On the Philosophical Significance of Frege's Theorem

Crispin Wright

in The Reason's Proper Study

Published in print March 2001 | ISBN: 9780198236399
Published online November 2003 | e-ISBN: 9780191597565 | DOI:
On the Philosophical Significance of Frege's Theorem

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Frege's theorem is the result that elementary arithmetic may be derived from the second‐ order sentence often referred to as ’Hume's Principle’ (HP). In this essay, Wright considers whether ’abstraction principles’ of this kind can give rise to a distinctively logicist philosophy of mathematics; in particular, he considers whether HP can underpin a satisfactory epistemology of arithmetic. After outlining the neo‐Fregean project as it arises out of Frege's Grundlagen, Wright moves to consider the bad company objection: the distinctive abstractive form of HP is shared by numerous other second‐order sentences, some of which (such as Frege's Basic Law V) are inconsistent. After examining various forms of this challenge and potential responses to them, Wright recommends that the neo‐Fregean restrict attention to those abstraction principles which are conservative. The final sections of the essay survey prospects for extending the neo‐Fregean story to other areas of mathematics.

Keywords: abstraction principle; Boolos; conservativeness; Dummett; Frege's theorem; Hume's Principle; logicism; mathematics

Chapter.  18410 words. 

Subjects: Philosophy of Mathematics and Logic

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