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Exploring the Philosophy of Mathematics: Beyond Logicism and Platonism
Open Journal of Philosophy, Volume: 14, Issue: 02, Pages: 219 - 243
Swansea University Author: Richard Startup
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DOI (Published version): 10.4236/ojpp.2024.142017
Abstract
A perspective in the philosophy of mathematics is developed from a consideration of the strengths and limitations of both logicism and platonism, with an early focus on Frege’s work. Importantly, although many set-theoretic structures may be developed each of which offers limited isomorphism with th...
| Published in: | Open Journal of Philosophy |
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| ISSN: | 2163-9434 2163-9442 |
| Published: |
Scientific Research Publishing, Inc.
2024
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| Online Access: |
Check full text
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa66085 |
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| Abstract: |
A perspective in the philosophy of mathematics is developed from a consideration of the strengths and limitations of both logicism and platonism, with an early focus on Frege’s work. Importantly, although many set-theoretic structures may be developed each of which offers limited isomorphism with the system of natural numbers, no one of them may be identified with it. Furthermore, the timeless, ever present nature of mathematical concepts and results itself offers direct access, in the face of a platonist account which generates a supposed problem of access. Crucially too, pure mathematics has its own distinctive method of confirming or validating results - mathematical proof - which supplies a higher level of confidence and objectivity than that available elsewhere. The dichotomy of invention and discovery is too jejune a framework for analysing creative mathematical activity. The Gödelian platonist perspective is evaluated and queried through scrutiny of the part played by mathematical resources and constraints in relation to human activity. It appears that there can be non-causal mathematical explanations and mathematical constraint on purely natural processes. Valuable implications of Quine’s naturalism are explored, but one must be cautious of his thesis of confirmational holism. The distinction between algebraic and non-algebraic mathematical theories usefully contributes to our understanding of the internally differentiated nature of the subject. |
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| Keywords: |
Logicism, Platonism, Gödel’s Platonism, Quine’s Naturalism, Confirmational Holism, Algebraic and Non-Algebraic Mathematical Theories |
| College: |
Faculty of Humanities and Social Sciences |
| Issue: |
02 |
| Start Page: |
219 |
| End Page: |
243 |