The hyperreals are deeply problematic because using them in normative contexts means that hugely normatively important things depend on which arbitrary Axiom-of-Choice-constructed ultrafilter you chose!
(It also has the feature that it endorses Richard Chappell's point that each person's utility matters, but there is no thing we call "total utility" that matters - that "total" concept was just a convenient fiction for keeping track of the tradeoffs between the things that actually matter, which are the individuals.)
My approach does have the feature that it doesn't do a whole lot of useful non-Pareto comparisons among infinite circumstances without some sort of arbitrary permutation-dependence. But I think that's just a logical fact that you can't get out of no matter what sorts of machinery you try to use.
The basic idea is that we evaluate actions by thinking of them as functions from states and persons to utilities - the choice to build a nuclear power plant assigns some people extremely negative utility in the states of the world where the power plant has a serious meltdown, but otherwise generally assigns people higher utilities than the choice not to build the nuclear power plant due to better access to electricity (though there's some complexity about benefits and costs to people who were making their income working for competing power plants, or supplying services to people who can now use electricity instead). What I claim we need in ethics is a way to say when one of these actions is better than another, or worse, or equally good (or perhaps incomparable).
I start with the basic idea - if one act gives every single person a higher utility in every single state than another, then the first act is better. This of course leaves most acts incomparable. Then we need a way to start to assess tradeoffs. If we can compare changes in utility among people in the same state, then we can say that if two acts differ just in how much utility finitely many people have in some states, then if the same number of people are better off in one than in the other, and by the same amount, then the two acts are equal. This then allows us to say that a bunch of options are better or worse than each other by transitivity.
If there's only finitely many people, and no uncertainty about the state, this yields exactly the same ordering as additive utilitarianism.
We can then extend further - if we decide some sets of states are equal importance to others, then we can do these tradeoffs between a given set of people in one of these states with the same people in the other state, and I show that in the case with finitely many people, this actually ends up with the same ordering as expected utility.
But these particular comparisons still make sense when the set of people is infinite, and when sets of states can't be compared by probability, so we can still evaluate acts in the infinite case. But there's often more incomparability. I give some attempts to extend these tradeoff rules to make decisions between lots of plausible cases that people in the literature consider, but I cite some results by certain economists showing that there's no way to make this a total ordering among acts without doing some arbitrary Axiom of Choice work (perhaps via hyperreals, or perhaps some other way).
I guess ultimately, I do want an ethical theory under consideration to extend some judgment on "controversial" cases where there's deep tradeoffs. (sorry, would write a longer reply, but it's getting quite late here).
Although I don't think I fully understand the tradeoff rules you're proposing or how far they go.
I haven't actually looked in detail at the recent paper of his - I should probably do that!
But first, I think it's hard to make conceptual sense of this - the hyperreals you get from one ultrafilter aren't comparable to the hyperreals you get from another ultrafilter; they live in different fields, and any attempt to identify members of one field with members of another field are going to end up with lots of conflicting comparisons.
Second, if you're averaging between a finite or countable set of ultrafilters, you're still going to have the arbitrariness worry, since there are uncountably many ultrafilters.
The most promising approach seems to me to be supervaluating over what is shared under *all* ultrafilters - but you can do this by skipping the detour through ultrafilters and just working with the cofinite filter, and accepting all the incomparability you get where the different ultrafilters conflict with each other. I suspect this will be basically equivalent to what I endorse, though with a weird metaphysics of "numbers" that don't quite have identity conditions. I think Jeff Russell has a chapter doing a version of this in the book he's been working on (I haven't checked if he's finished it, or even had a chance to do much more than look at that chapter and one other fairly quickly last year).
Interesting, yeah I don't know enough about the math to evaluate what you're saying. I just remember that in the extensive Claude chat I link, as I say in footnote 7, I was "left with the rough impression that the “averaging over ultrafilters” thing generally works for “non-weird” sequences".
IIRC, he doesn't generally average over "all" ultrafilters, but just over e.g. the two possible results for the sequence that an ultrafilter can give, and that intuitively seem to cut the space into two.
I'm wondering what a concrete counterintuitive example of this could be. Of a reordering that is "not that bad" and still offsets a Pareto worsening. (by "not that bad", I mean one that only changes the order and not the relative frequency, e.g. a change in the point of origin of integration. Obviously you can cause complete haywire if you allow changing the relative frequency of values, but then I wouldn't find that counterintuitive due to the intuition in EDIT 2).
I think order dependence is a serious problem - where would you get an order from, thats not ultimately space-time? Because once you admit space-time dependence, its not clear what the motivation even is doing it with summation. Hyperreals make a lot of comparison worth taking a look at and wondering if you really do think thats better. For example, consider the utility streams:
⟨1,-1,2,-2,...⟩ vs ⟨-1,1,-2,2,...⟩
and
⟨0,1,-1,2,-2,...⟩ vs ⟨0,-1,1,-2,2,...⟩
One of these is 0 vs 0, and the other is +ω/2 vs -ω/2 (which is which depends on the ultrafilter). Basically everything about this is bad: Putting a zero in front of two options changes the comparison. We're drawing infinite value differences from a difference in ordering thats trivially created by different points of origen. And... the things thats better about the better option here is apparently just that the better things happen earlier. So, in order to avoid temporal discounting, which would have found a finite difference between these worlds, we have adopted a theory which finds an infinite difference, and many more complications to boot.
Interesting, so I think if you do the "averaging over ultrafilters" approach, you get different values here, I'm not sure where you're getting yours from. Opus 4.8 gives me these (https://x.com/el1assss/status/2068399850891125159) (and they make sense to me):
⟨1,-1,2,-2,...⟩ : (ω+1)/4
⟨-1,1,-2,2,...⟩ : -(ω+1)/4
⟨0,1,-1,2,-2,...⟩ : ω/4
⟨0,-1,1,-2,2,...⟩ : -ω/4
I think these are not as terrible as you make it sound:
1) adding a 0 only makes a finite difference (and doesn't change the comparison), so this doesn't clash with my intuition that hard since the leading term is infinite.
2) I could see myself being okay with putting the negative terms first changing the sign - the negative terms are first *forever*, *for eternity*. It could be seen as a deep structural difference. There's in some way "always more" of the negative terms, projected out to infinity, forever - therefore the world as a whole is negative. idk.
Also, as you said temporal discounting views would also find a sign change here - all the partial sums change sign - so I don't understand why you think it's disqualifying.
3) In practice, we probably don't face choices like this.
It is counterintuitive though, and definitely a point against this approach. It does show that it's not just about relative frequencies, that sheer ordering can also matter.
I treated the even-included/odd-included ultrafilters seperately (and forgot the 1 in the first case).
1) Yes, the averaging version avoids this. I took the "less ambitious" version as the target.
2) I think its a problem because I thought *not* doing stuff like that was kind of the point of a summation-based approach.
3) Im at (1,0) youre at (-1,0). At (0.5,n) is a person with utility n, at (-0.5,n) is a person with utility -n. If we list agents in order of distance from us, one of us will see one world an one the other.
Also, if you think theres no way we can ever make more than finite changes, then there never was a need for any of this, and the pareto rule + finitary arithmetic tells you all you need to know.
1) Well, the averaging is pretty core to the whole thing IMO.
2) No, the point is to avoid infinitarian paralysis. Not to avoid all counterintuitive things, that's impossible.
3) Hm, that's a good point. I guess averaging over points of origin could solve that one? You should check out the Claude chat I link in the Edit, especially the second part of the last message.
No, I didn't say we would never be able to make infinite changes, just weird ones like that ("everybody swap places!"), it's not immediately clear that it will happen.
1) And as Claude tells you, the intuitive version of it doesnt exist. And where it works, it agrees with other summing approaches. I think if there is a way to do what you want here, its natural presentation wont hyperreals.
2) I guess I dont see the point of having specifically a summing notion then
3) I cant actually see what examples Claude is talking about.
I think that if we can instantiate one as a time series going forward, likely we can also decide on the other.
Another interesting case for positional dependence is:
⟨...,-1,-1,1,1...⟩ vs ⟨...,-1,-1,0,1,1...⟩
I think this points against the idea that "starting good later" is bad, because its unclear if the good starts later, or the bad ends earlier. More generally, positional matching between possible worlds isnt obviously meaningful.
I think that the positional dependence and the hyperfilter dependence are closely related.
>we might be able to form an average across all ways of specifying the hyperreals. In this case, the average value for our alternating sum SS is 1441. Interestingly, this is precisely the value assigned to this series by Abel summation, Euler summation and Borel summation
Of course, theres not a rigourous proof that these methods work in all the same cases, since ours isnt formal yet, and I expect not all of them do - just, this, together with the formal difficulty for the intuition, makes me expect that hyperreals wont do much work in a formalisation, if we find one.
I've been rather skeptical about bijections as a fine-grained measure of size for years. There are sets which have bijections to a subset of themselves.
Some people want to use this as a definition of infinity (Dedekind infinite), but this is sufficiently weird to me, that it feels like the kind of thing you should only believe if you've been forced into it by incredibly strong reasons, not something that you build into your system axiomatically.
2. (Infinity + one) minus (the same infinity) = one.
The struggle comes from trying to consider two different universes and decide which is preferable over the other by giving each an overall aggregate score. But what do we need overall scores for? We just need the ability to compare.
Let's say we took the digits of pi and said every digit is a happy life except the digit 1. Every digit 1 is a life in suffering.
Let's compare two infinite worlds. One of them will just be pi. The other will be pi with one modification:
World A: 3.141592653589
World B: 3.241592653589
World B is better than World A.
No need for hyperreals, nor need to entertain the false notion that one infinite set can have "more" members than another (as I explain here:
The biggest problem imo is that it implies that you might improve things by making everyone worse off and moving people around.
I prefer a solution that holds that there’s ubiquitous incompatibility but still uses hypereeals to evaluate effects of actions
The hyperreals are deeply problematic because using them in normative contexts means that hugely normatively important things depend on which arbitrary Axiom-of-Choice-constructed ultrafilter you chose!
You might be interested in my paper on infinite ethics, where I get a lot of incomparability, but get the intuitive good judgments without needing anything as mathematically powerful as the hyperreals: https://academic.oup.com/aristotelian/article-abstract/121/3/299/6367834
(It also has the feature that it endorses Richard Chappell's point that each person's utility matters, but there is no thing we call "total utility" that matters - that "total" concept was just a convenient fiction for keeping track of the tradeoffs between the things that actually matter, which are the individuals.)
My approach does have the feature that it doesn't do a whole lot of useful non-Pareto comparisons among infinite circumstances without some sort of arbitrary permutation-dependence. But I think that's just a logical fact that you can't get out of no matter what sorts of machinery you try to use.
I don't think I understand your approach from the abstract, would be interested for you to briefly summarize it.
The basic idea is that we evaluate actions by thinking of them as functions from states and persons to utilities - the choice to build a nuclear power plant assigns some people extremely negative utility in the states of the world where the power plant has a serious meltdown, but otherwise generally assigns people higher utilities than the choice not to build the nuclear power plant due to better access to electricity (though there's some complexity about benefits and costs to people who were making their income working for competing power plants, or supplying services to people who can now use electricity instead). What I claim we need in ethics is a way to say when one of these actions is better than another, or worse, or equally good (or perhaps incomparable).
I start with the basic idea - if one act gives every single person a higher utility in every single state than another, then the first act is better. This of course leaves most acts incomparable. Then we need a way to start to assess tradeoffs. If we can compare changes in utility among people in the same state, then we can say that if two acts differ just in how much utility finitely many people have in some states, then if the same number of people are better off in one than in the other, and by the same amount, then the two acts are equal. This then allows us to say that a bunch of options are better or worse than each other by transitivity.
If there's only finitely many people, and no uncertainty about the state, this yields exactly the same ordering as additive utilitarianism.
We can then extend further - if we decide some sets of states are equal importance to others, then we can do these tradeoffs between a given set of people in one of these states with the same people in the other state, and I show that in the case with finitely many people, this actually ends up with the same ordering as expected utility.
But these particular comparisons still make sense when the set of people is infinite, and when sets of states can't be compared by probability, so we can still evaluate acts in the infinite case. But there's often more incomparability. I give some attempts to extend these tradeoff rules to make decisions between lots of plausible cases that people in the literature consider, but I cite some results by certain economists showing that there's no way to make this a total ordering among acts without doing some arbitrary Axiom of Choice work (perhaps via hyperreals, or perhaps some other way).
Hm, I see! Thanks for the detailed reply!
I guess ultimately, I do want an ethical theory under consideration to extend some judgment on "controversial" cases where there's deep tradeoffs. (sorry, would write a longer reply, but it's getting quite late here).
Although I don't think I fully understand the tradeoff rules you're proposing or how far they go.
Ah, just saw you have the paper on your website - might give it a skim tomorrow
What do you think of the approach that Toby gestures at, of averaging between ultrafilters in some way? It seems promising to me.
I haven't actually looked in detail at the recent paper of his - I should probably do that!
But first, I think it's hard to make conceptual sense of this - the hyperreals you get from one ultrafilter aren't comparable to the hyperreals you get from another ultrafilter; they live in different fields, and any attempt to identify members of one field with members of another field are going to end up with lots of conflicting comparisons.
Second, if you're averaging between a finite or countable set of ultrafilters, you're still going to have the arbitrariness worry, since there are uncountably many ultrafilters.
The most promising approach seems to me to be supervaluating over what is shared under *all* ultrafilters - but you can do this by skipping the detour through ultrafilters and just working with the cofinite filter, and accepting all the incomparability you get where the different ultrafilters conflict with each other. I suspect this will be basically equivalent to what I endorse, though with a weird metaphysics of "numbers" that don't quite have identity conditions. I think Jeff Russell has a chapter doing a version of this in the book he's been working on (I haven't checked if he's finished it, or even had a chance to do much more than look at that chapter and one other fairly quickly last year).
Oh and by the way, the paper is a pleasure to read! Would definitely recommend
Interesting, yeah I don't know enough about the math to evaluate what you're saying. I just remember that in the extensive Claude chat I link, as I say in footnote 7, I was "left with the rough impression that the “averaging over ultrafilters” thing generally works for “non-weird” sequences".
IIRC, he doesn't generally average over "all" ultrafilters, but just over e.g. the two possible results for the sequence that an ultrafilter can give, and that intuitively seem to cut the space into two.
Actually, what do you mean by this? How would you use hyperreals to evaluate actions if their outcome states are incomparable?
I'm wondering what a concrete counterintuitive example of this could be. Of a reordering that is "not that bad" and still offsets a Pareto worsening. (by "not that bad", I mean one that only changes the order and not the relative frequency, e.g. a change in the point of origin of integration. Obviously you can cause complete haywire if you allow changing the relative frequency of values, but then I wouldn't find that counterintuitive due to the intuition in EDIT 2).
I'm also curious to hear you say more about your preferred solution!
I think order dependence is a serious problem - where would you get an order from, thats not ultimately space-time? Because once you admit space-time dependence, its not clear what the motivation even is doing it with summation. Hyperreals make a lot of comparison worth taking a look at and wondering if you really do think thats better. For example, consider the utility streams:
⟨1,-1,2,-2,...⟩ vs ⟨-1,1,-2,2,...⟩
and
⟨0,1,-1,2,-2,...⟩ vs ⟨0,-1,1,-2,2,...⟩
One of these is 0 vs 0, and the other is +ω/2 vs -ω/2 (which is which depends on the ultrafilter). Basically everything about this is bad: Putting a zero in front of two options changes the comparison. We're drawing infinite value differences from a difference in ordering thats trivially created by different points of origen. And... the things thats better about the better option here is apparently just that the better things happen earlier. So, in order to avoid temporal discounting, which would have found a finite difference between these worlds, we have adopted a theory which finds an infinite difference, and many more complications to boot.
Interesting, so I think if you do the "averaging over ultrafilters" approach, you get different values here, I'm not sure where you're getting yours from. Opus 4.8 gives me these (https://x.com/el1assss/status/2068399850891125159) (and they make sense to me):
⟨1,-1,2,-2,...⟩ : (ω+1)/4
⟨-1,1,-2,2,...⟩ : -(ω+1)/4
⟨0,1,-1,2,-2,...⟩ : ω/4
⟨0,-1,1,-2,2,...⟩ : -ω/4
I think these are not as terrible as you make it sound:
1) adding a 0 only makes a finite difference (and doesn't change the comparison), so this doesn't clash with my intuition that hard since the leading term is infinite.
2) I could see myself being okay with putting the negative terms first changing the sign - the negative terms are first *forever*, *for eternity*. It could be seen as a deep structural difference. There's in some way "always more" of the negative terms, projected out to infinity, forever - therefore the world as a whole is negative. idk.
Also, as you said temporal discounting views would also find a sign change here - all the partial sums change sign - so I don't understand why you think it's disqualifying.
3) In practice, we probably don't face choices like this.
It is counterintuitive though, and definitely a point against this approach. It does show that it's not just about relative frequencies, that sheer ordering can also matter.
I treated the even-included/odd-included ultrafilters seperately (and forgot the 1 in the first case).
1) Yes, the averaging version avoids this. I took the "less ambitious" version as the target.
2) I think its a problem because I thought *not* doing stuff like that was kind of the point of a summation-based approach.
3) Im at (1,0) youre at (-1,0). At (0.5,n) is a person with utility n, at (-0.5,n) is a person with utility -n. If we list agents in order of distance from us, one of us will see one world an one the other.
Also, if you think theres no way we can ever make more than finite changes, then there never was a need for any of this, and the pareto rule + finitary arithmetic tells you all you need to know.
1) Well, the averaging is pretty core to the whole thing IMO.
2) No, the point is to avoid infinitarian paralysis. Not to avoid all counterintuitive things, that's impossible.
3) Hm, that's a good point. I guess averaging over points of origin could solve that one? You should check out the Claude chat I link in the Edit, especially the second part of the last message.
No, I didn't say we would never be able to make infinite changes, just weird ones like that ("everybody swap places!"), it's not immediately clear that it will happen.
1) And as Claude tells you, the intuitive version of it doesnt exist. And where it works, it agrees with other summing approaches. I think if there is a way to do what you want here, its natural presentation wont hyperreals.
2) I guess I dont see the point of having specifically a summing notion then
3) I cant actually see what examples Claude is talking about.
I think that if we can instantiate one as a time series going forward, likely we can also decide on the other.
Another interesting case for positional dependence is:
⟨...,-1,-1,1,1...⟩ vs ⟨...,-1,-1,0,1,1...⟩
I think this points against the idea that "starting good later" is bad, because its unclear if the good starts later, or the bad ends earlier. More generally, positional matching between possible worlds isnt obviously meaningful.
I think that the positional dependence and the hyperfilter dependence are closely related.
what other summing approaches are you talking about concretely that get the same result?
As per Ord:
>we might be able to form an average across all ways of specifying the hyperreals. In this case, the average value for our alternating sum SS is 1441. Interestingly, this is precisely the value assigned to this series by Abel summation, Euler summation and Borel summation
Of course, theres not a rigourous proof that these methods work in all the same cases, since ours isnt formal yet, and I expect not all of them do - just, this, together with the formal difficulty for the intuition, makes me expect that hyperreals wont do much work in a formalisation, if we find one.
I've been rather skeptical about bijections as a fine-grained measure of size for years. There are sets which have bijections to a subset of themselves.
Some people want to use this as a definition of infinity (Dedekind infinite), but this is sufficiently weird to me, that it feels like the kind of thing you should only believe if you've been forced into it by incredibly strong reasons, not something that you build into your system axiomatically.
I argue:
1. Infinity + one IS still infinity.
2. (Infinity + one) minus (the same infinity) = one.
The struggle comes from trying to consider two different universes and decide which is preferable over the other by giving each an overall aggregate score. But what do we need overall scores for? We just need the ability to compare.
Let's say we took the digits of pi and said every digit is a happy life except the digit 1. Every digit 1 is a life in suffering.
Let's compare two infinite worlds. One of them will just be pi. The other will be pi with one modification:
World A: 3.141592653589
World B: 3.241592653589
World B is better than World A.
No need for hyperreals, nor need to entertain the false notion that one infinite set can have "more" members than another (as I explain here:
https://ramblingafter.substack.com/p/can-you-overbook-an-infinite-hotel )