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 R3?
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 R3, and even for some functions which are not integrable over R3 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.
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 R3?
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 R3, and even for some functions which are not integrable over R3 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.