Remarks on the Foundations of Mathematics

This substantially revised edition of Wittgenstein's Remarks on theFoundations of Mathematics contains one section, an essay of fiftypages, not previously published, as well as considerable additionsto others sections. In Parts I, II and III, Wittgenstein discussesamongst other things the idea that all strict reasoning, and so allmathematics, are built on the 'fundamental calculus' which islogic. These parts give the most thorough discussion of Russell'slogic. He writes on mathematical proof and the question of wherethe proofs of mathematics get their force and cogency, if they arenot reducible to proofs in logic. Thsi leads him todiscuss'contradiction in mathematics' and 'consistency proofs'. Heworks against the view that there is a sharp division between'grammatical propositions' and 'empirical prepositions'. He asks usat one point to imagine a people who made no distinction betweenthe applied mathematics and pure mathematics, although they countedand calculated. Could we say they had proofs? Here is a feature ofhis method which becomes more imporatnt; what Wittgenstein calls,at least half seriously, 'the anthropological method inphilosophy'. This emerges in Parts V, VI and VIII.

In Part VI, published here for the first time, Wittgenstein bringstogeher 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 thamand are convinced by them. Part VIII now has a fuller discussion ofdifficulties in the notion of 'following a rule' in calculation andthe notion of logical necessity.