For Wittgenstein mathematics is a human activity characterizing ways of seeing conceptual possibilities and empirical situations, proof and logical methods central to its progress. Sentences exhibit differing 'aspects', or dimensions of meaning, projecting mathematical 'realities'. Mathematics is an activity of constructing standpoints on equalities and differences of these. Wittgenstein's Later Philosophy of Mathematics (1934β1951) grew from his Early (1912β1921) and Middle (1929β33) philosophies, a dialectical path reconstructed here partly as a response to the limitative results of GΓΆdel and Turing.