I have an argument for a way in which infinity can be used but which doesn’t imply any of the negative conclusions. I’m not convinced of its reasonableness or correctness though.
I propose that infinity ethics should only be reasoned about by use of proof through induction. When done this way, the only way to reason about HEAVEN and HELL is by matching up galaxies in each universe, and doing induction across all of the elements:
Theorem: The universe HEAVEN that contains n galaxies is a better universe than HELL which contains n galaxies. We will formalize this as HEAVEN(n) > HELL(n). We will prove this by induction.
Base case, HEAVEN(1) > HELL(1):
The first galaxy in HEAVEN (which contains billions of happy people and one miserable person) is better than the first galaxy in HELL (which contains billions of miserable people and one happy person), by our understanding of morality.
HEAVEN(n) > HELL(n) (given) HEAVEN(n) + billions of happy people + 1 happy person > HELL(n) + billions of miserable people + 1 miserable person (by understanding of morality) HEAVEN(n) + billions of happy people + 1 miserable person > HELL(n) + billions of miserable people + 1 happy person (moving people around does not improve things if it changes nothing else.) HEAVEN(n + 1) > HELL(n + 1) □
A downside of this approach is that you lose the ability to reason about uncountably infinite numbers. However, I think that’s a bullet that I am willing to bite, to only be able to reason about a countably infinite number of moral entities.
That implies that order matters! If you rearange heaven, you get hell. There are other problems with ordering—some series can sum to any number depending on arrangement,.
I’m claiming that we should only ever reason about infinity by induction-type proofs. Due to the structure of the thought experiment, the only thing that is possible to use for to count in this way is galaxies, so (I claim) counting galaxies is the only thing that you’re allowed to use for moral reasoning. Since all of the galaxies in each universe are moral equivalents (either all happy but one or all miserable but one), how you rearrange galaxies doesn’t affect the outcome.
(To be clear, I agree that if you rearrange people under the concepts of infinity that mathematicians like to use, you can turn HEAVEN into HELL, but I’m claiming that we’re simply not allowed to use that type of infinity logic for ethics.)
Obviously this is taking a stance about the ways in which infinity can be used in ethics, but I think this is a reasonable way to do so without giving up the concept of infinity entirely.
Why is the only thing that we can use galaxies? We can compare people in any ways.
If you rearrange people, standard mathematics says that you can turn HEAVEN into HELL. Infinity/1 billion = infinity. You have to change the math of infinity, not just the math of ethics where you add up infinity.
I have an argument for a way in which infinity can be used but which doesn’t imply any of the negative conclusions. I’m not convinced of its reasonableness or correctness though.
I propose that infinity ethics should only be reasoned about by use of proof through induction. When done this way, the only way to reason about HEAVEN and HELL is by matching up galaxies in each universe, and doing induction across all of the elements:
Theorem: The universe HEAVEN that contains n galaxies is a better universe than HELL which contains n galaxies. We will formalize this as HEAVEN(n) > HELL(n). We will prove this by induction.
Base case, HEAVEN(1) > HELL(1):
The first galaxy in HEAVEN (which contains billions of happy people and one miserable person) is better than the first galaxy in HELL (which contains billions of miserable people and one happy person), by our understanding of morality.
Induction step HEAVEN(n) > HELL(n) ⇒ HEAVEN(n+1) > HELL(n+1):
HEAVEN(n) > HELL(n) (given)
HEAVEN(n) + billions of happy people + 1 happy person > HELL(n) + billions of miserable people + 1 miserable person (by understanding of morality)
HEAVEN(n) + billions of happy people + 1 miserable person > HELL(n) + billions of miserable people + 1 happy person (moving people around does not improve things if it changes nothing else.)
HEAVEN(n + 1) > HELL(n + 1) □
A downside of this approach is that you lose the ability to reason about uncountably infinite numbers. However, I think that’s a bullet that I am willing to bite, to only be able to reason about a countably infinite number of moral entities.
That implies that order matters! If you rearange heaven, you get hell. There are other problems with ordering—some series can sum to any number depending on arrangement,.
I don’t think that it does? There are infinitely many arrangements, but the same proof by induction applies to any possible arrangement.
Wait, do you agree that rearranged heaven gets hell? If so, you either have to deny that HEAVEN>HELL or that arrangement matters.
You’re assuming we’re comparing them by galaxies. But there’s no natural way to individuate that explains why we should do that.
I’m claiming that we should only ever reason about infinity by induction-type proofs. Due to the structure of the thought experiment, the only thing that is possible to use for to count in this way is galaxies, so (I claim) counting galaxies is the only thing that you’re allowed to use for moral reasoning. Since all of the galaxies in each universe are moral equivalents (either all happy but one or all miserable but one), how you rearrange galaxies doesn’t affect the outcome.
(To be clear, I agree that if you rearrange people under the concepts of infinity that mathematicians like to use, you can turn HEAVEN into HELL, but I’m claiming that we’re simply not allowed to use that type of infinity logic for ethics.)
Obviously this is taking a stance about the ways in which infinity can be used in ethics, but I think this is a reasonable way to do so without giving up the concept of infinity entirely.
Why is the only thing that we can use galaxies? We can compare people in any ways.
If you rearrange people, standard mathematics says that you can turn HEAVEN into HELL. Infinity/1 billion = infinity. You have to change the math of infinity, not just the math of ethics where you add up infinity.