[FOM] Foundational Issues: Carneiro

[Alan Weir]

Mario Carneiro (FOM Digest 160 Issue 13) wrote

>I think we have a difference of understanding of the word "exists" here.
When I say "the natural numbers exist", I mean no more or less than "? N,
(0 in N /\ ? n in N, n+1 in N) is a theorem in the theory of interest".<

Ok, but what you mean by 'the natural numbers exist' is not what almost all other competent users of the utterance mean, it's not what it means in English. And, unless we want to write off their understanding as illusory, it's natural to ask about the meaning of such a sentence, as usually understood, to ask   whether it has a truth value and, if so, what grounds its truth value and so forth.

Moreover we need also to consider sentences like (I use here an example of Neil Tennant's) '(2^(2^(2^(2^(2^2) ...) + 1 is prime'. If you still are thinking in terms of 'very finite' or 'feasible proof', as you seemed to be initially, and this string means in your idiolect that there exists a 'very finite' concrete proof of that very string  (in some standard arithmetic calculus understood from context)  then it may very well be false and not only it but also  '(2^(2^(2^(2^(2^2) ...) + 1 is not prime'. However, I'll be so bold as to claim, it doesn't mean that in English; perhaps there is no non-trivial gloss on its meaning. What's more interesting is what makes it true or false, if it has a truth value.  If the formalist says  it is made true or false not by an infinite abstract realm of numbers but only by the existence of a concrete, feasible, 'very finite' proof, here, or in similar examples, we have bivalence failing for such simple statements and classical logic is threatened even for Delta_0 Arithmetic.
>This sounds less like excluded middle and more like incompleteness.<

The point though is that incompleteness threatens bivalence if truth is identified with, or tied closely to, proof and thence the threat to LEM.

If the formalist says, of the primality example, 'there exists a proof or disproof in principle', then the formalist has descended to the last refuge of the philosophical scoundrel. She just really means that an abstract proof exists in a formal system she is happy with, however much she may play  around with fairy stories about superheroes and genies and the like. And for a formalist to identify truth with abstract proof seems a waste of time, if the motivation is anti-platonism. But there are, I have argued, ways round  this.


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

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