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Philosophia Mathematica Advance Access originally published online on March 4, 2008
Philosophia Mathematica 2008 16(3):354-373; doi:10.1093/philmat/nkn002
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© The Author [2008]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Wittgenstein on Circularity in the Frege-Russell Definition of Cardinal Number{dagger}

Boudewijn De Bruin*

* University of Groningen, Faculty of Philosophy, Oude Boteringestraat 52, 9712 GL Groningen, The Netherlands. b.p.de.bruin{at}rug.nl

Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out to be valid on its own terms, even though it depends on two epistemological principles the logicist may find too ‘constructivist’.

{dagger} This paper has benefited from detailed written comments of Juliet Floyd, Bob Hale, and Allard Tamminga, as well as from discussions with Charles Chihara, Hans Sluga, Martin Stokhof, and audiences in Amsterdam, Groningen, and Kirchberg.


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