Logic/Syntax versus Arithmetic

[Alan Weir]

Tim Chow wrote (FOM Vol 207 Issue 4.1):

'My hypothesis is that *if* you can come up with some kind of nominalistically satisfactory account of infinitely many wffs, and/or of syntactic operations, then you're going to be able to come up with a nominalistically satisfactory account of infinitely many numbers, and/or of arithmetic operations, essentially by mimicking whatever philosophical moves you make in the syntactic case.  I haven't seen anything in your account that would let me give a philosophically satisfactory development of syntax that wouldn't also let me give a philosophically satisfactory development of arithmetic (to an analogous degree).

Do you disagree?  Do you see a fundamental difference between syntax and arithmetic, and if so, what?'

As regards formal syntax and arithmetic, I am non-revisionary so I agree that, since Godel, we have known that the theories of syntax (for standard countable languages) and arithmetic are intertranslatable.

As regards a theory of actual concrete utterances (past, present, future), or perhaps actual plus possible feasible ones, on the one hand,  compared  to some sort of concretist ultra-finitist arithmetic on the other-  well I'm not sure that the question of whether we can establish some interesting link, an isomorphism between them perhaps,  is even all that well-posed.  In both cases the usual closure properties will fail: there will be pairs of wffs but no conjunction thereof (Sorites issues arise here of course), complex wffs, containing abbreviatory locutions, which  lack sub-constituents and the like. A theory of actual concrete representatives of the numbers will be similarly patchy; it's not clear how we could systematically relate them in an interesting way. (Similarly, even leaving performance errors aside, the linguist does not think syntax for a natural language consists simply in listing every utterance speakers of the language have uttered, or ever will utter. Some idealised formal structure has to be imposed.)

As to your hypothesis, yes I agree with the conditional and believe the antecedent too. But to give a nominalistically satisfactory account of either formal syntax or of arithmetic is, in my view, not to reject the formal theories of either but to do something essentially metalinguistic: give an account which can be applied to any concrete utterance of either theory, an account of what makes those utterances true or false, which does not take them as representing abstract structures, which thus makes no appeal to anything non-physical.

The original worry (looking back to try to remember what that was!) was whether it made sense to challenge platonism in, say arithmetic, by appeal to logical consequence, read for example as  derivability, a syntactic notion. For these logico-syntactic notions- proof, wff etc.- seem to be as abstract, and the syntactic theory of the pretty much the same power, e.g. entailing infinitely many distinct elements in a structure, as arithmetic. That's a good objection which nominalists/anti-platonists/formalists need to respond to, I agreed. 

Trying to sum up my responses, they were  that a) strict or ultrafinitism, radically revising standard maths so as to deny the claim that there are infinitely many numbers , (or infinitely many wffs) won't work. But  b) if the anti-platonist can give an account of what makes true actual utterances in standard mathematics, including tokens of theorems to the effect that there are infinitely many numbers, infinitely many wffs, that every Delta_0 sentence is either provable or else it is refutable, (proof formally defined) and so on, one which makes no claim that  these  utterances (or the propositions expressed) represent an acausal non-spatio-temporal structure, then job done for the anti-platonist. 

Alan Weir
Roinn na Feallsanachd/Philosophy
Sgoil nan Daonnachdan/School of Humanities 
Oilthigh Ghlaschu/University of Glasgow 
GLASGOW G12 8QQ