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## 15. Philosophy of mathematics

Platonism in mathematics involves two claims, that there is a realm of necessary facts independent of human thought and that these facts may outrun our ability to get access to them by proofs. Platonism is attractive because it accounts for several striking features of mathematical experience: first that proofs are compelling and yet may have conclusions which are surprising, and second that we seem to be able to understand some mathematical propositions without having any guarantee that proofs of them exist.

Wittgenstein never accepted Platonism because he always took the view that making substantive statements is one thing, while articulating the rules for making them is another. So-called necessary truths clearly do present rules of language, inasmuch as accepting them commits one to allowing and disallowing certain linguistic moves. Wittgenstein holds that it is therefore a muddle to think that such formulations describe some particularly hard and immovable states of affairs. Thus in the Tractatus mathematical propositions are treated together with tautologies as sets of signs which say nothing, but show the logic of the world.

Nevertheless the Tractatus view has some kind of affinity with at least the first claim in Platonism, inasmuch as the rules of our language, on which mathematics rest, are rules of the only logically possible language. But when Wittgenstein comes to see linguistic rules as features internal to our (possibly varying) practices, the resulting picture is unwelcoming even to this. We cannot now assume there to be such a thing as ‘the logic of the world’, whether to be shown or said. Instead, in Remarks on the Foundations of Mathematics, he explores ideas of the following kinds.

At a given time we have linguistic practices directed by certain rules. Someone may now produce a proof of a formula which if accepted would be a new rule - for example, ‘14+3=17‘. It is natural to think that to accept this is to unpack what we were already committed to by our understanding of ‘17’, ‘+’, and so on. But the rule-following considerations unsettle this assumption because they undermine the idea of an intellectual confrontation with an abstract item which forces awareness of its nature upon us and they also bring to our attention the element of spontaneity in any new application of a given term. Rather to accept the proof and its outcome is to change our practices of applying signs like ‘17’, because it is to adopt a new criterion for judging that seventeen things are present, namely that there are two groups of fourteen and three. Hence to accept the proof is to alter our concepts. What makes mathematics possible is that we nearly all agree in our reaction to proofs, and in finding them compelling. But to seek to explain this by pointing to Platonic structures is to fall back into incoherent mythology.

The present author’s own view is that it is persistent uneasiness with the first claim in Platonism which primarily motivates Wittgenstein’s reflections on mathematics. But those who see him as an antirealist will put more stress on hostility to the second claim (the idea of verification transcendence) and certainly some of Wittgenstein’s remarks (for example, his suspicion of the application of the law of excluded middle to mathematical propositions) have affinities with ideas in intuitionistic logic. A third reading will bring out the conventionalist-sounding elements, on which we choose what linguistic rules to adopt on pragmatic grounds.

In addition to reflections on the nature and use of elementary arithmetical claims, Wittgenstein also applies his ideas to some more complex constructs in mathematical logic, such as the Frege-Russell project of deriving mathematics from logic, Cantor’s diagonal argument to the non-denumerability of the real numbers, consistency proofs and Gödel’s theorem. His general line here is not that there is anything wrong with the mathematics but that the results have been misconstrued, because they have been interpreted against a mistaken background Platonism. Some mathematical logicians claim that Wittgenstein has not understood properly what he is discussing. His views on consistency and Gödel in particular have aroused annoyance (see Antirealism in the philosophy of mathematics §2; Intuitionism; Realism in the philosophy of mathematics §2).

Heal, Jane. Philosophy of mathematics. Wittgenstein, Ludwig Josef Johann (1889–1951), 2011, doi:10.4324/9780415249126-DD072-2. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/biographical/wittgenstein-ludwig-josef-johann-1889-1951/v-2/sections/philosophy-of-mathematics-1.

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