[FOM] extramathematical notions and the CH

[Timothy Y. Chow]

On Sat, 2 Feb 2013, Joe Shipman wrote:
> I did not use the statement "mathematical knowledge" to mean 
> "mathematically knowing" in the sense of having a humanly verified 
> proof

In that case, I don't see why you refuse to invite CH to the party.  Why 
couldn't someone come up with a physical theory that we somehow come to 
accept as correct, that predicts that some experimentally measurable value 
tells us the truth value of CH?  Once we leave the well-defined territory 
of finite verification and allow ourselves to "know" physical *theories*, 
how do we prevent the physicists from working CH into their theories? 
Granted, there's no plausible proposal on the horizon, but if you are 
urging us not to draw arbitrary lines, then I don't see why we should draw 
an arbitrary line between absolute and non-absolute statements.

The epistemological role that finite verifications play in mathematical 
and physical knowledge is well-established and well-understood and is not 
going to change; in mathematics it is the line between proved and not 
proved, and in physics it is the line between theory and experiment.  The 
line between absolute and non-absolute in physical theories, as far as I 
can see, is one that is rarely crossed, but there doesn't seem to be any 
principled reason why it can't be crossed, should it prove useful to do 
so one day.

> However, mathematicians recognize case 4a as a valid form of knowledge, 
> because the error probability can be bounded, which distinguishes it 
> from the Zeilbergerian scenario you cite which involves speculations 
> about probabilities of mistakes; furthermore, your acceptance of case 4b 
> because of the existence "in principle" of a proof object which can't 
> possibly be surveyed in the traditional way, while justified because of 
> the possibility of transforming such a proof into a PCP, also makes it 
> more unreasonable to reject case 4a.

Case 4a (probabilistic proof) is indeed "recognized," but it is also 
recognized as being qualitatively different from traditional mathematical 
knowledge.  Nobody has any trouble drawing a sharp distinction between 
IsProvablyPrime from IsProbablyPrime, despite the difference of only one 
letter in the spelling.

I also think you make too much of the alleged parallel between 4a and 4b 
(unsurveyable proofs).  In 4b, there's a giant finite object that I have 
good, albeit fallible, reasons to believe has been fully explored by 
someone or something or some collection of people or things.  In 4a, I 
*know* that the finite object hasn't been fully explored.  While I'm not 
going to reject 4a as useless, I have no difficulty making a principled 
distinction between knowledge of type 4a and knowledge of type 4b.

> I claim that if you accept case 4a, the step to case 5 is sound, and it 
> has already been taken by all the mathematicians who work in Quantum 
> Complexity theory.

Yes, I'm happy lumping 4a together with 5.

> Do you reject case 5?

What do you mean by "reject"?  I don't believe that 4a or 5 represents 
mathematical knowledge *as that term is usually understood*.  They 
represent knowledge of a useful sort that we can loosely call 
"mathematical knowledge" if we're not being too fussy, but which is 
nevertheless sharply distinguishable from knowledge based on complete 
finite proofs.

> Or do you accept case 5 as describing valid "knowledge" of mathematical 
> statements, but still reject case 6?

I do think it's possible to draw another line between 5 and 6.  To 
summarize:

1, 2, 3, 4b: Finite and completely verified, as far as we can tell.
4a, 5: Finite but definitely not completely verified.
6. Infinite.

Or maybe I should add a seventh category, and say instead:

6. Infinite, but absolute.
7. Infinite and non-absolute.

I'd prefer to stick to 1, 2, 3, 4b when using the term "mathematical 
knowledge."  If you are happy slipping down the slope to 6 then I don't 
see why we might not slip down to 7 some day.

Tim