Chrysoula Gitsoulis

Cantor’s proof that the reals are uncountable forms a central pillar in the edifices of higher order recursion theory and set theory. It also has important applications in model theory, and in the foundations of topology and analysis. Due partly to these factors, and to the simplicity and elegance of the proof, it has come to be accepted as part of the ABC’s of mathematics. But even if as an Archimedean point it supports tomes of mathematical theory, there is a question that demands clarification: What, exactly, does Cantor’s proof show? One of few places where this question is addressed is Appendix II of Wittgenstein’s Remarks on the Foundations of Mathematics. This essay is devoted to clarifying Wittgenstein’s remarks in that section.

In Wittgenstein on Rules and Private Language, Saul Kripke presents a controversial skeptical argument, which he attributes to Wittgenstein’s interlocutor in the Philosophical Investigations [PI]. The argument purports to show that there are no facts that correspond to what we mean by our words. Kripke maintains, moreover, that the conclusion of Wittgenstein’s so-called private language argument is a corollary of results Wittgenstein establishes in §§137-202 of PI concerning the topic of following-a-rule, and not the conclusion of an independently developed argument in §§243ff of PI, as most commentators take it to be. In this work, I assess Kripke’s skeptical argument both in its own right, and as an interpretation of the rule-following sections of PI. In its own right, I try to show that it is critically flawed. However, as an interpretation of the rule-following sections of PI, I try to show that it is essentially correct. I do this by showing that Kripke’s interpretation squares with and supports the meta-philosophical framework developed by Wittgenstein in §§107-136 of PI, which immediately precedes his remarks on following-a-rule. (Oct 16, 2008. Committee: Paul Horwich, Galen Strawson, Stephen Neale, Michael Levin)

There are two aspects to Wittgenstein’s method of deconstructing pseudo-philosophical problems that need to be distinguished: (1) describing actual linguistic practice, and (2) constructing hypothetical ‘language-games’. Both methods were, for Wittgenstein, indispensable means of clarifying the ‘grammar’ of expressions of our language -- i.e., the appropriate contexts for using those expressions – and thereby dissolving pseudo-philosophical problems. Though (2) is often conflated with (1), it is important to recognize that it differs from it in important respects. (1) can be seen as functioning as a direct method of ‘proof’ (i.e., attempt to convince the reader of some thesis), and (2) as an indirect method of ‘proof’ -- proof by reductio ad absurdum. This essay will be devoted to clarifying (2) by forging an analogy with surrealism in art.