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 of

For 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 that an account of numbers should start from their use in counting (so a structuralist understanding that explains the nature of arithmetical truths prior to explaining their application is going wrong). But, Parsons argues, our structuralist in fact can resist that further thought.

I’m not sure I fully have the measure of Parsons thinking here. Part of the trouble is that he slips back into talking of numbers as objects (e.p. pp. 74–75), while I thought the attraction of the eliminative structuralism was to get rid of numbers as a special kind of object. But I take it the thought is something like this. Counting some objects involves putting them into one-one correspondence with an initial segment of some paradigm simply infinite system (of numerals, say). That involves setting up some external relations between some members of the relevant simply infinite system, over an above the internal relations which constitute their being a such a system. But now, via the Dedekind categoricity theorem, we see that these external relations will engender a one-one correspondence with an isomorphic initial segment of any simply infinite system. So, in counting, we automatically get an implicit generalization over simply infinite systems — which is what, according to the eliminative structuralist, talk of numbers amounts to. Hence, as Frege wanted, even on the structuralist view, we do after all have an essential connection between numbers and their application in counting.

That, I think, does deal with the supposed general problem. Now, Dummett has raised a more specific problem — roughly, defining a simply infinite system doesn’t tell us whether its initial element is to be treated as 0 or 1 (or indeed, I suppose, 42). But Parsons (rightly in my view) doesn’t find this worry a telling one for the structuralist. He can regard it as just a matter of pragmatic convention whether, in applications, we start counting at 0 or 1, depending on how much we care about having a number for empty collections.

One final comment on this section. Having quieted worries about the structuralist view, Parsons remarks that as well as the natural number 3, we have the integer 3, the rational 3, the real number 3 and the complex number 3 (not to mention more exotic constructions). And the structuralist can say that the use of “3” each time signifies not the same entity but the same structural role, a point congenial to his general account of the significance of number words. But, contra Parsons, I don’t see that the multiple use of “3” counts at all against the Fregean view that numbers are specific objects. The Fregean can just say that there are here a number of different terms, (“natural number 3”, “rational number 3”, etc.) with different objects as reference — with the common elements of the referring terms justified by the likeness of the role of the denoted objects in the respective families.

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