[FOM] extramathematical notions and the CH

[Timothy Y. Chow]

On Fri, 1 Feb 2013, Sam Sanders wrote:
>> You have not really addressed my crucial question: How do you rule out 
>> the possibility that your proposed experiment, instead of giving us new 
>> mathematical knowledge, simply serves to disprove the physical theory?
>
> We rule this out because in the past, we have never encountered such a 
> thing.  Can you give an example of an experiment which has predicted new 
> mathematical knowledge (along the line of "ZFC is consistent" or "CH is 
> false") and was discarded because it is at odds with a physical theory 
> which later turned out to be incorrect?

I can't give such an example, even in principle, and that is my point.  I 
think you misunderstand my point.

Let's review the scenario: Physical theory X predicts that such-and-such 
a physical quantity will be 1 if ZFC is consistent and 0 if ZFC is not 
consistent.  We measure the quantity and it comes out to be 1.  What do we 
conclude?

I'm prepared to grant that X is a very well-confirmed theory with the 
highest credentials and experimental confirmation.  Do I conclude that we 
have obtained new mathematical knowledge?  No!  What we've done, instead, 
is to begin an experiment to test the validity of X.  X has made a 
prediction and now we're in the process of carrying out a finite 
experiment to check whether the prediction is correct.  Like everyone 
else, I have a lot of faith in X and I'm willing to bet a lot of money 
that X's experimental prediction will be confirmed.  But measuring the 
physical quantity and getting the answer 1 is only Part A of the 
experiment.  In order to complete the experiment and declare another 
dramatic victory for the predictive power of X, we need to carry out
Part B: Namely, we need to verify experimentally that ZFC is in fact 
consistent.  Once we do that, I'll join everyone else in sipping the 
champagne.

Physicists, of course, already have enough evidence that ZFC is consistent 
that they don't need to do anything further before the corks start 
popping.  If I put on my physicist's hat then I'll be happy to join them. 
But nowhere in all this process have we acquired any new mathematical 
knowledge.  All that's happened is that people who already believe that 
ZFC is consistent have acquired further confidence in the predictive power 
of X.  And a few diehard believers in the inconsistency of ZFC can now 
toss X onto the junk-heap of invalidated physical theories.

Tim