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The Contemporary Interest of an old Doctrine

Published online by Cambridge University Press:  28 February 2022

William Demopoulos
William Demopoulos*
Affiliation:
The University of Western Ontario
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Extract

My purpose in this talk is to give an overview of the rediscovery of Frege's theorem together with certain of the issues that this rediscovery has raised concerning the evaluation of Frege's logicism—the ‘old doctrine’ of my title.

The contextual definition of the cardinality operator, suggested in §63 of Grundlagen— what, after George Boolos, has come to be known as Hume's principle—asserts

The number of Fs = the number of Gs if, and only if, F ≈ G,

where F ≈ G (the Fs and the Gs are in one-to-one correspondence) has its usual, second order, explicit definition. The importance of this principle for the derivation of Peano's second postulate (‘Every natural number has a successor’) was emphasized by Crispin Wright (1983, §xix) who presented an extended argument showing that, in the context of the system of second-order logic of Frege's Begriffsschrift, Peano's second postulate is derivable from Hume's principle.

Type
Part VII. Foundational Projects in Mathematics at the Beginning of the 20th Century in Their Systematic and Historical Contexts
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1994 , Issue 2: Volume Two: Symposia and Invited Papers 1994 , 1994 , pp. 208 - 216
Copyright
Copyright © 1995 by the Philosophy of Science Association

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Footnotes

1

Support from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged.

References

Boolos, G. (1987), ‘The consistency of Frege's Foundations of arithmetic,’ in Thomson, Judith Jarvis, ed., On being and saying: Essays for Richard Cartwright, Cambridge: M. I. T. Press, pp. 3 - 20, reprinted in (Demopoulos 1995).Google Scholar
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