[FOM] extramathematical notions and the CH (corrected version)
joeshipman at aol.com
joeshipman at aol.com
Fri Feb 1 13:08:48 EST 2013
Tim, we would "know" the mathematical statement in question because we believe our physical theories to be true and trust our experimental results to a certain degree of precision, in the same way we know a mathematical statement to be true that we don't have a humanly readable proof of but only a computer-generated proof that is too long to verify by hand, but which we accept because we understand the physics of our computers and know how they work.
Your example on CH is not apt. The point of my example is that we could have a physical theory that we ALREADY believe to be correct in the usual way physical theories come to be accepted as correct, but which also involve certain experimental outcomes that, although they can be DEFINED in ZFC, cannot be DECIDED by ZFC. (That is, you might prove in ZFC that the theory entails the result of a suitable experiment being the Kth bit of some version of Chaitin's Omega number with a certain level of statistical confidence, but you would have to actually conduct the experiment to find out whether that bit is 1 or 0 because ZFC can't prove what the Kth bit actually is except for the first few bits, even though it can define all of the bits).
And it is not so implausible that a number defined in a physical theory and experimentally measurable could be non-computable. For example, some versions of quantum gravity involve summing amplitudes or probabilities over spacetime topologies represented as homeomorphism classes of 4-dimensional manifolds, which we know are not recursively classifiable so the theory does not come with an algorithm for predicting the result.
And the finite "grain" of the universe does not prevent the measurement of dimensionless quantities to arbitrary precision, because the quantities can be related to probabilities which are in principle capable of bring estimated to arbitrary precision by repeating the experiment often enough. Though there is an exponential slowdown, you can still get lots of bits--for example a bulk sample of a radioactive substance might have 10^24 ~ 2^80 atoms, and by counting radioactive decays you can get an 80-bit estimate which will give roughly 40 bits of precision due to the square-root behavior of the statistical error.We don't need infinite precision either, finitely many bits would get us beyond what ZFC can prove and thus provide new mathematical knowledge.-- JS
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