In his Remarks on the Foundations of Mathematics, Wittgenstein claims, puzzlingly, that ‘the proof creates a new concept’ (RFM III-41). This paper aims to contribute to clarifying this idea, and to showing how it marks a major break with the traditional conception of proof. Moreover, since the most natural way to understand his claim is open to criticism, a secondary goal of what follows is to offer an interpretation of it that neutralizes the objection. The discussion proceeds by analysing a (...) well-known geometrical proof. (shrink)

Samuel J. Wheeler defends Wittgenstein's criticism of Cantor's set theory against the objections raised by Hilary Putnam. Putnam claims that Wittgenstein's dismissal of the basic tenets of this set theory concerning the noncountability of the set of real numbers was unfounded and ill‐conceived. In Wheeler's view, Putnam's charges result from his failure to grasp Wittgenstein's intention and, in particular, to consider the difference between empirical and logical impossibility. In my paper, I argue that Wheeler's defence is unsuccessful and, at the (...) same time, that Putnam's objections go too far. (shrink)

This is a reply to ‘Defending Wittgenstein’, Piotr Dehnel's critique of my article, ‘Defending Wittgenstein's Remarks on Cantor from Putnam’. I first show that my position is much more in agreement with Felix Mühlhölzer than Dehnel takes it to be, and that his criticism of me is nothing more than a failure to recognize this. I then show how Dehnel incorrectly reads Wittgenstein as rejecting set theory as false. It is an overemphasis on and a much too narrow picture of (...) ‘applicability’ which leads him to this view. Finally, I conclude by rejecting Dehnel's view that Wittgenstein was a finitist about mathematics. (shrink)

We analyze two problems in mathematics – the first (stated in our title) is extracted from Wittgenstein’s “Philosophy for Mathematicians”; the second (“What set of numbers is non-denumerable?”) is taken from Cantor. We then consider, by way of comparison, a problem in musical aesthetics concerning a Brahms variation on a theme by Haydn. Our aim is to bring out and elucidate the essentially riddle-like character of these problems.