[FOM] Thoughts on CH

[Timothy Y. Chow]

Joe Shipman wrote:

> I don't understand why so few mathematicians are willing to say "of 
> course CH is either true or false, but we don't have access to the 
> answer because it makes no difference to physics, and with no direct 
> access to uncountable sets we can?t prove anything about it."
>
> No one seems to think it's a problem we can't answer my question about 
> pi. Why is it such a problem that we can't answer CH?

Obviously, different people will have different answers to this question, 
but I think that the general line of thought goes something like this.

In the good ol' days, pre-independence, people tacitly assumed that 
all mathematical statements were not just either true or false, but also 
either provable or disprovable.

Independence forced people to revise their beliefs.  One popular attitude 
was to flip from "innocent until proven guilty" all the way to "guilty 
until proven innocent"; i.e., mathematical statements were not accepted as 
being bivalent (either true or false) unless there was a convincing reason 
to do so.

Thus, depending on which statements people were able to convince 
themselves were bivalent (or perhaps more accurately, couldn't bear the 
thought that they weren't bivalent), they landed on different points on 
the spectrum from strict formalist to dyed-in-the-wool Platonist.

Here's an analogy.  I own too many books.  Occasionally I go through my 
bookshelves and try to find books that I am willing to part with.  Some 
books I'm unwilling to part with, while others I am willing to toss out 
without much wincing.

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.

Tim