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

[Alan Weir]

Dennis Hamilton writes (FOM Digest Vol 205 Issue 52):

' I submit that the proposition EE is not a mathematical question at all.'

Yes indeed:  I intended it to be an empirical proposition. I fear my example EE- 'An even number of elephants died on this day 1000 years ago'-  has confused things by bringing in the even numbers. I take EE to be an empirical proposition and used a concept from applied maths 'even number of elephants' just to get a snappy example of an unknowable empirical proposition. Maybe 'Caesar blew his nose when crossing the Rubicon' would do if there was no prospect of evidence either way, no documentation surviving for example.

My point was that if mathematicians accept bivalence for empirical unknowables they don't care about but not CH then it can't be not caring alone, nor unknowability combined with some sort of general verificationism,  which (rationally) explains the different treatment in set theory. 

Tim Chow's writes (same issue) 

'when I say that people's attitude toward CH is grounded in "not caring," it's really the entire framework of infinite set theory that they feel is dispensable, not just their lack of interest in specific statements within that framework.'...

Yes that seems right but I think my point still stands: not caring about any question specifically about a particular domain seems no reason to reject bivalence, and unknowability a good reason only if you have some sort of anti-realist stance for that particular the domain in question.

For another empirical example, suppose you and I both believe that there are galaxies beyond the observable horizon about which a whole range of questions will always be unknowable (by us), perhaps because we believe that the universe is finite in time and they will never come into our observable horizon. You might nonetheless care about a host of questions regarding them: 'Is there life on any of them?' etc, though know you'll never know the answer. I may care not a whit about any matters concerning them. If I still, as a realist about the empirical world,  think it's either true or false that there is life out that far, but don't adopt bivalence throughout 'higher' set theory, then it cannot be because I don't care or because I think undecidability robs a proposition of a determinate truth value. Not if I'm consistent in these matters.  

Tim's observations on his conversations with other mathematicians are interesting. He would have a much better idea about their views than a philosopher like myself. The existence of a differential attitude, not between empirical propositions on the one hand and mathematical on the other, but between arithmetic and infinitistic set theory on the other, is particularly interesting. It reminds one of Kronecker's supposed remark that God made the integers (not, it seems, just the non-negative ones) and humans are responsible for all the rest.

But why this difference in attitude, given the negation-incompleteness (relative to Godelian notions of proof) of arithmetic? I suggested tentatively there might be a view that arithmetic is in fact negation-complete because informal proofs cannot be completely captured in finitistic proof theory (Yehuda Rav was of the latter view). Another reason might be constructivism: a constructivist about mathematics who is a realist about the empirical world will likely take currently undecidable propositions to be (currently) such that bivalence should not be affirmed of them. That still will not yield a neat arithmetic/set theory distinction in terms of determinacy in one not the other but the constructivist may well argue for the coherence of some form of intuitionistic arithmetic  but also for the incoherence of all infinitistic set theory on that basis. 

Another suggestion: perhaps the differential attitude arises because informal arithmetic is, as it were, the cultural property of most educated humans. Most can add and multiply numbers, especially given a handy system of numerals such as the positional Hindu-Arabic system. Few can do anything much similar with set theory, the concept of sum is much more widespread than that of power set. The Peano-Dedekind axioms, in second-order form, capture categorically the natural numbers system which so many have a good handle on. So perhaps, even if it is true that all of maths is the work of humanity, one part, arithmetic, with its special status, feels more determinate than others. 

The important thing, of course, is whether such a  reason for differentiating arithmetic from infinitistic set theory can be made clearer and be coherent.  I,  for one, am sceptical. 

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