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> Why isn't this approach to calculus more common? Does it lead to more complex proofs or have any serious drawback? Simply put, because there aren't more text

by JadeNB 5y ago

> Why isn't this approach to calculus more common? Does it lead to more complex proofs or have any serious drawback? Simply put, because there aren't more textbooks for it and because professors aren't as familiar with it, leading to a vicious circle where it's not taught because it's not as easy as teaching the standard way, and then it's not routinely learned because it's not taught …. (The advanced logic that goes into the underlying constructions, but that isn't necessary to use non-standard analysis, also causes many non-logician mathmaticians to give it an unjustified bit of side-eye.)

btilly 5y ago
Unjustified bit of side-eye??? 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? I personally like understanding infinitesmals, and knowing what the d in dy/dx means. And knowing that the notation for a second derivative, d^2y / dx^2, is not just arbitrary. This occasionally has uses. For example if you have implemented a numerical d function, for example as lambda x: f(x+h/2) - f(x-h/2), then d(d(y))/d(x)^2 is an excellent approximation to the second derivative. But conceptually I think it is far better to understand approximation a lot more directly than using a complex construction for the infinitesmals.
- 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

Unjustified bit of side-eye??? 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? I personally like understanding infinitesmals, and knowing what the d in dy/dx means. And knowing that the notation for a second derivative, d^2y / dx^2, is not just arbitrary. This occasionally has uses. For example if you have implemented a numerical d function, for example as lambda x: f(x+h/2) - f(x-h/2), then d(d(y))/d(x)^2 is an excellent approximation to the second derivative. But conceptually I think it is far better to understand approximation a lot more directly than using a complex construction for the infinitesmals.

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.