[FOM] Logic/syntax versus arithmetic

Timothy Y. Chow tchow at math.princeton.edu
Sun Feb 23 15:33:12 EST 2020

Alan Weir wrote:

> Ok, let's call the ink on a specific page a 'concrete token'. Leaving 
> aside the question of interpretation and meaning (how can an 
> anti-platonist account for that is a good question, but here arguably 
> nobody has a good answer as yet) the nominalist needs to give a 
> nominalist account of symbol types which are not abstract but can have 
> many tokens (in different fonts, spoken as well as written, electronic 
> and so on).

I think I'm with you so far.

> Quine and Goodman in their 'Steps towards a Constructive Nominalism' 
> (JSL 12 1947 pp. 97-122) attempt to carry out a program of nominalist 
> syntax using mereological sums of tokens to play the role of types. I 
> don't think this quite works and tried to do better in my aforementioned 
> 'Truth through Proof' (OUP, 2010) Chapter Six. (I called my 'concretist' 
> simulacra for 'types' 'tipes'. Unfortunately the copy-editor thought 
> this was a tipo, sorry typo, and changed the lot back to 'types' causing 
> a lot of problems fixing it back!)

Maybe I need to go read your book, but I confess that 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?

> So I think one can give a nominalist theory of 'concrete syntax' in 
> which we have complex wffs built out of simpler. However one is still 
> left with the problem that, on naturalistic assumptions, there are only 
> finitely many concrete wffs and proofs. Moreover they aren't built up in 
> a recursive fashion enabling one to prove things about them, e.g. using 
> induction.

I'm actually not too bothered, at least at this stage of the discussion, 
if nominalism drives us to ultrafinitism and/or a formalism that evacuates 
mathematics of everything except the ability to verify concrete proofs 
that are explicitly presented to us (what I've called "strict formalism"). 
I'm more bothered by the claim that nominalists can come up with a 
satisfactory theory of formal syntax that does not immediately yield a 
satisfactory theory of natural numbers.  If you can get finitely many wffs 
then surely you can get finitely many natural numbers by following exactly 
the same strategy?

And surely the notion that syntax can be faithfully arithmetized was one 
of the key ideas behind the advances in mathematical logic in the early 
20th century?


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