Back, after rather a gap, to Charles Parsons’s book and on to the first half of his second chapter, “Structuralism and nominalism”.
(Sec. 8) Parsons says that he himself thinks that “something close to the structuralist view is true”. But structuralist in what sense? It is often said, perhaps in a Bourbachiste spirit, that mathematics is the study of structures. But — as Parsons stresses — that leaves it wide open what picture we should adopt of the ontology of mathematical objects. He is more concerned with structuralism(s) with more ontological bite — something along the lines suggested by “the objects of mathematics are positions in structures, [and] have no identity or features outside of a structure” (to quote from Michael Resnik’s well-known 1981 Nous paper).
(Sec. 9) But what are structures? The usual modern mathematical story sees these as sets (or classes) with distinguished elements, equipped with relations and/or functions. So it looks as though an account of mathematical objects as positions in structures already presupposes familiar kinds of objects (sets, classes) to build structures out of, and explaining their nature in structuralist terms threatens circularity. But Parsons puts this worry on hold for the moment.
(Sec. 10) So go with the set-theoretic conception of structure, just pro tem, and consider as an exemplar Dedekind’s treatment of the natural numbers. Dedekind defines what it is for a set N, with distinguished element 0, and a mapping S: N -> N – {0} to be “simply infinite”. Abbreviate those (categorical) conditions Ω(N, 0, S). With some effort, an ordinary statement of arithmetic can be correlated with a version A(N, 0, S) whose primitives are again N, 0, S. And on one reading of Dedekind — the eliminative reading — the suggestion is that the ordinary statement can be treated as elliptical for
For any N, 0, S, if Ω(N, 0, S) then A(N, 0, S).
This is ‘eliminative’ in that a statement apparently about one kind of thing, numbers, is treated as in fact a disguised generalization about other kinds of things. The suggestion neatly sidesteps “multiple reduction” problems for more straightforward attemps to reduce arithmetic to set theory. But (on the face of it) it faces the worry that if there are no simply infinite systems then any ordinary arithmetical statement comes out as vacuously true and arithmetic is inconsistent. True, that first worry won’t be pressing if we already buy into a background universe with enough sets, but it will become more urgent when we try to repeat the trick and give an eliminative structuralist account of them. And there’s a related second worry. Ω(N, 0, S) will involve quantification over sets, as indeed will a typical A(N, 0, S) as we give explicit definitions of e.g. recursive arithmetical functions. Do we want really want a structuralist account of a particular familiar kind of mathematical object, numbers, to tells us that we’ve been generalizing about some other rather less familiar kind of object all along? (Parsons wonders: Maybe we need to generalize over structures to state structuralism as a general thesis: but does a structuralist account of a particular kind of object have to similarly generalize over structures?)
(Sec. 11) Well, we can sidestep the second of those worries, and the worries of Sec. 9, perhaps, by trading in an explicitly set-theoretic presentation of Dedekind’s eliminative structuralism for a version couched in second-order logical terms. We get a new second-order definition of being simply infinite, Ω'(N, 0, S), a new correlate of an ordinary arithmetical claim, A‘(N, 0, S), and correspondingly a new suggestion that the ordinary statement can be treated as elliptical for
For any N, 0, S, if Ω'(N, 0, S) then A’(N, 0, S).
where now ‘any N‘ and ‘any S‘ are treated as second-order. If we are relaxed enough about second-order quantification, we might find this easier to swallow that the previous version (though that’s quite a big “if”). However, this kind of ‘if-thenism’ is still threatened by the possibility of vacuity. What to do?
One option is to read the conditional as stronger-than-material, e.g. by discerning a governing modal operator. But that opens up another set of problems. What kind of modality is involved here? Can we e.g. give a modest possibility-as-consistency reading? Perhaps “we interpret the theories in an if-thenist way, but deal with the problem of possibility by appealing to consistency, nominalistically interpreted.” The suggestion is to be pursued critically in Sec. 12.
OK, so much by way of brisk summary of these sections (I didn’t find them entirely easy to follow, but I hope I’ve fairly represented the way the discussion develops). I don’t think I have much to add by way of commentary: in fact, the dialectic so far is a pretty familiar one.