Mathematical Modalities and Mathematical Explanations

Krzysztof Wójtowicz

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Chapter 3 Mathematical Modalities and Mathematical Explanations

In: Philosophical Approaches to the Foundations of Logic and Mathematics

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Krzysztof Wójtowicz

Krzysztof Wójtowicz
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Type:: Chapter

Pages:: 123–149

DOI:: https://doi.org/10.1163/9789004445956_5

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Abstract

In the philosophy of mathematics, modal concepts appear in the realism-antirealism debate in different contexts and roles. (1) They are used by advocates of the anti-realistic position as a tool for giving anti-realistic (re)interpretations of mathematics. Geoffrey Hellman’s modal structuralism (discussed in this text) is a prominent example. (2) The notion of a mathematical modality is also used by supporters of realism in the context of contemporary versions of the indispensability argument (which is based on the claim that mathematics per se has some explanatory contribution to science). An interesting tension appears as both sides of the debate refer to modal concepts, albeit in a different way (and obviously drawing different conclusions). The aim of the article is to discuss this matter. I formulate several hypotheses concerning the possibility of giving an account of the mathematical explanation based on Hellman’s anti-realist account of mathematics. Hellman considers the indispensability argument to be a serious challenge to the anti-realist camp, so he tries to give a non-realistic account of the applicability of mathematics using modal notions. However, he does not discuss the problem of mathematical explanation, which has become popular some years after the publication of Hellman’s main work. I claim that Hellman’s approach could be used to provide a better understanding of the mathematical modality in question – and also to undermine the realistic thesis. It allows one to reconstruct (or: reinterpret) the mathematical tools needed in science in an anti-realistic manner, making it coherent with accounts of explanation formulated in terms of a mathematical modality. In this paper, I discuss the possibility of pursuing such program, and formulate some tentative hypotheses hoping to initiate a discussion. The results might be a challenge for the mathematical realist.

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Philosophical Approaches to the Foundations of Logic and Mathematics

In Honor of Stanisław Krajewski

Series: Poznań Studies in the Philosophy of the Sciences and the Humanities, Volume: 114

Cover Philosophical Approaches to the Foundations of Logic and Mathematics

E-Book ISBN:: 9789004445956

Publisher:: Brill

Print Publication Date:: 13 Jan 2021

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- Philosophy
  - 19th & 20th Century Philosophy
  - Logic

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Chapter 3 Mathematical Modalities and Mathematical Explanations

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Philosophical Approaches to the Foundations of Logic and Mathematics

In Honor of Stanisław Krajewski

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