Results for 'Categories (Mathematics'

103 found
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  1.  37
    Between Logic and Mathematics: Al-Kindī's Approach to the Aristotelian Categories.Ahmad Ighbariah - 2012 - Arabic Sciences and Philosophy 22 (1):51-68.
    What is the function of logic in al-Kind's theory of categories as it was presented in his epistle On the Number of Aristotle's Books and Quality, whereas the rest of the categories are thought to be no more than different combinations of these two categories with the category Substance. The discussion will pay special attention to the function of the categories of Quantity and Quality as mediators between logic and mathematics.
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  2. Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-Mathematics.Stewart Shapiro - 2005 - Philosophia Mathematica 13 (1):61-77.
    There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the (...)
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  3. Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-Mathematics &Dagger.Stewart Shapiro - 2005 - Philosophia Mathematica 13 (1):61-77.
    There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the (...)
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  4.  6
    Inconsistent Mathematics.Category Theory.Closed Set Sheaves and Their Categories.Foundations: Provability, Truth and Sets. [REVIEW]Newton C. A. da Costa, Otavio Bueno, Chris Mortensen, Peter Lavers, William James & Joshua Cole - 1997 - Journal of Symbolic Logic 62 (2):683.
  5. Review of F. W. Lawvere and S. H. Schanuel, Conceptual Mathematics: A First Introduction to Categories; and J. L. Bell, A Primer of Infinitesimal Analysis. [REVIEW]D. Corfield - 2002 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 33 (2):359-366.
     
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  6.  21
    Conceptual Mathematics: A First Introduction to Categories.David Corfield - 2002 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 33 (2):359-366.
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  7.  2
    Lawvere F. William. Algebraic Theories, Algebraic Categories, and Algebraic Functors. The Theory of Models, Proceedings of the 1963 International Symposium at Berkeley, Edited by Addison J. W., Henkin Leon, and Tarski Alfred, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam 1965, Pp. 413–418. [REVIEW]Anne Preller - 1971 - Journal of Symbolic Logic 36 (2):336-337.
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  8.  2
    Conceptual Mathematics: A First Introduction to Categories.David Corfield - 2002 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 33 (2):359-366.
  9.  1
    Lawvere F. William. The Category of Categories as a Foundation for Mathematics. Proceedings of the Conference on Categorical Algebra, La Jolla 1965, Edited by Eilenberg S., Harrison D. K., MacLane S., and Röhrl H., 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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  10.  1
    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.
  11. Makkai Michael and Paré Robert. Accessible Categories: The Foundations of Categorical Model Theory. Contemporary Mathematics, Vol. 104. American Mathematical Society, Providence 1989, Viii + 176 Pp. [REVIEW]Andreas Blass - 1993 - Journal of Symbolic Logic 58 (1):355-357.
  12. Hiż Henry. Congrammaticality, Batteries of Transformations and Grammatical Categories. Structure of Language and its Mathematical Aspects, Proceedings of Symposia in Applied Mathematics, Vol. 12, American Mathematical Society, Providence 1961, Pp. 43–50.Hiż H.. The Intuitions of Grammatical Categories. Methodos, Vol. 12 , Pp. 311–319. [REVIEW]G. H. Matthews - 1967 - Journal of Symbolic Logic 32 (1):115-116.
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  13. Van Benthem Johan. Language in Action. Categories, Lambdas and Dynamic Logic. Studies in Logic and the Foundations of Mathematics, Vol. 130. North-Holland, Amsterdam, New York, Etc., 1991, X + 349 Pp. [REVIEW]Richard T. Oehrle - 1993 - Journal of Symbolic Logic 58 (4):1472-1475.
  14. Lawvere F. William. Introduction to Part I. Model Theory and Topoi, A Collection of Lectures by Various Authors, Edited by Lawvere F. W., Maurer C., and Wraith G. C., Lecture Notes in Mathematics, Vol. 445, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, Pp. 3–14.Keane Orville. Abstract Horn Theories. Model Theory and Topoi, A Collection of Lectures by Various Authors, Edited by Lawvere F. W., Maurer C., and Wraith G. C., Lecture Notes in Mathematics, Vol. 445, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, Pp. 15–50.Volger Hugo. Completeness Theorem for Logical Categories. Model Theory and Topoi, A Collection of Lectures by Various Authors, Edited by Lawvere F. W., Maurer C., and Wraith G. C., Lecture Notes in Mathematics, Vol. 445, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, Pp. 51–86.Volger Hugo. Logical Categories, Semantical Categories and Topoi. Model Theory and Topoi, A Collection of Lectures by Various Authors, Edited by Lawvere F. W., Maurer C.,. [REVIEW]M. E. Szabo - 1981 - Journal of Symbolic Logic 46 (1):158-161.
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  15.  4
    Purity, Spectra and Localisation.Mike Prest - 2009 - Cambridge University Press.
    The central aim of this book is to understand modules and the categories they form through associated structures and dimensions, which reflect the complexity of these, and similar, categories.
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  16.  8
    Algebra of Proofs.M. E. Szabo - 1978 - Sole Distributors for the U.S.A. And Canada, Elsevier North-Holland.
    Provability, Computability and Reflection.
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  17. Fundamentals of Computation Theory: Proceedings of the 1977 International Fct-Conference, Poznán-Kórnik, Poland, September 19-23, 1977. [REVIEW]Marek Karpiński (ed.) - 1977 - Springer Verlag.
     
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  18.  95
    Category Theory: The Language of Mathematics.Elaine Landry - 1999 - Philosophy of Science 66 (3):27.
    In this paper I argue that category theory ought to be seen as providing the language for mathematical discourse. Against foundational approaches, I argue that there is no need to reduce either the content or structure of mathematical concepts and theories to the constituents of either the universe of sets or the category of categories. I assign category theory the role of organizing what we say about the content and structure of both mathematical concepts and theories. Insofar, then, as (...)
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  19.  12
    Categories of Large Numbers in Line Estimation.Landy David, Charlesworth Arthur & Ottmar Erin - 2017 - Cognitive Science 41 (2):326-353.
    How do people stretch their understanding of magnitude from the experiential range to the very large quantities and ranges important in science, geopolitics, and mathematics? This paper empirically evaluates how and whether people make use of numerical categories when estimating relative magnitudes of numbers across many orders of magnitude. We hypothesize that people use scale words—thousand, million, billion—to carve the large number line into categories, stretching linear responses across items within each category. If so, discontinuities in position and (...)
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  20.  94
    From Sets to Types to Categories to Sets.Steve Awodey - 2009 - Philosophical Explorations.
    Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each system into the others and examining the preservation and loss of mathematical content thereby. In order to stay focused on the “big picture”, we merely sketch the overall form of each construction, referring to the (...)
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  21.  38
    Boltzmann on Mathematics.Setsuko Tanaka - 1999 - Synthese 119 (1-2):203-232.
    Boltzmann’s lectures on natural philosophy point out how the principles of mathematics are both an improvement on traditional philosophy and also serve as a necessary foundation of physics or what the English call “Natura Philosophy”, a title which he will retain for his own lectures. We start with lecture #3 and the mathematical contents of his lectures plus a few philosophical comments. Because of the length of the lectures as a whole we can only give the main points of each (...)
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  22.  2
    Non-Well-Founded Trees in Categories.Benno van den Berg & Federico De Marchi - 2007 - Annals of Pure and Applied Logic 146 (1):40-59.
    Non-well-founded trees are used in mathematics and computer science, for modelling non-well-founded sets, as well as non-terminating processes or infinite data structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call M-types. We derive existence results for M-types in locally cartesian closed pretoposes with a natural numbers object, using their internal logic. These are then used to prove stability of such categories with M-types under various topos-theoretic constructions; namely, slicing, formation of coalgebras , and sheaves for (...)
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  23.  23
    Le Pragmatisme Peircéen, la Théorie des Catégories Et le Programme de Thiel.Ralf Krömer - 2005 - Philosophia Scientiae 9 (2):79-96.
    Category theory is important by its mathematical applications and by the philosophical debates it causes. It is used to express in algebraic topology, to deduce in homological algebra and, as an alternative to the theory of sets, to construct objects in Grothendieck’s conception of algebraic geometry. Category theory is a fundamental discipline in Christian Thiel’s sense, because it is a theory of some typical operations of structural mathematics. This thesis is defended through a particular interpretation of peircean pragmatism; in this (...)
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  24. Kant: Studies on Mathematics in the Critical Philosophy.Emily Carson & Lisa Shabel (eds.) - 2015 - Routledge.
    There is a long tradition, in the history and philosophy of science, of studying Kant’s philosophy of mathematics, but recently philosophers have begun to examine the way in which Kant’s reflections on mathematics play a role in his philosophy more generally, and in its development. For example, in the Critique of Pure Reason , Kant outlines the method of philosophy in general by contrasting it with the method of mathematics; in the Critique of Practical Reason , Kant compares the Formula (...)
     
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  25. Mathematics in Kant's Critical Philosophy.Emily Carson & Lisa Shabel (eds.) - 2015 - Routledge.
    There is a long tradition, in the history and philosophy of science, of studying Kant’s philosophy of mathematics, but recently philosophers have begun to examine the way in which Kant’s reflections on mathematics play a role in his philosophy more generally, and in its development. For example, in the Critique of Pure Reason , Kant outlines the method of philosophy in general by contrasting it with the method of mathematics; in the Critique of Practical Reason , Kant compares the Formula (...)
     
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  26. The Role of Intuition in Kant's Philosophy of Mathematics and Theory of Magnitudes.Daniel Sutherland - 1998 - Dissertation, University of California, Los Angeles
    The way in which mathematics relates to experience has deeply engaged philosophers from the scientific revolution to the present. It has strongly influenced their views on epistemology, mathematics, science, and the nature of reality. Kant's views on the nature of mathematics and its relation to experience both influence and are influenced by his epistemology, and in particular the distinction Kant draws between concepts and intuitions. My dissertation contributes to clarifying the role of intuition in Kant's theory of mathematical cognition. It (...)
     
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  27.  5
    Book Review: On the Real Possibilities of Continuity. [REVIEW]Thomas Riese - 2008 - Biosemiotics 1 (2):271-279.
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  28. Idee Und Menge: Der Aufbau E. Kategorialen Ontologie Als Folge Aud D. Paradozien D Bergriffsrealismus in D Griech.Rainer Carls - 1974 - Berchmannskolleg-Verlag.
     
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  29. Part II. Themes: Introduction.Riccardo Martinelli - 2015 - In Denis Fisette & Riccardo Martinelli (eds.), Philosophy from an Empirical Standpoint. Essays on Carl Stumpf. Brill. pp. 145-149.
    An Introduction to the main themes of Carl Stumpf's Philosophy.
     
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  30. A Conceptual Construction of Complexity Levels Theory in Spacetime Categorical Ontology: Non-Abelian Algebraic Topology, Many-Valued Logics and Dynamic Systems. [REVIEW]R. Brown, J. F. Glazebrook & I. C. Baianu - 2007 - Axiomathes 17 (3-4):409-493.
    A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Super-complex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item’s essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures that (...)
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  31. 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 is explicitly (...)
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  32.  70
    Proof-Theoretic Semantics for Classical Mathematics.William W. Tait - 2006 - Synthese 148 (3):603-622.
    We discuss the semantical categories of base and object implicit in the Curry-Howard theory of types and we derive derive logic and, in particular, the comprehension principle in the classical version of the theory. Two results that apply to both the classical and the constructive theory are discussed. First, compositional semantics for the theory does not demand ‘incomplete objects’ in the sense of Frege: bound variables are in principle eliminable. Secondly, the relation of extensional equality for each type is (...)
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  33. Sets, Classes, and Categories.F. A. Muller - 2001 - British Journal for the Philosophy of Science 52 (3):539-573.
    This paper, accessible for a general philosophical audience having only some fleeting acquaintance with set-theory and category-theory, concerns the philosophy of mathematics, specifically the bearing of category-theory on the foundations of mathematics. We argue for six claims. (I) A founding theory for category-theory based on the primitive concept of a set or a class is worthwile to pursue. (II) The extant set-theoretical founding theories for category-theory are conceptually flawed. (III) The conceptual distinction between a set and a class can be (...)
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  34.  84
    Categories Without Structures.Andrei Rodin - 2011 - Philosophia Mathematica 19 (1):20-46.
    The popular view according to which category theory provides a support for mathematical structuralism is erroneous. Category-theoretic foundations of mathematics require a different philosophy of mathematics. While structural mathematics studies ‘invariant form’ (Awodey) categorical mathematics studies covariant and contravariant transformations which, generally, have no invariants. In this paper I develop a non-structuralist interpretation of categorical mathematics.
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  35.  67
    Categories for the Working Mathematician: Making the Impossible Possible.Jessica Carter - 2008 - Synthese 162 (1):1-13.
    This paper discusses the notion of necessity in the light of results from contemporary mathematical practice. Two descriptions of necessity are considered. According to the first, necessarily true statements are true because they describe ‘unchangeable properties of unchangeable objects’. The result that I present is argued to provide a counterexample to this description, as it concerns a case where objects are moved from one category to another in order to change the properties of these objects. The second description concerns necessary (...)
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  36.  59
    Catégories formelles, nombres et conceptualisme. La première philosophie de l’arithmétique de Husserl.Arkadiusz Chrudzimski - 2009 - Philosophiques 36 (2):427-445.
    Résumé -/- Dans son premier livre (Philosophie de l’arithmétique 1891), Husserl élabore une très intéressante philosophie des mathématiques. Les concepts mathématiques sont interprétés comme des concepts de « deuxième ordre » auxquels on accède par une réflexion sur nos opérations mentales de numération. Il s’ensuit que la vérité de la proposition : « il y a trois pommes sur la table » ne consiste pas dans une relation mythique quelconque avec la réalité extérieure au psychique (où le nombre trois doit (...)
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  37.  44
    Categories of Space and of Quantity.F. William Lawvere - 1992 - In Javier Echeverria, Andoni Ibarra & Thomas Mormann (eds.), The Space of Mathematics: Philosophical, Epistemological, and Historical Explorations. De Gruyter. pp. 14--30.
  38.  35
    Algebraic Biology: Creating Invariant Binding Relations for Biochemical and Biological Categories[REVIEW]Jerry L. R. Chandler - 2009 - Axiomathes 19 (3):297-320.
    The desire to understand the mathematics of living systems is increasing. The widely held presupposition that the mathematics developed for modeling of physical systems as continuous functions can be extended to the discrete chemical reactions of genetic systems is viewed with skepticism. The skepticism is grounded in the issue of scientific invariance and the role of the International System of Units in representing the realities of the apodictic sciences. Various formal logics contribute to the theories of biochemistry and molecular biology (...)
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  39.  6
    Memory Evolutive Systems.Ronald Brown - 2009 - Axiomathes 19 (3):271-280.
    This is a review of the book ‘Memory Evolutive Systems; Hierarchy, Emergence, Cognition’, by A. Ehresmann and J.P. Vanbremeersch. I welcome the use of category theory and the notion of colimit as a way of describing how complex hierarchical systems can be organised, and the notion of categories varying with time to give a notion of an evolving system. In this review I also point out the relation of the notion of colimit to ideas of communication; the necessity of (...)
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  40.  3
    The Problem of the Formation of New Categories in Dialectical Logic.E. P. Sitkovskii - 1976 - Russian Studies in Philosophy 15 (2):94-106.
    The process of the formation of new scientific concepts conditioned by scientific and technological progress often has the consequence that many concepts formed in special sciences - such as mathematics, physics, and biology - are immediately introduced by certain authors into the sphere of philosophy.
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  41. Some Reflections about Alain Badiou’s Approach to Platonism in Mathematics.Miriam Franchella - 2007 - Analytica 1:67-81.
    A reproach has been done many times to post-modernism: its picking up mathematical notions or results, mostly by misrepresenting their real content, in order to strike the readers and obtaining their assent only by impressing them . In this paper I intend to point out that although Alain Badiou’s approach to philosophy starts with taking distance both from analytic philosophy and from French post-modernism, the categories that he uses for labelling logicism, formalism and intuitionism do not reflect the real (...)
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  42.  91
    Does Category Theory Provide a Framework for Mathematical Structuralism?Geoffrey Hellman - 2003 - Philosophia Mathematica 11 (2):129-157.
    Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried (...)
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  43.  11
    What is a Higher-Level Set?Dimitris Tsementzis - forthcoming - Philosophia Mathematica:nkw032.
    Structuralist foundations of mathematics aim for an ‘invariant’ conception of mathematics. But what should be their basic objects? Two leading answers emerge: higher groupoids or higher categories. I argue in favor of the former over the latter. First, I explain why to choose between them we need to ask the question of what is the correct ‘categorified’ version of a set. Second, I argue in favor of groupoids over categories as ‘categorified’ sets by introducing a pre-formal understanding of (...)
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  44. Jakob Friedrich Fries (1773-1843): Eine Philosophie der Exakten Wissenschaften.Kay Herrmann - 1994 - Tabula Rasa. Jenenser Zeitschrift Für Kritisches Denken (6).
    Jakob Friedrich Fries (1773-1843): A Philosophy of the Exact Sciences -/- Shortened version of the article of the same name in: Tabula Rasa. Jenenser magazine for critical thinking. 6th of November 1994 edition -/- 1. Biography -/- Jakob Friedrich Fries was born on the 23rd of August, 1773 in Barby on the Elbe. Because Fries' father had little time, on account of his journeying, he gave up both his sons, of whom Jakob Friedrich was the elder, to the Herrnhut Teaching (...)
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  45.  90
    Models of Deduction.Kosta Dosen - 2006 - Synthese 148 (3):639-657.
    In standard model theory, deductions are not the things one models. But in general proof theory, in particular in categorial proof theory, one finds models of deductions, and the purpose here is to motivate a simple example of such models. This will be a model of deductions performed within an abstract context, where we do not have any particular logical constant, but something underlying all logical constants. In this context, deductions are represented by arrows in categories involved in a (...)
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  46.  60
    Category Theory and Set Theory as Theories About Complementary Types of Universals.David P. Ellerman - 2017 - Logic and Logical Philosophy 26 (2):1-18.
    Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_{F}={x|F(x)} for a property F() could never be self-predicative in the sense of u_{F}∈u_{F}. But the mathematical theory of (...), dating from the mid-twentieth century, includes a theory of always-self-predicative universals--which can be seen as forming the "other bookend" to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. (shrink)
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  47.  98
    Logic and Formal Ontology.Barry Smith - 2000 - Manuscrito 23 (2):275-323.
    Revised version of chapter in J. N. Mohanty and W. McKenna (eds.), Husserl’s Phenomenology: A Textbook, Lanham: University Press of America, 1989, 29–67. -/- Logic for Husserl is a science of science, a science of what all sciences have in common in their modes of validation. Thus logic deals with universal laws relating to truth, to deduction, to verification and falsification, and with laws relating to theory as such, and to what makes for theoretical unity, both on the side of (...)
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  48. Peirce: A Guide for the Perplexed.Cornelis de Waal - 2013 - Continuum.
    Charles Sanders Peirce, the founder of pragmatism, is a hugely important and influential thinker in the history of American philosophy. His philosophical interests were broad and he made significant contributions in several different areas of thought. Moreover, his contributions are intimately connected and his philosophy designed to form a coherent and systematic whole. Contents: 1: Life and Work; Chapter 2: Logic; Chapter 3: The Doctrine of the Categories; Chapter 4: Semiotics; Chapter 5: Philosophy of Science; Chapter 6: Pragmatism but (...)
     
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  49.  96
    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 (...)
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  50. Meillassoux's Virtual Future.Graham Harman - 2011 - Continent 1 (2):78-91.
    continent. 1.2 (2011): 78-91. This article consists of three parts. First, I will review the major themes of Quentin Meillassoux’s After Finitude . Since some of my readers will have read this book and others not, I will try to strike a balance between clear summary and fresh critique. Second, I discuss an unpublished book by Meillassoux unfamiliar to all readers of this article, except those scant few that may have gone digging in the microfilm archives of the École normale (...)
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