Results for ' Mathematical Relativism and the Ontology of Mathematics: Platonism'

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  1.  64
    Plato's Problem: An Introduction to Mathematical Platonism.Marco Panza & Andrea Sereni - 2013 - New York: Palgrave-Macmillan. Edited by Andrea Sereni & Marco Panza.
    What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the the (...)
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  2. (1 other version)Mathematical platonism meets ontological pluralism?Matteo Plebani - 2017 - Inquiry: An Interdisciplinary Journal of Philosophy:1-19.
    Mathematical platonism is the view that abstract mathematical objects exist. Ontological pluralism is the view that there are many modes of existence. This paper examines the prospects for...
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  3. Mathematical Platonism.Stuart Cornwell - 1991 - Dissertation, University of Southern California
    The present dissertation includes three chapters: chapter one 'Challenges to platonism'; chapter two 'counterparts of non-mathematical statements'; chapter three 'Nominalizing platonistic accounts of the predictive success of mathematics'. The purpose of the dissertation is to articulate a fundamental problem in the philosophy of mathematics and explore certain solutions to this problem. The central problematic is that platonistic mathematics is involved in the explanation and prediction of physical phenomena and hence its role in such explanations gives (...)
     
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  4.  12
    Mathematical Fictionalism Revisited.Otávio Bueno - 2023 - In Cristián Soto (ed.), Current Debates in Philosophy of Science: In Honor of Roberto Torretti. Springer Verlag. pp. 103-122.
    Mathematical fictionalism is the view according to which mathematical objects are ultimately fictions, and, thus, need not be taken to exist. This includes fictional objects, whose existence is typically not assumed to be the case. There are different versions of this view, depending on the status of fictions and on how they are connected to the world. In this paper, I critically examine the various kinds of fictionalism that Roberto Torretti identifies, determining to what extent they provide independent, (...)
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  5.  80
    Mathematical apriorism and warrant: A reliabilist-platonist account.Mark Mcevoy - 2005 - Philosophical Forum 36 (4):399–417.
    Mathematical apriorism holds that mathematical truths must be established using a priori processes. Against this, it has been argued that apparently a priori mathematical processes can, under certain circumstances, fail to warrant the beliefs they produce; this shows that these warrants depend on contingent features of the contexts in which they are used. They thus cannot be a priori. -/- In this paper I develop a position that combines a reliabilist version of mathematical apriorism with a (...)
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  6. Mathematical Platonism.Nicolas Pain - 2011 - In Michael Bruce & Steven Barbone (eds.), Just the Arguments: 100 of the Most Important Arguments in Western Philosophy. Malden, MA: Wiley-Blackwell.
     
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  7. Mathematical Contingentism.Kristie Miller - 2012 - Erkenntnis 77 (3):335-359.
    Platonists and nominalists disagree about whether mathematical objects exist. But they almost uniformly agree about one thing: whatever the status of the existence of mathematical objects, that status is modally necessary. Two notable dissenters from this orthodoxy are Hartry Field, who defends contingent nominalism, and Mark Colyvan, who defends contingent Platonism. The source of their dissent is their view that the indispensability argument provides our justification for believing in the existence, or not, of mathematical objects. This (...)
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  8.  79
    On what exists mathematically: Indispensability without platonism.Anne Newstead & James Franklin - forthcoming - In Brian Ellis (ed.), Metaphysical Realism.
    According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this criterion with our own brand of metaphysics, (...)
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  9.  9
    Relativism in Set Theory and Mathematics.Otávio Bueno - 2010 - In Steven D. Hales (ed.), A Companion to Relativism. Malden, MA: Wiley-Blackwell. pp. 553–568.
    This chapter contains sections titled: Abstract Introduction Mathematical Relativism: Does Everything Go In Mathematics? Conceptual, Structural and Logical Relativity in Mathematics Mathematical Relativism and Mathematical Objectivity Mathematical Relativism and the Ontology of Mathematics: Platonism Mathematical Relativism and the Ontology of Mathematics: Nominalism Conclusion: The Significance of Mathematical Relativism References.
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  10. Mathematical Platonism.Massimo Pigliucci - 2011 - Philosophy Now 84:47-47.
    Are numbers and other mathematical objects "out there" in some philosophically meaningful sense?
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  11.  44
    On Thought Experiments, Theology, and Mathematical Platonism.Yiftach Fehige & Andrea Vestrucci - 2022 - Axiomathes 32 (1):43-54.
    In our contribution to this special issue on thought experiments and mathematics, we aim to insert theology into the conversation. There is a very long tradition of substantial inquiries into the relationship between theology and mathematics. Platonism has been provoking a consolidation of that tradition to some extent in recent decades. Accordingly, in this paper we look at James R. Brown’s Platonic account of thought experiments. Ultimately, we offer an analysis of some of the merits and perils (...)
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  12.  68
    Ontology, Modality, and Mathematics: Remarks on Chihara's Constructibility Theory.Stephen Puryear - 2000 - Dissertation, Texas a&M University
    Chihara seeks to avoid commitment to mathematical objects by replacing traditional assertions of the existence of mathematical objects with assertions about possibilities of constructing certain open-sentence tokens. I argue that Chihara's project can be defended against several important objections, but that it is no less epistemologically problematic than its platonistic competitors.
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  13.  14
    Mathematical Platonism.Nicolas Pain - 2011 - In Michael Bruce & Steven Barbone (eds.), Just the Arguments. Chichester, West Sussex, U.K.: Wiley‐Blackwell. pp. 373–375.
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  14.  48
    (1 other version)Descartes on Mathematical Essences.Raffaella De Rosa & Otávio Bueno - 2008 - ProtoSociology 25:160-177.
    Descartes seems to hold two inconsistent accounts of the ontological status of mathematical essences. Meditation Five apparently develops a platonist view about such essences, while the Principles seems to advocate some form of “conceptualism”. We argue that Descartes was neither a platonist nor a conceptualist. Crucial to our interpretation is Descartes’ dispositional nativism. We contend that his doctrine of innate ideas allows him to endorse a hybrid view which avoids the drawbacks of Gassendi’s conceptualism without facing the difficulties of (...)
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  15.  26
    The Self in Logical-Mathematical Platonism.Ulrich Blau - 2009 - Mind and Matter 7 (1):37-57.
    A non-classical logic is proposed that extends classical logic and set theory as conservatively as possible with respect to three domains: the logic of natural language, the logcal foundations of mathematics, and the logical-philosophical paradoxes. A universal mechanics of consciousness connects these domains, and its best witness is the liar paradox. Its solution rests formally on a subject-object partition, mentally arising and disappearing perpetually. All deep paradoxes are paradoxes of consciousness. There are two kinds, solvable ones and unsolvable ones. (...)
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  16. Epistemological Challenges to Mathematical Platonism.Øystein Linnebo - 2006 - Philosophical Studies 129 (3):545-574.
    Since Benacerraf’s “Mathematical Truth” a number of epistemological challenges have been launched against mathematical platonism. I first argue that these challenges fail because they unduely assimilate mathematics to empirical science. Then I develop an improved challenge which is immune to this criticism. Very roughly, what I demand is an account of how people’s mathematical beliefs are responsive to the truth of these beliefs. Finally I argue that if we employ a semantic truth-predicate rather than just (...)
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  17.  93
    Mathematical Relativism.Hugly Philip & Sayward Charles - 1989 - History and Philosophy of Logic 10 (1):53-65.
    We set out a doctrine about truth for the statements of mathematics—a doctrine which we think is a worthy competitor to realist views in the philosophy of mathematics—and argue that this doctrine, which we shall call ‘mathematical relativism’, withstands objections better than do other non-realist accounts.
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  18. Ontology and mathematical truth.Michael Jubien - 1977 - Noûs 11 (2):133-150.
    The main goal of this paper is to urge that the normal platonistic account of mathematical truth is unsatisfactory even if pure abstract entities are assumed to exist (in a non-Question-Begging way). It is argued that the task of delineating an interpretation of a formal mathematical theory among pure abstract entities is not one that can be accomplished. An important effect of this conclusion on the question of the ontological commitments of informal mathematical theories is discussed. The (...)
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  19. What is Wrong with the Intuitionist Ontology in Mathematics.Vadim Batitsky - 1997 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 4 (2):111-116.
    The main aspect of the intuitionist ontology of mathematicsis the conception of mathematical objects as products of the human mind. This paper argues that so long as the existence of mathematical objects is made dependent on thehuman mind , the intuitionist ontology is refutable in that it is inconsistent with our well-confirmed beliefs about what is physically possible. At the same time, it is also argued that the intuitionistś attempt to remove this inconsistency by endowing the (...)
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  20.  15
    Mathematics, relativism and David Bloor.Richard W. Hadden - 1988 - Philosophy of the Social Sciences 18 (4):433-445.
  21.  62
    Mathematical Platonism and Dummettian Anti‐Realism.John McDowell - 1989 - Dialectica 43 (1‐2):173-192.
    SummaryThe platonist, in affirming the principle of bivalence for sentences for which there is no decision procedure, disconnects their truth‐conditions from conditions that would enable us to prove them ‐ as if Goldbach's conjecture, say, might just happen to be true. According to Dummett, what has gone wrong here is that the meaning of the relevant sentences has been conceived so as to go beyond what could be learned in learning to use them, or displayed in using them competently. Dummett (...)
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  22. Confirming mathematical theories: An ontologically agnostic stance.Anthony Peressini - 1999 - Synthese 118 (2):257-277.
    The Quine/Putnam indispensability approach to the confirmation of mathematical theories in recent times has been the subject of significant criticism. In this paper I explore an alternative to the Quine/Putnam indispensability approach. I begin with a van Fraassen-like distinction between accepting the adequacy of a mathematical theory and believing in the truth of a mathematical theory. Finally, I consider the problem of moving from the adequacy of a mathematical theory to its truth. I argue that the (...)
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  23. Is an Unpictorial Mathematical Platonism Possible?Charles W. Sayward - 2002 - Journal of Philosophical Research 27:201-214.
    In his book Wittgenstein on the Foundations of Mathematics, Crispin Wright notes that remarkably little has been done to provide an unpictorial, substantial account of what mathematical platonism comes to. Wright proposes to investigate whether there is not some more substantial doctrine than the familiar images underpinning the platonist view. He begins with the suggestion that the essential platonist claim is that mathematical truth is objective. Although he does not demarcate them as such, Wright proposes several (...)
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  24. Indispensability Without Platonism.Anne Newstead & James Franklin - 2011 - In Alexander Bird, Brian David Ellis & Howard Sankey (eds.), Properties, Powers and Structures: Issues in the Metaphysics of Realism. New York: Routledge. pp. 81-97.
    According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this criterion with our own brand of metaphysics, (...)
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  25.  12
    Mathematics in philosophy.Vesselin Petrov, François Beets & Katie Anderson (eds.) - 2017 - [Mazy]: Les Éditions Chromatika.
    The systematic mapping of the interplay of ontology and epistemology in the context of present day philosophy of mathematics constitutes an important heuristic goal. In order to achieve it, we must analyze and reinterpret the position of mathematics in philosophy." -- Back cover.
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  26.  32
    Abolishing Platonism in Multiverse Theories.Stathis Livadas - 2022 - Axiomathes 32 (2):321-343.
    A debated issue in the mathematical foundations in at least the last two decades is whether one can plausibly argue for the merits of treating undecidable questions of mathematics, e.g., the Continuum Hypothesis (CH), by relying on the existence of a plurality of set-theoretical universes except for a single one, i.e., the well-known set-theoretical universe V associated with the cumulative hierarchy of sets. The multiverse approach has some varying versions of the general concept of multiverse yet my intention (...)
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  27. Mathematical Platonism’ Versus Gathering the Dead: What Socrates teaches Glaucon &dagger.Colin McLarty - 2005 - Philosophia Mathematica 13 (2):115-134.
    Glaucon in Plato's _Republic_ fails to grasp intermediates. He confuses pursuing a goal with achieving it, and so he adopts ‘mathematical platonism’. He says mathematical objects are eternal. Socrates urges a seriously debatable, and seriously defensible, alternative centered on the destruction of hypotheses. He offers his version of geometry and astronomy as refuting the charge that he impiously ‘ponders things up in the sky and investigates things under the earth and makes the weaker argument the stronger’. We (...)
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  28. Naturalized platonism versus platonized naturalism.Bernard Linsky & Edward N. Zalta - 1995 - Journal of Philosophy 92 (10):525-555.
    In this paper, we develop an alternative strategy, Platonized Naturalism, for reconciling naturalism and Platonism and to account for our knowledge of mathematical objects and properties. A systematic (Principled) Platonism based on a comprehension principle that asserts the existence of a plenitude of abstract objects is not just consistent with, but required (on transcendental grounds) for naturalism. Such a comprehension principle is synthetic, and it is known a priori. Its synthetic a priori character is grounded in the (...)
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  29.  66
    Mathematical Facts in a Physicalist Ontology.Laszlo E. Szabo - unknown
    If physicalism is true, everything is physical. In other words, everything supervenes on, or is necessitated by, the physical. Accordingly, if there are logical/mathematical facts, they must be necessitated by the physical facts of the world. The aim of this paper is to clarify what logical/mathematical facts actually are and how these facts can be accommodated in a purely physical world.
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  30. What is applied mathematics?James Robert Brown - 1997 - Foundations of Science 2 (1):21-37.
    A number of issues connected with the nature of applied mathematics are discussed. Among the claims are these: mathematics "hooks onto" the world by providing models or representations, not by describing the world; classic platonism is to be preferred to structuralism; and several issues in the philosophy of science are intimately connected to the nature of applied mathematics.
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  31.  73
    What could mathematics be for it to function in distinctively mathematical scientific explanations?Marc Lange - 2021 - Studies in History and Philosophy of Science Part A 87 (C):44-53.
    Several philosophers have suggested that some scientific explanations work not by virtue of describing aspects of the world’s causal history and relations, but rather by citing mathematical facts. This paper investigates what mathematical facts could be in order for them to figure in such “distinctively mathematical” scientific explanations. For “distinctively mathematical explanations” to be explanations by constraint, mathematical language cannot operate in science as representationalism or platonism describes. It can operate as Aristotelian realism describes. (...)
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  32.  37
    (1 other version)Platonism And Mathematical Explanations.Fabrice Pataut - 2020 - Balkan Journal of Philosophy 12 (2):63-74.
    Ontological parsimony requires that if we can dispense with A when best explaining B, or when deducing a nominalistically statable conclusion B from nominalistically statable premises, we must indeed dispense with A. When A is a mathematical theory and it has been established that its conservativeness undermines the platonistic force of mathematical derivations (Field), or that a nonnumerical formulation of some explanans may be obtained so that the platonistic force of the best numerical-based account of the explanandum is (...)
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  33.  12
    Mathematics as Paideia in Proclus.John J. Cleary - 1998 - The Paideia Archive: Twentieth World Congress of Philosophy 3:79-84.
    I examine one aspect of the central role which mathematics plays in Proclus's ontology and epistemology, with particular reference to his Elements of Theology. I focus on his peculiar views about the ontological status of mathematical objects and the special faculties of the soul that are involved in understanding them. If they are merely abstract objects that are "stripped away" from sensible things, then they are unlikely to reorient the mind towards the intelligible realm, as envisioned by (...)
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  34. Some Reflections about Alain Badiou’s Approach to Platonism in Mathematics.Miriam Franchella - 2007 - Analytica 1:67-81.
    A reproach has been done many times to post-modernism: its picking up mathematical notions or results, mostly by misrepresenting their real content, in order to strike the readers and obtaining their assent only by impressing them . In this paper I intend to point out that although Alain Badiou’s approach to philosophy starts with taking distance both from analytic philosophy and from French post-modernism, the categories that he uses for labelling logicism, formalism and intuitionism do not reflect the real (...)
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  35. 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 (...)
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  36.  26
    Platonism in mathematics/Platonismo na matemática.O. Chateaubriand - 2007 - Manuscrito 30 (2):507-538.
    In this paper I examine arguments by Benacerraf and by Chihara against Gödel’s platonistic philosophy of mathematics.Neste artigo examino argumentos de Benacerraf e de Chihara contra a filosofia platônica da matemática de Gödel.
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  37.  9
    Mathematics and Ontology.Charles Castonguay - 1973 - In Mario Bunge (ed.), The methodological unity of science. Boston,: Reidel. pp. 15--22.
  38.  65
    Who's afraid of mathematical platonism? An historical perspective.Dirk Schlimm - 2024 - In Karine Chemla, José Ferreirós, Lizhen Ji, Erhard Scholz & Chang Wang (eds.), The Richness of the History of Mathematics. Springer. pp. 595-615.
    In "Plato's Ghost" Jeremy Gray presented many connections between mathematical practices in the nineteenth century and the rise of mathematical platonism in the context of more general developments, which he refers to as modernism. In this paper, I take up this theme and present a condensed discussion of some arguments put forward in favor of and against the view of mathematical platonism. In particular, I highlight some pressures that arose in the work of Frege, Cantor, (...)
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  39.  35
    Peirce's realistic approach to mathematics: or can one be a realist without being a platonist.Claudine Tiercelin - unknown
    Peirce's realism is a sophisticated realism inherited from the Avicennian Scotistic tradition, which may be briefly characterized by its opposition to metaphysical realism (Platonism) and various forms of nominalism. In this chapter, I consider how Peirce's realism fits his approach to mathematics, which is often presented as a somewhat incoherent mixture of Platonistic and conceptualistic elements. Without denying these, I claim that Peirce's subtle position not only helps to clear up some of these so-called inconsistencies but offers many (...)
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  40.  21
    (1 other version)Platonism in mathematics.O. Chateaubriand - 2005 - Manuscrito 28 (2):201-230.
    In this paper I examine arguments by Benacerraf and by Chihara against Gödel’s platonistic philosophy of mathematics.
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  41.  26
    One or Many Ontologies? Badiou’s Arguments for His Thesis ‘Mathematics is Ontology’.Oliver Feltham - 2021 - Filozofski Vestnik 41 (2).
    This article explores rival interpretations of Badiou’s strategy behind the claim ‘mathematics is ontology’, from his construction of an alternative history of being to that of Heidegger to his exposure of the radical contingency of the ‘decisions on being’ carried out by transformative practices in the four conditions of philosophy: art, politics, love and science. The goal of this exploration is to open up the possibility of another strategy that responds to Badiou’s initial intuition – that being is (...)
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  42.  25
    Richard Tieszen, After Gödel. Platonism and Rationalism in Mathematics and Logic.Paola Cantu - 2014 - Journal for the History of Analytical Philosophy 2 (8).
    Oxford: Oxford University Press 2011, x + 245 pp. £44.00 (hardcover). ISBN 978-0-19-960620-7.
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  43.  71
    Fundamental physical theories: mathematical structures grounded on a primitive ontology.Valia Allori - 2007 - Dissertation, Rutgers
    In my dissertation I analyze the structure of fundamental physical theories. I start with an analysis of what an adequate primitive ontology is, discussing the measurement problem in quantum mechanics and theirs solutions. It is commonly said that these theories have little in common. I argue instead that the moral of the measurement problem is that the wave function cannot represent physical objects and a common structure between these solutions can be recognized: each of them is about a clear (...)
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  44.  11
    Mathematical Plato.Roger Sworder - 2013 - Ranchos de Taos, New Mexico: Sophia Perennis.
    Plato is the first scientist whose work we still possess. He is our first writer to interpret the natural world mathematically, and also the first theorist of mathematics in the natural sciences. As no one else before or after, he set out why we should suppose a link between nature and mathematics, a link that has never been stronger than it is today. Mathematical Plato examines how Plato organized and justified the principles, terms, and methods of our (...)
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  45. Mathematical formalisms in scientific practice: From denotation to model-based representation.Axel Gelfert - 2011 - Studies in History and Philosophy of Science Part A 42 (2):272-286.
    The present paper argues that ‘mature mathematical formalisms’ play a central role in achieving representation via scientific models. A close discussion of two contemporary accounts of how mathematical models apply—the DDI account (according to which representation depends on the successful interplay of denotation, demonstration and interpretation) and the ‘matching model’ account—reveals shortcomings of each, which, it is argued, suggests that scientific representation may be ineliminably heterogeneous in character. In order to achieve a degree of unification that is compatible (...)
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  46.  57
    Kitcher’s Mathematical Naturalism.James Robert Brown - 2003 - Croatian Journal of Philosophy 3 (1):1-20.
    Recent years have seen a number of naturalist accounts of mathematics. Philip Kitcher’s version is one of the most important and influential. This paper includes a critical exposition of Kitcher’s views and a discussion of several issues including: mathematical epistemology, practice, history, the nature of applied mathematics. It argues that naturalism is an inadequate account and compares it with mathematical Platonism, to the advantage of the latter.
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  47. Mathematical Explanation: A Pythagorean Proposal.Sam Baron - 2024 - British Journal for the Philosophy of Science 75 (3):663-685.
    Mathematics appears to play an explanatory role in science. This, in turn, is thought to pave a way toward mathematical Platonism. A central challenge for mathematical Platonists, however, is to provide an account of how mathematical explanations work. I propose a property-based account: physical systems possess mathematical properties, which either guarantee the presence of other mathematical properties and, by extension, the physical states that possess them; or rule out other mathematical properties, and (...)
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  48. Existence, Mathematical Nominalism, and Meta-Ontology: An Objection to Azzouni on Criteria for Existence.Farbod Akhlaghi-Ghaffarokh - 2018 - Philosophia Mathematica 26 (2):251-265.
    Jody Azzouni argues that whilst it is indeterminate what the criteria for existence are, there is a criterion that has been collectively adopted to use ‘exist’ that we can employ to argue for positions in ontology. I raise and defend a novel objection to Azzouni: his view has the counterintuitive consequence that the facts regarding what exists can and will change when users of the word ‘exist’ change what criteria they associate with its usage. Considering three responses, I argue (...)
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  49.  80
    Michelangelo’s stone: an argument against platonism in mathematics.Carlo Rovelli - 2017 - European Journal for Philosophy of Science 7 (2):285-297.
    If there is a ‘platonic world’ \ of mathematical facts, what does \ contain precisely? I observe that if \ is too large, it is uninteresting, because the value is in the selection, not in the totality; if it is smaller and interesting, it is not independent of us. Both alternatives challenge mathematical platonism. I suggest that the universality of our mathematics may be a prejudice and illustrate contingent aspects of classical geometry, arithmetic and linear algebra, (...)
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  50. Thomistic Foundations for Moderate Realism about Mathematical Objects.Ryan Miller - forthcoming - In Serge-Thomas Bonino & Luca F. Tuninetti (eds.), Vetera Novis Augere: Le risorse della tradizione tomista nel contesto attuale II. Rome: Urbaniana University Press.
    Contemporary philosophers of mathematics are deadlocked between two alternative ontologies for numbers: Platonism and nominalism. According to contemporary mathematical Platonism, numbers are real abstract objects, i.e. particulars which are nonetheless “wholly nonphysical, nonmental, nonspatial, nontemporal, and noncausal.” While this view does justice to intuitions about numbers and mathematical semantics, it leaves unclear how we could ever learn anything by mathematical inquiry. Mathematical nominalism, by contrast, holds that numbers do not exist extra-mentally, which raises (...)
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