This substantially revised edition of Wittgenstein's Remarks on the Foundations of Mathematics contains one section, an essay
of fifty pages, not previously published, as well as considerable additions to others sections. In Parts I, II and III, Wittgenstein
discusses amongst other things the idea that all strict reasoning, and so all mathematics, are built on the 'fundamental calculus'
which is logic. These parts give the most thorough discussion of Russell's logic. He writes on mathematical proof and the
question of where the proofs of mathematics get their force and cogency, if they are not reducible to proofs in logic. Thsi
leads him to discuss'contradiction in mathematics' and 'consistency proofs'. He works against the view that there is a sharp
division between 'grammatical propositions' and 'empirical prepositions'. He asks us at one point to imagine a people who
made no distinction between the applied mathematics and pure mathematics, although they counted and calculated. Could we say
they had proofs? Here is a feature of his method which becomes more imporatnt; what Wittgenstein calls, at least half seriously,
'the anthropological method in philosophy'.
This emerges in Parts V, VI and VIII. In Part VI, published here for the
first time, Wittgenstein brings togeher the view that in mathematics proofs ae 'concept forming' and the view that language
and logic and mathematics 'presuppose' common ways of acting and of living among the people who give tham and are convinced
by them. Part VIII now has a fuller discussion of difficulties in the notion of 'following a rule' in calculation and the
notion of logical necessity.