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

[Timothy Y. Chow]

Alan Weir wrote:
> 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.

I think that a better analogy might be---to adapt an example used by 
William James---"Mirza Ghulam Ahmad was the Mahdi."  There are a lot of 
subscribers to the FOM mailing list, so probably some of them regard this 
statement as bivalent, but I would guess that its bivalence is unclear to 
most FOM subscribers, because they don't accept enough of the 
presuppositions behind it (notably, that there is a Guided One who will 
appear and rid the world of evil, and that any given person might or might 
not be the Guided One).  Moreover, I think it is fair to say that most 
people who are hesitant about its bivalence are hesitant because they 
don't really care about the domain of discourse in question.

On the other hand, the presuppositions behind EE are ones that probably 
all FOM subscribers accept---that elephants exist, that they live and die, 
and that time unfolds in a manner that makes it meaningful to talk about 
events that happened 1000 years ago.  I may not care about the specific 
statement EE, but I do care enough about the underlying framework and 
presuppositions to have definite views about them, and that's more or less 
all I need in order to conclude that EE is bivalent.

Similarly, 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.

> 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?

I have participated actively on MathOverflow (and prior to that, USENET) 
and have also engaged in various email conversations in connection with my 
expository article in the Mathematical Intelligencer about the consistency 
of PA, so I have been exposed to a fairly large cross-section of views in 
the mathematical community.  Admittedly my sample is still small and is 
biased in various ways, but I can say that there are many different views 
out there among practicing mathematicians.  In particular, there are lots 
of mathematicians who reject bivalence in set theory, but accept bivalence 
in arithmetic.  One prominent person who has written about this publicly 
is Scott Aaronson (who would describe himself as a theoretical computer 
scientist rather than a mathematician, but that's close enough in my 
book).  My personal impression is that the computer revolution has been 
highly influential in this regard.  People feel comfortable with 
syntactical entities because of their familiarity with computers, so they 
are quick to regard "ZFC is consistent" as bivalent, but the farther one 
strays from syntax, the more leery they become.  As it becomes more widely 
known that ZF is "way overkill" for most of mathematics, the sense that 
set theory is mere superstition---and largely dispensable---becomes more 
widespread.  I'm not saying that this viewpoint is rational, but it's what 
I see.

> 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).

I don't have a good sense for how widespread it is, but I do think that 
the overall trend is toward a more formal view of proof, again because of 
the computer revolution.  There are still those who agree with Reuben 
Hersh (who, I was sad to learn, died earlier this month), but even Hersh 
was forced to admit that the mathematical community was much more 
accepting of the verdict of Flyspeck than he had predicted.

Tim