In "Philosophy" (1929) Ramsey states that "Logic issues in tautologies, mathematics in identities, philosophy in definitions; all trivial but all part of the vital work of clarifying and organizing our thought. " I'm assuming that Ramsey means to identify the central product in each of these fields, otherwise the statement reads like a platitude -- sure, Ramsey was just twenty-five when he made the comment, but we're talking about someone who D.H. Mellor, in his introduction to Philosophical Papers, seems to place above the likes of Moore, Russell, Whitehead, and Wittgenstein. In any case, if Ramsey intended something along the lines of the former, then his statement strikes me as wrong, at least from a modern view. Sometimes an important mathematical contribution comes in the form of a definition, as the successful isolation of a powerful idea. For example, consider some of the basic definitions from computability theory or, perhaps closer to mathematics proper, some of the fundamental ideas from category theory (e.g. natural transformation, adjoint functor).
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Or, e.g., real numbers as Dedekind cuts.
It’s such an odd thing to say because one of the lasting contributions of the logicists (of which Ramsey was one) is the definitions of mathematical concepts they provided (e.g., the Frege/Russell definition of natural number).
I don’t think Ramsey had a clear understanding in his own mind of what he himself meant by ‘tautologies’ and ‘identities’ in this sentence. For example, he describes the multiplicative axiom as a ‘tautology’ in The Foundations of Mathematics (Mellor, p. 221). Clearly it isn’t. And the statement that mathematics issues in identities presumably derives from the Tractatus. But Wittgenstein’s point there was that mathematical propositions are really pseudo-propositions. (“A proposition of mathematics does not express a thought,” Tractatus 6.21.) But I don’t think Ramsey really understood that point. He seemed to think that Wittgenstein was merely arguing that the identity symbol is dispensable (see, e.g., Mellor, p. 194-95).
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