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  1. Jeremy Avigad & Jeffrey Helzner (2002). Transfer Principles in Nonstandard Intuitionistic Arithmetic. Archive for Mathematical Logic 41 (6):581-602.
    Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of these rules destroy (...)
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  2. S. Awodey (1996). Structure in Mathematics and Logic: A Categorical Perspective. Philosophia Mathematica 4 (3):209-237.
    A precise notion of ‘mathematical structure’ other than that given by model theory may prove fruitful in the philosophy of mathematics. It is shown how the language and methods of category theory provide such a notion, having developed out of a structural approach in modern mathematical practice. As an example, it is then shown how the categorical notion of a topos provides a characterization of ‘logical structure’, and an alternative to the Pregean approach to logic which is continuous with the (...)
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  3. Steve Awodey (2010). Category Theory. OUP Oxford.
    Category theory is a branch of abstract algebra with incredibly diverse applications. This text and reference book is aimed not only at mathematicians, but also researchers and students of computer science, logic, linguistics, cognitive science, philosophy, and any of the other fields in which the ideas are being applied. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and (...)
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  4. Steve Awodey (2009). From Sets to Types to Categories to Sets. .
    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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  5. Steve Awodey (2004). An Answer to Hellman's Question: ‘Does Category Theory Provide a Framework for Mathematical Structuralism?’. Philosophia Mathematica 12 (1):54-64.
    An affirmative answer is given to the question quoted in the title.
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  6. Jonathan Bain (2013). Category-Theoretic Structure and Radical Ontic Structural Realism. Synthese 190 (9):1621-1635.
    Radical Ontic Structural Realism (ROSR) claims that structure exists independently of objects that may instantiate it. Critics of ROSR contend that this claim is conceptually incoherent, insofar as, (i) it entails there can be relations without relata, and (ii) there is a conceptual dependence between relations and relata. In this essay I suggest that (ii) is motivated by a set-theoretic formulation of structure, and that adopting a category-theoretic formulation may provide ROSR with more support. In particular, I consider how a (...)
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  7. J. L. Bell (1986). From Absolute to Local Mathematics. Synthese 69 (3):409 - 426.
    In this paper (a sequel to [4]) I put forward a "local" interpretation of mathematical concepts based on notions derived from category theory. The fundamental idea is to abandon the unique absolute universe of sets central to the orthodox set-theoretic account of the foundations of mathematics, replacing it by a plurality of local mathematical frameworks - elementary toposes - defined in category-theoretic terms.
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  8. J. L. Bell (1981). Category Theory and the Foundations of Mathematics. British Journal for the Philosophy of Science 32 (4):349-358.
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  9. John L. Bell (2001). Observations on Category Theory. 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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  10. Jean Bénabou (1985). Fibered Categories and the Foundations of Naive Category Theory. Journal of Symbolic Logic 50 (1):10-37.
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  11. Otavio Bueno, Outline of a Paraconsistent Category Theory.
    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 modified to give rise to a (...)
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  12. Jessica Carter (2008). Categories for the Working Mathematician: Making the Impossible Possible. 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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  13. D. Corfield (2002). 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] Studies in History and Philosophy of Science Part B 33 (2):359-366.
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  14. David Ellerman, Category Theory and Universal Models: Adjoints and Brain Functors.
    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 "heteromorphic" theory (...)
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  15. David P. Ellerman (1988). Category Theory and Concrete Universals. Erkenntnis 28 (3):409 - 429.
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  16. Solomon Feferman, Foundations of Category Theory: What Remains to Be Done.
    • Session on CF&FCT proposed by E. Landry; participants: G. Hellman, E. Landry, J.-P. Marquis and C. McLarty..
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  17. Solomon Feferman, Enriched Stratified Systems for the Foundations of Category Theory.
    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 should allow us to establish the (...)
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  18. Michael John Healy & Thomas Preston Caudell (2006). Ontologies and Worlds in Category Theory: Implications for Neural Systems. [REVIEW] 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 of theories, with an operation on (...)
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  19. Geoffrey Hellman (2003). Does Category Theory Provide a Framework for Mathematical Structuralism? 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 out (...)
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  20. Chris Heunen, Klaas Landsman & Bas Spitters, The Principle of General Tovariance.
    We tentatively propose two guiding principles for the construction of theories of physics, which should be satisfied by a possible future theory of quantum gravity. These principles are inspired by those that led Einstein to his theory of general relativity, viz. his principle of general covariance and his equivalence principle, as well as by the two mysterious dogmas of Bohr's interpretation of quantum mechanics, i.e. his doctrine of classical concepts and his principle of complementarity. An appropriate mathematical language for combining (...)
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  21. David G. Holdsworth (1977). Category Theory and Quantum Mechanics (Kinematics). Journal of Philosophical Logic 6 (1):441 - 453.
  22. C. J. Isham (2005). Quantising on a Category. Foundations of Physics 35 (2):271-297.
    We review the problem of finding a general framework within which one can construct quantum theories of non-standard models for space, or space-time. The starting point is the observation that entities of this type can typically be regarded as objects in a category whose arrows are structure-preserving maps. This motivates investigating the general problem of quantising a system whose ‘configuration space’ (or history-theory analogue) is the set of objects Ob(Q) in a category Q. We develop a scheme based on constructing (...)
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  23. C. J. Isham & J. Butterfield (2000). Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity. Foundations of Physics 30 (10):1707-1735.
    We discuss some ways in which topos theory (a branch of category theory) can be applied to interpretative problems in quantum theory and quantum gravity. In Sec.1, we introduce these problems. In Sec.2, we introduce topos theory, especially the idea of a topos of presheaves. In Sec.3, we discuss several possible applications of topos theory to the problems in Sec.1. In Sec.4, we draw some conclusions.
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  24. Chris Isham & Jeremy Butterfield, A Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalised Valuations.
    Any attempt to construct a realist interpretation of quantum theory founders on the Kochen-Specker theorem, which asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context (...)
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  25. C. Barry Jay (1991). Coherence in Category Theory and the Church-Rosser Property. Notre Dame Journal of Formal Logic 33 (1):140-143.
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  26. C. Barry Jay (1989). A Note on Natural Numbers Objects in Monoidal Categories. Studia Logica 48 (3):389 - 393.
    The internal language of a monoidal category yields simple proofs of results about a natural numbers object therein.
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  27. Paul C. Kainen (2009). On the Ehresmann–Vanbremeersch Theory and Mathematical Biology. Axiomathes 19 (3):225-244.
    Category theory has been proposed as the ultimate algebraic model for biology. We review the Ehresmann–Vanbremeersch theory in the context of other mathematical approaches.
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  28. Molly Kao, Nicolas Fillion & John Bell (2010). J Ean -P Ierre M Arquis . From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory. Philosophia Mathematica 18 (2):227-234.
    (No abstract is available for this citation).
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  29. M. Kary (2009). (Math, Science, ?). Axiomathes 19 (3):61-86.
    In science as in mathematics, it is popular to know little and resent much about category theory. Less well known is how common it is to know little and like much about set theory. The set theory of almost all scientists, and even the average mathematician, is fundamentally different from the formal set theory that is contrasted against category theory. The latter two are often opposed by saying one emphasizes Substance, the other Form. However, in all known systems of mathematics (...)
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  30. Goro Kato & D. Struppa (2002). Category Theory and Consciousness. In Kunio Yasue, Marj Jibu & Tarcisio Della Senta (eds.), No Matter, Never Mind. John Benjamins.
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  31. Goro Kato & Tsunefumi Tanaka (2006). Double-Slit Interference and Temporal Topos. Foundations of Physics 36 (11):1681-1700.
    The electron double-slit interference is re-examined from the point of view of temporal topos. Temporal topos (or t-topos) is an abstract algebraic (categorical) method using the theory of sheaves. A brief introduction to t-topos is given. When the structural foundation for describing particles is based on t-topos, the particle-wave duality of electron is a natural consequence. A presheaf associated with the electron represents both particle-like and wave-like properties depending upon whether an object in the site (t-site) is specified (particle-like) or (...)
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  32. Jerzy Król (2006). A Model for Spacetime: The Role of Interpretation in Some Grothendieck Topoi. [REVIEW] Foundations of Physics 36 (7):1070-1098.
    We analyse the proposition that the spacetime structure is modified at short distances or at high energies due to weakening of classical logic. The logic assigned to the regions of spacetime is intuitionistic logic of some topoi. Several cases of special topoi are considered. The quantum mechanical effects can be generated by such semi-classical spacetimes. The issues of: background independence and general relativity covariance, field theoretic renormalization of divergent expressions, the existence and definition of path integral measures, are briefly discussed (...)
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  33. Luis M. Laita (1976). A Study of Algebraic Logic From the Point of View of Category Theory. Notre Dame Journal of Formal Logic 17 (1):89-118.
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  34. J. Lambek (1989). On Some Connections Between Logic and Category Theory. 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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  35. Elaine Landry, Reconstructing Hilbert to Construct Category Theoretic Structuralism.
    This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, (...)
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  36. Elaine Landry (1999). Category Theory: The Language of Mathematics. 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 the (...)
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  37. Elaine Landry & Jean-Pierre Marquis (2005). Categories in Context: Historical, Foundational, and Philosophical. Philosophia Mathematica 13 (1):1-43.
    The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various category-theoretic foundational debates and to point to some common elements found among those who advocate adopting a foundational stance. We then use these elements to argue for the philosophical position that category theory provides a framework for an algebraic in re interpretation of mathematical structuralism. In each context, what we aim to show (...)
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  38. F. W. Lawvere (1994). Cohesive Toposes and Cantor's 'Lauter Einsen'. Philosophia Mathematica 2 (1):5-15.
    For 20th century mathematicians, the role of Cantor's sets has been that of the ideally featureless canvases on which all needed algebraic and geometrical structures can be painted. (Certain passages in Cantor's writings refer to this role.) Clearly, the resulting contradication, 'the points of such sets are distinc yet indistinguishable', should not lead to inconsistency. Indeed, the productive nature of this dialectic is made explicit by a method fruitful in other parts of mathematics (see 'Adjointness in Foundations', Dialectia 1969). This (...)
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  39. Øystein Linnebo & Richard Pettigrew (2011). Category Theory as an Autonomous Foundation. 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 is customary in other (...)
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  40. Jean-Pierre Marquis (2013). Mathematical Forms and Forms of Mathematics: Leaving the Shores of Extensional Mathematics. Synthese 190 (12):2141-2164.
    In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according (...)
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  41. Jean-Pierre Marquis (2012). Categorical Foundations of Mathematics. Review of Symbolic Logic.
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  42. Jean-Pierre Marquis (2010). Mathematical Conceptware: Category Theory: R Alf K R Ö Mer . Tool and Object: A History and Philosophy of Category Theory. Philosophia Mathematica 18 (2):235-246.
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  43. Jean-Pierre Marquis (2009). From a Geometrical Point of View: A Study in the History and Philosophy of Category Theory. 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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  44. Jean-Pierre Marquis, Category Theory. Stanford Encyclopedia of Philosophy.
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  45. Jean-Pierre Marquis (1995). Category Theory and the Foundations of Mathematics: Philosophical Excavations. 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 into 5 (...)
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  46. C. Mclarty (2004). Exploring Categorical Structuralism. Philosophia Mathematica 12 (1):37-53.
    Hellman [2003] raises interesting challenges to categorical structuralism. He starts citing Awodey [1996] which, as Hellman sees, is not intended as a foundation for mathematics. It offers a structuralist framework which could denned in any of many different foundations. But Hellman says Awodey's work is 'naturally viewed in the context of Mac Lane's repeated claim that category theory provides an autonomous foundation for mathematics as an alternative to set theory' (p. 129). Most of Hellman's paper 'scrutinizes the formulation of category (...)
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  47. Colin McLarty (2005). Learning From Questions on Categorical Foundations. Philosophia Mathematica 13 (1):44-60.
    We can learn from questions as well as from their answers. This paper urges some things to learn from questions about categorical foundations for mathematics raised by Geoffrey Hellman and from ones he invokes from Solomon Feferman.
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  48. Colin Mclarty (1994). Category Theory in Real Time. 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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  49. Colin McLarty (1993). Numbers Can Be Just What They Have To. Noûs 27 (4):487-498.
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  50. Robert Paré & Leopoldo Román (1989). Monoidal Categories with Natural Numbers Object. Studia Logica 48 (3):361 - 376.
    The notion of a natural numbers object in a monoidal category is defined and it is shown that the theory of primitive recursive functions can be developed. This is done by considering the category of cocommutative comonoids which is cartesian, and where the theory of natural numbers objects is well developed. A number of examples illustrate the usefulness of the concept.
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