If physics is deterministic then conditional on the state of the world at the time you make the bet, the probability of heads is either 0 or 1. The only disanalogy with your example is that you may not already have sufficient information to determine how the coin will land (which isn’t even a disanalogy if we assume that the person doesn’t know what the Collatz conjecture says). But suppose you did have that information—there would be vastly more of it than you could process in the time available, so it wouldn’t affect your probability assignments. (Note: The case where the Collatz conjecture turns out to be true but unprovable is analogous to the case where the laws of physics are deterministic but ‘uncomputable’ in some sense.)
Anyway, the real reason why I want to resist your line of argument here is due to Chaitin’s number “omega”, the “halting probability”. One can prove that the bits in the binary expansion of omega are algorithmically incompressible. More precisely: In order to deduce n bits’ worth of information about omega you need at least n—k bits’ worth of axioms, for some constant k. Hence, if you look sufficiently far along the binary expansion of omega, you find yourself looking at an infinite string of “random mathematical facts”. One “ought” to treat these numbers as having probability 1⁄2 of being 1 and 1⁄2 of being 0 (if playing games against opponents who lack oracle powers).
Deterministic world and “too much information to process” are uninteresting. All that simply means that due to practical constraints, sometimes the only reasonably thing is to assign no probability. As if we didn’t know that already. But probabilities still might be assignable in theory.
Except uncomputability means it won’t work even in theory. You are always Dutch bookable.
Anyway, the real reason why I want to resist your line of argument here is due to Chaitin’s number “omega”, the “halting probability”. One can prove that the bits in the binary expansion of omega are algorithmically incompressible.
Chaitin’s number is not a mathematical entity—it’s creation of pure metaphysics.
The claim that kth bit of Chaitin’s number is 0 just doesn’t mean anything once k becomes big enough to include a procedure to compute Chaitin’s number.
Deterministic world and “too much information to process” are uninteresting. All that simply means that due to practical constraints, sometimes the only reasonably thing is to assign no probability.
Better: Sometimes the only reasonable thing is to assign a probability that’s strictly speaking “wrong”, but adequate if you’re only facing opponents who are (approximately) as hampered as you in terms of how much they know and much they can feasible compute. (E.g. Like humans playing poker, where the cards are only pseudo-random.)
If you want to say this is uninteresting, fine. I’m not trying to argue that it’s interesting.
Except uncomputability means it won’t work even in theory. You are always Dutch bookable.
Sorry, you’ve lost me.
Chaitin’s number is not a mathematical entity—it’s creation of pure metaphysics.
Chaitin’s number is awfully tame by the standards of descriptive set theory. So what you’re really saying here is that you personally regard a whole branch of mathematics as “pure metaphysics”. Maybe a few philosophers of mathematics agree with you—I suspect most do not—but actual mathematicians will carry on studying mathematics regardless.
The claim that kth bit of Chaitin’s number is 0 just doesn’t mean anything once k becomes big enough to include a procedure to compute Chaitin’s number.
I’m not sure what you’re trying to say here but what you’ve actually written is false. Why do you think Chaitin’s number isn’t well defined?
For what it’s worth, I think you made a very interesting contribution to this thread, and I find it somewhat baffling that EY responded in the way he did (though perhaps there’s some ‘history’ here that I’m not aware of) and equally baffling that this has apparently caused others to downvote you.
If physics is deterministic then conditional on the state of the world at the time you make the bet, the probability of heads is either 0 or 1. The only disanalogy with your example is that you may not already have sufficient information to determine how the coin will land (which isn’t even a disanalogy if we assume that the person doesn’t know what the Collatz conjecture says). But suppose you did have that information—there would be vastly more of it than you could process in the time available, so it wouldn’t affect your probability assignments. (Note: The case where the Collatz conjecture turns out to be true but unprovable is analogous to the case where the laws of physics are deterministic but ‘uncomputable’ in some sense.)
Anyway, the real reason why I want to resist your line of argument here is due to Chaitin’s number “omega”, the “halting probability”. One can prove that the bits in the binary expansion of omega are algorithmically incompressible. More precisely: In order to deduce n bits’ worth of information about omega you need at least n—k bits’ worth of axioms, for some constant k. Hence, if you look sufficiently far along the binary expansion of omega, you find yourself looking at an infinite string of “random mathematical facts”. One “ought” to treat these numbers as having probability 1⁄2 of being 1 and 1⁄2 of being 0 (if playing games against opponents who lack oracle powers).
Deterministic world and “too much information to process” are uninteresting. All that simply means that due to practical constraints, sometimes the only reasonably thing is to assign no probability. As if we didn’t know that already. But probabilities still might be assignable in theory.
Except uncomputability means it won’t work even in theory. You are always Dutch bookable.
Chaitin’s number is not a mathematical entity—it’s creation of pure metaphysics.
The claim that kth bit of Chaitin’s number is 0 just doesn’t mean anything once k becomes big enough to include a procedure to compute Chaitin’s number.
Better: Sometimes the only reasonable thing is to assign a probability that’s strictly speaking “wrong”, but adequate if you’re only facing opponents who are (approximately) as hampered as you in terms of how much they know and much they can feasible compute. (E.g. Like humans playing poker, where the cards are only pseudo-random.)
If you want to say this is uninteresting, fine. I’m not trying to argue that it’s interesting.
Sorry, you’ve lost me.
Chaitin’s number is awfully tame by the standards of descriptive set theory. So what you’re really saying here is that you personally regard a whole branch of mathematics as “pure metaphysics”. Maybe a few philosophers of mathematics agree with you—I suspect most do not—but actual mathematicians will carry on studying mathematics regardless.
I’m not sure what you’re trying to say here but what you’ve actually written is false. Why do you think Chaitin’s number isn’t well defined?
For what it’s worth, I think you made a very interesting contribution to this thread, and I find it somewhat baffling that EY responded in the way he did (though perhaps there’s some ‘history’ here that I’m not aware of) and equally baffling that this has apparently caused others to downvote you.