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Exploring the Philosophy of Mathematics: Beyond Logicism and Platonism

Richard Startup

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...

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Published in: Open Journal of Philosophy
ISSN: 2163-9434 2163-9442
Published: Scientific Research Publishing, Inc. 2024
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URI: https://cronfa.swan.ac.uk/Record/cronfa66085
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spelling v2 66085 2024-04-18 Exploring the Philosophy of Mathematics: Beyond Logicism and Platonism d86a8b1f7833763cea35d2b88386d0d4 Richard Startup Richard Startup true false 2024-04-18 FGHSS 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. Journal Article Open Journal of Philosophy 14 02 219 243 Scientific Research Publishing, Inc. 2163-9434 2163-9442 Logicism, Platonism, Gödel’s Platonism, Quine’s Naturalism, Confirmational Holism, Algebraic and Non-Algebraic Mathematical Theories 9 4 2024 2024-04-09 10.4236/ojpp.2024.142017 COLLEGE NANME Humanities and Social Sciences - Faculty COLLEGE CODE FGHSS Swansea University 2024-04-23T10:36:49.0617303 2024-04-18T16:41:51.2367827 Faculty of Humanities and Social Sciences School of Social Sciences - Criminology, Sociology and Social Policy Richard Startup 1 66085__30066__c106e62e92114a18a0d2990ccd20b8dc.pdf 66085.VoR.pdf 2024-04-18T16:46:56.5263027 Output 374579 application/pdf Version of Record true Copyright © 2024 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). true eng http://creativecommons.org/licenses/by/4.0/
title Exploring the Philosophy of Mathematics: Beyond Logicism and Platonism
spellingShingle Exploring the Philosophy of Mathematics: Beyond Logicism and Platonism
Richard Startup
title_short Exploring the Philosophy of Mathematics: Beyond Logicism and Platonism
title_full Exploring the Philosophy of Mathematics: Beyond Logicism and Platonism
title_fullStr Exploring the Philosophy of Mathematics: Beyond Logicism and Platonism
title_full_unstemmed Exploring the Philosophy of Mathematics: Beyond Logicism and Platonism
title_sort Exploring the Philosophy of Mathematics: Beyond Logicism and Platonism
author_id_str_mv d86a8b1f7833763cea35d2b88386d0d4
author_id_fullname_str_mv d86a8b1f7833763cea35d2b88386d0d4_***_Richard Startup
author Richard Startup
author2 Richard Startup
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container_volume 14
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publishDate 2024
institution Swansea University
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publisher Scientific Research Publishing, Inc.
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hierarchy_top_title Faculty of Humanities and Social Sciences
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description 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.
published_date 2024-04-09T10:36:46Z
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