[FOM] extramathematical notions and the CH

[Timothy Y. Chow]

On Mon, 4 Feb 2013, joeshipman at aol.com wrote:
> You seem to think I am trying to erase the distinction between 4a and 4b 
> despite my making that very distinction; if you accept that 
> 4a represents a VALID but DIFFERENT TYPE OF knowledge of mathematical 
> statements, I have made my point

All right, fine.  I still think that as a practical matter, it is highly 
misleading to state that your thought experiments will give us new 
mathematical knowledge simpliciter, without any warning that you're 
talking about a DIFFERENT TYPE of mathematical knowledge from what most 
people mean by that term.  But we can let this part of the discussion 
rest.

> The problem with the distinction you draw between cases 4a/5 and 6 is 
> that the "finite object" in cases 4a and  5 is too big to be contained 
> in this universe, rather than merely being too big to be humanly 
> feasible but manageable by a computer, as in cases 3 and 4b. This is an 
> important distinction between 3/4b and 4a/5, which renders the further 
> step to case 6 of no additional practical importance.

I don't see that there is any "problem" with the distinction I'm drawing. 
It may be a distinction of no *practical* importance but it is a clear and 
familiar distinction in *principle*.  But again, I'm not going to quibble.

> In more straightforward terms, our physical theories can't be pushed 
> beyond arithmetical statements and still be experimentally testable, 
> because other theories with the same arithmetical consequences will be 
> indistinguishable from them.

But here I still part ways with you.  If the main argument is that there's 
always going to be an alternative explanation consistent with the 
empirical evidence, then the same can be said of our current physical 
theories.  We have only a finite amount of experimental evidence of 
*anything*, and there will always to be infinitely many explanations 
consistent with a finite amount of information, even if we restrict 
ourselves to theories that are limited to "countable mathematics."  That 
fact doesn't stop us from saying that the evidence confirms Theory A over 
Theory B when Theory B is some baroque and contrived (should I say 
Ptolemaic?) theory that technically fits the evidence but fails some 
other criterion (Occam's razor perhaps).

Similarly, I don't see why we couldn't, in principle, come to favor 
Nonabsolute Physical Theory A over Nonabsolute Physical Theory B because 
our intuition tells us that A makes better sense and explains the facts 
better, even though the baroque alternative B is technically consistent 
with the facts as well.  Say, just for the sake of argument, that A 
postulates that there are uncountably many things out there interacting in 
some way that usually cancels out, but that if somehow there were an 
intermediate cardinality between aleph_0 and 2^(aleph_0) then it wouldn't 
cancel out a 2^(aleph_0) thing and some finite residue would pop out.  You 
would have your experimentally measurable prediction and it might give you 
"knowledge" of CH in some newfangled sense of the word "knowledge."

Tim