[FOM] extramathematical notions and the CH

Timothy Y. Chow tchow at alum.mit.edu
Wed Jan 30 22:58:47 EST 2013

Joe Shipman wrote:

> It's possible we could use physics to gain access to noncomputable 
> objects, if an experimentally measurable dimensionless physical quantity 
> is a noncomputable real number; in that case ZFC would only decide 
> finitely many bits of the number and if we could measure more than that 
> we would have new mathematical knowledge coming from physics.

As I've argued before, this makes no sense to me.  In your scenario, we 
have a physical theory that predicts that such-and-such a physical 
quantity equals such-and-such a noncomputable real number.  The theory 
predicts (let's say) that such-and-such a measurement will be 1 if ZFC is 
consistent and it will be 0 if ZFC is inconsistent.  We make the 
measurement and it comes out to be 1.

According to you, we now have "new mathematical knowledge"; we now "know" 
that ZFC is consistent.  But do we really?  How do you rule out the 
following possibility: ZFC is really inconsistent and it's the physical 
theory that is wrong?  Physical theories by their very nature are supposed 
to be tentative hypotheses that can be disproved at any time by 
observation.  They can at most give us some *evidence* that ZFC is 
consistent---e.g., our current physical theories about how computers work 
give us some such evidence whenever we program a computer to search for a 
contradiction in ZFC and let it run for a while.

If you allow us to acquire "mathematical certainty" of physical theories, 
then I don't see why CH is immune.  For example, I've just come up with a 
brilliant new physical theory, which predicts that if CH is true, then 
under the surface of the Moon is a layer of green cheese, and if CH is 
false, then there is no such layer.  All we have to do to settle CH is to 
go to the Moon again and start digging.  My brilliant physical theory will 
give us new mathematical knowledge of CH by physical means.

The "physics boundary" is not between absolute and non-absolute, but 
between finite and infinite.


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