Full-blooded Bolshevism: Wittgenstein's Philosophy of Mathematics

It is often maintained that Wittgenstein's later philosophy is fundamentally conservative, a claim apparently supported by one of the most frequently quoted passages from Philosophical Investigations: paragraph 124. Philosophy, Wittgenstein says there, 'may in no way interfere with the actual use of language; it can in the end only describe it. For it cannot give it any foundation either. It leaves
everything as it is'. He goes on: It also leaves mathematics as it is, and no mathematical discovery can advance it. A 'leading problem of mathematical logic is for us a problem of mathematics like any other. In the face of such direct and unequivocable statements of nonrevisionism, it is not surprising that commentators have taken it that it is no part of Wittgenstein's purpose to change the way we
study and teach mathematics. I want to argue that this is a misunderstanding. The changes Wittgenstein wished to see are, on the contrary, I believe, so radical that the name 'full-blooded Bolshevism' suggests itself as a natural way to describe the militant tendency of his remarks.