Logic Matters 2008-08-18

Parsons's Mathematical Thought: Sec 14, Structuralism and application

We're considering the schematic idea that an ordinary arithmetical statement is elliptical for something generalizing over structures, along the lines ofFor any N, 0, S, if Ω(N, 0, S) then A(N, 0, S),where Ω(N, 0, S) lays down the conditions for a set N (equipped with a distinguished element 0, and a mapping S: N -> N - {0}) to be "simply infinite", and A(N, 0, S) is appropriately correlated with the ordinary statement. (Parsons, you'll recall, associates such a view with Dedekind. That doesn't seem historically correct. But let that pass.)Does this "eliminative structuralist" view have a problem accounting for the application of numbers as cardinals? Recall Frege's remark: "It is applicability alone that raises arithmetic from the rank of a game to that of a science. Applicability therefore belongs to it of necessity." And Frege further takes it ...

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