Monk comments that Wittgenstein’s innovation in the philosophy of mathematics has to do with technique, not necessarily theory (although there are theoretical repercussions). His disdain was for the latent Platonism in contemporary (Russell, Cantor, Turing) mathematics which saw mathematical ‘discoveries’ where the Witt saw only ‘grammatical relations’. For example,
A proof in mathematics does not establish the truth of a conclusion; it fixes, rather, the meaning of certain signs. The ‘inexorability’ of mathematics, therefore, does not consist in certain knowledge of mathematical truths, but in the fact that mathematical propositions are grammatical. To deny, for example, that two plus two equals four is not to disagree with a widely held view about a matter of fact; it is to show ignorance of the meanings of the terms involved.
This seems somewhat similar (and possibly derived from) the idea in the first set of propositions in the Tractatus:
1. The world is all that is the case.
1.1 The world is the totality of facts, not of things.
1.11 The world is determined by the facts, and by their being all the facts.
1.12 For the totality of facts determines what is the case, and also whatever is not the case.
1.13 The facts in logical space are the world.
1.2 The world divides into facts.
1.21 Each item can be the case or not the case while everything else remains the same.
This is the idea that relations between things (i.e. facts) are atomic, and not the things themselves. Perhaps we could say that the Mathematical world is every fact which is the case. There are no mathematical ‘discoveries’.
What is interesting here, in my opinion, is the possible relation of Wittgenstein’s thinking on Mathematics with Husserl’s phenomenology. Husserl’s great addition to philosophy was the idea that consciousness is always intending towards something. Consciousness is not a static, neutral object, but is very much subjective. This places the subject in direct relation with the object being considered, thereby introducing him/her into Wittgenstein’s world of ‘facts’.