Conservative Extensions Are Permissible, But Non-Conservative Ones Can Be Mandatory

18 03 2005

On the face of it, Hartry Field’s insistence that the conservativity of mathematics (together with the existence of acceptable nominalistic scientific theories) means that we shouldn’t believe in numbers flies in the face of mathematical history. It seems that historically, it’s precisely when complex numbers were shown to be conservative over the reals that they were first fully accepted. However, I think that this can be explained in a Fieldian way. The mathematician who believes in complex numbers believes in them exactly the way she believes in real numbers – which I think a hermeneutic reading of Field’s fictionalism suggests is just fictional. The established conservativity of the complex numbers means that they are a consistent extension that is conservative over the nominalistic physical theory we already have, and therefore it is just as acceptable to talk about complex numbers as it is to talk about real numbers. This sort of acceptability doesn’t mean that the claims should be literally read as true (on Field’s account), but they are still legitimate to talk about as fictional entities.

This legitimacy is meant to contrast with the non-acceptability of the talk of a set of all sets, or Frege’s Basic Law V. Once they were established to be conservative, complex numbers became just as much a permissible subject of mathematical discourse as the real numbers. The Russell set or Fregean extensions have never become permissible, because of the particular form of their non-conservativity (ie, contradictoriness).

However, non-conservativity of an extension sometimes makes the extension mandatory rather than impermissible. The parts of our physical theories that talk about subatomic particles are not conservative over the nicely formulated parts that don’t, but we must accept them and their implied entities anyway. This is because the extra statements that get proven by the non-conservative extension have explanatory power for facts that were observable at the macroscopic level. Similarly, Donald Martin (in “Mathematical Explanation”, in eds. Dales and Oliveri, Truth in Mathematics, 1998, Oxford University Press) suggests that once one countenances the entities whose existence can be derived from ZFC, one should countenance large cardinals as well, because they are a non-conservative addition in precisely a way that explains certain facts about the existence of Turing cones and Wadge degrees in all known sets of a certain complexity. So in whatever sense one accepts ZFC, one must accept certain large cardinal axioms, and in whatever sense one accepts one’s observations of macrosopic objects one must accept subatomic particles.

However, just by accepting macroscopic objects and subatomic particles, one is not forced to accept sets. They can be taken (fictionally, according to Field) or left.


Logical Consequence

11 03 2005

I’ve recently read both Dummett’s article “The Justification of Deduction” and Field’s article “Is Mathematical Knowledge Just Logical Knowledge?” and both seem to have a similar attitude towards the notion of logical consequence. In fact, the more I read of both Field and Dummett, the more similar they start to seem on a lot of issues (though they definitely have a very different way of talking about things). Basically, both of them argue that logical consequence is a notion that is prior to both the syntactic and semantic definitions that are usually given, and the reasons aren’t actually all that different. Field needs this to explain how we can do without the concepts that talk about mathematical objects (like models or formal proofs) and Dummett wants to show how it is that this practice is both justified and contentful and suggests that we can’t really be explaining either the syntactic or semantic notion in terms of the other. I think I should check out the Etchemendy book on this subject.

On Conservativeness and Incompleteness

8 03 2005

Field addresses many of the points raised in my previous post in his essay by this title (reprinted in “Realism, Mathematics, and Modality”) and I realize that I should have had an idea of what he would do anyway, having read the introduction to that book. He believes in a primitive modal notion of logical consequence that may in fact be quite powerful, to which both the syntactic and semantic consequence relations described by proof theorists and model theorists are approximations. The fact that one is an upper bound and the other is a lower bound, and that in some languages they coincide, is evidence that these approximations are exact for those languages. However, these results (and even the theories themselves) only really make sense from a platonist perspective, because they discuss abstract entities like deductions and models.

Shapiro on Conservativeness

1 03 2005

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.