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  1. What We Talk About When We Talk About Numbers.Richard Pettigrew - 2018 - Annals of Pure and Applied Logic 169 (12):1437-1456.
    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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  • Two Types of Abstraction for Structuralism.Øystein Linnebo & Richard Pettigrew - 2014 - Philosophical Quarterly 64 (255):267-283.
    If numbers were identified with any of their standard set-theoretic realizations, then they would have various non-arithmetical properties that mathematicians are reluctant to ascribe to them. Dedekind and later structuralists conclude that we should refrain from ascribing to numbers such ‘foreign’ properties. We first rehearse why it is hard to provide an acceptable formulation of this conclusion. Then we investigate some forms of abstraction meant to purge mathematical objects of all ‘foreign’ properties. One form is inspired by Frege; the other (...)
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  • Generic Structures†.Leon Horsten - 2019 - Philosophia Mathematica 27 (3):362-380.
    In this article ideas from Kit Fine’s theory of arbitrary objects are applied to questions regarding mathematical structuralism. I discuss how sui generis mathematical structures can be viewed as generic systems of mathematical objects, where mathematical objects are conceived of as arbitrary objects in Fine’s sense.
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  • Structuralism and Its Ontology.Marc Gasser - 2015 - Ergo: An Open Access Journal of Philosophy 2:1-26.
    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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  • 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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  • Motivating Wittgenstein's Perspective on Mathematical Sentences as Norms.S. Friederich - 2011 - Philosophia Mathematica 19 (1):1-19.
    The later Wittgenstein’s perspective on mathematical sentences as norms is motivated for sentences belonging to Hilbertian axiomatic systems where the axioms are treated as implicit definitions. It is shown that in this approach the axioms are employed as norms in that they function as standards of what counts as using the concepts involved. This normative dimension of their mode of use, it is argued, is inherited by the theorems derived from them. Having been motivated along these lines, Wittgenstein’s perspective on (...)
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  • Are the Natural Numbers Fundamentally Ordinals?Bahram Assadian & Stefan Buijsman - 2019 - Philosophy and Phenomenological Research 99 (3):564-580.
    Philosophy and Phenomenological Research, Volume 99, Issue 3, Page 564-580, November 2019.
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  • Inferentialism and Structuralism: A Tale of Two Theories.Ryan Mark Nefdt - 2018 - Logique Et Analyse 61 (244):489-512.
    This paper aims to unite two seemingly disparate themes in the philosophy of mathematics and language respectively, namely ante rem structuralism and inferentialism. My analysis begins with describing both frameworks in accordance with their genesis in the work of Hilbert. I then draw comparisons between these philosophical views in terms of their similar motivations and similar objections to the referential orthodoxy. I specifically home in on two points of comparison, namely the role of norms and the relation of ontological dependence (...)
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  • Warum die Mathematik keine ontologische Grundlegung braucht.Simon Friederich - 2014 - Wittgenstein-Studien 5 (1).
    Einer weit verbreiteten Auffassung zufolge ist es eine zentrale Aufgabe der Philosophie der Mathematik, eine ontologische Grundlegung der Mathematik zu formulieren: eine philosophische Theorie darüber, ob mathematische Sätze wirklich wahr sind und ob mathematischen Gegenstände wirklich existieren. Der vorliegende Text entwickelt eine Sichtweise, der zufolge diese Auffassung auf einem Missverständnis beruht. Hierzu wird zunächst der Grundgedanke der Hilbert'schen axiomatischen Methode orgestellt, die Axiome als implizite Definitionen der in ihnen enthaltenen Begriffe zu behandeln. Anschließend wird in Anlehnung an einen Wittgenstein'schen Gedanken (...)
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  • The Last Mathematician From Hilbert's Göttingen: Saunders Mac Lane as Philosopher of Mathematics.Colin McLarty - 2007 - British Journal for the Philosophy of Science 58 (1):77-112.
    While Saunders Mac Lane studied for his D.Phil in Göttingen, he heard David Hilbert's weekly lectures on philosophy, talked philosophy with Hermann Weyl, and studied it with Moritz Geiger. Their philosophies and Emmy Noether's algebra all influenced his conception of category theory, which has become the working structure theory of mathematics. His practice has constantly affirmed that a proper large-scale organization for mathematics is the most efficient path to valuable specific results—while he sees that the question of which results are (...)
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  • Chasing Individuation: Mathematical Description of Physical Systems.Zalamea Federico - 2016 - Dissertation, Paris Diderot University
    This work is a conceptual analysis of certain recent developments in the mathematical foundations of Classical and Quantum Mechanics which have allowed to formulate both theories in a common language. From the algebraic point of view, the set of observables of a physical system, be it classical or quantum, is described by a Jordan-Lie algebra. From the geometric point of view, the space of states of any system is described by a uniform Poisson space with transition probability. Both these structures (...)
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  • Structuralism and Isomorphism.C. McCarty - 2015 - Philosophia Mathematica 23 (1):1-10.
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  • De la posibilidad a la existencia matemática: los casos de Shapiro y de Balaguer.Max Fernández de Castro - 2009 - Signos Filosóficos 11 (21):73-101.
    En este artículo me gustaría concentrarme en al forma de tratar el problema de Benacerraf respecto de la inaccesibilidad de los objetos abstractos. Este es el principio (llamado FBP por Balaguer) que caracteriza a los objetos por axiomas de una teoría de la existencia consistente. Analizo los argume..
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  • Is the Continuum Hypothesis a Definite Mathematical Problem?Solomon Feferman - manuscript
    The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.
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  • An Inferentialist Conception of the A Priori.Ralph Wedgwood - 2015 - Oxford Studies in Epistemology 5:295–314.
    This paper offers an account of the a priori. According to this account, the fundamental notion is not that of a priori knowledge, or even of a priori justified belief, but a notion of an a priori justified inferential disposition. The rationality or justification of such a priori justified inferential dispositions is explained purely by some of the basic cognitive capacities that the thinker possesses, independently of any further experiences or other conscious mental states that the thinker happens to have (...)
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  • On Non-Eliminative Structuralism. Unlabeled Graphs as a Case Study, Part B.Hannes Leitgeb - forthcoming - Philosophia Mathematica:nkaa009.
    This is Part B of an article that defends non-eliminative structuralism about mathematics by means of a concrete case study: a theory of unlabeled graphs. Part A motivated an understanding of unlabeled graphs as structures sui generis and developed a corresponding axiomatic theory of unlabeled graphs. Part B turns to the philosophical interpretation and assessment of the theory: it points out how the theory avoids well-known problems concerning identity, objecthood, and reference that have been attributed to non-eliminative structuralism. The part (...)
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  • Lautman on Problems as the Conditions of Existence of Solutions.Simon B. Duffy - 2018 - Angelaki 23 (2):79-93.
    Albert Lautman (b. 1908–1944) was a philosopher of mathematics whose views on mathematical reality and on the philosophy of mathematics parted with the dominant tendencies of mathematical epistemology of the time. Lautman considered the role of philosophy, and of the philosopher, in relation to mathematics to be quite specific. He writes that: ‘in the development of mathematics, a reality is asserted that mathematical philosophy has as a function to recognize and describe’ (Lautman 2011, 87). He goes on to characterize this (...)
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  • Naturalism and Abstract Entities.Feng Ye - 2010 - International Studies in the Philosophy of Science 24 (2):129-146.
    I argue that the most popular versions of naturalism imply nominalism in philosophy of mathematics. In particular, there is a conflict in Quine's philosophy between naturalism and realism in mathematics. The argument starts from a consequence of naturalism on the nature of human cognitive subjects, physicalism about cognitive subjects, and concludes that this implies a version of nominalism, which I will carefully characterize. The indispensability of classical mathematics for the sciences and semantic/confirmation holism does not affect the argument. The disquotational (...)
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  • Say My Name. An Objection to Ante Rem Structuralism.Tim Räz - 2015 - Philosophia Mathematica 23 (1):116-125.
    I raise an objection to Stewart Shapiro's version of ante rem structuralism: I show that it is in conflict with mathematical practice. Shapiro introduced so-called ‘finite cardinal structures’ to illustrate features of ante rem structuralism. I establish that these structures have a well-known counterpart in mathematics, but this counterpart is incompatible with ante rem structuralism. Furthermore, there is a good reason why, according to mathematical practice, these structures do not behave as conceived by Shapiro's ante rem structuralism.
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  • Mathematical Structuralism, Modal Nominalism, and the Coherence Principle.James S. J. Schwartz - 2015 - Philosophia Mathematica 23 (3):367-385.
    According to Stewart Shapiro's coherence principle, structures exist whenever they can be coherently described. I argue that Shapiro's attempts to justify this principle are circular, as he relies on criticisms of modal nominalism which presuppose the coherence principle. I argue further that when the coherence principle is not presupposed, his reasoning more strongly supports modal nominalism than ante rem structuralism.
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  • Frege Meets Aristotle: Points as Abstracts.Stewart Shapiro & Geoffrey Hellman - 2015 - Philosophia Mathematica:nkv021.
    There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake (...)
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  • Discernibility by Symmetries.Davide Rizza - 2010 - Studia Logica 96 (2):175 - 192.
    In this paper I introduce a novel strategy to deal with the indiscernibility problem for ante rem structuralism. The ante rem structuralist takes the ontology of mathematics to consist of abstract systems of pure relata. Many of such systems are totally symmetrical, in the sense that all of their elements are relationally indiscernible, so the ante rem structuralist seems committed to positing indiscernible yet distinct relata. If she decides to identify them, she falls into mathematical inconsistency while, accepting their distinctness, (...)
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