[FOM] extramathematical notions and the CH
JoeShipman at aol.com
Wed Feb 6 15:07:11 EST 2013
Most of our disagreements are not worth quibbling over, but there is one point I would like you to address that is not speculative at all.
The structure of our current best theories of fundamental physics is such that many of the physically measurable real numbers have a mathematical definition that is pi^0_3 or else pi^0_3 relative to some parameter which is itself physically measurable. They have the form "summation of a countably infinite sequence of terms, each of which is a computable (or computable relative to the parameter) real number", where there is no computable modulus of convergence that comes along with the definition.
Whether or not it is possible to prove that these pi^0_3 objects are equal to computable reals (which would have definitions of lower pi^0_1 or even delta^0_0 type) or not, it's still the case that a physical measurement of such a quantity is going to correspond to high confidence that its value lies in some interval, which would be strong evidence for a sigma^0_2 statement that we would not normally expect to be able to prove or disprove using pure mathematics. (If the computation is relative to a measurable parameter rather than parameter-free this is still the case though the details are more complicated.)
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?
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.
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