Results for 'category-theory'

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  1.  41
    Category Theory in Physics, Mathematics, and Philosophy.Marek Kuś & Bartłomiej Skowron (eds.) - 2019 - Springer Verlag.
    The contributions gathered here demonstrate how categorical ontology can provide a basis for linking three important basic sciences: mathematics, physics, and philosophy. Category theory is a new formal ontology that shifts the main focus from objects to processes. The book approaches formal ontology in the original sense put forward by the philosopher Edmund Husserl, namely as a science that deals with entities that can be exemplified in all spheres and domains of reality. It is a dynamic, processual, and (...)
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  2.  52
    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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  3. 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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  4.  40
    Strong Homomorphisms, Category Theory, and Semantic Paradox.Jonathan Wolfgram & Roy T. Cook - 2022 - Review of Symbolic Logic 15 (4):1070-1093.
    In this essay we introduce a new tool for studying the patterns of sentential reference within the framework introduced in [2] and known as the language of paradox $\mathcal {L}_{\mathsf {P}}$ : strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms. In particular, we show that (i) strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms between $\mathcal {L}_{\mathsf {P}}$ constructions preserve paradoxicality, (ii) many (but not all) earlier results regarding the paradoxicality of $\mathcal {L}_{\mathsf {P}}$ constructions can be recast as special cases of our central result regarding (...)
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  5. 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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  6.  7
    Category theory for the sciences.David I. Spivak - 2014 - Cambridge, Massachusetts: The MIT Press.
    An introduction to category theory as a rigorous, flexible, and coherent modeling language that can be used across the sciences. Category theory was invented in the 1940s to unify and synthesize different areas in mathematics, and it has proven remarkably successful in enabling powerful communication between disparate fields and subfields within mathematics. This book shows that category theory can be useful outside of mathematics as a rigorous, flexible, and coherent modeling language throughout the sciences. (...)
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  7. 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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  8. 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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  9.  52
    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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  10.  30
    Category Theory.Steve Awodey - 2006 - Oxford, England: 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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  11. 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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  12.  22
    Diagrammatic Immanence: Category Theory and Philosophy.Rocco Gangle - 2015 - Edinburgh, UK: Edinburgh University Press.
    Rocco Gangle addresses the methodological questions raised by a commitment to immanence in terms of how diagrams may be used both as tools and as objects of philosophical investigation. Gangle integrates insights from Spinoza, Pierce and Deleuze in conjunction with the formal operations of category theory.
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  13. 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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  14.  65
    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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  15.  21
    Category Free Category Theory and Its Philosophical Implications.Michael Heller - 2016 - Logic and Logical Philosophy 25 (4):447-459.
    There exists a dispute in philosophy, going back at least to Leibniz, whether is it possible to view the world as a network of relations and relations between relations with the role of objects, between which these relations hold, entirely eliminated. Category theory seems to be the correct mathematical theory for clarifying conceptual possibilities in this respect. In this theory, objects acquire their identity either by definition, when in defining category we postulate the existence of (...)
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  16. 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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  17.  90
    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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  18. 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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  19. Category theory and the foundations of mathematics.J. L. Bell - 1981 - British Journal for the Philosophy of Science 32 (4):349-358.
  20.  20
    Category Theory in the hands of physicists, mathematicians, and philosophers. [REVIEW]Mariusz Stopa - 2020 - Philosophical Problems in Science 69:283-293.
    Book review: Category Theory in Physics, Mathematics, and Philosophy, Kuś M., Skowron B., Springer Proc. Phys. 235, 2019, pp.xii+134.
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  21. Category theory and concrete universals.David P. Ellerman - 1988 - Erkenntnis 28 (3):409 - 429.
  22. Category Theory.S. Awodey - 2007 - Bulletin of Symbolic Logic 13 (3):371-372.
  23.  46
    Category theory for linear logicians.Richard Blute & Philip Scott - 2004 - In Thomas Ehrhard (ed.), Linear logic in computer science. New York: Cambridge University Press. pp. 316--3.
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  24. Hyperintensional Category Theory and Indefinite Extensibility.David Elohim - manuscript
    This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility for set-theoretic truths in category theory is identifiable with the Grothendieck Universe Axiom and the elementary embeddings in Vopenka's principle. The interaction between the interpretational and objective modalities of indefinite extensibility is defined via the epistemic interpretation of two-dimensional semantics. The semantics can be defined intensionally or hyperintensionally. By characterizing the modal (...)
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  25.  81
    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. Category Theory.[author unknown] - 2007 - Studia Logica 86 (1):133-135.
  27. 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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  28.  52
    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: Assessing Philosophy of Logic and Mathematics Today. Dordrecht, Netherland: Springer. pp. 163--179.
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  29.  27
    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 (...)
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  30.  61
    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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  31.  18
    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 (...)
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  32.  49
    Axiomatic Method and Category Theory.Rodin Andrei - 2013 - Cham: Imprint: Springer.
    This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. (...)
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  33. 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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  34. 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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  35.  31
    Ontologies and Worlds in Category Theory: Implications for Neural Systems.Michael John Healy & Thomas Preston Caudell - 2006 - Axiomathes 16 (1-2):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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  36.  5
    REVIEWS-Category theory.S. Awodey & Jiri Rosicky - 2007 - Bulletin of Symbolic Logic 13 (3):371-372.
  37. Category Theory and the Representation of Geometrical Information.G. Graham White - 1994 - In F. D. Anger & R. V. Rodriguez (eds.), Spatial and Temporal Reasoning. Aaai.
     
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  38. Category theory.Jean-Pierre Marquis - 2008 - Stanford Encyclopedia of Philosophy.
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  39. Category theory based on combinatory logic.M. W. Bunder - 1984 - Archive for Mathematical Logic 24 (1):1-16.
     
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  40.  71
    Category theory and quantum mechanics (kinematics).David G. Holdsworth - 1977 - Journal of Philosophical Logic 6 (1):441 - 453.
  41. Category theory and consciousness.Goro Kato & D. Struppa - 2002 - In Kunio Yasue, Mari Jibu & Tarcisio Della Senta (eds.), No Matter, Never Mind: Proceedings of Toward a Science of Consciousness : Fundamental Approaches (Tokyo '99). John Benjamins.
  42.  15
    Category theory and family resemblances.Alan Ford - 1987 - In Basil J. Hiley & D. Peat (eds.), Quantum Implications: Essays in Honour of David Bohm. Methuen. pp. 361.
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  43.  50
    Category theory, logic and formal linguistics: Some connections, old and new.Jean Gillibert & Christian Retoré - 2014 - Journal of Applied Logic 12 (1):1-13.
  44.  14
    An invitation to applied category theory: seven sketches in compositionality.Brendan Fong - 2019 - New York, NY: Cambridge University Press. Edited by David I. Spivak.
    Category theory reveals commonalities between structures of all sorts. This book shows its potential in science, engineering, and beyond.
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  45. Category Theory and Quantum Mechanics.P. Mittelstaedt - 1977 - Journal of Philosophical Logic 6 (4):441.
  46.  26
    Categorial theory and political philosophy.Terry Pinkard - 1980 - Journal of Value Inquiry 14 (2):105-118.
  47. 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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  48.  30
    Mathematical Category Theory and Mathematical Philosophy.F. William Lawvere - unknown
    Explicit concepts and sufficiently precise definitions are the basis for further advance of a science beyond a given level. To move toward a situation where the whole population has access to the authentic results of science (italics mine) requires making explicit some general philosophical principles which can help to guide the learning, development, and use of mathematics, a science which clearly plays a pivotal role regarding the learning, development and use of all the sciences. Such philosophical principles have not come (...)
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  49.  24
    Category Theory and Structuralism in Mathematics: Syntactical Considerations.Jean-Pierre Marquis - 1997 - In Evandro Agazzi & György Darvas (eds.), Philosophy of Mathematics Today. Kluwer Academic Publishers. pp. 123--136.
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  50. 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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