FOM Digest, Vol 205, Issue 50: Tim Chow on Bivalence and Unknowability

[Alan Weir]

Tim Chow wrote :

"If the feeling that they need to clean house comes over them, then mathematicians are much more willing to toss out the continuum hypothesis than the Riemann hypothesis, because they don't really care about CH. 
Who needs infinite set theory anyway?
Unknowability has relatively little to do with it, in my opinion."

Most mathematicians will surely care very little about whether or not an even number of elephants died on this day 1000 years ago. But they may well not reject (or even be agnostic on) bivalence with respect to the question  just because they don't care or just because the answer is unknowable. (Some may claim it's "in principle possible" to know  the proposition's  truth value, but I've scare quoted the phrase because I think it is one of the most unhelpful and arguably incoherent phrases in philosophy of mathematics.) Of course one might demur from bivalence with respect to this proposition EE (for Even number of Elephants) because of vague boundaries: of 'elephant', 'death', 'day' and so on. But a realist will still want to affirm some sort of bivalence claim, perhaps to the effect that every precisification of EE is either true or false, even for propositions whose truth value they are totally uninterested in. 

If that's the reaction of most mathematicians to EE, why the difference with CH, even if most don't care about it much? One obvious conjecture is that  those who withhold bivalence or (not the same but relatedly) excluded middle from CH but not EE do so for philosophical reasons: perhaps because they are verificationists in mathematics but not about ordinary physical matters, verificationism  often surfacing in mathematics as some sort of constructivism; or because, though realists about the physical world, they are anti-platonists for other, non-verificationist reasons and this leads them to claim that mathematical truth does not consist in representing an external reality but is determined in some other way, is closely tied to provability for example. 

As against this, mathematicians who reject bivalence for CH for those sort of reasons would seem (to go back to Joe Shipman's original query) to have to restrict it also in arithmetic, in particular with respect to  'concrete undecidables' as well as rather esoteric Godelian propositions originally used to exemplify the negation-incompleteness of arithmetic. Godel's incompleteness results assume, of course, that proofhood is decidable and that the theorems are recursively enumerable (and so rules out such things as omega rules as genuine rules of proof: though simply moving to infinitary logic will not of itself decide all questions in set theory, of course). This Godelian constraint on provability is usually claimed to follow from the epistemic nature of truth, though in my view it's just a form of finitism which has seeped into the minds of many mathematicians quite happy with infinite set theory and model theory. 

At any rate, if most mathematicians would, on being forced to think about this, take a different view on arithmetical undecidables from CH and the like, that would seem to support Tim's conjecture that unknowability has nothing to do with it and tell against my suspicion that philosophical beliefs, perhaps not reflectively held, are at the root of the hostility to CH here. On the other hand, it could be interpreted as showing a widespread feeling that, notwithstanding Godel's results, provability in arithmetic for a proposition P is not to be equated with formal derivability, in a formal system satisfying recursive enumerability constraints,  of a representation S of P in from some version of the PA axioms. Rather, the feeling might be, proof is a wider, informal, though still rigorous notion. If this feeling is coupled with the view that (some, at least) set-theoretic undecidables look to be undecidable  on any reasonable notion of informal proof and from any set of axioms reasonably taken as constitutive (or something like that) of 'set' that would be an alternative explanation of the asymmetry re bivalence (supposing there is such), one on which it has little or nothing to do with caring more about arithmetic than set theory.  

So a sociological question, I suppose, which arises is what is the general position of informed mathematicians about bivalence in arithmetic vis a vis bivalence in set theory? And how widespread is the view that proof, which in mathematical journals is nearly always still an informal notion  (despite the increasing use of automatic theorem provers) is nonetheless closely linked to formal derivability in the Godelian sense? (It's a tricky matter indeed, of course, to say what 'closely linked' might amount to). 


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