David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Synthese 105 (2):141 - 176 (1995)
In this paper, elementary but hitherto overlooked connections are established between Wittgenstein's remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of theTractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgenstein's rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolem's primitive recursive arithmetic and Goodstein's equational calculus, Wittgenstein represented generality by the use of free variables. This has the effect that negation of unbounded universal and existential propositions cannot be expressed. This is claimed in the second section to be the basis for Wittgenstein's criticism of the universal validity of the law of excluded middle. In the last section, there is a brief discussion of Wittgenstein's remarks on real numbers. These show a preference, in line with finitism, for a recursive version of the continuum.
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References found in this work BETA
Michael A. E. Dummett (1978). Truth and Other Enigmas. Harvard University Press.
Ludwig Wittgenstein (1922/1999). Tractatus Logico-Philosophicus. Dover Publications.
Frank Plumpton Ramsey (1960). The Foundations of Mathematics and Other Logical Essays. Paterson, N.J.,Littlefield, Adams.
Ludwig Wittgenstein (1980). Culture and Value. University of Chicago Press.
Citations of this work BETA
Victor Rodych (2003). Misunderstanding Gödel: New Arguments About Wittgenstein and New Remarks by Wittgenstein. Dialectica 57 (3):279–313.
Montgomery Link (2009). Wittgenstein and Logic. Synthese 166 (1):41 - 54.
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