Using modern mathematics, we can now prove the intuition of Mills and Bentham: because addition is so special, any ethical system which is in a certain technical sense “reasonable” is equivalent to total utilitarianism.
but then your actual argument includes steps like these:
The most obvious way of defining an ethics of populations is to just take an ordering of individual lives and “glue them together” in an order-preserving way, like I did above.
which, please note, does not amount to any sort of argument that we must or even should just glue values-of-lives together in this sort of way.
I do not see any sign in what you have written that Hölder’s theorem is doing any real work for you here. It says that an archimedean totally ordered group is isomorphic to a subset of (R,+) -- but all the contentious stuff about total utilitarianism is already there by the time you suppose that utilities form an archimedean totally ordered group and that combining people is just a matter of applying the group operation to their individual utilities.
which, please note, does not amount to any sort of argument that we must or even should just glue values-of-lives together in this sort of way.
Thanks for the feedback, I should’ve used clearer terminology.
I do not see any sign in what you have written that Hölder’s theorem is doing any real work for you here
This seems to be the consensus. It’s very surprising to me that we get such a strong result from only the l-group axioms, and the fact that his result is so celebrated seems to indicate that other mathematicians find it surprising too, but the commenters here are rather blase.
Do you think giving examples of how many things completely unrelated to addition are groups (wallpaper groups, rubik’s cube, functions under composition, etc.) would help show that the really restrictive axiom is the archimedean one?
It doesn’t seem to me like the issue is one of terminology, but maybe I’m missing something.
Do you think giving examples [...] would help show that the really restrictive axiom is the archimedean one?
I’m not convinced that it is. The examples you give aren’t ordered groups, after all.
It’s unclear to me whether your main purpose here is to exhibit a surprising fact about ethics (which happens to be proved by means of Hölder’s theorem) or to exhibit an interesting mathematical theorem (which happens to have a nice illustration involving ethics). From the original posting it looked like the former but what you’ve now written seems to suggest the latter.
My impression is that the blasé-ness is aimed more at the alleged application to ethics rather than denying that the theorem, quite mathematical theorem, is interesting and surprising.
Near the beginning you write this:
but then your actual argument includes steps like these:
which, please note, does not amount to any sort of argument that we must or even should just glue values-of-lives together in this sort of way.
I do not see any sign in what you have written that Hölder’s theorem is doing any real work for you here. It says that an archimedean totally ordered group is isomorphic to a subset of (R,+) -- but all the contentious stuff about total utilitarianism is already there by the time you suppose that utilities form an archimedean totally ordered group and that combining people is just a matter of applying the group operation to their individual utilities.
Thanks for the feedback, I should’ve used clearer terminology.
This seems to be the consensus. It’s very surprising to me that we get such a strong result from only the l-group axioms, and the fact that his result is so celebrated seems to indicate that other mathematicians find it surprising too, but the commenters here are rather blase.
Do you think giving examples of how many things completely unrelated to addition are groups (wallpaper groups, rubik’s cube, functions under composition, etc.) would help show that the really restrictive axiom is the archimedean one?
It doesn’t seem to me like the issue is one of terminology, but maybe I’m missing something.
I’m not convinced that it is. The examples you give aren’t ordered groups, after all.
It’s unclear to me whether your main purpose here is to exhibit a surprising fact about ethics (which happens to be proved by means of Hölder’s theorem) or to exhibit an interesting mathematical theorem (which happens to have a nice illustration involving ethics). From the original posting it looked like the former but what you’ve now written seems to suggest the latter.
My impression is that the blasé-ness is aimed more at the alleged application to ethics rather than denying that the theorem, quite mathematical theorem, is interesting and surprising.