Re Logic/Syntax versus Arithmetic

[Alan Weir]

Tim Chow wrote (Vol 206 Issue 25)

"I find it very difficult to believe that one can come up with a theory of "symbol types"
or "symbol tipes" that are

1. convincingly "not abstract," yet
2. can have many tokens,

that does not also yield a completely analogous theory of natural numbers.
If one tries to mimic the theory in your book to develop a theory of "natural number tipes," at what point does the argument break down?"

I discuss (in my book chapter six) the concept of equiform tokens, noting how complex and resistant to definition it is (a sound and an ink mark in some font can be equiform) but assume it is acceptable to take it as primitive for the purposes of syntax. Then with, for example, a bit of mereology, a 'tipe' can be taken to be the mereological sum of all (actual) tokens equiform to a given actual token. This gives us finitely many tipes but nothing like numbers because the tipes aren't structured in the right sort of way.

I then attempt, in the tradition of  Quine and Goodman, to give a physicalist account of syntactic structure, atomic wffs, one wff being a constituent of another and so forth (ditto with proofs). This might seem to give the materials to construct something isomorphic to an initial segment of omega. But it doesn't because of abbreviation and notational innovation. This: (10^10032 + 57)- might be the first token of that large number (to speak with the platonist) which was ever constructed, followed now by lots of equiform tokens on different displays. But most of the smaller numbers will never be represented by concrete tokens (barring very improbable hypotheses about superhumans existing in other galaxies etc.)

You can't do formal syntax without inductive characterisation, recursive structure; not necessarily set-theoretically (I don't want to wade into that big FOM debate) but with something like the intersection of inductive sets generated from a base set by  some operations.

This is fatal for strict finitism it seems to me (and I shed no tears, not wishing to cramp any actual mathematical endeavours). It might then seem game, set and match for the platonist. But there I disagree from the perspective of 'cake and eat it' anti-platonism (I think also nowadays philosophers in the fictionalist tradition embrace this). Roughly the view is that the theorems of standard mathematics, including the existential ones, are true/correct but not made true by an independent reality. If that can work for arithmetic then why not metamathematics? And then the strategy is to apply one's anti-platonist account to metamathematics and use that to tack back to ordinary number theory, topology, whatever; the idea is that this  can render the anti-platonist stance, in my case a formalist identification of  truth with proof, more plausible. Identification of concrete proof and the truth of concrete 'tipes' in some cases, typically perhaps,  but in the general systematic theory (specified using concrete tipes and furnished with concrete proofs or proof sketches), truth abstractly specified with formal proof.

But defending those large claims takes more argument than can go in a blog post. I hope, however, that this gives some answer to  the more detailed point-  how can you get physicalistic/concrete syntax without thereby getting  concrete number theory, given the links between the syntax of finitary languages and arithmetic?

Alan Weir

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

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