Oh great, ethics and set theory together, what could possibly go wrong.
Ahem.
OK, so personally I reject totalism as nonsense to begin with so I get to laugh at your problems from my “of course HEAVEN is better than HELL if you stop trying to just sum things” pedestal. But if I accepted totalism, I wouldn’t really see a problem with saying that HEAVEN and HELL are equivalent. They just are, in that sense. If a subtler answer exists, it probably rests in some deep mathematics that I do not know how to use. But as things stand, the only unsettling thing is that you generalised a finite problem by making it infinite and it feels like the answer changed, but IMO that’s underestimating just how deeply the problem changed too. You want to make a true HEAVEN, you need an uncountable infinity of happy people. Make one happy person for every possible set of the (infinite) unhappy ones.
You don’t have to be a totalist to think that HEAVEN>HELL. The problem is that it’s intuitively obvious that if we discovered that every galaxy in the universe was filled with almost all happy people, that would seem better than if they were filled almost exclusively with miserable people.
No, it’s the opposite, you have to be a totalist to even doubt it. For example, if there are more happy than unhappy people, obviously the average is positive. It’s only if you compare raw summed totals that you get “oops they’re both infinities, can’t tell the difference” confusion.
You have not understood the problem. There are not more happy people than unhappy people in any rigorous sense—the infinities are of the same cardinality. And the pasadena game scenario gives indeterminate averages. Also, average utilitarianism is crazy, and implies you should create lots of miserable people in hell as long as they’re slightly less miserable than existing people.
I’m not an average utilitarian either—I don’t think it’s easy to define a good utility function at all, and I wrote a whole post to jokingly talk of this problem. My point was that only totalists would encounter this specific issue. If the galaxy has 1 trillion people, of which only one is unhappy, you can easily get the average for a single galaxy, which is finite. And since all galaxies have the same average, it can’t really change if you just take more of them, no? Even numbers have the same cardinality as natural numbers, but we can still say that the density of even numbers on the natural numbers line is 1/2. This is not a Pasadena scenario, this is just a regular old limit of the ratio of two linear functions. Average utilitarianism has other issues, but on this, it captures our intuition exactly right.
No, you don’t. This is like saying that if you rearrange the even numbers, they stop being roughly half of all naturals. They’re still one every two. If you pick a large enough ensemble, you notice that. The arrangement with one unhappy person per galaxy is very convenient, but it’s the other way around—if the arrangement was inconvenient but the ratio was given, we could group them this way to make the calculation simpler. Relevant concept: Natural Density.
Yes, but the natural density of even numbers is 0.5. And that is the natural extension to infinity of your intuition that there are more happy than unhappy people in the HEAVEN universe.
If you were born as a person at random in HEAVEN, you’d be most likely happy!
Oh great, ethics and set theory together, what could possibly go wrong.
Ahem.
OK, so personally I reject totalism as nonsense to begin with so I get to laugh at your problems from my “of course HEAVEN is better than HELL if you stop trying to just sum things” pedestal. But if I accepted totalism, I wouldn’t really see a problem with saying that HEAVEN and HELL are equivalent. They just are, in that sense. If a subtler answer exists, it probably rests in some deep mathematics that I do not know how to use. But as things stand, the only unsettling thing is that you generalised a finite problem by making it infinite and it feels like the answer changed, but IMO that’s underestimating just how deeply the problem changed too. You want to make a true HEAVEN, you need an uncountable infinity of happy people. Make one happy person for every possible set of the (infinite) unhappy ones.
You don’t have to be a totalist to think that HEAVEN>HELL. The problem is that it’s intuitively obvious that if we discovered that every galaxy in the universe was filled with almost all happy people, that would seem better than if they were filled almost exclusively with miserable people.
No, it’s the opposite, you have to be a totalist to even doubt it. For example, if there are more happy than unhappy people, obviously the average is positive. It’s only if you compare raw summed totals that you get “oops they’re both infinities, can’t tell the difference” confusion.
You have not understood the problem. There are not more happy people than unhappy people in any rigorous sense—the infinities are of the same cardinality. And the pasadena game scenario gives indeterminate averages. Also, average utilitarianism is crazy, and implies you should create lots of miserable people in hell as long as they’re slightly less miserable than existing people.
I’m not an average utilitarian either—I don’t think it’s easy to define a good utility function at all, and I wrote a whole post to jokingly talk of this problem. My point was that only totalists would encounter this specific issue. If the galaxy has 1 trillion people, of which only one is unhappy, you can easily get the average for a single galaxy, which is finite. And since all galaxies have the same average, it can’t really change if you just take more of them, no? Even numbers have the same cardinality as natural numbers, but we can still say that the density of even numbers on the natural numbers line is 1/2. This is not a Pasadena scenario, this is just a regular old limit of the ratio of two linear functions. Average utilitarianism has other issues, but on this, it captures our intuition exactly right.
You can also get the total of a single galaxy—the problem is how you count up things in an infinite world.
Yes but the total accumulates, the average does not.
UTOT=∑GUG=∞⟨U⟩=limN→∞NUGN=UGIf you rearrange heaven to hell, you get a different average. So you either have to think rearrangement matters or that they’re equal.
No, you don’t. This is like saying that if you rearrange the even numbers, they stop being roughly half of all naturals. They’re still one every two. If you pick a large enough ensemble, you notice that. The arrangement with one unhappy person per galaxy is very convenient, but it’s the other way around—if the arrangement was inconvenient but the ratio was given, we could group them this way to make the calculation simpler. Relevant concept: Natural Density.
There are as many even numbers as there are total numbers. They are the same cardinality.
Yes, but the natural density of even numbers is 0.5. And that is the natural extension to infinity of your intuition that there are more happy than unhappy people in the HEAVEN universe.
If you were born as a person at random in HEAVEN, you’d be most likely happy!