Rough take on this: to me lot of this reasoning seems not paying close enough attention to the relation of maths and reality; in practice the problems of infinite ethics are more likely to be solved at the level of maths, as opposed on the level of ethics and thinking about what this means for actual decisions.
Why:
General problems with how “infinities” are used in this text is it seems it is importing the assumption that something like ZFC tells us something fundamental and true about reality. Subsequently, lot of problems with infinities seem to be basically “imported from math” (how to sum infinite series?).
I’m happy to bite this bullet: - our default math being based on ZFC axioms is to a large extend random historical fact - how ZFC deals with infinities tells us very little about real infinities— default infinite ethics tells us something about ethical problems in ZFC-based matemathical universes; as I don’t assume ZFC is some fundamental base of my reality, it’s problems, questions and answers about infinities do not seem particularly relevant. Ad absurdum: if we postulated as an axioms reality is based on wiggling of big elephants standing on the back of an enormous turtle, we would likely arrive at weird ethical problems depending on obscure details of turtleology.
I still do agree with the overall conclusion that infinite ethics is in a way a lesson in humility, and there is a lot of what we don’t know.
Sidenote/reasoning transparency: Few weeks ago, I attended a seminar about alternative set theory proposed by Petr Vopenka from the “Vopenka’s principle” in the diagram.
Part of what I got from this was:
Vopenka was someone, who understood some of the problems with infinities in standard set theory, and actually took them seriously. At least what I got from some of his collaborators is, due to his research on ultrafilters, he become worried the whole math based on ZFC, as we are using it, puts a lot of what we assume to “know” on more shaky grounds / more in tension with reality than people ordinarily think. (Actually this post was really helpful for me to understand Vopenka’s deep philosophical horror). He decided to fix it, spending decades trying to develop mentioned alternative set theory, which would be somehow closer to reality. In my impression, from a math perspective, his program was not really successful. From a more meta- perspective, in my view this seems an approach to “infinite ethics” more likely to lead to progress.
I think a lot of these problems don’t depend on the axiom of choice in particular. I think you can still construct incomparable options under most or all approaches, with explicit bijections. Maybe there are still problems with ZF, but I’m more skeptical.
I don’t think ZF(C) is the problem. If whatever alternative you come up with doesn’t have equivalent results, then I think it isn’t expressive enough (or you’ve possibly even assumed away infinites, which would be empirically questionable). And whatever solution you might come up with can probably be expressed in ZF, and with less work than trying to build new foundations for math. I think it’s better to work within ZF, but with additional structure on sets or using different ethical axioms.
I confess my comment was motivated by seeing something where it looked like I could make a quick “gotcha” point, which is a bad way to converse.
Reading the original comment more carefully, I’m seeing how I disagree with it. It says (emphasis mine)
in practice the problems of infinite ethics are more likely to be solved at the level of maths, as opposed on the level of ethics and thinking about what this means for actual decisions.
I highly doubt this problem will be solved purely on the level of math, and expect it will involve more work on the level of ethics than on the level of foundations of mathematics. However, I think taking an overly realist view on the conventions mathematicians have chosen for dealing with infinities is an impediment to thinking about these issues, and studying alternative foundations is helpful to ward against that. The problems of infinite ethics, especially for uncountable infinities, seem to especially rely on such realism. I do expect a solution to such issues, to the extent it is mathematical at all, could be formalized in ZFC. The central thing I liked about the comment is the call to rethink the relationship of math and mathematical infinity to reality, and that doesn’t necessary require changing our foundations, just changing our attitude towards them.
Rough take on this: to me lot of this reasoning seems not paying close enough attention to the relation of maths and reality; in practice the problems of infinite ethics are more likely to be solved at the level of maths, as opposed on the level of ethics and thinking about what this means for actual decisions.
Why:
General problems with how “infinities” are used in this text is it seems it is importing the assumption that something like ZFC tells us something fundamental and true about reality. Subsequently, lot of problems with infinities seem to be basically “imported from math” (how to sum infinite series?).
I’m happy to bite this bullet:
- our default math being based on ZFC axioms is to a large extend random historical fact
- how ZFC deals with infinities tells us very little about real infinities—
default infinite ethics tells us something about ethical problems in ZFC-based matemathical universes; as I don’t assume ZFC is some fundamental base of my reality, it’s problems, questions and answers about infinities do not seem particularly relevant.
Ad absurdum: if we postulated as an axioms reality is based on wiggling of big elephants standing on the back of an enormous turtle, we would likely arrive at weird ethical problems depending on obscure details of turtleology.
I still do agree with the overall conclusion that infinite ethics is in a way a lesson in humility, and there is a lot of what we don’t know.
Sidenote/reasoning transparency: Few weeks ago, I attended a seminar about alternative set theory proposed by Petr Vopenka from the “Vopenka’s principle” in the diagram.
Part of what I got from this was:
Vopenka was someone, who understood some of the problems with infinities in standard set theory, and actually took them seriously. At least what I got from some of his collaborators is, due to his research on ultrafilters, he become worried the whole math based on ZFC, as we are using it, puts a lot of what we assume to “know” on more shaky grounds / more in tension with reality than people ordinarily think. (Actually this post was really helpful for me to understand Vopenka’s deep philosophical horror). He decided to fix it, spending decades trying to develop mentioned alternative set theory, which would be somehow closer to reality. In my impression, from a math perspective, his program was not really successful. From a more meta- perspective, in my view this seems an approach to “infinite ethics” more likely to lead to progress.
I think a lot of these problems don’t depend on the axiom of choice in particular. I think you can still construct incomparable options under most or all approaches, with explicit bijections. Maybe there are still problems with ZF, but I’m more skeptical.
If the only alternative you can conceive of for ZFC is removing the axiom of choice then you are proving Jan_Kulveit’s point.
I don’t think ZF(C) is the problem. If whatever alternative you come up with doesn’t have equivalent results, then I think it isn’t expressive enough (or you’ve possibly even assumed away infinites, which would be empirically questionable). And whatever solution you might come up with can probably be expressed in ZF, and with less work than trying to build new foundations for math. I think it’s better to work within ZF, but with additional structure on sets or using different ethical axioms.
On further thought I want to walk back a bit:
I confess my comment was motivated by seeing something where it looked like I could make a quick “gotcha” point, which is a bad way to converse.
Reading the original comment more carefully, I’m seeing how I disagree with it. It says (emphasis mine)
I highly doubt this problem will be solved purely on the level of math, and expect it will involve more work on the level of ethics than on the level of foundations of mathematics. However, I think taking an overly realist view on the conventions mathematicians have chosen for dealing with infinities is an impediment to thinking about these issues, and studying alternative foundations is helpful to ward against that. The problems of infinite ethics, especially for uncountable infinities, seem to especially rely on such realism. I do expect a solution to such issues, to the extent it is mathematical at all, could be formalized in ZFC. The central thing I liked about the comment is the call to rethink the relationship of math and mathematical infinity to reality, and that doesn’t necessary require changing our foundations, just changing our attitude towards them.