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 (1934тАУ1951) grew from his Early (1912тАУ1921) and Middle (1929тАУ33) philosophies, a dialectical path reconstructed here partly as a response to the limitative results of G├╢del and Turing.