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In his Remarks on the Foundations of Mathematics, Wittgenstein says repeatedly that only through its applications does mathematics become more than a game. He also says that proofs provide new criteria for concepts, fix the meanings of words used in the statement of theorems, and the like. This suggests the following picture: that mathematical propositions begin as empirical assertions that are transformed into necessary truths by means of proofs. This requires, of course, that propositions should not be thought of as fixed abstract objects with determinate modalities but rather as entities whose sense can evolve. Although Wittgenstein’s thoughts about following rules have many contexts, their point here is not scepticism, but rather to block a regress argument which would say that all meanings are fully determinate once rules and axioms have been laid down. This lecture is not intended as an “interpretation” of Wittgenstein, but as one line of thought suggested by his Remarks. Many of his observations are prefaced by “one would like to say” (or similar); he was trying out ideas and furnishing alternative ways of thinking. He changed his mind often. There is no such thing as his “theory” about mathematics, but the alternatives he considered do make us think again about what we often take for granted.