*Charles S. Chihara*

- Published in print:
- 1991
- Published Online:
- November 2003
- ISBN:
- 9780198239758
- eISBN:
- 9780191597190
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198239750.003.0007
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

The first of six chapters in which rival views are critically evaluated and compared with the Constructibility view described in earlier chapters. The views considered here (forms of ‘Structuralism’) ...
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The first of six chapters in which rival views are critically evaluated and compared with the Constructibility view described in earlier chapters. The views considered here (forms of ‘Structuralism’) are those of Stewart Shapiro and Michael Resnik. A number of difficulties with these two views are detailed and it is explained how the Constructibility Theory is not troubled by the problems that Structuralism was explicitly developed to resolve.Less

The first of six chapters in which rival views are critically evaluated and compared with the Constructibility view described in earlier chapters. The views considered here (forms of ‘Structuralism’) are those of Stewart Shapiro and Michael Resnik. A number of difficulties with these two views are detailed and it is explained how the Constructibility Theory is not troubled by the problems that Structuralism was explicitly developed to resolve.

*Michael D. Resnik*

- Published in print:
- 1999
- Published Online:
- November 2003
- ISBN:
- 9780198250142
- eISBN:
- 9780191598296
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198250142.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

Mathematics is regarded as our most developed science, and yet philosophical troubles surface as soon as we inquire about its subject matter partly because mathematics itself says nothing about the ...
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Mathematics is regarded as our most developed science, and yet philosophical troubles surface as soon as we inquire about its subject matter partly because mathematics itself says nothing about the metaphysical nature of its objects. Taking mathematics at face value seems to favour the Platonist view according to which mathematics concerns causally inert objects existing outside space‐time, but this view seems to preclude any account of how we acquire mathematical knowledge without using some mysterious intellectual intuition. In this book, I defend a version of mathematical realism, motivated by the indispensability of mathematics in science, according to which (1) mathematical objects exist independently of us and our constructions, (2) much of contemporary mathematics is true, and (3) mathematical truths obtain independently of our beliefs, theories, and proofs.The ontological component of my realism is a form of structuralism according to which mathematical objects are featureless, abstract positions in structures, or patterns, and like geometric points, their identities are fixed only through their relationships to each other. Structuralism is also part of my epistemology in that material objects ‘fit’ simple patterns, and in doing so, they ‘fill’ the positions of simple mathematical structures. We may perceive the arrangements of objects but we cannot perceive their positions i.e. the abstract, non‐spatiotemporal mathematical objects, and the problem then consists in explaining how we can form beliefs about them.Answering this question introduces a central notion of my epistemology, that of a posit: by representing and designing patterned objects our ancestors posited geometric objects as sui generis and started describing them by describing the patterns in which they are positions. Since positing mathematical objects, like positing new scientific entities, is an activity similar to making up a story, one might wonder how such an activity can lead to mathematical knowledge and truth, but I believe that our ancestors were justified in introducing mathematical objects and we are justified in retaining them, by pragmatic and global considerations: mathematics has proved immensely fruitful for science, technology, and practical life, and doing without it is now virtually impossible.This account of justification introduces a further problem: if our justification for believing in mathematical truths is global and pragmatic, then it might turn out that one is not justified in accepting a mathematical claim unless it is accepted by science, and this is clearly at odds with the practice of mathematics where we hardly ever invoke such global considerations in order to justify a mathematical claim. In mathematics, we usually employ a local conception of evidence made up mainly of a priori proofs. However, arguing from the perspective of a Quinean epistemic holism, I claim that this feature of the practice should not make us conclude that mathematics is an a priori science, disconnected evidentially from both observation and natural science, for observation is relevant to mathematics, and technological and scientific success forms a vital part of our justification for believing in the truth of mathematics.Less

Mathematics is regarded as our most developed science, and yet philosophical troubles surface as soon as we inquire about its subject matter partly because mathematics itself says nothing about the metaphysical nature of its objects. Taking mathematics at face value seems to favour the Platonist view according to which mathematics concerns causally inert objects existing outside space‐time, but this view seems to preclude any account of how we acquire mathematical knowledge without using some mysterious intellectual intuition. In this book, I defend a version of mathematical realism, motivated by the indispensability of mathematics in science, according to which (1) mathematical objects exist independently of us and our constructions, (2) much of contemporary mathematics is true, and (3) mathematical truths obtain independently of our beliefs, theories, and proofs.

The ontological component of my realism is a form of structuralism according to which mathematical objects are featureless, abstract positions in structures, or patterns, and like geometric points, their identities are fixed only through their relationships to each other. Structuralism is also part of my epistemology in that material objects ‘fit’ simple patterns, and in doing so, they ‘fill’ the positions of simple mathematical structures. We may perceive the arrangements of objects but we cannot perceive their positions i.e. the abstract, non‐spatiotemporal mathematical objects, and the problem then consists in explaining how we can form beliefs about them.

Answering this question introduces a central notion of my epistemology, that of a posit: by representing and designing patterned objects our ancestors posited geometric objects as *sui generis* and started describing them by describing the patterns in which they are positions. Since positing mathematical objects, like positing new scientific entities, is an activity similar to making up a story, one might wonder how such an activity can lead to mathematical knowledge and truth, but I believe that our ancestors were justified in introducing mathematical objects and we are justified in retaining them, by pragmatic and global considerations: mathematics has proved immensely fruitful for science, technology, and practical life, and doing without it is now virtually impossible.

This account of justification introduces a further problem: if our justification for believing in mathematical truths is global and pragmatic, then it might turn out that one is not justified in accepting a mathematical claim unless it is accepted by science, and this is clearly at odds with the practice of mathematics where we hardly ever invoke such global considerations in order to justify a mathematical claim. In mathematics, we usually employ a local conception of evidence made up mainly of *a priori* proofs. However, arguing from the perspective of a Quinean epistemic holism, I claim that this feature of the practice should not make us conclude that mathematics is an *a priori* science, disconnected evidentially from both observation and natural science, for observation is relevant to mathematics, and technological and scientific success forms a vital part of our justification for believing in the truth of mathematics.

*Mary Leng*

- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199280797
- eISBN:
- 9780191723452
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199280797.003.0010
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology

This chapter sums up the argument of the book, and ties up a few loose ends. In particular, it considers Michael Resnik's alternative ‘pragmatic’ version of the indispensability argument, and shows ...
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This chapter sums up the argument of the book, and ties up a few loose ends. In particular, it considers Michael Resnik's alternative ‘pragmatic’ version of the indispensability argument, and shows how this argument, while not relying on confirmational holism, can be responded to from the fictionalist perspective developed in this book. It also considers the question of whether one should reject the existence of mathematical objects or merely remain agnostic about their existence, and argues for the stronger position which rejects the existence of such things. Finally, it considers a sense in which the view of the axioms of set theory with non‐mathematical objects as urelements as generative of a fiction can be considered to be a revival of a version of conventionalism about our mathematical hypotheses, as analytic ‘truths’ which are not in fact truths.Less

This chapter sums up the argument of the book, and ties up a few loose ends. In particular, it considers Michael Resnik's alternative ‘pragmatic’ version of the indispensability argument, and shows how this argument, while not relying on confirmational holism, can be responded to from the fictionalist perspective developed in this book. It also considers the question of whether one should reject the existence of mathematical objects or merely remain agnostic about their existence, and argues for the stronger position which rejects the existence of such things. Finally, it considers a sense in which the view of the axioms of set theory with non‐mathematical objects as urelements as generative of a fiction can be considered to be a revival of a version of conventionalism about our mathematical hypotheses, as analytic ‘truths’ which are not in fact truths.