The Logical Must

Abstract

This chapter is structured, following preliminaries, mainly around a debate in recent academic philosophy of mathematics, with reference to work by present-day philosophers as well as Wittgenstein himself and his contemporary Gottlob Frege. Frege is mentioned in the schematic list of topics in the summary of Chap. 2. A contrast with Frege’s views is useful here in Chap. 4 too, particularly in the light of (i) Frege’s own insistence on the applicability of mathematics, and (ii) Wittgenstein’s relation to Frege as a philosopher. So the chapter begins with some remarks about Frege and the objectivity of mathematics. The development then moves to consider objectivity and necessity, including Hartry Field’s striking notion of ‘necessary truth without truth’. Bringing in Wittgenstein again at this stage and relating his views once again to Field’s, the chapter considers Wittgenstein’s own view of necessity, and explains how what he says might seem to lead him to a conventionalist view of mathematics, thereby denying the necessary nature of mathematical claims. Michael Dummett, in his review of Wittgenstein’s Remarks on the Foundation of Mathematics, famously saddled Wittgenstein with an extreme kind of conventionalism about mathematics; in fact Dummett labelled Wittgenstein a ‘full-blooded conventionalist’. Hilary Putnam is another distinguished philosopher of mathematics who backed Dummett on this. I take Putnam to set the terms of the debate here, moving to give a reading of Wittgenstein deriving in overall terms from an article by Barry Stroud. Dummett responded to Stroud, as I explain. I argue, however, that in the end both Dummett and Putnam miss an important aspect of Wittgenstein’s thought. In short, as I explain in support of Stroud, Wittgenstein does not offer to justify, explain, or give an account of necessity at all. What he looks for, as ever, is an overview of the philosophical terrain enabling a clear sight of the conceptual difficulties to which we are prone. The chapter ends by taking up and developing another aspect of Stroud’s exegesis of Wittgenstein on necessity, namely the stress Wittgenstein places on sense and possible lack of the same we find in our philosophical excavations. In turn this leads to an analysis of some remarks Wittgenstein made in his Lectures on the Foundations of Mathematics about mathematics and its ‘responsibility to reality’. This, then, brings us to the final chapter, which parlays the overview we have gained so far into a solution to the problem with which we began.