The Problem Solved

Abstract

This Chapter begins with an exegesis of §241–2 of Wittgenstein’s Philosophical Investigations, developing its relevance to some of what he says in his Lectures on the Foundations of Mathematics. Then, to assist with clarification on how this applies to the issue of mathematical ‘responsibility to reality’, the chapter takes a brief foray into some actual mathematics—the beginnings of the theory of groups. Group theory, at least in its beginnings, has an ideal simplicity and brevity for our purpose here. To keep the narrative thread going strong, however, I relegate explanation of the basic mathematics involved to an Appendix. No specific knowledge of mathematics beyond that attained in a decent high school is presupposed or required to understand what is going on, here and elsewhere in the book. The example of ‘groups of small order’ satisfyingly illuminates what Wittgenstein has to say about the sense of talk of the responsibility of mathematics to reality. The chapter moves on from this example to relate this illumination, together with the work covered in earlier chapters, to a specific contemporary dialogue between two distinguished philosophers of mathematics, Mary Leng and John Burgess. The dialogue centres on some work of Rudolf Carnap, and a distinction he makes between questions internal and external to a given theory. Carnap himself recognised a debt to Wittgenstein in his work; the chapter draws this out to apply what we have learned about Wittgenstein and the philosophy of mathematics to the controversy at issue in this dialogue between Burgess and Leng. There is an interim conclusion to be drawn at the close of this section: it is that mathematical fictionalism, like other brands of nominalist non- or anti-realism, illegitimately assumes for the realism it denies a sense it has yet to be given. Baldly put in that way, the conclusion may seem gnomic; clarified as it is by its position in the overview that has been emphasised throughout, nevertheless it points, within this overview, to a resolution of Einstein’s enigma and the overall problem of mathematical applicability—of mathematics and world. It remains to nail down this final point. An example helps, again. The chapter returns to Steiner and his characterisation of the problem of mathematical applicability, before moving to consider a way in which some (more advanced) group theory seems to epitomise the problem. (Again, explanation of some mathematical aspects of the story here is relegated to an Appendix in order to avoid breaking the narrative thread.) A striking and surprising way in which group theory seems to predict the physics of elementary particles has deeply puzzled philosophers and physicists. By locating this surprise within how, as Wittgenstein has it, we may or may not make sense of mathematics relating to reality, we can develop a way of seeing that enables a clear view of why such surprise is inappropriate. This way of seeing is made explicit in relation to the example to hand. Taken as a whole, the book offers the reader an understanding of how to do philosophy of mathematics following Wittgenstein. This chapter ends with the claim that Einstein’s enigma, the philosophical problem of mathematical applicability—of mathematics and world—has been determinately solved through this understanding.