(1) Empirical vs Non-empirical is, I think, a bit of a red herring because insofar as empirical data (e.g. the output of a computer program) bears on mathematical questions, what we glean from it could all, in principle, have been deduced ‘a priori’ (i.e. entirely in the thinker’s mind, without any sensory engagement with the world.)
(2) You ought to read about Chaitin’s constant ‘Omega’, the ‘halting probability’, which is a number between 0 and 1.
I think we should be able to prove something along these lines: Assume that there is a constant K such that your “mental state” does not contain more than K bits of information (this seems horribly vague, but if we assume that the mind’s information is contained in the body’s information then we just need to assume that your body never requires more than K bits to ‘write down’).
Then it is impossible for you to ‘compress’ the binary expansion of Omega by more than K + L bits, for some constant L (the same L for all possible intelligent beings.)
This puts some very severe limits on how closely your ‘subjective probabilities’ for the bits of Omega can approach the real thing. For instance, either there must be only finitely many bits b where your subjective probability that b = 0 differs from 1⁄2, or else, if you guess something other than 1⁄2 infinitely many times, you must ‘guess wrongly’ exactly 1⁄2 of the time (with the pattern of correct and incorrect guesses being itself totally random).
Basically, it sounds like you’re saying: “If we’re prepared to let go of the demand to have strict, formal proofs, we can still acquire empirical evidence, even very convincing evidence, about the truth or falsity of mathematical statements.” This may be true in some cases, but there are others (like the bits of Omega) where we find mathematical facts (expressible as propositions of number theory) that are completely inaccessible by any means. (And in some way that I’m not quite sure yet how to express, I suspect that the ‘gap’ between the limits of ‘formal proof’ and ‘empirical reasoning’ is insignificant compared to the vast ‘terra incognita’ that lies beyond both.)
Comments:
(1) Empirical vs Non-empirical is, I think, a bit of a red herring because insofar as empirical data (e.g. the output of a computer program) bears on mathematical questions, what we glean from it could all, in principle, have been deduced ‘a priori’ (i.e. entirely in the thinker’s mind, without any sensory engagement with the world.)
(2) You ought to read about Chaitin’s constant ‘Omega’, the ‘halting probability’, which is a number between 0 and 1.
I think we should be able to prove something along these lines: Assume that there is a constant K such that your “mental state” does not contain more than K bits of information (this seems horribly vague, but if we assume that the mind’s information is contained in the body’s information then we just need to assume that your body never requires more than K bits to ‘write down’).
Then it is impossible for you to ‘compress’ the binary expansion of Omega by more than K + L bits, for some constant L (the same L for all possible intelligent beings.)
This puts some very severe limits on how closely your ‘subjective probabilities’ for the bits of Omega can approach the real thing. For instance, either there must be only finitely many bits b where your subjective probability that b = 0 differs from 1⁄2, or else, if you guess something other than 1⁄2 infinitely many times, you must ‘guess wrongly’ exactly 1⁄2 of the time (with the pattern of correct and incorrect guesses being itself totally random).
Basically, it sounds like you’re saying: “If we’re prepared to let go of the demand to have strict, formal proofs, we can still acquire empirical evidence, even very convincing evidence, about the truth or falsity of mathematical statements.” This may be true in some cases, but there are others (like the bits of Omega) where we find mathematical facts (expressible as propositions of number theory) that are completely inaccessible by any means. (And in some way that I’m not quite sure yet how to express, I suspect that the ‘gap’ between the limits of ‘formal proof’ and ‘empirical reasoning’ is insignificant compared to the vast ‘terra incognita’ that lies beyond both.)