Samuel J. Wheeler

I begin by attempting to get a perspicuous overview of what Wittgenstein means by saying that a mathematical proof forms concepts. I then distinguish these sorts of cases from those we might call concept‐extending proofs, which, rather than introducing new concepts, function to enrich those concepts that have already been given a home in our mathematical practice. At the same time, I also want to argue that the line between these two sorts of proofs is not always clear and will sometimes be blurred. I go on to compare paradigmatic examples of each, concluding with a case in which it is not immediately clear whether we ought to all the proof concept‐forming or extending. This last section includes a discussion of Sorin Bangu's recent account of Wittgenstein on mathematical proof.

In Remarks on the Foundations of Mathematics, Wittgenstein notes that he has ‘not yet made the role of miscalculating clear’ and that ‘the role of the proposition: “I must have miscalculated”…is really the key to an understanding of the “foundations” of mathematics.’ In this paper, I hope to get clear on how this is the case. First, I will explain Wittgenstein's understanding of a ‘foundation’ for mathematics. Then, by showing how the proposition ‘I must have miscalculated’ differentiates mathematics from the physical sciences, we will see how this proposition is the key to understanding the foundations of mathematics.

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.