Rediscovering Mathematics

Philosophy of Mathematics in the Philosophical Remarks: the Case of Induction

Abstract

Everything indicates that the detonator of Wittgenstein’s renewed interest in philosophy was Brouwer’s lecture in Vienna, in 1928. In this paper I give a quick account of the philosophical background in Cambridge and describe some of the difficulties Wittgenstein had to overcome in order to work as a lecturer at the university. I reconstruct his views on mathematical induction, one of many topics in the philosophy of mathematics he dealt with. Contrary to many other philosophers, he centres round the utility of recursive proofs and clarifies what this kind of proof it consists in. He highlights the connections between induction and the infinite, on the one hand, and induction and the notion of law, on the other. He makes clear the the proof doesn’t take us through the endless series of a set, but is rather the indication that a proof in the ordinary mathematical sense can be carried out. Wittgenstein rescues the peculiarity of recursive proofs which show that something can be proved taking as a basis the regularity of a law. Finally, I briefly consider a couple of criticisms and show that they leave Wittgenstein’s stance untouched.