# djbinder comments on On infinite ethics

• I think Section XIII is too dismissive of the view that infinities are not “real”, conflating it with ultrafinitism. But the sophisticated version of this view is that infinities should only be treated as “idealized limits” of finite processes. This is, as far as understand, the default view amongst practicing mathematicians and physicists. If you stray from it, and use infinities without specifying the limiting process, it is very easy to produce paradoxes, or at least, indeterminancy in the problem. The sophisticated view, then, is not that infinities don’t exist, but that, since they only exist as limiting cases of finite processes. One must always specify the limiting process, and in doing so any paradoxes or indeterminancies will disappear.

As Jaynes’ summarizes in Chapter 15 of Probability: The Logic of Science:

[P]aradoxes caused by careless dealing with infinite sets or limits can be mass-produced by the following simple procedure:

(1) Start from a mathematically well-defined situation, such as a finite set, a normalized probability distribution, or a convergent integral, where everything is well-behaved and there is no question about what is the correct solution.

(2) Pass to a limit – infinite magnitude, infinite set, zero measure, improper pdf, or some other kind – without specifying how the limit is approached.

(3) Ask a question whose answer depends on how the limit was approached.

• Agree with djbinder on this, that “infinities should only be treated as ‘idealized limits’ of finite processes”.

To explain what I mean:

Infinites outside of limiting sequences are not well defined (at least that is how I would describe it). Sure you can do some funky set theory maths on them but from the point of view of physics they don’t work, cannot be used.

(My favorite example (HT Jacob Hilton) is a man throws tennis balls into a room once every 1 second numbered 1,2,3,4,… and you throw them out once every 2 seconds, how many balls are in the room after infinite time and which balls are they? Well if you throw out the odd balls (2,4,6...) then 1,3,5,.. are left but if you throw out the balls in the order they are thrown in (1,2,3,4...) then after infinite time no balls are left. The lesson: “infinite” is not a precise enough term to answer the question.)

As far as I understand it, from a philosophy of science point of view doing physics is something like writing the simplest set of mathematical formulas to describe the universe. These formulas need to work. So you will never find a physicist that uses infinites in the way this post does. If physics is ever able to succeed at mapping the universe it will have to do it without using infinite sets, except where they can be well defined as limits (unless there is drastic change to what physics is).

As such doing infinite ethics of the type done in this post makes as much sense as doing any other poorly defined by physics thought experiment (see example in my other reply about what if time travel paradoxes are true).

Of course there could still be infinites in limits. E.g. one happy person a day tending forever (as Joe flags in his comment). But hopefully they are better defined and may avoid some of the problems of the post above (certainly it breaks the zones of happiness/​suffering thought experiment). I am not sure.

• But the sophisticated version of this view is that infinities should only be treated as “idealized limits” of finite processes. This is, as far as understand, the default view amongst practicing mathematicians and physicists.

I think this is false in general (at least for mathematicians), but true for many specific applications. Mathematicians frequently deal with infinite sets, and they don’t usually treat them like limits of finite processes, especially if they’re uncountable.

How would you handle the possibility of a spatially unbounded universe, e.g. if our space looks like ?

Expansionism is an approach that basically treats the universe as a limit of bounded universes to add up ethical value, since you take limits of partial sums expanding out from a point. But it still runs into problems, as discussed in the post.

• I think you are right about infinite sets (most of the mathematicians I’ve talked to have had distinctly negative views about set theory, in part due to the infinities, but my guess is that such views are more common amongst those working on physics-adjacent areas of research). I was thinking about infinities in analysis (such as continuous functions, summing infinite series, integration, differentiation, and so on), which bottom out in some sort of limiting process.

On the spatially unbounded universe example, this seems rather analogous to me to the question of how to integrate functions over the same space. There are a number of different sets of functions which are integrable over , and even for some functions which are not integrable over there are natural regularization schemes which allows the integral to be defined. In some cases these regularizations may even allow a notion of comparing different “infinities”, as in cases where the integral diverges as the regularizer is taken to zero one integral may strictly dominate the other. When dealing with situations in ethics, perhaps we should always be restricting to these cases? There are a lot of different choices here, and it isn’t clear to me what the correct restriction is, but it seems plausible to me that some form of restriction is needed. Note that such a restrictions include ultrafinitism, as an extreme case, but in general allows a much richer set of possibilities.

Expansionism is neceessarily incomplete, it assumes that the world has a specific causal structure (ie, one that is locally that of special relativity) which is an empirical observation about our universe rather than a logically necessary fact. I think it is plausible that, given the right causal assumptions, expansionism follows (at least for individual observers making decisions that respect causality).

• So you’re saying a utilitarian needs both a utility function, and a measure with which to integrate over any sets of interest (OK)? And also some transformations to regularise infinite sets (giving up the dream of impartiality)? And still there are some that cannot be regularised, so utilitarian ethics can’t order them (but isn’t that the problem we were trying to solve)?

• I agree with with your first question, the utilitarian needs a measure (they don’t need a separate utility function from their measure, but there may be other natural measures to consider in which case you do need a utility function).

With respect to your second question, I think you can either give up on the infinite cases (because you think they are “metaphysically” impossible, perhaps) or you demand that a regularization must exist (because with this the problem is “metaphysically” underspecified). I’m not sure what the correct approach is here, and I think it is an interesting question to try and understand this in more detail. On the latter case you have to give up impartiality, but only in a fairly benign way, and that our intuitions about impartiality are probably wrong here (analogous situations occur in physics with charge conservation, as I noted in another comment).

With respect to your third question, I think it is likely that problems with no regularization are non-sensical. This is not to say that all problems involving infinities are themselves non-sense, nor to say that correct choice of regularization is obvious.

As an intuition pump maybe we can consider cases that don’t involve infinities. Say we are in (rather contrived world) in which utility is literally a function of space-time, and we integrate to get the total utility. How should I assign utility for a function which has support on a non-measurable set? Should I even think such a thing is possible? After all, the existence of non-measuarable sets follows not from ZF alone, but requires also the axiom of choice. As another example, maybe my utility function depends on whether or not the continuum hypothesis is true or false. How should I act in this case?

My own guess is that such questions likely have no meaningful answer, and I think the same is true for questions involving infinities without specified ways to operationalize the infinities. I think it would be odd to give up on the utilitarian dream due to unmeasurable sets, and that the same is true for ill-defined infinities.

1. Does this view imply that it is actually not possible to have a world where e.g. a machine creates one immortal happy person per day, forever, who then form an ever-growing line?

2. How does this view interpret cosmological hypotheses on which the universe is infinite? Is the claim that actually, on those hypotheses, the universe is finite after all?

3. It seems like lots of the (countable) worlds and cases discussed in the post can simply be reframed as never-ending processes, no? And then similar (identical?) questions will arise? Thus, for example, w5 is equivalent to a machine that creates a1 at −1, then a3 at −1, then a5 at −1, etc. w6 is equivalent to a machine that creates a1 at −1, then a2 at −1, a3 at −1, etc. What would this view say about which of these machines we should create, given the opportunity? How should we compare these to a w8 machine that creates b1 at −1, b2 at −1, b3 at −1, b4 at −1, etc?

Re: the Jaynes quote: I’m not sure I’ve understood the full picture here, but in general, to me it doesn’t feel like the central issues here have to do with dependencies on “how the limit is approached,” such that requiring that each scenario pin down an “order” solves the problems. For example, I think that a lot of what seems strange about Neutrality-violations in these cases is that even if we pin down an order for each case, the fact that you can re-arrange one into the other makes it seem like they ought to be ethically equivalent. Maybe we deny that, and maybe we do so for reasons related to what you’re talking about—but it seems like the same bullet.

• My take (think I am less of an expert than djbinder here)

1. This view allows that.

2. This view allows that. (Although entirely separately consideration of entropy etc would not allow infinite value.)

3. No I don’t think identical questions arise. Not sure. Skimming the above post it seems to solve most of the problematic examples you give. At any point a moral agent will exist in a universe with finite space and finite time that will tend infinite going forward. So you cannot have infinite starting points so no zones of suffering etc. Also I think you don’t get problems with “welfare-preserving bijections” when well defined it time but struggle to explain why. It seems that for example w1 below is less bad than w2

Time t1 t2 t3 t4 t5 t6 t7
Agent a1 a2 a3 a4 a5 a6 a7
w1 −1 −1 −1 −1….
w2 −1 −1 −1 −1 −1 −1 −1….

• As an aside, while neutrality-violations are a necessary consequence of regularization, a weaker form of neutrality is preserved. If we regularize with some discounting factor so that everything remains finite, it is easy to see that “small rearrangments” (where the amount that a person can move in time is finite) do not change the answer, because the difference goes to zero as . But “big rearrangments” can cause differences that grow with . Such situations do arise in various physical situations, and are interpretted as changes to boundary conditions, whereas the “small rearrangments” manifestly preserve boundary conditions and manifestly do not cause problems with the limit. (The boundary is most easily seen by mapping the infinite interval sequence onto a compact interval, so that “infinity” is mapped to a finite point. “Small rearrangments” leave infinity unchanged, whereas “large” ones will cause a flow of utility across infinity, which is how the two situations are able to give different answers.)

• I think what is true is probably something like “neverending process don’t exist, but arbitrarily long ones do”, but I’m not confident. My more general claim is that there can be intermediate positions between ultrafinitism (“there is a biggest number”), and any laissez faire “anything goes” attitude, where infinities appear without care or scrunity. I would furthermore claim (but on less solid ground), that the views of practicing mathematicians and physicists falls somewhere in here.

As to the infinite series examples you give, they are mathematically ill-defined without giving a regularization. There is a large literature in mathematics and physics on the question of regularizing infinite series. Regularization and renormalization are used through physics (particular in QFT), and while poorly written textbooks (particularly older ones) can make this appear like voodoo magic, the correct answers can always be rigorously be obtained by making everything finite.

For the situation you are considering, a natural regularization would be to replace your sum with a regularized sum where you discount each time step by some discounting factor . Physically speaking, this is what would happen if we thought the universe had some chance of being destroyed at each timestep; that is, it can be arbitrarily long-lived, yet with probability 1 is finite. You can sum the series and then take and thus derive a finite answer.

There may be many other ways to regulate the series, and it often turns out that how you regulate the series doesn’t matter. In this way, it might make sense to talk about this infinite universe without reference to a specific limiting process, but rather potentially with only some weaker limiting process specification. This is what happens, for instance, in QFT; the regularizations don’t matter, all we care about are the things that are independent of regularization, and so we tend to think of the theories as existing without a need for regularization. However, when doing calculations it is often wise to use a specific (if arbitrary) regularization, because it guarantees that you will get the right answer. Without regularizations it is very easy to make mistakes.

This is all a very long-winded way to say that there are at least two intermediate views one could have about these infinite sequence examples, between the “ultrafinitist” and the “anything goes”:

1. The real world (or your priors) demands some definitive regularization, which determines the right answer. This would be the case if the real world had some probability of being destroyed, even if it is arbitrarily small.

2. Maybe infinite situations like the one you described are allowed, but require some “equivalence class of regularizations” to be specified in order to be completely specified. Otherwise the answer is as indeterminant as if you’d given me the situation without specifiying the numbers. I think this view is a little weirder, but also the one that seems to be adopted in practice by physicists.