Wittgenstein versus Hilbert

Abstract

A crucial link between language and the world was identified by Wittgenstein,
when he discussed the extra-linguistic paradigms around which our understanding
of concepts must centre. The place of such paradigms has been forgotten in the
twentieth century tradition of Meta-mathematics. Remembering their place enables
two very basic criticisms to be made of this tradition: it obscured the Kantian,
i.e. synthetic nature of the rest of Mathematics, but also it obscured the fact
that it itself was equally synthetic.