Philip Bold

With respect to the metaphysics of infinity, the tendency of standard debates is to either endorse or to deny the reality of ‘the infinite’. But how should we understand the notion of ‘reality’ employed in stating these options? Wittgenstein’s critical strategy shows that the notion is grounded in a confusion: talk of infinity naturally takes hold of one’s imagination due to the sway of verbal pictures and analogies suggested by our words. This is the source of various philosophical pictures that in turn give rise to the standard metaphysical debates: that the mathematics of infinity corresponds to a special realm of infinite objects, that the infinite is profoundly huge or vast, or that the ability to think about infinity reveals mysterious powers in human beings. First, I explain Wittgenstein’s general strategy for undermining philosophical pictures of ‘the infinite’ – as he describes it in Zettel; and then show how that critical strategy is applied to Cantor’s diagonalization proof in Remarks on the Foundations of Mathematics II.

This dissertation provides a careful reading of the later Wittgenstein’s philosophy of mathematics centered around three major themes: reality, determination, and infinity. The reading offered gives pride of place to Wittgenstein’s therapeutic conception of philosophy. This conception views questions often taken as fundamental in the philosophy of mathematics with suspicion and attempts to diagnose the confusions which lead to them. In the first essay, I explain Wittgenstein’s approach to perennial issues regarding the alleged reality to which mathematical truths or propositions correspond. Wittgenstein diagnoses exotic pictures of mathematical reality as stemming from misleading analogies formed across empirical and mathematical propositions. The second essay explains Wittgenstein’s treatment of perplexity regarding the ability of a mathematical rule to determine its applications in advance. This too is found to depend on a misleading analogy, in this case across behavioral and mathematical senses of ‘determine’. The third and final essay discusses Wittgenstein’s general critical approach to “the infinite”. Philosophical perplexity about infinity is shown to depend on the verbal imagery regularly associated with proofs and their power to take hold of one’s imagination. Wittgenstein dampens the imaginative excesses often associated with “the infinite” by offering a sober redescription of Cantor’s famous method of diagonalization.

I show that Wittgenstein's critique of G.H. Hardy's mathematical realism naturally extends to Paul Benacerraf's influential paper, ‘Mathematical Truth’. Wittgenstein accuses Hardy of hastily analogizing mathematical and empirical propositions, thus leading to a picture of mathematical reality that is somehow akin to empirical reality despite the many puzzles this creates. Since Benacerraf relies on that very same analogy to raise problems about mathematical ‘truth’ and the alleged ‘reality’ to which it corresponds, his major argument falls prey to the same critique. The problematic pictures of mathematical reality suggested by Hardy and Benacerraf can be avoided, according to Wittgenstein, by disrupting the analogy that gives rise to them. I show why Tarskian updates to our conception of ‘truth’ discussed by Benacerraf do not answer Wittgenstein's concerns. That is, because they merely presuppose what Wittgenstein puts into question, namely, the essential uniformity of ‘truth’ and ‘proposition’ in ordinary discourse.

In PI 189, Wittgenstein's interlocutor asks, ‘But are the steps then not determined by the algebraic formula?’. Wittgenstein responds, ‘The question contains a mistake’. What is the mistake contained in the interlocutor's question? Wittgenstein's elaboration is neither explicit nor its intended upshot transparent. In this paper, I offer a reading on which the interlocutor's question arises from illicitly crossing different pictures of ‘determination’. I begin by working through Wittgenstein's machine analogy in PI 193, which illustrates picture‐crossing in our ways of talking about a machine. Using the lessons from this analogy, I show how the interlocutor's ‘mistake’ can be diagnosed in similar terms: their confusion about the power of a rule to determine its applications rests on mistakenly crossing a behavioural and a mathematical sense of ‘determine’—thereby concocting a mystifying picture of rule‐following.