Re Logic/Syntax versus Arithmetic.

[Alan Weir]

Re Sam Sanders citation of some remarks of Church (Vol 206 Issue 26):

Alonzo Church's comparison of nominalists with extreme misogynists might have seemed funny in the 1950s, likewise perhaps a comparison with white supremacists, but I confess comparing the platonism/anti-platonism dispute with much more important questions doesn't personally have me rolling in the aisles. But then we don't value Church as a humourist but as a brilliant logician whose philosophical views are also well worth attending to.

Church's remarks  criticising nominalism and Quine and Goodman are interesting but not at all decisive. And anyway they are irrelevant to the point at issue in the thread started by Tim Chow, one of whose questions was whether a nominalist restricting herself solely to concreta could make sense of even the most basic of syntactic distinctions, for example between symbol tokens and symbol types. I argued that the nominalist could, that Quine and Goodman had pointed the way even though their account was not totally successful; nothing in Church's comments leads me to alter that view. 

As to linguistics: 

'As Church thought likely, the 'finitistic nominalism of Quine and Goodman has been useless for linguistics, both theoretical and computational.

That's exactly in line with what I said in my comments on the inadequacy of strict finitism and the need for idealisation (FOM Volume 206 Issue 21) so I'm a bit puzzled about the point of this posting.

The dispute between platonists and 'cake and eat it' anti-platonists, like myself (and, I think, the more recent work of those in the 'fictionalist' camp), is over whether one can hold that mathematical theorems, including existential claims, are true/correct without believing abstract objects exist in mind-independent reality. One anti-platonist strategy  here is conditional: if such anti-platonism is plausible for e.g. number theory can it also be maintained for formal syntax and metamathematics?  I say yes, and moreover that this extension makes this sort of anti-platonism more plausible. 

The hard work lies in establishing that 'cake and eat' anti-platonism is not incoherent or at least hopelessly implausible. If successful though, I see no need to restrict the work of mathematicians at all, something which I think is resented by mathematicians, naturally enough. I'm not a philosophical doormat-type (David Lewis's  Parts of Classes pp. 58-9 is often taken as verse for the day by them) who thinks philosophers should never intervene in other disciplines and challenge the coherence of certain parts of them. But I agree with the sentiment that we have to have very good reason to do so. In maths, in particular, not having the least sympathy with constructivism or finitism, which seem to me to be based on verificationist philosophy,  I don't see any need to do so at all, including for example in transfinite set theory with ever-so-large cardinals. And similarly I don't know of any good philosophical reason to object to the applied mathematics as used in linguistics, computer science and so on; the proposal is not that these disciplines should be recast as some sort of mereology over actual concrete tokens, that's a total non-starter.

Alan Weir

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