2 ms·

> You really think that it makes sense to require the axiom of choice to prove that the derivative of x^2 is 2x as Robinson's ultrafilter construction does? Th

by JadeNB 5y ago

> You really think that it makes sense to require the axiom of choice to prove that the derivative of x^2 is 2x as Robinson's ultrafilter construction does? This was just what I meant to say—the details of the construction shouldn't matter, only the axiomatics of the structure that's been constructed. You can use infinitesimals perfectly well without having to get into the weeds of how they're constructed. To be sure, your results only apply within that structure, but that's the way of mathematics, that things are proven only within some structure. I think one wouldn't expect, for example, a constructivist mathematician to disparage classical mathematics because it relies on such inelegant logical machinery as the law of the excluded middle. Well, maybe some constructivists do that, but more often they offer the better response of showing how many of the same results you can recover without requiring that logical machinery. The same response seems appropriate here: there's no reason to cast aspersions on people whose work rests on the axiom of choice; but, if it bothers you, then see how much of their work you can do without the axiom of choice. If you can prove the equivalent, then great; no need to complain! If part of their work genuinely requires choice, then that's an interesting fact, too.

btilly 5y ago
Here is the thing. Robinson's construction required the axiom of choice and proved that every result that you can prove with infinitesmals has a proof with limits without the axiom of choice. But he went on to show that infinitesmals allowed abandoned proofs to be made rigorous, and claimed that they were good for intuition. In other words infinitesmals were trying to replace something that had become established. The axioms behind the established thing are much weaker than the axioms required for infinitesmals. And infinitesmals provably don't let you prove anything fundamentally new. At that point any interest in infinitesmals has to come down to curiosity and the claim that it helps intuition. As for curiosity, I'm glad to have understood them. As for helping intuition, that is in the eye of the beholder, and has not been a compelling enough argument to get people to switch.

by JadeNB 5y ago

btilly 5y ago

Here is the thing. Robinson's construction required the axiom of choice and proved that every result that you can prove with infinitesmals has a proof with limits without the axiom of choice. But he went on to show that infinitesmals allowed abandoned proofs to be made rigorous, and claimed that they were good for intuition. In other words infinitesmals were trying to replace something that had become established. The axioms behind the established thing are much weaker than the axioms required for infinitesmals. And infinitesmals provably don't let you prove anything fundamentally new. At that point any interest in infinitesmals has to come down to curiosity and the claim that it helps intuition. As for curiosity, I'm glad to have understood them. As for helping intuition, that is in the eye of the beholder, and has not been a compelling enough argument to get people to switch.