Wittgenstein and infinity (Ludwig Wittgenstein)

Wittgenstein's complex understanding of infinity developed through three major phases. In the early phase Wittgenstein began with the world: infinity was compatible with logic insofar as logic "says nothing". This was to be a pure approach to infinity. In the middle phase infinity was no longer to be understood through logic, which constituted a rejection of any logical analysis of the propositions of the Principia Mathematica on Wittgenstein's part. In the later phase Wittgenstein acknowledged infinity to be deeply entrenched in mathematical language although never in some perfect or perfectable approximation to language beyond what possibilities a mark or symbol presents. Since language is an aspect of human life and the philosophical analysis of infinity is by means of language, Wittgenstein throughout is presenting infinity sub specie humanitatis, to quote Burton Dreben. For Wittgenstein overall the philosophical problem was to understand infinity, not to deny it.