Cian Dorr‘s talk, “Of Numbers and Electrons” on Thursday morning made me realize that we’ve got a lot of the same metaphysical goals. The point of his talk was to show that a weakened (and therefore tractable) version of Hartry Field’s program will be able to support realism about theoretical entities of physics and anti-realism about mathematical entities. The scientific anti-realist might suggest a theory like the following:

BAD: As far as observable matters are concerned, it is just as if T

where T is our actual scientific theory, that talks about electrons and other unobservables. However, almost everyone agrees that such a theory is bad (hence the name Dorr has given it). The mathematical realist then claims that the mathematical anti-realist would have to give a theory like:

AS-IF: As far as the concrete world is concerned, it is just as if T

where T is our actual scientific theory, that talks about functions and numbers and other abstract entities. Dorr proposes an alternative.

One might at first try to find a relevant difference between BAD and AS-IF – perhaps “observable” is a vague term, where “concrete” isn’t? But Dorr points out that this would allow the possibility of

BAD’: As far as the positions of molecules are concerned, it is just as if T/center>

which is clearly just as bad (if not worse).

Hartry Field tries to avoid these issues by coming up with a theory that proceeds directly, without having to use the terms of a mathematical theory. This works for a version of Newtonian gravity, but it hasn’t been able to be extended, and there are even a few logical worries about the strength of his theory. Dorr instead proposes

NEC: Necessarily, if [ZFC], and the concrete world is intrinsically just as it actually is, then T.

(Note that NEC seems to be entailed by a theory of the form of Field’s, and seems quite plausible – if the world is actually purely concrete, then if there *were* numbers, then they would have the appropriate relations to one another to support counting.) To support this theory, he needs to defend a distinction between AS-IF and BAD on the one hand, and NEC on the other.

He suggests that one way to get this distinction is to say that having a large amount of theory in the scope of an existential quantifier is much worse than having a similarly large amount of theory in the scope of a universal quantifier. Then, BAD and AS-IF are cashed out as possibilities, and NEC is a necessity, which are analogous to existentials and universals, and thus NEC is not as bad as the other two.

(He was running low on time at this point, so I wasn’t entirely clear on his argument in this next section.) To motivate the difference between existential and universal quantification, he provides several examples of theories like

For some function x from spacetime points to R^{4} which respects the topological facts, T(x).

Such a theory says that spacetime can be coordinatized in a way that our theory (say special relativity) holds. But a better theory says

For every function x from spacetime points to R^{4} that respects the underlying geometric facts, T(x).

(I think he doesn’t mean anything by the switch between topological and geometric facts.) An example of the latter theory is general relativity, which Einstein thought was more appealing even before experimental verification. (Of course, the latter is no good unless we also postulate that “There is a function x from spacetime ponits to R^{4} that respects the underlying geometric facts”, but this means that “T(x)” only ever appears in the scope of a universal, rather than an existential, so this existential claim is relatively harmless.)

There are other such examples of theory improvement by replacing an existential with a universal. Basically, the reason seems to be that if some such function exists, we should be able to characterize the class of all such functions. A theory that requires existential rather than universal quantification would leave unexplained the fact that not every such function works, because we couldn’t characterize the ones that don’t in any simpler way than just saying that they don’t work.

The scientific anti-realist might then try to turn Dorr’s trick against him and convert BAD into a better theory, the way he converted AS-IF into NEC. However, there’s no nice independent axiomatization of theoretical particles that would entail them being connected to the macroscopic world in the same way that ZFC entails numbers being connected to physical things. (That is, ZFC where separation and replacement allow non-mathematical predicates and relations to occur, as well as mathematical ones.) An attempt might be

Necessarily, if the laws [were] as simple as they could be consistent with the facts about observablee [sic] matters, T.

But of course, the predicate “simple” is quite complicated, and we are talking about laws rather than just things, and all this occurs inside the scope of “consistent”, which is a sort of possibility operator, so this theory is still quite bad.

Thus, if Dorr is right, then he can get a scientific realist, mathematical anti-realist theory that is not as hard as Field’s and better than the instrumentalist’s (BAD). Of course, I think he would in fact be happier with an extension of Field’s theory, but this one would be a good approach.

In her commentary, Karen Bennett summarized Dorr’s paper (and clarified some aspects of the presentation that were lost due to time pressure), and then questioned a few of his points. For instance, she pointed out that it’s not clear that the best way to formulate BAD or AS-IF is as “It is possible for T, and observable matters to be just as they actually are” – instead, it seems to be a subjunctive conditional, which one might cash out as “Necessarily, if T, then observable matters are just as they actually are”. (She attributes this point to Gideon Rosen, who was the chair of the session.) If this is right, then the link between existential quantification and theoretical badness is not as strong as Dorr thinks, or is at least independent from the badness of these theories BAD and AS-IF. (For instance, they might be bad for being parasitic on some other theory, or the like.) Dorr responded by saying that subjunctive conditionals are better cashed out as an AE-type claim (“For every world in which T is true, there is a closer world where T is also true, where things are just as they actually are”) rather than a purely universal one. Of course, this matter is far from resolved.

In addition, it seems that since ZFC includes existential quantifiers, she suggests that NEC is just as bad as the anti-realist alternative to BAD – but Dorr points out that the badness is proportional to the size of the scope of the existential, which includes “simple” in the alternative to BAD, but only a couple lines of symbols in ZFC. (I certainly didn’t understand this point at first, but I’ve tried to clarify it here, after reading my notes. I had thought he just counted individual existential quantifiers as bad, but it’s actually symbols in the scope of existential quantifiers that he counts as bad, which is a rather awkward convention to be sure.)

And finally, Bennett pointed out that it’s not clear just what sort of modality is needed here. Dorr needs to say that it’s possible for ZFC to be true while everything is just as it actually is, and that it’s necessary that if ZFC is true and everything is just as it actually is, then T would be too. For the former, it seems that he needs logical possibility, because most nominalists would deny that it’s metaphysically possible that ZFC is true (I think I, and Field, would disagree, but she’s probably right in general), but for the latter it seems that he needs metaphysical necessity. And of course, both types of modality would need to be cashed out in a nominalistically acceptable way. Dorr seemed to agree that there were some issues left here.

At any rate, though it was inconclusive, I think it was a very interesting idea. I’d of course ideally like to see an extension of Field’s program, but this intermediate approach does suggest a possibility of some amount of progress. I wouldn’t want nominalism to get too easy though – that would make it quite questionable.

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