 Another persistent theme in Wittgenstein’s thought is that the meaning of mathematics resides entirely in its utility in non-mathematical applications. But there is no systematic theory of how this applicability comes about, no proof of a conservative extension theorem, for example, showing how application of mathematical calculi to empirical premisses will never lead us to derive an empirical conclusion which does not follow from those premisses. And there is no resolution of the problem of the metatheory. On the other hand, we should observe that these notes of Wittgenstein on the philosophy of mathematics were not published by him, but by others after his death. For an overall view of Wittgenstein’s philosophy of mathematics as a whole see  Wittgenstein’s philosophy of mathematics.