Logic/syntax versus arithmetic

[Alan Weir]

Timothy Y. Chow (FOM Vol 206 Issue 23) raises some further points about nominalism arising from the thread mentioned in the subject header. He asks what the nominalist means by 'concrete' and 'abstract'.

I agree these aren't precise notions, perhaps not exhaustive ones, but I think we have enough handle on them to progress the debate in foundations of mathematics.

I don't think the issue of quantum physics, raised, albeit only tentatively, by Tim, is all that relevant here. Even in classical physics, gravitational and electromagnetic force fields pose problems for crude accounts of what it is to be concrete:- as having compact (in the non-mathematical sense) spatiotemporal location for example. (Moreover, as an aside, some quantum physicists talk a tremendous load of baloney about quantum physics. As a non-(philosopher of physics) it's maybe a bit presumptuous to say that, but a lot of them talk a mix of empirical science and philosophy without realising they are doing philosophy, absolutely terrible crude verificationist philosophy (aka the Copenhagen interpretation(s)) which would get them failed in 101 Philosophy, at that.)

But ink splotches, voltage highs and lows in condensers, these sort of things seem to me unproblematic examples of concreta, even the voltage highs since we aren't going too quantum (at the moment). And so the debate about 'concretist nominalism' can work with them.

Tim, rightly focussing on such things not quantum phenomena,  objects that

'The ink on the page is not the symbol "0".  The symbol "0" is an abstraction.  If I write "0" on a blackboard with chalk, or display a "0" using pixels on a computer screen, then these are distinct concrete physical entities.'

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).

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!)

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. Open-ended notational innovations, for example adding notation for exponentiation, then for super-exponentiation and so on mean that, to speak with the platonist, we create concrete numeral expression tokens which aren't 'downward closed': we can't possibly create concrete numeral expressions for all smaller numbers.

So we have to idealise. I claim not through imagining what superhumans "might" be capable of, but using metamathematics, expressed in concrete tokens of wffs and proofs of results. But how that  might work is a bit too long a story for a blog posting with any hope of being persuasive I think.

Alan Weir

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

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