Results for 'Category theory'

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  1.  47
    How Category Theory Works.David Ellerman - manuscript
    The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and functions. The analysis extends directly to other concrete categories (groups, rings, vector spaces, etc.) where the objects are sets with a certain type of structure and the morphisms are functions that preserve that structure. Then the elements & distinctions-based definitions can (...)
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  2. On the Self-Predicative Universals of Category Theory.David Ellerman - manuscript
    This paper shows how the universals of category theory in mathematics 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. The paper also shows how the always-self-predicative universals of category theory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus (...)
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  3. 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 (...)
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  4. Category Theory is a Contentful Theory.Shay Logan - 2015 - Philosophia Mathematica 23 (1):110-115.
    Linnebo and Pettigrew present some objections to category theory as an autonomous foundation. They do a commendable job making clear several distinct senses of ‘autonomous’ as it occurs in the phrase ‘autonomous foundation’. Unfortunately, their paper seems to treat the ‘categorist’ perspective rather unfairly. Several infelicities of this sort were addressed by McLarty. In this note I address yet another apparent infelicity.
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  5. On Concrete Universals: A Modern Treatment Using Category Theory.David Ellerman - 2014 - AL-Mukhatabat.
    Today it would be considered "bad Platonic metaphysics" to think that among all the concrete instances of a property there could be a universal instance so that all instances had the property by virtue of participating in that concrete universal. Yet there is a mathematical theory, category theory, dating from the mid-20th century that shows how to precisely model concrete universals within the "Platonic Heaven" of mathematics. This paper, written for the philosophical logician, develops this category-theoretic (...)
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  6. 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 (...)
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  7.  32
    Category Theory and Physical Structuralism.Benjamin Eva - 2016 - European Journal for Philosophy of Science 6 (2):231-246.
    As a metaphysical theory, radical ontic structural realism is characterised mainly in terms of the ontological primacy it places on relations and structures, as opposed to the individual relata and objects that inhabit these relations/structures. The most popular criticism of ROSR is that its central thesis is incoherent. Bain attempts to address this criticism by arguing that the mathematical language of category theory allows for a coherent articulation of ROSR’s key thesis. Subsequently, Wüthrich and Lam and Lal (...)
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  8.  41
    Axiomatizing Category Theory in Free Logic.Christoph Benzmüller & Dana Scott - manuscript
    Starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. Our axiom sets have been formalized in the Isabelle/HOL interactive proof assistant, and this formalization utilizes a semantically correct embedding of free logic in classical higher-order logic. The modeling and formal analysis of our axiom sets has been significantly supported by series of experiments with automated reasoning tools integrated with Isabelle/HOL. We also (...)
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  9. Diagrammatic Immanence: Category Theory and Philosophy.Rocco Gangle - 2016 - Edinburgh, UK: Edinburgh University Press.
  10.  94
    Category Theory: A Gentle Introduction.Peter Smith - manuscript
    This Gentle Introduction is very much still work in progress. Roughly aimed at those who want something a bit more discursive, slower-moving, than Awodey's or Leinster's excellent books. -/- The current [Jan 2018] version is 291pp.
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  11. Category Theory in Physics, Mathematics, and Philosophy.Marek Kuś & Bartłomiej Skowron (eds.) - 2019 - Springer Verlag.
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  12. Category Theory.Steve Awodey - 2010 - Oxford University Press.
    A comprehensive reference to category theory for students and researchers in mathematics, computer science, logic, cognitive science, linguistics, and philosophy. Useful for self-study and as a course text, the book includes all basic definitions and theorems, as well as numerous examples and exercises.
     
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  13. 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 (...)
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  14. Category Theory as an Autonomous Foundation.Øystein Linnebo & Richard Pettigrew - 2011 - Philosophia Mathematica 19 (3):227-254.
    Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as (...)
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  15. Outline of a Paraconsistent Category Theory.Otavio Bueno - unknown
    The aim of this paper is two-fold: (1) To contribute to a better knowledge of the method of the Argentinean mathematicians Lia Oubifia and Jorge Bosch to formulate category theory independently of set theory. This method suggests a new ontology of mathematical objects, and has a profound philosophical significance (the underlying logic of the resulting category theory is classical iirst—order predicate calculus with equality). (2) To show in outline how the Oubina-Bosch theory can be (...)
     
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  16.  47
    From a Geometrical Point of View: A Study in the History and Philosophy of Category Theory.Jean-Pierre Marquis - 2009 - Springer.
    A Study of the History and Philosophy of Category Theory Jean-Pierre Marquis. to say that objects are dispensable in geometry. What is claimed is that the specific nature of the objects used is irrelevant. To use the terminology already ...
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  17. Foundations of Unlimited Category Theory: What Remains to Be Done: Foundations of Unlimited Category Theory: What Remains to Be Done.Solomon Feferman - 2013 - Review of Symbolic Logic 6 (1):6-15.
    Following a discussion of various forms of set-theoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unlimited” or “naive” category theory. The author proposed four criteria for such some years ago. The article describes how much had previously been accomplished on one approach to meeting those criteria, then takes care (...)
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  18. Category Theory and the Foundations of Mathematics: Philosophical Excavations.Jean-Pierre Marquis - 1995 - Synthese 103 (3):421 - 447.
    The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It is divided (...)
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  19. 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 (...)
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  20.  82
    The Prospects of Unlimited Category Theory: Doing What Remains to Be Done.Michael Ernst - 2015 - Review of Symbolic Logic 8 (2):306-327.
    The big question at the end of Feferman is: Is it possible to find a foundation for unlimited category theory? I show that the answer is no by showing that unlimited category theory is inconsistent.
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  21. Observations on Category Theory.John L. Bell - 2001 - Axiomathes 12 (1-2):151-155.
    is a presentation of mathematics in terms of the fundamental concepts of transformation, and composition of transformations. While the importance of these concepts had long been recognized in algebra (for example, by Galois through the idea of a group of permutations) and in geometry (for example, by Klein in his Erlanger Programm), the truly universal role they play in mathematics did not really begin to be appreciated until the rise of abstract algebra in the 1930s. In abstract algebra the idea (...)
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  22. The Meaning of Category Theory for 21st Century Philosophy.Alberto Peruzzi - 2006 - Axiomathes 16 (4):424-459.
    Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific (...)
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  23.  70
    Enriched Stratified Systems for the Foundations of Category Theory.Solomon Feferman - unknown
    Four requirements are suggested for an axiomatic system S to provide the foundations of category theory: (R1) S should allow us to construct the category of all structures of a given kind (without restriction), such as the category of all groups and the category of all categories; (R2) It should also allow us to construct the category of all functors between any two given categories including the ones constructed under (R1); (R3) In addition, S (...)
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  24.  68
    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.  74
    Category Theory and Universal Models: Adjoints and Brain Functors.David Ellerman - unknown
    Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of an adjunction is what might be called "internalization through a universal" based on universal mapping properties. A recently developed (...)
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  26.  32
    Category Theory and Mathematical Structuralism.Andrei Rodin - 2008 - Proceedings of the Xxii World Congress of Philosophy 41:37-40.
    Category theory doesn't support Mathematical Structuralism but suggests a new philosophical view on mathematics, which differs both from Structuralism and from traditional Substantialism about mathematical objects. While Structuralism implies thinking of mathematical objects up to isomorphism the new categorical view implies thinking up to general morphism.
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  27. Review of Giandomenico Sica (Ed.) What is Category Theory? Polimetrica, 2006. [REVIEW]John Symons - unknown
    Giandomenico Sica’s volume is a collection of eleven papers on category theory by philosophers, mathematicians, and mathematical physicists. In addition to papers of direct interest to philosophers of mathematics, the volume contains some introductory expositions of category theory along with a valuable discussion of the relationship between category theory and physics by Bob Coecke. While there are several technically difficult papers, the volume as a whole is reasonably accessible to those with some familiarity with (...)
     
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  28. Enriched Meanings: Natural Language Semantics with Category Theory.Ash Asudeh & Gianluca Giorgolo - 2020 - Oxford University Press.
    This book develops a theory of enriched meanings for natural language interpretation that uses the concept of monads and related ideas from category theory. The volume is interdisciplinary in nature, and will appeal to graduate students and researchers from a range of disciplines interested in natural language understanding and representation.
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  29.  69
    Ontologies and Worlds in Category Theory: Implications for Neural Systems.Michael John Healy & Thomas Preston Caudell - 2006 - Axiomathes 16 (1):165-214.
    We propose category theory, the mathematical theory of structure, as a vehicle for defining ontologies in an unambiguous language with analytical and constructive features. Specifically, we apply categorical logic and model theory, based upon viewing an ontology as a sub-category of a category of theories expressed in a formal logic. In addition to providing mathematical rigor, this approach has several advantages. It allows the incremental analysis of ontologies by basing them in an interconnected hierarchy (...)
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  30. Categorical Foundations and Foundations of Category Theory.Solomon Feferman - 1977 - In Robert E. Butts & Jaakko Hintikka (eds.), Logic, Foundations of Mathematics, and Computability Theory. Springer. pp. 149-169.
     
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  31. An Answer to Hellman's Question: ‘Does Category Theory Provide a Framework for Mathematical Structuralism?’.Steve Awodey - 2004 - Philosophia Mathematica 12 (1):54-64.
    An affirmative answer is given to the question quoted in the title.
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  32. Complex Systems From the Perspective of Category Theory: I. Functioning of the Adjunction Concept.Elias Zafiris - 2005 - Axiomathes 15 (1):147-158.
    We develop a category theoretical scheme for the comprehension of the information structure associated with a complex system, in terms of families of partial or local information carriers. The scheme is based on the existence of a categorical adjunction, that provides a theoretical platform for the descriptive analysis of the complex system as a process of functorial information communication.
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  33.  21
    Internal Diagrams and Archetypal Reasoning in Category Theory.Eduardo Ochs - 2013 - Logica Universalis 7 (3):291-321.
    We can regard operations that discard information, like specializing to a particular case or dropping the intermediate steps of a proof, as projections, and operations that reconstruct information as liftings. By working with several projections in parallel we can make sense of statements like “Set is the archetypal Cartesian Closed Category”, which means that proofs about CCCs can be done in the “archetypal language” and then lifted to proofs in the general setting. The method works even when our archetypal (...)
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  34. Category Theory.S. Awodey - 2007 - Bulletin of Symbolic Logic 13 (3):371-372.
  35. Category Theory.[author unknown] - 2007 - Studia Logica 86 (1):133-135.
  36. Category Theory and the Foundations of Mathematics.J. L. Bell - 1981 - British Journal for the Philosophy of Science 32 (4):349-358.
  37.  99
    Category Theory and Concrete Universals.David P. Ellerman - 1988 - Erkenntnis 28 (3):409 - 429.
  38.  35
    The Categorical Imperative: Category Theory as a Foundation for Deontic Logic.Clayton Peterson - 2014 - Journal of Applied Logic 12 (4):417-461.
  39.  28
    Category Theory for Linear Logicians.Richard Blute & Philip Scott - 2004 - In Thomas Ehrhard (ed.), Linear Logic in Computer Science. Cambridge University Press. pp. 316--3.
  40.  69
    Fibered Categories and the Foundations of Naive Category Theory.Jean Bénabou - 1985 - Journal of Symbolic Logic 50 (1):10-37.
  41.  86
    Category Theory.Jean-Pierre Marquis - 2008 - Stanford Encyclopedia of Philosophy.
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  42.  46
    Andrew M. Pitts. Interpolation and Conceptual Completeness for Pretoposes Via Category Theory. Mathematical Logic and Theoretical Computer Science, Edited by Kueker David W., Lopez-Escobar Edgar G. K. And Smith Carl H., Lecture Notes in Pure and Applied Mathematics, Vol. 106, Marcel Dekker, New York and Basel1987, Pp. 301–327. - Andrew M. Pitts. Conceptual Completeness for First-Order Intuitionistic Logic: An Application of Categorical Logic. Annals of Pure and Applied Logic, Vol. 41 , Pp. 33–81. [REVIEW]Marek Zawadowski - 1995 - Journal of Symbolic Logic 60 (2):692-694.
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  43. Foundations of Category Theory: What Remains to Be Done.Solomon Feferman - unknown
    • Session on CF&FCT proposed by E. Landry; participants: G. Hellman, E. Landry, J.-P. Marquis and C. McLarty..
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  44.  97
    Mathematical Conceptware: Category Theory: Critical Studies/Book Reviews.Jean-Pierre Marquis - 2010 - Philosophia Mathematica 18 (2):235-246.
    (No abstract is available for this citation).
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  45.  63
    Hegel and Category Theory.Robert B. Pippin - 1990 - Review of Metaphysics 43 (4):839 - 848.
    THE IDEA OF A "PHILOSOPHICAL SCIENCE," something of a Fata Morgana in the West for several centuries, underwent a well-known revolutionary change when Kant argued that in all philosophical speculation about the nature of things, reason is really "occupied only with itself." Indeed, Kant argued convincingly that the possibility of any cognitive relation to objects presupposed an original and constitutive "relation to self." Thereafter, instead of an a priori science of substance, a science of "how the world must be", a (...)
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  46. Philosophical Relevance of Category Theory.Colin McLarty - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press.
     
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  47.  31
    Category Theory as a Framework for an in Re Interpretation of Mathematical Structuralism.Elaine Landry - 2006 - In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. pp. 163--179.
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  48. Category Theory as a Conceptual Tool in the Study of Cognition.François Magnan & Gonzalo E. Reyes - 1994 - In John Macnamara & Gonzalo E. Reyes (eds.), The Logical Foundations of Cognition. Oxford University Press USA. pp. 57-90.
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  49. Complex Systems From the Perspective of Category Theory: II. Covering Systems and Sheaves.Elias Zafiris - 2005 - Axiomathes 15 (2):181-190.
    Using the concept of adjunction, for the comprehension of the structure of a complex system, developed in Part I, we introduce the notion of covering systems consisting of partially or locally defined adequately understood objects. This notion incorporates the necessary and sufficient conditions for a sheaf theoretical representation of the informational content included in the structure of a complex system in terms of localization systems. Furthermore, it accommodates a formulation of an invariance property of information communication concerning the analysis of (...)
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  50. On Some Connections Between Logic and Category Theory.J. Lambek - 1989 - Studia Logica 48 (3):269 - 278.
    Categories may be viewed as deductive systems or as algebraic theories. We are primarily interested in the interplay between these two views and trace it through a number of structured categories and their internal languages, bearing in mind their relevance to the foundations of mathematics. We see this as a common thread running through the six contributions to this issue of Studia Logica.
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