In “Conservativeness and Incompleteness” (from 1983), Stewart Shapiro attacks Field’s program for nominalizing physics (and eliminating mathematics) by showing that the addition of ZFC to the nominalistic theory of gravity that he gives is in fact deductively non-conservative. That is, the system encodes arithmetic, and therefore has undecidable Gödel statements. These statements are provable when the full first-order theory of ZFC is added though, because of Field’s representation theorem. These statements are semantic consequences of Field’s theory (as in, every actual model of the second-order nominalistic theory satisfies these formulas), but because of the incompleteness of second-order deductive logic, these statements aren’t theorems of the system.

This sounds to me like a fairly decisive argument against Field’s program, unless Field wants to argue that a purely first-order theory is sufficient for physics.

It also seems to trade on the fact that there are second order semantic consequences that we can recognize independent of any deductive system. If one’s picture of mathematics is like Dummett’s view seems to be (that mathematics is the study of the a priori – a set of pre-packaged deductions to be applied wherever the premises are satisfied) then this suggests that either some of this second-order reasoning is not a priori valid (and therefore not part of mathematics) or that our deductive system for the nominalist theory is not fully recursively axiomatizable, because we’re willing to accept reasoning representable as the full second-order system here.

I’ll have to revisit these thoughts when I’ve had more sleep.

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