Collatz conjecture is true in every universe, or false in every universe.
You can slice it into a set of trivial statements which are trivially true or trivially false, like “Collatz conjecture is true for N=531” etc., connected by trivially true or trivially false logical statements.
There’s no way to meaningfully get any probability but 0 or 1 out of this, other than by claiming that some basic mathematical law is uncertain (and if you believe that, you are more Dutch bookable than entire Netherlands). I might not know how to Dutch book you yet, but logic dictates such a way exists.
Except thanks to Incompleteness Theorem, you have no way to find a definite answer to every such statement. No matter which strategy you choose, and how much time you have, you’ll either be inconsistent (Dutch bookable), or incomplete (not able to answer 0 or 1 - and as no other answer is valid, any answer you give makes you Dutch bookable).
Do you assign probability 1 to the proposition that 182547553 is prime? Right now, without doing an experiment on a calculator? (well, computer maybe. For most calculators testing this proposition would be somewhat tedious)
If yes, would you willing to pay me $10 if you ever found out it was not prime?
Conversely
Do you assign probability 0 to the proposition that 182547553 is prime? Right now, without doing an experiment on a calculator?
If yes, would you willing to pay me $10 if you ever found out it was prime?
EDIT: Actually, I suppose this counts as “doing it again”, even though I’m not Peter de Blanc. I think that makes me a bad bad person.
I suggest you look up the concept of “subjective Bayesian”. Probabilities are states of knowledge. If you don’t know an answer, it’s uncertain. If someone who doesn’t know anything you don’t can look over your odds and construct a knowably losing bet anyway, or construct a winning bet that you refuse, then you are Dutch-bookable.
Also, considering that you have apparently been reading this site for years and you still have not grasped the concept of subjective uncertainty, and you are still working with a frequentist notion of probability, nor yet have you even grasped the difference, I would suggest to you in all seriousness that you seek enlightenment elsewhere.
(Sorry, people, there’s got to be some point at which I can express that. Downvote if you must, I suppose, if you think such a concept is unspeakable or unallowed, but it’s not a judgment based on only one case of incomprehension, of course.)
I don’t know the background conflict. But at least one of taw’s points is correct. Any prior P, of any agent, has at least one of the following three properties:
It is not defined on all X—i.e. P is agnostic about some things
It has P(X) < P(X and Y) for at least one pair X and Y—i.e. P sometimes falls for the conjunction fallacy
It has P(X) = 1 for all mathematically provable statements X—i.e. P is an oracle.
You aren’t excused from having to pick one by rejecting frequentist theory.
To make use of a theory like probability one doesn’t have to have completely secure foundations. But it’s responsible to know what the foundational issues are. If you make a particularly dicey or weird application of probability theory, e.g. game theory with superintelligences, you should be prepared to explain (especially to yourself!) why you don’t expect those foundational issues to interfere with your argument.
About taw’s point in particular, I guess it’s possible that von Neumann gave a completely satisfactory solution when he was a teenager, or whatever, and that I’m just showcasing my ignorance. (I would dearly like to hear about this solution!) Otherwise your comment reads like you’re shooting the messenger.
Logical uncertainty is an open problem, of course (I attended a workshop on it once, and was surprised at how little progress has been made).
But so far as Dutch-booking goes, the obvious way out is 2 with the caveat that the probability distribution never has P(X) < (PX&Y) at the same time, i.e., you ask it P(X), it gives you an answer, you ask it P(X&Y), it gives you a higher answer, you ask it P(X) again and it now gives you an even higher answer from having updated its logical uncertainty upon seeing the new thought Y.
It is also clear from the above that taw does not comprehend the notion of subjective uncertainty, hence the notion of logical uncertainty.
How, exactly, to deal with logical uncertainty is an unsolved problem, no?
It’s not clear why it’s more of a problem for Bayesian than anyone else.
The post you linked to is a response to Eliezer arguing with Hanson, with Eliezer taking a pro-his-own-contrarianism stance. Of course he’s aware of that contrarianism.
How, exactly, to deal with logical uncertainty is an unsolved problem, no?
Your choice is either accepting that you will be sometimes inconsistent,
or accepting that you will sometimes answer “I don’t know” without providing a specific number, or both.
There’s nothing wrong with “I don’t know”.
It’s not clear why it’s more of a problem for Bayesian than anyone else.
For Perfect Bayesian or for Subjective Bayesian?
Subjective Bayesian does believe many statements of kind P(simple math step) = 1, P(X|conjunction of simple math steps) = 1, and yet P(X) < 1.
it does not believe math statements with probability 1 or 0 until it investigates them. As soon as it investigates whether (X|conjunction of simple math steps) is true and determines the answer, it sets P(X)=1.
The problem with “I don’t know” is that sometimes you have to make decisions. How do you propose to make decisions if you don’t know some relevant mathematical fact X?
For example, if you’re considering some kind of computer security system that is intended to last a long time, you really need an estimate for how likely it is that P=NP.
The problem with “I don’t know” is that sometimes you have to make decisions. How do you propose to make decisions if you don’t know some relevant mathematical fact X?
Then you need to fully accept that you will be inconsistent sometimes. And compartmentalize your belief system accordingly, or otherwise find a way to deal with these inconsistencies.
Well, they can wriggle out of this by denying P(simple math step) = 1
Doesn’t this imply you’d be willing to accept P(2+2=5) on good enough odds?
This might be pragmatically a reasonable thing to do, but if you accept that all math might be broken, you’ve already given up any hope of consistency.
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.
Not at all.
Collatz conjecture is true in every universe, or false in every universe.
You can slice it into a set of trivial statements which are trivially true or trivially false, like “Collatz conjecture is true for N=531” etc., connected by trivially true or trivially false logical statements.
There’s no way to meaningfully get any probability but 0 or 1 out of this, other than by claiming that some basic mathematical law is uncertain (and if you believe that, you are more Dutch bookable than entire Netherlands). I might not know how to Dutch book you yet, but logic dictates such a way exists.
Except thanks to Incompleteness Theorem, you have no way to find a definite answer to every such statement. No matter which strategy you choose, and how much time you have, you’ll either be inconsistent (Dutch bookable), or incomplete (not able to answer 0 or 1 - and as no other answer is valid, any answer you give makes you Dutch bookable).
Do you assign probability 1 to the proposition that 182547553 is prime? Right now, without doing an experiment on a calculator? (well, computer maybe. For most calculators testing this proposition would be somewhat tedious)
If yes, would you willing to pay me $10 if you ever found out it was not prime?
Conversely
Do you assign probability 0 to the proposition that 182547553 is prime? Right now, without doing an experiment on a calculator?
If yes, would you willing to pay me $10 if you ever found out it was prime?
EDIT: Actually, I suppose this counts as “doing it again”, even though I’m not Peter de Blanc. I think that makes me a bad bad person.
I suggest you look up the concept of “subjective Bayesian”. Probabilities are states of knowledge. If you don’t know an answer, it’s uncertain. If someone who doesn’t know anything you don’t can look over your odds and construct a knowably losing bet anyway, or construct a winning bet that you refuse, then you are Dutch-bookable.
Also, considering that you have apparently been reading this site for years and you still have not grasped the concept of subjective uncertainty, and you are still working with a frequentist notion of probability, nor yet have you even grasped the difference, I would suggest to you in all seriousness that you seek enlightenment elsewhere.
(Sorry, people, there’s got to be some point at which I can express that. Downvote if you must, I suppose, if you think such a concept is unspeakable or unallowed, but it’s not a judgment based on only one case of incomprehension, of course.)
I don’t know the background conflict. But at least one of taw’s points is correct. Any prior P, of any agent, has at least one of the following three properties:
It is not defined on all X—i.e. P is agnostic about some things
It has P(X) < P(X and Y) for at least one pair X and Y—i.e. P sometimes falls for the conjunction fallacy
It has P(X) = 1 for all mathematically provable statements X—i.e. P is an oracle.
You aren’t excused from having to pick one by rejecting frequentist theory.
To make use of a theory like probability one doesn’t have to have completely secure foundations. But it’s responsible to know what the foundational issues are. If you make a particularly dicey or weird application of probability theory, e.g. game theory with superintelligences, you should be prepared to explain (especially to yourself!) why you don’t expect those foundational issues to interfere with your argument.
About taw’s point in particular, I guess it’s possible that von Neumann gave a completely satisfactory solution when he was a teenager, or whatever, and that I’m just showcasing my ignorance. (I would dearly like to hear about this solution!) Otherwise your comment reads like you’re shooting the messenger.
Logical uncertainty is an open problem, of course (I attended a workshop on it once, and was surprised at how little progress has been made).
But so far as Dutch-booking goes, the obvious way out is 2 with the caveat that the probability distribution never has P(X) < (PX&Y) at the same time, i.e., you ask it P(X), it gives you an answer, you ask it P(X&Y), it gives you a higher answer, you ask it P(X) again and it now gives you an even higher answer from having updated its logical uncertainty upon seeing the new thought Y.
It is also clear from the above that taw does not comprehend the notion of subjective uncertainty, hence the notion of logical uncertainty.
Have any ideas?
Your full endorsement of evaporative cooling is quite disturbing.
Are you at least aware that epistemic position you’re promoting is highly contrarian?
How, exactly, to deal with logical uncertainty is an unsolved problem, no?
It’s not clear why it’s more of a problem for Bayesian than anyone else.
The post you linked to is a response to Eliezer arguing with Hanson, with Eliezer taking a pro-his-own-contrarianism stance. Of course he’s aware of that contrarianism.
Your choice is either accepting that you will be sometimes inconsistent, or accepting that you will sometimes answer “I don’t know” without providing a specific number, or both.
There’s nothing wrong with “I don’t know”.
For Perfect Bayesian or for Subjective Bayesian?
Subjective Bayesian does believe many statements of kind P(simple math step) = 1, P(X|conjunction of simple math steps) = 1, and yet P(X) < 1.
it does not believe math statements with probability 1 or 0 until it investigates them. As soon as it investigates whether (X|conjunction of simple math steps) is true and determines the answer, it sets P(X)=1.
The problem with “I don’t know” is that sometimes you have to make decisions. How do you propose to make decisions if you don’t know some relevant mathematical fact X?
For example, if you’re considering some kind of computer security system that is intended to last a long time, you really need an estimate for how likely it is that P=NP.
Then you need to fully accept that you will be inconsistent sometimes. And compartmentalize your belief system accordingly, or otherwise find a way to deal with these inconsistencies.
Well, they can wriggle out of this by denying P(simple math step) = 1, which is why I introduced this variation.
Doesn’t this imply you’d be willing to accept P(2+2=5) on good enough odds?
This might be pragmatically a reasonable thing to do, but if you accept that all math might be broken, you’ve already given up any hope of consistency.
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.