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Is Hume’s principle analytic? (English) Zbl 0968.03009
Summary: One recent ‘neologicist’ claim is that what has come to be known as “Frege’s Theorem” – the result that Hume’s Principle, plus second-order logic, suffices for a proof of the Dedekind-Peano postulate – reinstates Frege’s contention that arithmetic is analytic. This claim naturally depends upon the analyticity of Hume’s Principle itself. The present paper reviews five misgivings that developed in various of George Boolos’s writings. It observes that each of them really concerns not ‘analyticity’ but either the truth of Hume’s Principle or our entitlement to accept it and reviews possible neologicist replies. A two-part Appendix explores recent developments of the fifth of Boolos’s objections – the problem of Bad Company – and outlines a proof of the principle \(N^q\), an important part of the defense of the claim that what follows from Hume’s Principle is not merely a theory which allows of interpretation as arithmetic but arithmetic itself.

MSC:
03A05 Philosophical and critical aspects of logic and foundations
00A30 Philosophy of mathematics
03-03 History of mathematical logic and foundations
03F30 First-order arithmetic and fragments
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[1] Boolos, G., “Saving Frege from contradiction,” Proceedings of the Aristotelian Society , vol. 87 (1986), pp. 137–51; reprinted on pp. 438–52 in Frege’s Philosophy of Mathematics , edited by W. Demopoulos, Harvard University Press, Cambridge, 1995. · Zbl 0900.03021
[2] Boolos, G., “The consistency of Frege’s Foundations of Arithmetic ,” pp. 3–20, in On Being and Saying: Essays in Honor of Richard Cartwright , edited by J. J. Thompson, The MIT Press, Cambridge, 1987; reprinted on pp. 211–33 in Frege’s Philosophy of Mathematics , edited by W. Demopoulos, Harvard University Press, Cambridge, 1995. · Zbl 0900.03062
[3] Boolos, G.. “The standard of equality of numbers,” pp. 261–77 in Meaning and Method: Essays in Honor of Hilary Putnam , edited by G. Boolos, Cambridge University Press, Cambridge, 1990; reprinted on pp. 234–54, in Frege’s Philosophy of Mathematics , edited by W. Demopoulos, Harvard University Press, Cambridge, 1995. · Zbl 0900.03004
[4] Boolos, G., “Is Hume’s Principle analytic?,” pp. 245–61 in Language, Thought and Logic , edited by R. G. Heck, Jr., The Clarendon Press, Oxford, 1997. · Zbl 0938.03506
[5] Boolos, G., and R. G. Heck, Jr., “Die Grundlagen der Arithmetik §§82-83,” pp. 407–28 in Philosophy of Mathematics Today , edited by M. Schirn, The Clarendon Press, Oxford, 1998. · Zbl 0935.03008
[6] Clark, P., “Dummett’s argument for the indefinite extensibility of set and real number,” pp. 51–63 in Grazer Philosophische Studien 55, New Essays on the Philosophy of Michael Dummett , edited by J. Brandl and P. Sullivan, Rodopi, Vienna, 1998. · Zbl 0970.03007
[7] Demopoulos, W., Frege’s Philosophy of Mathematics , Harvard University Press, Cambridge, 1995. · Zbl 0915.03004
[8] Dummett, M., “The philosophical significance of Gödel’s Theorem,” Ratio , vol. 5 (1963), pp. 140–55.
[9] Dummett, M., Frege: Philosophy of Mathematics , Duckworth, London, 1991.
[10] Dummett, M., The Seas of Language , The Clarendon Press, Oxford, 1993. · Zbl 0875.03033
[11] Dummett, M., Truth and Other Enigmas , Duckworth, London, 1978.
[12] Field, H., “Critical notice of Crispin Wright Frege’s Conception of Numbers as Objects ,” Canadian Journal of Philosophy , vol. 14 (1984), pp. 637–62.
[13] Field, H., “Platonism for cheap? Crispin Wright on Frege’s context principle,” pp. 147–70 in Realism, Mathematics and Modality , Basil Blackwell, Oxford, 1989. · Zbl 1098.00500
[14] Hale, B., “ Grundlagen §64,” Proceedings of the Aristotelian Society , vol. 97 (1997), pp. 243–61.
[15] Hale, B., “Reals by Abstraction,” Philosophia Mathematica , vol. 8 (2000), pp. 100–23. · Zbl 0968.03010
[16] Heck, R. G., Jr., “Finitude and Hume’s Principle,” Journal of Philosophical Logic , vol. 26 (1997), pp. 589–617. · Zbl 0885.03045
[17] Oliver, A., “Hazy totalities and indefinitely extensible concepts: an exercise in the interpretation of Dummett’s Philosophy of Mathematics ,” pp. 25–50 in Grazer Philosophische Studien 55, New Essays on the Philosophy of Michael Dummett , edited by J. Brandl and P. Sullivan, Rodopi, Vienna, 1998. · Zbl 0973.03004
[18] Parsons, C., “Frege’s theory of number,” pp. 180–203, in Philosophy in America , edited by M. Black, Allen and Unwin, London; reprinted on pp. 182–210 in Frege’s Philosophy of Mathematics , edited by W. Demopoulos, Harvard University Press, Cambridge, 1995. · Zbl 0900.03011
[19] Shapiro, S., “Induction and indefinite extensibility: the Gödel sentence is true but did someone change the subject,” Mind , vol. 107 (1998), pp. 597–624. JSTOR:
[20] Shapiro S., and A. Weir, “New V, ZF and abstraction,” Philosophia Mathematica , vol. 7 (1999), pp. 293–321. · Zbl 0953.03061
[21] Wright, C., Frege’s Conception of Numbers as Objects , Aberdeen University Press, Aberdeen, 1983. · Zbl 0524.03005
[22] Wright, C., “On the philosophical significance of Frege’s Theorem,” pp. 201–44 in Language, Thought and Logic , edited by R. G. Heck, Jr., The Clarendon Press, Oxford, 1997. · Zbl 0938.03508
[23] Wright, C., “On the harmless impredicativity of N\(^=\) (‘Hume’s Principle’), ” pp. 339–68 in Philosophy of Mathematics Today , edited by M. Schirn, The Clarendon Press, Oxford, 1998. · Zbl 0925.03022
[24] Wright, C., “Response to Dummett,” pp. 389–405 in Philosophy of Mathematics Today , edited by M. Schirn, The Clarendon Press, Oxford, 1998. · Zbl 0937.03005
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