Ontology of Mathematics

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  1. Natural Cybernetics and Mathematical History: The Principle of Least Choice in History.Vasil Penchev - 2020 - Cultural Anthropology (Elsevier: SSRN) 5 (23):1-44.
    The paper follows the track of a previous paper “Natural cybernetics of time” in relation to history in a research of the ways to be mathematized regardless of being a descriptive humanitarian science withal investigating unique events and thus rejecting any repeatability. The pathway of classical experimental science to be mathematized gradually and smoothly by more and more relevant mathematical models seems to be inapplicable. Anyway quantum mechanics suggests another pathway for mathematization; considering the historical reality as dual or “complimentary” (...)
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  2. Rez. „Adam Drozdek: In the Beginning Was the Apeiron: Infinity in Greek Philosophy, Stuttgart: Steiner, 2008“. [REVIEW]Sergiusz Kazmierski - 2010 - Bryn Mawr Classical Review 2010.
    Es ist das Verdienst der Arbeit von Adam Drozdek, in einem noch grösseren historischen Umfang sowie mit einer noch stärkeren thematischen Gewichtung und Stringenz als dies bereits Sinnige getan hat, nicht nur die entscheidendste Phase der griechischen Philosophie, sondern auch der Mathematik, ausgehend vom physikalischen und mathematischen Infinitätsgedanken dargestellt zu haben.
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  3. Mathematics - an Imagined Tool for Rational Cognition.Boris Culina - manuscript
    Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) mathematical truths are not (...)
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  4. Numbers, Empiricism and the A Priori.Olga Ramirez Calle - 2020 - Logos and Episteme: An International Journal of Epistemology 11 (2):149-177.
    The present paper deals with the ontological status of numbers and considers Frege ́s proposal in Grundlagen upon the background of the Post-Kantian semantic turn in analytical philosophy. Through a more systematic study of his philosophical premises, it comes to unearth a first level paradox that would unset earlier still than it was exposed by Russell. It then studies an alternative path, that departin1g from Frege’s initial premises, drives to a conception of numbers as synthetic a priori in a more (...)
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  5. Does Set Theory Really Ground Arithmetic Truth?Alfredo Roque Freire - manuscript
    We consider the foundational relation between arithmetic and set theory. Our goal is to criticize the construction of standard arithmetic models as providing grounds for arithmetic truth (even in a relative sense). Our method is to emphasize the incomplete picture of both theories and treat models as their syntactical counterparts. Insisting on the incomplete picture will allow us to argue in favor of the revisability of the standard model interpretation. We then show that it is hopeless to expect that the (...)
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  6. Maddy On The Multiverse.Claudio Ternullo - 2019 - In Deniz Sarikaya, Deborah Kant & Stefania Centrone (eds.), Reflections on the Foundations of Mathematics. Berlin: Springer Verlag. pp. 43-78.
    Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...)
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  7. Forms of Structuralism: Bourbaki and the Philosophers.Jean-Pierre Marquis - 2020 - Structures Meres, Semantics, Mathematics, and Cognitive Science.
    In this paper, we argue that, contrary to the view held by most philosophers of mathematics, Bourbaki’s technical conception of mathematical structuralism is relevant to philosophy of mathematics. In fact, we believe that Bourbaki has captured the core of any mathematical structuralism.
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  8. The Ontological Import of Adding Proper Classes.Alfredo Roque Freire & Rodrigo de Alvarenga Freire - 2019 - Manuscrito 42 (2):85-112.
    In this article, we analyse the ontological import of adding classes to set theories. We assume that this increment is well represented by going from ZF system to NBG. We thus consider the standard techniques of reducing one system to the other. Novak proved that from a model of ZF we can build a model of NBG (and vice versa), while Shoenfield have shown that from a proof in NBG of a set-sentence we can generate a proof in ZF of (...)
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  9. Mathematical Models of Abstract Systems: Knowing Abstract Geometric Forms.Jean-Pierre Marquis - 2013 - Annales de la Faculté des Sciences de Toulouse 22 (5):969-1016.
    Scientists use models to know the world. It i susually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a central role, namely homotopy theory. (...)
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  10. Access Problems and Explanatory Overkill.Silvia Jonas - 2017 - Philosophical Studies 174 (11):2731-2742.
    I argue that recent attempts to deflect Access Problems for realism about a priori domains such as mathematics, logic, morality, and modality using arguments from evolution result in two kinds of explanatory overkill: the Access Problem is eliminated for contentious domains, and realist belief becomes viciously immune to arguments from dispensability, and to non-rebutting counter-arguments more generally.
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  11. ¿ES LA MATEMÁTICA LA NOMOGONÍA DE LA CONCIENCIA? REFLEXIONES ACERCA DEL ORIGEN DE LA CONCIENCIA Y EL PLATONISMO MATEMÁTICO DE ROGER PENROSE / Is Mathematics the “nomogony” of Consciousness? Reflections on the origin of consciousness and mathematical Platonism of Roger Penrose.Miguel Acosta - 2016 - Naturaleza y Libertad. Revista de Estudios Interdisciplinares 7:15-39.
    Al final de su libro “La conciencia inexplicada”, Juan Arana señala que la nomología, explicación según las leyes de la naturaleza, requiere de una nomogonía, una consideración del origen de las leyes. Es decir, que el orden que observamos en el mundo natural requiere una instancia previa que ponga ese orden específico. Sabemos que desde la revolución científica la mejor manera de explicar dicha nomología ha sido mediante las matemáticas. Sin embargo, en las últimas décadas se han presentado algunas propuestas (...)
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  12. Ortega y Gasset on Georg Cantor’s Theory of Transfinite Numbers.Lior Rabi - 2016 - Kairos (15):46-70.
    Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and articulated a (...)
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  13. Reid, Constance. Hilbert (a Biography). Reviewed by Corcoran in Philosophy of Science 39 (1972), 106–08.John Corcoran - 1972 - Philosophy of Science 39 (1):106-108.
    Reid, Constance. Hilbert (a Biography). Reviewed by Corcoran in Philosophy of Science 39 (1972), 106–08. -/- Constance Reid was an insider of the Berkeley-Stanford logic circle. Her San Francisco home was in Ashbury Heights near the homes of logicians such as Dana Scott and John Corcoran. Her sister Julia Robinson was one of the top mathematical logicians of her generation, as was Julia’s husband Raphael Robinson for whom Robinson Arithmetic was named. Julia was a Tarski PhD and, in recognition of (...)
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  14. Corcoran Recommends Hambourger on the Frege-Russell Number Definition.John Corcoran - 1978 - MATHEMATICAL REVIEWS 56.
    It is widely agreed by philosophers that the so-called “Frege-Russell definition of natural number” is actually an assertion concerning the nature of the numbers and that it cannot be regarded as a definition in the ordinary mathematical sense. On the basis of the reasoning in this paper it is clear that the Frege-Russell definition contradicts the following three principles (taken together): (1) each number is the same entity in each possible world, (2) each number exists in each possible world, (3) (...)
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  15. The Construction of Logical Space, by Augustin Rayo. [REVIEW]S. Berry - 2015 - Mind 124 (496):1375-1379.
    Review of "The Construction of Logical Space", by Augustin Rayo. Oxford: OxfordUniversity Press, 2013. Pp. xix+220. H/b$35.00.
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  16. Normativity and Mathematics: A Wittgensteinian Approach to the Study of Number.J. Robert Loftis - 1999 - Dissertation, Northwestern University
    I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year's resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view can (...)
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  17. Dialectics and Catastrophe.Martin Zwick - 1978 - In F. Geyer & J. Van der Zouwen (ed.), Sociocybernetics. Martinus Nijhoff. pp. 129-154.
    The Catastrophe Theory of Rene Thom and E. C. Zeeman suggests a mathematical interpretation of certain aspects of Hegelian and Marxist dialectics. Specifically, the three 'classical' dialectical principles, (1) the transformation of quantity into quality, (2) the unity and struggle of opposites, and (3) the negation of negation, can be modeled with the seven 'elementary catastrophes' given by Thorn, especially the catastrophes known as the 'cusp' and the 'butterfly'. Far from being empty metaphysics or scholasticism, as critics have argued, the (...)
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  18. Historicity, Value and Mathematics.Barry Smith - 1975 - In A. T. Tymieniecka (ed.), Analecta Husserliana, vol. 4. Dordrecht: Reidel. pp. 219-239.
    At the beginning of the present century, a series of paradoxes were discovered within mathematics which suggested a fundamental unclarity in traditional mathemati­cal methods. These methods rested on the assumption of a realm of mathematical idealities existing independently of our thinking activity, and in order to arrive at a firmly grounded mathematics different attempts were made to formulate a conception of mathematical objects as purely human constructions. It was, however, realised that such formulations necessarily result in a mathematics which lacks (...)
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  19. Parts: A Study in Ontology.Peter Simons - 1987 - Oxford University Press.
    Although the relationship of part to whole is one of the most fundamental there is, this is the first full-length study of this key concept. Showing that mereology, or the formal theory of part and whole, is essential to ontology, Simons surveys and critiques previous theories--especially the standard extensional view--and proposes a new account that encompasses both temporal and modal considerations. Simons's revised theory not only allows him to offer fresh solutions to long-standing problems, but also has far-reaching consequences for (...)
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Mathematical Fictionalism
  1. Mathematical Anti-Realism and Explanatory Structure.Bruno Whittle - forthcoming - Synthese:1-15.
    Plausibly, mathematical claims are true, but the fundamental furniture of the world does not include mathematical objects. This can be made sense of by providing mathematical claims with paraphrases, which make clear how the truth of such claims does not require the fundamental existence of mathematical objects. This paper explores the consequences of this type of position for explanatory structure. There is an apparently straightforward relationship between this sort of structure, and the logical sort: i.e. logically complex claims are explained (...)
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  2. Fictionalist Strategies in Metaphysics.Lukas Skiba & Richard Woodward - 2020 - In Ricki Bliss & James Miller (eds.), Routledge Handbook of Metametaphysics. New York, NY, USA:
    This paper discusses the nature of, problems for, and benefits delivered by fictionalist strategies in metaphysics.
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  3. Bunge’s Mathematical Structuralism Is Not a Fiction.Jean-Pierre Marquis - 2019 - In Michael Robert Matthews (ed.), Mario Bunge: A Centenary Festschrift. New York, NY, USA: Springer Verlag. pp. 587-608.
    In this paper, I explore Bunge’s fictionism in philosophy of mathematics. After an overview of Bunge’s views, in particular his mathematical structuralism, I argue that the comparison between mathematical objects and fictions ultimately fails. I then sketch a different ontology for mathematics, based on Thomasson’s metaphysical work. I conclude that mathematics deserves its own ontology, and that, in the end, much work remains to be done to clarify the various forms of dependence that are involved in mathematical knowledge, in particular (...)
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  4. Optimal Representations and the Enhanced Indispensability Argument.Manuel Barrantes - 2019 - Synthese 196 (1):247-263.
    The Enhanced Indispensability Argument appeals to the existence of Mathematical Explanations of Physical Phenomena to justify mathematical Platonism, following the principle of Inference to the Best Explanation. In this paper, I examine one example of a MEPP—the explanation of the 13-year and 17-year life cycle of magicicadas—and argue that this case cannot be used defend the EIA. I then generalize my analysis of the cicada case to other MEPPs, and show that these explanations rely on what I will call ‘optimal (...)
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  5. Conversational Exculpature.Daniel Hoek - 2018 - Philosophical Review 127 (2):151-196.
    Conversational exculpature is a pragmatic process whereby information is subtracted from, rather than added to, what the speaker literally says. This pragmatic content subtraction explains why we can say “Rob is six feet tall” without implying that Rob is between 5'0.99" and 6'0.01" tall, and why we can say “Ellen has a hat like the one Sherlock Holmes always wears” without implying Holmes exists or has a hat. This article presents a simple formalism for understanding this pragmatic mechanism, specifying how, (...)
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  6. Can Mathematical Objects Be Causally Efficacious?Seungbae Park - 2019 - Inquiry: An Interdisciplinary Journal of Philosophy 62 (3):247–255.
    Callard (2007) argues that it is metaphysically possible that a mathematical object, although abstract, causally affects the brain. I raise the following objections. First, a successful defence of mathematical realism requires not merely the metaphysical possibility but rather the actuality that a mathematical object affects the brain. Second, mathematical realists need to confront a set of three pertinent issues: why a mathematical object does not affect other concrete objects and other mathematical objects, what counts as a mathematical object, and how (...)
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  7. In Defense of Mathematical Inferentialism.Seungbae Park - 2017 - Analysis and Metaphysics 16:70-83.
    I defend a new position in philosophy of mathematics that I call mathematical inferentialism. It holds that a mathematical sentence can perform the function of facilitating deductive inferences from some concrete sentences to other concrete sentences, that a mathematical sentence is true if and only if all of its concrete consequences are true, that the abstract world does not exist, and that we acquire mathematical knowledge by confirming concrete sentences. Mathematical inferentialism has several advantages over mathematical realism and fictionalism.
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  8. The Inaccuracy of Partial Truth in Yablovian If-Thenism.Joseph Ulatowski - 2017 - Australasian Philosophical Review 1 (2):206-211.
    Yablo has argued for an alternative form of if-thenism that is more conducive with his figurative fictionalism. This commentary sets out to challenge whether the remainder, ρ, tends to be an inaccurate representation of the conditions that are supposed to complete the enthymeme from φ to Ψ. Whilst by some accounts the inaccuracies shouldn't set off any alarm bells, the truth of ρ is too inexact. The content of ρ, a partial truth, must display a sensitivity to the contextual background (...)
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  9. From Mathematical Fictionalism to Truth‐Theoretic Fictionalism.Bradley Armour-Garb & James A. Woodbridge - 2014 - Philosophy and Phenomenological Research 88 (1):93-118.
    We argue that if Stephen Yablo (2005) is right that philosophers of mathematics ought to endorse a fictionalist view of number-talk, then there is a compelling reason for deflationists about truth to endorse a fictionalist view of truth-talk. More specifically, our claim will be that, for deflationists about truth, Yablo’s argument for mathematical fictionalism can be employed and mounted as an argument for truth-theoretic fictionalism.
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  10. Fictionalism and Mathematical Objectivity.Iulian D. Toader - 2012 - In Metaphysics and Science. University of Bucharest Press. pp. 137-158.
    This paper, written in Romanian, compares fictionalism, nominalism, and neo-Meinongianism as responses to the problem of objectivity in mathematics, and then motivates a fictionalist view of objectivity as invariance.
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  11. 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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  12. Abstract Expressionism and the Communication Problem.David Liggins - 2014 - British Journal for the Philosophy of Science 65 (3):599-620.
    Some philosophers have recently suggested that the reason mathematics is useful in science is that it expands our expressive capacities. Of these philosophers, only Stephen Yablo has put forward a detailed account of how mathematics brings this advantage. In this article, I set out Yablo’s view and argue that it is implausible. Then, I introduce a simpler account and show it is a serious rival to Yablo’s. 1 Introduction2 Yablo’s Expressionism3 Psychological Objections to Yablo’s Expressionism4 Introducing Belief Expressionism5 Objections and (...)
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  13. Mathematics as Make-Believe: A Constructive Empiricist Account.Sarah Elizabeth Hoffman - 1999 - Dissertation, University of Alberta (Canada)
    Any philosophy of science ought to have something to say about the nature of mathematics, especially an account like constructive empiricism in which mathematical concepts like model and isomorphism play a central role. This thesis is a contribution to the larger project of formulating a constructive empiricist account of mathematics. The philosophy of mathematics developed is fictionalist, with an anti-realist metaphysics. In the thesis, van Fraassen's constructive empiricism is defended and various accounts of mathematics are considered and rejected. Constructive empiricism (...)
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  14. Weaseling and the Content of Science.David Liggins - 2012 - Mind 121 (484):997-1005.
    I defend Joseph Melia’s nominalist account of mathematics from an objection raised by Mark Colyvan.
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  15. Musil's Imaginary Bridge.Achille C. Varzi - 2014 - The Monist 97 (1):30-46.
    In a calculation involving imaginary numbers, we begin with real numbers that represent concrete measures and we end up with numbers that are equally real, but in the course of the operation we find ourselves walking “as if on a bridge that stands on no piles”. How is that possible? How does that work? And what is involved in the as-if stance that this metaphor introduces so beautifully? These are questions that bother Törless deeply. And that Törless is bothered by (...)
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  16. What Mathematicians' Claims Mean : In Defense of Hermeneutic Fictionalism.Gábor Forrai - 2010 - Hungarian Philosophical Review 54 (4):191-203.
    Hermeneutic fictionalism about mathematics maintains that mathematics is not committed to the existence of abstract objects such as numbers. Mathematical sentences are true, but they should not be construed literally. Numbers are just fictions in terms of which we can conveniently describe things which exist. The paper defends Stephen Yablo’s hermeneutic fictionalism against an objection proposed by John Burgess and Gideon Rosen. The objection, directed against all forms of nominalism, goes as follows. Nominalism can take either a hermeneutic form and (...)
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  17. On 'Average'.Christopher Kennedy & Jason Stanley - 2009 - Mind 118 (471):583 - 646.
    This article investigates the semantics of sentences that express numerical averages, focusing initially on cases such as 'The average American has 2.3 children'. Such sentences have been used both by linguists and philosophers to argue for a disjuncture between semantics and ontology. For example, Noam Chomsky and Norbert Hornstein have used them to provide evidence against the hypothesis that natural language semantics includes a reference relation holding between words and objects in the world, whereas metaphysicians such as Joseph Melia and (...)
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  18. Mathematics and Conceptual Analysis.Antony Eagle - 2008 - Synthese 161 (1):67–88.
    Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number of (...)
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Mathematical Nominalism
  1. Mathematical Anti-Realism and Explanatory Structure.Bruno Whittle - forthcoming - Synthese:1-15.
    Plausibly, mathematical claims are true, but the fundamental furniture of the world does not include mathematical objects. This can be made sense of by providing mathematical claims with paraphrases, which make clear how the truth of such claims does not require the fundamental existence of mathematical objects. This paper explores the consequences of this type of position for explanatory structure. There is an apparently straightforward relationship between this sort of structure, and the logical sort: i.e. logically complex claims are explained (...)
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  2. The Idea of Continuity as Mathematical-Philosophical Invariant.Eldar Amirov - 2019 - “Metafizika” Journal 2 (8):p. 87-100.
    The concept of ‘ideas’ plays central role in philosophy. The genesis of the idea of continuity and its essential role in intellectual history have been analyzed in this research. The main question of this research is how the idea of continuity came to the human cognitive system. In this context, we analyzed the epistemological function of this idea. In intellectual history, the idea of continuity was first introduced by Leibniz. After him, this idea, as a paradigm, formed the base of (...)
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  3. Mathematical Explanation by Law.Sam Baron - 2019 - British Journal for the Philosophy of Science 70 (3):683-717.
    Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...)
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  4. The Enhanced Indispensability Argument, the Circularity Problem, and the Interpretability Strategy.Jan Heylen & Lars Arthur Tump - 2019 - Synthese 198 (4):3033-3045.
    Within the context of the Quine–Putnam indispensability argument, one discussion about the status of mathematics is concerned with the ‘Enhanced Indispensability Argument’, which makes explicit in what way mathematics is supposed to be indispensable in science, namely explanatory. If there are genuine mathematical explanations of empirical phenomena, an argument for mathematical platonism could be extracted by using inference to the best explanation. The best explanation of the primeness of the life cycles of Periodical Cicadas is genuinely mathematical, according to Baker (...)
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  5. Deflationary Nominalism and Puzzle Avoidance.David Mark Kovacs - 2019 - Philosophia Mathematica 27 (1):88-104.
    In a series of works, Jody Azzouni has defended deflationary nominalism, the view that certain sentences quantifying over mathematical objects are literally true, although such objects do not exist. One alleged attraction of this view is that it avoids various philosophical puzzles about mathematical objects. I argue that this thought is misguided. I first develop an ontologically neutral counterpart of Field’s reliability challenge and argue that deflationary nominalism offers no distinctive answer to it. I then show how this reasoning generalizes (...)
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  6. Optimal Representations and the Enhanced Indispensability Argument.Manuel Barrantes - 2019 - Synthese 196 (1):247-263.
    The Enhanced Indispensability Argument appeals to the existence of Mathematical Explanations of Physical Phenomena to justify mathematical Platonism, following the principle of Inference to the Best Explanation. In this paper, I examine one example of a MEPP—the explanation of the 13-year and 17-year life cycle of magicicadas—and argue that this case cannot be used defend the EIA. I then generalize my analysis of the cicada case to other MEPPs, and show that these explanations rely on what I will call ‘optimal (...)
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  7. An Intrinsic Theory of Quantum Mechanics: Progress in Field's Nominalistic Program, Part I.Eddy Keming Chen - 2017
    In this paper, I introduce an intrinsic account of the quantum state. This account contains three desirable features that the standard platonistic account lacks: (1) it does not refer to any abstract mathematical objects such as complex numbers, (2) it is independent of the usual arbitrary conventions in the wave function representation, and (3) it explains why the quantum state has its amplitude and phase degrees of freedom. -/- Consequently, this account extends Hartry Field’s program outlined in Science Without Numbers (...)
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  8. Can Mathematical Objects Be Causally Efficacious?Seungbae Park - 2019 - Inquiry: An Interdisciplinary Journal of Philosophy 62 (3):247–255.
    Callard (2007) argues that it is metaphysically possible that a mathematical object, although abstract, causally affects the brain. I raise the following objections. First, a successful defence of mathematical realism requires not merely the metaphysical possibility but rather the actuality that a mathematical object affects the brain. Second, mathematical realists need to confront a set of three pertinent issues: why a mathematical object does not affect other concrete objects and other mathematical objects, what counts as a mathematical object, and how (...)
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  9. Number Words as Number Names.Friederike Moltmann - 2017 - Linguistics and Philosophy 40 (4):331-345.
    This paper criticizes the view that number words in argument position retain the meaning they have on an adjectival or determiner use, as argued by Hofweber :179–225, 2005) and Moltmann :499–534, 2013a, 2013b). In particular the paper re-evaluates syntactic evidence from German given in Moltmann to that effect.
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  10. Numbers and Functions in Hilbert's Finitism.Richard Zach - 1998 - Taiwanese Journal for History and Philosophy of Science 10:33-60.
    David Hilbert's finitistic standpoint is a conception of elementary number theory designed to answer the intuitionist doubts regarding the security and certainty of mathematics. Hilbert was unfortunately not exact in delineating what that viewpoint was, and Hilbert himself changed his usage of the term through the 1920s and 30s. The purpose of this paper is to outline what the main problems are in understanding Hilbert and Bernays on this issue, based on some publications by them which have so far received (...)
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  11. Truth and Existence.Jan Heylen & Leon Horsten - 2017 - Thought: A Journal of Philosophy 6 (1):106-114.
    Halbach has argued that Tarski biconditionals are not ontologically conservative over classical logic, but his argument is undermined by the fact that he cannot include a theory of arithmetic, which functions as a theory of syntax. This article is an improvement on Halbach's argument. By adding the Tarski biconditionals to inclusive negative free logic and the universal closure of minimal arithmetic, which is by itself an ontologically neutral combination, one can prove that at least one thing exists. The result can (...)
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  12. Boarding Neurath's Boat: The Early Development of Quine's Naturalism.Sander Verhaegh - 2017 - Journal of the History of Philosophy 55 (2):317-342.
    W. V. Quine is arguably the intellectual father of contemporary naturalism, the idea that there is no distinctively philosophical perspective on reality. Yet, even though Quine has always been a science-minded philosopher, he did not adopt a fully naturalistic perspective until the early 1950s. In this paper, I reconstruct the genesis of Quine’s ideas on the relation between science and philosophy. Scrutinizing his unpublished papers and notebooks, I examine Quine’s development in the first decades of his career. After identifying three (...)
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  13. Numerical Cognition and Mathematical Realism.Helen De Cruz - 2016 - Philosophers' Imprint 16.
    Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and propose that they prima (...)
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