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  1. added 2019-06-06
    Aristotelianism in the Philosophy of Mathematics.James Franklin - 2011 - Studia Neoaristotelica 8 (1):3-15.
    Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementarymathematics: the ratio of two heights, for example, is a (...)
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  2. added 2019-04-26
    Stairway to Heaven: The Abstract Method and Levels of Abstraction in Mathematics.Jean Pierre Marquis & Jean-Pierre Marquis - 2016 - The Mathematical Intelligencer 38 (3):41-51.
    In this paper, following the claims made by various mathematicians, I try to construct a theory of levels of abstraction. I first try to clarify the basic components of the abstract method as it developed in the first quarter of the 20th century. I then submit an explication of the notion of levels of abstraction. In the final section, I briefly explore some of main philosophical consequences of the theory.
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  3. added 2019-01-23
    Circularities In The Contemporary Philosophical Accounts Of The Applicability Of Mathematics In The Physical Universe.Catalin Barboianu - 2015 - Revista de Filosofie 61 (5):517-542.
    Contemporary philosophical accounts of the applicability of mathematics in physical sciences and the empirical world are based on formalized relations between the mathematical structures and the physical systems they are supposed to represent within the models. Such relations were constructed both to ensure an adequate representation and to allow a justification of the validity of the mathematical models as means of scientific inference. This article puts in evidence the various circularities (logical, epistemic, and of definition) that are present in these (...)
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  4. added 2018-11-21
    What Are Structural Properties?†.Johannes Korbmacher & Georg Schiemer - 2018 - Philosophia Mathematica 26 (3):295-323.
    Informally, structural properties of mathematical objects are usually characterized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. We present two formal explications corresponding to these two informal characterizations of structural properties. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathematical properties. (...)
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  5. added 2018-11-15
    The Applicability of Mathematics to Physical Modality.Nora Berenstain - 2017 - Synthese 194 (9):3361-3377.
    This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of (...)
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  6. added 2018-06-06
    Poincaré on the Foundation of Geometry in the Understanding.Jeremy Shipley - 2017 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta. Springer. pp. 19-37.
    This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study (...)
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  7. added 2017-11-28
    What We Talk About When We Talk About Numbers.Richard Pettigrew - manuscript
    In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.
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  8. added 2017-10-21
    Mathematics and Its Applications, A Transcendental-Idealist Perspective.Jairo Da Silva - 2017 - Springer.
    This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what (...)
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  9. added 2017-01-29
    Review Of: Hodesdon, K. “Mathematica Representation: Playing a Role”. Philosophical Studies (2014) 168:769–782. Mathematical Reviews. MR 3176431.John Corcoran - 2015 - MATHEMATICAL REVIEWS 2015:3176431.
    This 4-page review-essay—which is entirely reportorial and philosophically neutral as are my other contributions to MATHEMATICAL REVIEWS—starts with a short introduction to the philosophy known as mathematical structuralism. The history of structuralism traces back to George Boole (1815–1864). By reference to a recent article various feature of structuralism are discussed with special attention to ambiguity and other terminological issues. The review-essay includes a description of the recent article. The article’s 4-sentence summary is quoted in full and then analyzed. The point (...)
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  10. added 2016-12-08
    The Epistemology of Geometry I: The Problem of Exactness.Anne Newstead & Franklin James - 2010 - Proceedings of the Australasian Society for Cognitive Science 2009.
    We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which the (...)
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  11. added 2016-11-04
    Is Mathematics the Theory of Instantiated Structural Universals?Iulian D. Toader - 2013 - Transylvanian Review 22:132-142.
    The paper argues against defending realism about numbers on the basis of realism about instantiated structural universals. After presenting Armstrong’s theory of structural properties as instantiated universals and Lewis’s devastating criticism of it, I argue that several responses to this criticism are unsuccessful, and that one possible construal of structural universals via non-well-founded sets should be resisted by the mathematical realist.
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  12. added 2016-11-01
    Modal Structuralism and Theism.Silvia Jonas - forthcoming - In Fiona Ellis (ed.), New Models of Religious Understanding. Oxford: Oxford University Press.
    Drawing an analogy between modal structuralism about mathematics and theism, I o er a structuralist account that implicitly de nes theism in terms of three basic relations: logical and metaphysical priority, and epis- temic superiority. On this view, statements like `God is omniscient' have a hypothetical and a categorical component. The hypothetical component provides a translation pattern according to which statements in theistic language are converted into statements of second-order modal logic. The categorical component asserts the logical possibility of the (...)
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  13. added 2016-08-11
    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 mathematical theory of categories, (...)
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  14. added 2016-04-17
    Frege, the Complex Numbers, and the Identity of Indiscernibles.Wenzel Christian Helmut - 2010 - Logique Et Analyse 53 (209):51-60.
    There are mathematical structures with elements that cannot be distinguished by the properties they have within that structure. For instance within the field of complex numbers the two square roots of −1, i and −i, have the same algebraic properties in that field. So how do we distinguish between them? Imbedding the complex numbers in a bigger structure, the quaternions, allows us to algebraically tell them apart. But a similar problem appears for this larger structure. There seems to be always (...)
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  15. added 2016-02-25
    Personality Model.Miro Brada - 2000 - Problem Paradise:42-43.
    In 1995, as a student of psychology inspired by natural science, I defined a logical model of personality explaining psychosis. I created (for my MA thesis, 1998 and grant research, 1999) new kind of tests assessing intelligence, creativity, prejudices, expectations to show more exact methods in psychology. During my Phd study in economics, I developed 'Maximization of Uniqueness (Originality)' model enhancing the classic utility to explain irrational motivations linking economics and psychology. Later I became computer programmer developing functional programming. According (...)
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  16. added 2015-06-18
    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 that the paradoxes arose from having (...)
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  17. added 2015-06-09
    Global and Local.James Franklin - 2014 - Mathematical Intelligencer 36 (4).
    The global/local contrast is ubiquitous in mathematics. This paper explains it with straightforward examples. It is possible to build a circular staircase that is rising at any point (locally) but impossible to build one that rises at all points and comes back to where it started (a global restriction). Differential equations describe the local structure of a process; their solution describes the global structure that results. The interplay between global and local structure is one of the great themes of mathematics, (...)
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  18. added 2014-09-15
    Structuralism and Its Ontology.Marc Gasser - 2015 - Ergo: An Open Access Journal of Philosophy 2 (1):1-26.
    A prominent version of mathematical structuralism holds that mathematical objects are at bottom nothing but "positions in structures," purely relational entities without any sort of nature independent of the structure to which they belong. Such an ontology is often presented as a response to Benacerraf's "multiple reductions" problem, or motivated on hermeneutic grounds, as a faithful representation of the discourse and practice of mathematics. In this paper I argue that there are serious difficulties with this kind of view: its proponents (...)
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  19. added 2014-04-03
    Mathematical Necessity and Reality.James Franklin - 1989 - Australasian Journal of Philosophy 67 (3):286 – 294.
    Einstein, like most philosophers, thought that there cannot be mathematical truths which are both necessary and about reality. The article argues against this, starting with prima facie examples such as "It is impossible to tile my bathroom floor with regular pentagonal tiles." Replies are given to objections based on the supposedly purely logical or hypothetical nature of mathematics.
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  20. added 2014-04-02
    Mathematical Representation: Playing a Role.Kate Hodesdon - 2014 - Philosophical Studies 168 (3):769-782.
    The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Non-eliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the features to an ontology of structural objects conflicts with claims often made by structuralists to the effect that their structuralist theses are versions of Quine’s ontological relativity or Putnam’s internal realism. I describe and argue for an alternative explanation for these features which instead explains the (...)
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  21. added 2013-08-24
    Names and Objects.Dan Kurth - manuscript
    In this paper I try to fortify the nominalistic objectology (cf. Meinong's 'Gegenstandstheorie') with essentialist means. This also is intended as a preparation for introducing Information Monism.
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  22. added 2011-06-23
    The Cultural Phenomenology of Qualitative Quantity - Work in Progress - Introduction Autobiographical.Borislav Dimitrov - manuscript
    This study is about the Quality. Here I have dealt with the quality that differs significantly from the common understanding of quality /as determined quality/ that arise from the law of dialectics. This new quality is the quality of the quantity /quality of the quantitative changes/, noticed in philosophy by Plato as “quality of numbers”, and later developed by Hegel as “qualitative quantity. The difference between the known determined quality and qualitative quantity is evident in the exhibit form of these (...)
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  23. added 2011-03-03
    Aristotle and Modern Mathematical Theories of the Continuum.Anne Newstead - 2001 - In Demetra Sfendoni-Mentzou & James Brown (eds.), Aristotle and Contemporary Philosophy of Science. Peter Lang.
    This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open versus closed intervals (...)
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