[FOM] extramathematical notions and the CH

[Timothy Y. Chow]

On Wed, 6 Feb 2013, Joe Shipman wrote:
> Do you have a reason to believe that all the numbers measurable in this 
> way (to theoretically arbitrary precision) are in fact either computable 
> as real numbers, or computable relative to the small number of 
> physically measurable free parameters in our current theories?

No, I have no reason to believe that.

> Before Archimedes figured out how to calculate Pi to arbitrary precision 
> (his best-known numerical estimate was an interval of width 0.002 but he 
> made it clear how to go as far as you like), physical measurements of 
> its value provided knowledge that the mathematics of the time could not. 
> I don't see a difference in principle. We might become able to calculate 
> the predictions of our physics as effectively as Archimedes did, but 
> right now there is no algorithm, only pi^0_3 definitions, for all the 
> quantities whose definitions involve summations over infinitely many 
> Feynman diagrams or some similar enumeration of ways things can happen.

Good analogy.  Here's how I would describe what the Greeks did.  Their 
physical measurements produced knowledge, but it was not what I would 
strictly speaking call *mathematical* knowledge.  Rather, they tacitly had 
a physical theory that we would, in modern terms, express as, "spacetime 
is locally flat."  Based on that theory, they predicted that certain 
physical measurements would yield digits of pi.  To the extent that they 
"knew" that spacetime was locally flat, they also "knew" the digits of pi. 
But strictly speaking they had to wait until mathematics had advanced to 
the point where the value of pi could be proved mathematically (or perhaps 
until they had some other physical way of estimating the value of pi) 
before they could take the final step, i.e., of declaring that they had 
experimentally confirmed their physical theory that spacetime is locally 
flat.

Similarly, if we acquire the ability to measure those "pi^0_3 reals," then 
we will be poised to declare new experimental confirmation of our physical 
theories as soon as mathematics advances to the point where the values of 
those reals can be derived mathematically (or perhaps we develop some 
other physical method of estimating those numbers).

Tim