Wittgenstein, Mathematics, and the Temporality of Technique

One of the stated commitments of the later Wittgenstein’ philosophy is that, just as philosophy must not in any way “interfere” with the practice of mathematicians, conversely and equally, “no mathematical discovery” can by itself “advance” philosophy in its quest to clarify the forms of our lives and language. It would thus appear ab initio that, for Wittgenstein, the mathematician and the philosopher of mathematics, operating with different methods and in distinct regions of inquiry and insight, have very little to say to each other. On the other hand, however, Wittgenstein is committed equally strongly to the idea that philosophy can and should take a different kind of interest in mathematics: not as a body of results to be explicated or methods to be emulated but as a set of techniques or practices within human life, to be understood in general terms only in the context of related practices that are not simply or exclusively mathematical and thereby as illuminating our practices and ways of life, much more broadly. This includes the characteristic practices of mathematics “itself”—activities such as calculating and problem-solving, developing proofs, and making conjectures. But it also, crucially, includes those practices that characterize our practical, lived, social, emotional, and educational experience much more generally—practices, for example, of teaching and learning, of understanding and being convinced, of “seeing” a relationship, and of “knowing” what is the right way to proceed. The philosopher’s interest in these practices, and in particular in the ways that they are involved in (what we call) “doing” mathematics, extends to the illumination of what is meant by (or what we understand by) such “ordinary” phenomena and experience as those of following a rule, practicing a regular method, developing a technique, arguing rationally for a conclusion, and convincing someone of something (whether by means of a “formal” or “informal” “proof”). With respect to each of these, Wittgenstein argues the philosopher’s attention to mathematical practice provides a decisive guideline for the broader kinds of clarification and illumination that philosophical reflection itself produces more generally. It does so, in part, by directing our attention to those features of (specifically) mathematical practice that mark its role within the broader and multiple contexts of what we may call, using Wittgenstein’s terminology, our collective and shared human and linguistic “form of life.”