If you answered 1 to the first, and anything but 0 or 1 to the second, you’re inconsistent.
1 to the first for reasonable definitions of “true.” .8 to the second- it seems like the sort of thing that should be true.
To assess the charge of inconsistency, though, we have unpack what you mean by that. Do you mean that I can’t see the mathematical truth of a statement without reasoning through it? Then, yes, I very much agree with you. That is not a power I have. (My reasoning is also finite; I doubt I will solve the Collatz conjecture.)
But what I mean by an uncertainty of .8 is not “in the exterior world, a die is rolled such that the Collatz conjecture is true in 80% of universes but not the rest.” Like you point out, that would be ridiculous. I’m not measuring math; I’m measuring my brain. What I mean is “I would be willing to wager at 4-1 odds that the Collatz conjecture is true for sufficiently small dollar amounts.” Inconsistency, to me, is allowing myself to be Dutch Booked- which those two probabilities do not do.
You can be “Dutch booked” by someone who can solve the conjecture. (I am not sure whether this can be referred to as Dutch booking, but it would be the case where you both would have access to the same information and one would be in a better position due to imperfections in the other’s reasoning.)
it seems like the sort of thing that should be true
It seems also a bit like the sort of thing that might be undecidable.
I’m pretty sure that a Dutch Book is only a Dutch Book if it’s pure arbitrage- that is, you beat someone using only the odds they publish. If you know more than someone else and win a bet against them, that seems different.
It seems also a bit like the sort of thing that might be undecidable.
Quite possibly. I’m not a good judge of mathematical truth- I tend to be more trusting than I should be. It looks to me like if you can prove “every prime can be expressed as the output of algorithm X”, where X is some version of the Collatz conjecture in reverse, then you’re done. (Heck, that might even map onto the Sieve of Eratosthenes.) That it isn’t solved already drops my credence down from ~.95 to ~.8.
I’m pretty sure that a Dutch Book is only a Dutch Book if it’s pure arbitrage- that is, you beat someone using only the odds they publish. If you know more than someone else and win a bet against them, that seems different.
They publish probability of axioms of arithmetics being roughly 1 and probability of Collatz conjecture being 0.8, you see that the conjecture is logically equivalent to the axioms and thus that their odds are mutually inconsistent. You don’t “know” more in the sense of having observed more evidence. (I’d agree that this is a tortured interpretation of Dutch booking, but it’s probably what you get if you systematically distinguish external evidence from own reasoning.)
I’m pretty sure that a Dutch Book is only a Dutch Book if it’s pure arbitrage- that is, you beat someone using only the odds they publish. If you know more than someone else and win a bet against them, that seems different.
Yes. Crudely speaking they have to be stupid, not just ignorant!
1 to the first for reasonable definitions of “true.” .8 to the second- it seems like the sort of thing that should be true.
To assess the charge of inconsistency, though, we have unpack what you mean by that. Do you mean that I can’t see the mathematical truth of a statement without reasoning through it? Then, yes, I very much agree with you. That is not a power I have. (My reasoning is also finite; I doubt I will solve the Collatz conjecture.)
But what I mean by an uncertainty of .8 is not “in the exterior world, a die is rolled such that the Collatz conjecture is true in 80% of universes but not the rest.” Like you point out, that would be ridiculous. I’m not measuring math; I’m measuring my brain. What I mean is “I would be willing to wager at 4-1 odds that the Collatz conjecture is true for sufficiently small dollar amounts.” Inconsistency, to me, is allowing myself to be Dutch Booked- which those two probabilities do not do.
You can be “Dutch booked” by someone who can solve the conjecture. (I am not sure whether this can be referred to as Dutch booking, but it would be the case where you both would have access to the same information and one would be in a better position due to imperfections in the other’s reasoning.)
It seems also a bit like the sort of thing that might be undecidable.
I’m pretty sure that a Dutch Book is only a Dutch Book if it’s pure arbitrage- that is, you beat someone using only the odds they publish. If you know more than someone else and win a bet against them, that seems different.
Quite possibly. I’m not a good judge of mathematical truth- I tend to be more trusting than I should be. It looks to me like if you can prove “every prime can be expressed as the output of algorithm X”, where X is some version of the Collatz conjecture in reverse, then you’re done. (Heck, that might even map onto the Sieve of Eratosthenes.) That it isn’t solved already drops my credence down from ~.95 to ~.8.
They publish probability of axioms of arithmetics being roughly 1 and probability of Collatz conjecture being 0.8, you see that the conjecture is logically equivalent to the axioms and thus that their odds are mutually inconsistent. You don’t “know” more in the sense of having observed more evidence. (I’d agree that this is a tortured interpretation of Dutch booking, but it’s probably what you get if you systematically distinguish external evidence from own reasoning.)
Yes. Crudely speaking they have to be stupid, not just ignorant!
Not being able to decide upon the Collatz conjecture is stupidity, not ignorance. A very widespread sort of stupidy, but still.
Grandparent is self contained and entirely Collatz-independent.