[FOM] extramathematical notions and the CH
JoeShipman at aol.com
Fri Feb 8 19:46:50 EST 2013
Tim, I agree; but you can also turn that around and say that, rather than waiting for better numerical estimates of pi before they declared that measurements of pi confirmed their theory that space was locally flat, they could have declared that the other evidence of the local flatness of space was strong enough that the mathematical pi must be close to its physically measured value of 22/7. When Archimedes proved that the mathematical pi was between 22/7 and 22/7 - 1/497, I am pretty certain he took the close agreement as evidence his math was right, not evidence that space was flat. Similarly, there is no reason to rule out the possibility that our physical measurements of the pi^0_3 parameter will be regarded as gaining true information about its mathematically defined value, once we have reached the level of precision too fine to decide using mathematical proof (you agreed that there is no reason to believe such a level doesn't exist).
On Feb 7, 2013, at 2:42 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> 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....
.... 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).
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