Samuel J. Wheeler

The aim of our paper is to examine John McDowell’s theory of perceptual experience with reference to its inheritance of the early Wittgenstein. We begin by showing that several fundamental tenets of McDowell’s account emerge from an encounter with the Tractatus, most notably the requirement that there be an isomorphism between thought and the world in our claims to perceptual knowledge. We argue that those passages from the early Wittgenstein McDowell exploits already contain an internal criticism of an essential element of the latter’s account, namely, his distinction between perceptual apprehension and perceptual judgment. We show how this distinction in McDowell must be understood as an extension of the Fregean distinction between the force and content of a judgment, and how Wittgenstein’s critique of Frege in the Tractatus is equally applicable to McDowell’s account of perceptual judgment. Following Wittgenstein’s lead, we show how McDowell’s commitment to the force/content distinction threatens to undermine his claim that perceptual experiences non-inferentially justify the judgments in which they figure. To do justice to the idea of perception as a capacity for knowledge, we conclude by proposing an alternative view in which perception alone is capable of constituting a judgment so as to the way things are.

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.