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. (shrink)

The relation between Wittgenstein's philosophy of mathematics and mathematical Intuitionism has raised a considerable debate. My attempt is to analyse if there is a commitment in Wittgenstein to themes characteristic of the intuitionist movement in Mathematics and if that commitment is one important strain that runs through his Remarks on the foundations of mathematics. The intuitionistic themes to analyse in his philosophy of mathematics are: firstly, his attacks on the unrestricted use of the Law of Excluded Middle; secondly, his distrust (...) of non-constructive proofs; and thirdly, his impatience with the idea that mathematics stands in need of a foundation. These elements are Fogelin's starting point for the systematic reconstruction of Wittgenstein's conception of mathematics. (shrink)

In one of his meetings with members of the Vienna Circle, Wittgenstein discusses Hermann Weyl’s brief conversion to intuitionism and criticizes his arguments against applying the law of the excluded middle to generalizations over the natural numbers. Like Weyl, however, Wittgenstein rejects the classical model theoretic conception of generality when it comes to infinite domains. Nonetheless, he disagrees with him about the reasons for doing so. This paper provides an account of Wittgenstein’s criticism of Weyl that is based on his (...) differing understanding of what a general statement over infinite domains consists in. This difference in their conception of generality is argued to be central to the middle Wittgenstein’s overall stance on intuitionism as well. While Weyl (and other intuitionists) reject the law of the excluded middle on grounds of constructivity, Wittgenstein argues that general statements over infinite domains do not express propositions in the first place. The origin of this position as well as its consequences for contemporary debates on generality are further assessed. (shrink)