The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

13
On the Harmless Impredicativity of N=
('Hume's Principle')

CRISPIN WRIGHT


I. INTRODUCTION: DUMMETT'S OBJECTIONS TO
'NEO-FREGEANISM'

Michael Dummett Frege: Philosophy of Mathematics and its famous precursor, Frege: Philosophy of Language, contrast sharply in their estimates of Frege's achievements in those respective subjects.1 The guiding conviction of Frege: Philosophy of Language was that even admirers of Frege's writings have tended to a merely superficial appreciation of his contribution, that he is justly viewed as the father of analytical philosophy itself, the first philosopher to perceive the centrality of the philosophy of language, and especially the theory of meaning, in all philosophy, and the inventor of the tradition of systematic thought about meaning within which so much of important recent and contemporary philosophy belongs. By contrast, Frege: Philosophy of Mathematics--notwithstanding the kindest possible concluding accolade2--returns a predominantly negative verdict on Frege's accomplishment in relation to the specific questions which were most important to him. In particular, to the questions at which Grundlagen and Grundgesetze are primarily directed--the character of the subject-matter of number theory and real analysis and the nature of our knowledge about it--Frege's distinctive Platonist-cum-logicist answer was, according to Dummett, demonstrably 'catastrophically wrong'.3

My thanks to all who participated in the discussion at the Munich conference and especially to Bob Hale and Stewart Shapiro.

____________________
1
Michael Dummett, Frege: Philosophy of Language, London, Duckworth, 1973; Frege: Philosophy of Mathematics, London, Duckworth, 1991.
2
'[T]he greatest philosopher of mathematics yet to have written', Frege: Philosophy of Mathematics, 321.
3
Frege: Philosophy of Mathematics, 292. The emphatic tone of this assessment is sustained in numerous passages in Dummett's text. Compare 'the central flaw in [ Frege's] entire philosophy of arithmetic', p. 232; 'the big error . . . in [ Frege's] philosophy of mathematics' and '[ Frege's] philosophy of arithmetic was, indeed, fatally flawed', p. 321; 'The inconsistency of Frege's . . . system was not a mere accident [but] . . . an error affecting his entire philosophy', p. 223; 'the task at which Frege so lamentably failed', p. 317; and so on.

-339-

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