Logic/syntax versus arithmetic (Timothy Y. Chow)

Alan Weir Alan.Weir at glasgow.ac.uk
Tue Feb 18 08:33:36 EST 2020

Tim Chow wrote (FOM Vol. 206 Issue 14)

'A nominalist presumably balks at abstract entities, and an "axiom" is surely an abstract entity, and "logical consequence" (or its syntactic counterpart, provability) is surely an abstract relation.  So I don't see why a nominalist should be any more comfortable with logic than with mathematics.'

This is a very good question. Mary Leng, in her book-length treatment Mathematics and Reality (Oxford, 2010) deals with the issue of a nominalist account of logical consequence by reading 'P is a consequence of axioms A' as, roughly, Nec(if A then P), where the conditional is material and Nec is a modal necessity operator taken to be primitive. (Hartry Field has taken a similar line.) But (to toot my own trumpet- or perhaps blow my own bagpipes) I argued in  a review of that book (British Journal for the Philosophy of Science,  2014, pp. 657-664) that that still leaves, as Tim's comments also emphasise, the question of the nominalist treatment of syntax problematic, even if consequence is not read as derivability. For, on the face of it, the nominalist has to believe that there exists, in mind-independent reality, infinitely many wffs and proofs of arbitrarily high complexity, so most with way more constituents than the estimated number of neutrinos in the observable universe.

One thing which has been helpful in this debate, I think, is a move away from talk about 'fictionalism' to framing things in terms of nominalism or 'anti-platonism'. 'Fictionalism' suggests drawing an analogy between the ontology of maths and the ontology of fiction, with its debates about the status of so-called 'fictional objects'. But there is no consensus philosophical position on fiction, so the analogy isn't clear. Moreover few nominalists now want to say that '2+3=5' or 'there are infinitely many primes' are false (in line with crude views in the ontology of fiction). They tend to urge, rather, that it is correct to say 2+3=5, wrong (in ordinary, non-modular arithmetic) to say 2+3 = 1. Given a fairly deflationary view of truth there seems no reason not to say that 'there are infinitely many primes' is true (or true in an "internal sense") the debate now switching to what makes it true: the structure of an abstract realm of causally inert objects or, for example, it following (in some sense) from certain axioms.

But this development in fictionalism/nominalism does nothing to answer the question of the status of syntax. One route is to go strict finitist. There are only finitely many wffs and proofs, all concrete, all 'feasible' in something like the sense that more recent interesting work investigating the strict finitist tradition suggests;  see e.g. Walter Dean 'Strict Finitism, Feasibility and the Sorites' Journal of Symbolic Logic, Vol 11, Number 2, 2018.

However, to blow on my pipes once more, I argued, at book-length in 'Truth through Proof' (Oxford 2010) and at article length in 'Informal Proof, Formal Proof and Formalism' (Review of Symbolic Logic,  Volume 9, March 2016, pp. 23-43) that strict finitism is untenable, that everyone, including the 'concretist'  who takes  only feasible concrete utterances as genuine syntactic denizens of reality, has to idealise from our actual concrete practice.

One way to idealise is to engage in 'supernaturalised epistemology', to make play with supertaskers, genies who can do things which are 'in principle possible', in that exceptionally dodgy phrase, and so forth. But no naturalistically inclined philosopher should take that route.

Alternatively, idealisation can be done metamathematically. For example, there are, or at any rate 'could be' in a fairly ordinary sense, concrete metamathematical proofs that every sentence of delta_zero arithmetic (a system we have concrete proofs exists, in the same sense as we have concrete proofs of the infinity of the primes) is either (formally) provable or else is  refutable. That is enough, I claim, for the nominalist to hold each such wff is either true or false. More controversially, I have made the same claim about Peano arithmetic which is (concretely provably) negation-complete in omega logic, criticising (as based on a baseless finitism) the idea that formal proofs with more steps than there are neutrinos are perfectly ok, as idealisations of concrete practice, if they are finite, but not if they are infinite.

Alan Weir

PS: I can't find the contribution from Panu Raatikainen on deductivism, excerpts from which Tim introduced into the debate. I don't seem to have missed a FOM posting, was it a private communication? 

Emeritus Professor Alan Weir
Roinn na Feallsanachd/Philosophy
Sgoil nan Daonnachdan/School of Humanities 
Oilthigh Ghlaschu/University of Glasgow 

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