon B. Duffy - 2013 - Bloomsbury Academic.
    Gilles Deleuze’s engagements with mathematics, replete in his work, rely upon the construction of alternative lineages in the history of mathematics, which challenge some of the self imposed limits that regulate the canonical concepts of the discipline. For Deleuze, these challenges provide an opportunity to reconfigure particular philosophical problems – for example, the problem of individuation – and to develop new concepts in response to them. The highly original research presented in this book explores the mathematical construction of (...)
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  47.  29
    The Conceptual Roots of Mathematics: An Essay on the Philosophy of Mathematics.J. R. Lucas - 2000 - Routledge.
    The Conceptual Roots of Mathematics is a comprehensive study of the foundation of mathematics. Lucas, one of the most distinguished Oxford scholars, covers a vast amount of ground in the philosophy of mathematics, showing us that it is actually at the heart of the study of epistemology and metaphysics.
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  48.  8
    Arguments and Elements of Realistic Interpretation of Mathematics: Arithmetical Component.E. I. Arepiev & V. V. Moroz - 2015 - Liberal Arts in Russiaроссийский Гуманитарный Журналrossijskij Gumanitarnyj Žurnalrossijskij Gumanitaryj Zhurnalrossiiskii Gumanitarnyi Zhurnal 4 (3):198.
    The prospects for realistic interpretation of the nature of initial mathematical truths and objects are considered in the article. The arguments of realism, reasons impeding its recognition among philosophers of mathematics as well as the ways to eliminate these reasons are discussed. It is proven that the absence of acceptable ontological interpretation of mathematical realism is the main obstacle to its recognition. This paper explicates the introductory positions of this interpretation and presents a realistic interpretation of the arithmetical component (...)
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  49. Second-Order Logic and Foundations of Mathematics.Jouko Väänänen - 2001 - Bulletin of Symbolic Logic 7 (4):504-520.
    We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are (...)
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  50. Does Homotopy Type Theory Provide a Foundation for Mathematics?James Ladyman & Stuart Presnell - 2016 - British Journal for the Philosophy of Science:axw006.
    Homotopy Type Theory is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory. This article explains and motivates an account of how to define, justify, and think about HoTT in a way that is self-contained, and argues that, so construed, it is a candidate for being an autonomous foundation for mathematics. We first consider various questions that a (...) for mathematics might be expected to answer, and find that many of them are not answered by the standard formulation of HoTT as presented in the ‘HoTT Book’. More importantly, the presentation of HoTT given in the HoTT Book is not autonomous since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon ideas from other parts of mathematics, and in particular makes no reference to homotopy theory, and argue that it is a candidate autonomous foundation for mathematics. Our elaboration of HoTT is based on a new interpretation of types as mathematical concepts, which accords with the intensional nature of the type theory. 1 Introduction2 What Is a Foundation for Mathematics?2.1 A characterization of a foundation for mathematics2.2 Autonomy3 The Basic Features of Homotopy Type Theory3.1 The rules3.2 The basic ways to construct types3.3 Types as propositions and propositions as types3.4 Identity3.5 The homotopy interpretation4 Autonomy of the Standard Presentation?5 The Interpretation of Tokens and Types5.1 Tokens as mathematical objects?5.2 Tokens and types as concepts6 Justifying the Elimination Rule for Identity7 The Foundations of Homotopy Type Theory without Homotopy7.1 Framework7.2 Semantics7.3 Metaphysics7.4 Epistemology7.5 Methodology8 Possible Objections to this Account8.1 A constructive foundation for mathematics?8.2 What are concepts?8.3 Isn’t this just Brouwerian intuitionism?8.4 Duplicated objects8.5 Intensionality and substitution salva veritate9 Conclusion9.1 Advantages of this foundation. (shrink)
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