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

Philosophia Mathematica 16 (3):354-373 (2008)
Abstract
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 on philosophy of mathematics 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 logicist philosophers of mathematics may find too ‘constructivist’.
Keywords wittgenstein  mathematics  hume's principle  number  cardinal number  philosophy of mathematics  Frege-Russell definition of cardinal number  logicism  constructivism
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