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  1. Scientific Theories as Intervening Representations.Andoni Ibarra & Thomas Mormann - 2010 - Theoria : An International Journal for Theory, History and Fundations of Science 21 (1):21-38.
    In this paper we use some ideas of Hertz and Duhem to propose a new concept of representation that overcomes the dichotomy between representation and intervention. It is based on the category-theoretical notion of adjoint situations and is used to explain the relation between empirical facts and theoretical laws in a new way.
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  • Intuitionistic Quantum Logic of an N-Level System.Martijn Caspers, Chris Heunen, Nicolaas P. Landsman & Bas Spitters - 2009 - Foundations of Physics 39 (7):731-759.
    A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the (...)
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  • Categories, Toposes and Sets.J. L. Bell - 1982 - Synthese 51 (3):293 - 337.
    This paper is an introduction to topos theory which assumes no prior knowledge of category theory. It includes a discussion of internal logic in a topos, A characterization of the category of sets, And an investigation of the notions of topology and sheaf in a topos.
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  • Hyperdoctrines, Natural Deduction and the Beck Condition.Robert A. G. Seely - 1983 - Mathematical Logic Quarterly 29 (10):505-542.
  • From Absolute to Local Mathematics.J. L. Bell - 1986 - 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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  • Philosophical Reflections on the Foundations of Mathematics.Jocelyne Couture & Joachim Lambek - 1991 - Erkenntnis 34 (2):187 - 209.
    This article was written jointly by a philosopher and a mathematician. It has two aims: to acquaint mathematicians with some of the philosophical questions at the foundations of their subject and to familiarize philosophers with some of the answers to these questions which have recently been obtained by mathematicians. In particular, we argue that, if these recent findings are borne in mind, four different basic philosophical positions, logicism, formalism, platonism and intuitionism, if stated with some moderation, are in fact reconcilable, (...)
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  • An Interpretation of Classical Proofs.Glen Helman - 1983 - Journal of Philosophical Logic 12 (1):39 - 71.
  • The Three Arrows of Zeno.Craig Harrison - 1996 - Synthese 107 (2):271 - 292.
    We explore the better known paradoxes of Zeno including modern variants based on infinite processes, from the point of view of standard, classical analysis, from which there is still much to learn (especially concerning the paradox of division), and then from the viewpoints of non-standard and non-classical analysis (the logic of the latter being intuitionist).The standard, classical or Cantorian notion of the continuum, modeled on the real number line, is well known, as is the definition of motion as the time (...)
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  • Structural Universals as Structural Parts: Toward a General Theory of Parthood and Composition.Thomas Mormann - 2010 - Axiomathes 20 (2-3):229 - 253.
    David Lewis famously argued against structural universals since they allegedly required what he called a composition “sui generis” that differed from standard mereological com¬position. In this paper it is shown that, although traditional Boolean mereology does not describe parthood and composition in its full generality, a better and more comprehensive theory is provided by the foundational theory of categories. In this category-theoretical framework a theory of structural universals can be formulated that overcomes the conceptual difficulties that Lewis and his followers (...)
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  • Upsetting the Foundations for Mathematics.Lawrence Neff Stout - 2005 - Philosophia Scientiae 9 (2):5-21.
    Starting with a review of the kinds of questions a foundation for mathematics should address, this paper provides a critique of set theoretical foundations, a proposal that multiple interconnected categorical foundations would be an improvement, and a way of recovering set theory within a categorical approach.RésuméCommençant par une revue sommaire des types de questions qu’une fondation des mathématiques devrait poser, cet article présente premièrement une critique des fondements basés sur la théorie des ensembles, puis propose l’idée que plusieurs fondements catégoriques, (...)
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  • Matematický Realismus a Naturalismus Penelope Maddy1.Vít Punčochář - 2012 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 19:199-226.
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  • The Dynamics of Belief as a Basis for Logic.Peter Gärdenfors - 1984 - British Journal for the Philosophy of Science 35 (1):1-10.
  • Issues in Commonsense Set Theory.Varol Akman - manuscript
    The success of set theory as a foundation for mathematics inspires its use in arti cial intelligence, particularly in commonsense reasoning. In this survey, we brie y review classical set theory from an AI perspective, and then consider alternative set theories. Desirable properties of a possible commonsense set theory are investigated, treating di erent aspects like cumulative hierarchy, self-reference, cardinality, etc. Assorted examples from the ground-breaking research on the subject are also given.
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  • A Topos Perspective on the Kochen-Specker Theorem: II. Conceptual Aspects, and Classical Analogues.Jeremy Butterfield & Chris Isham - unknown
    In a previous paper, we have proposed assigning as the value of a physical quantity in quantum theory, a certain kind of set (a sieve) of quantities that are functions of the given quantity. The motivation was in part physical---such a valuation illuminates the Kochen-Specker theorem; and in part mathematical---the valuation arises naturally in the topos theory of presheaves. This paper discusses the conceptual aspects of this proposal. We also undertake two other tasks. First, we explain how the proposed valuations (...)
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  • Information Processing as an Intrinsic Property of Biological Systems: Origin and Dynamics of Information.Abir Igamberdiev - 1997 - World Futures 50 (1):571-581.
    (1997). Information processing as an intrinsic property of biological systems: Origin and dynamics of information. World Futures: Vol. 50, No. 1-4, pp. 571-581.
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  • Topos Based Semantic for Constructive Logic with Strong Negation.Barbara Klunder & B. Klunder - 1992 - Mathematical Logic Quarterly 38 (1):509-519.
    The aim of the paper is to show that topoi are useful in the categorial analysis of the constructive logic with strong negation. In any topos ϵ we can distinguish an object Λ and its truth-arrows such that sets ϵ have a Nelson algebra structure. The object Λ is defined by the categorial counterpart of the algebraic FIDEL-VAKARELOV construction. Then it is possible to define the universal quantifier morphism which permits us to make the first order predicate calculus. The completeness (...)
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  • The Future of Mathematics in Economics: A Philosophically Grounded Proposal.Ricardo Crespo & Fernando Tohmé - 2017 - Foundations of Science 22 (4):677-693.
    The use of mathematics in economics has been widely discussed. The philosophical discussion on what mathematics is remains unsettled on why it can be applied to the study of the real world. We propose to get back to some philosophical conceptions that lead to a language-like role for the mathematical analysis of economic phenomena and present some problems of interest that can be better examined in this light. Category theory provides the appropriate tools for these analytical approach.
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  • The Example of the Unicorn: A Knowledge-Based Approach to Scientific Creativity and the Growth of Knowledge. [REVIEW]Kenneth R. Blochowiak - 1993 - AI and Society 7 (1):52-61.
    In the course of researching the question ‘What does it mean for knowledge to grow?’, the author has developed a large and unique compendium of components, some of which are knowledge systems that serve as research and creativity support systems. The self-modifying, self-effecting creative process and the results of developing and working with these systems, using novel methods and drawing on eclectic sources, is discussed.
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