Why isn't this approach to calculus more common? Does it lead to more complex proofs or have any serious drawback?

I understand the ε,δ way of defining the limit is important because it extends to other context such as continuity in topological spaces etc; however, as a physicist, infinitesimal quantities reflect the way we really think about calculus intuitively, so it makes sense make them "first-class" numbers.

From what I read on the subject, this _was_ the main way of dealing with calculus, but it was not rigorous. The first rigorous definition of a limit was given using the epsilon-delta approach, by Weierstrass, with of course many many contributions by his predecessors.

Everybody switched during the 19th century, because epsilon-delta was in fact the only rigorous method.

Abraham Robinson gave a rigorous foundation for infinitesimals only in 1960.

Work done by Schmieden&Laugwitz in the 1950s helped to formalize and retroactively apply rigor to the earlier uses of infinitesimals. Their Omega Calculus approach was also constructive.

cf. Detlef Laugwitz, Curt Schmieden's Approach to Infinitesimals. https://doi.org/10.1007/978-94-015-9757-9_12

However this turned out less useful going forward.

Sure - this is a great way to think about calculus intuitively, but imprecisely; rigorous, mathematical proofs based on the hyperreals, on the other hand, may become unwieldy, and, as you rightly noted, they do not extend to the modern treatments of, say, differential geometry which, too, plays a huge role, as a framework, in the modern theoretical physics.

> 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.)

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.

> 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.

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.

One factor I've heard cited is that if you can prove something in calculus using nonstandard analysis, you can always prove the same thing with standard analysis.

If we switched to teaching calculus based on nonstandard analysis we'd have to also teach the standard approach because of all the existing material that uses it and all the people who already know the standard approach but not the nonstandard approach.

Not many people are willing to commit to a few generations of teaching dual approaches and using both in their work until the people who only know standard are all dead or retired and all the old material has either been translated or is obsolete, when the advantage of the nonstandard approach is just that it might be conceptually easier or more intuitive.

Learning math isn't just about solving problems. It's about solving problems in ways that other people can follow, evaluate, and understand. Since nearly no one knows nonstandard analysis, but most people with college mathematics know standard calculus, there is much less utility in learning nonstandard analysis. It might make it easier for you to solve certain problems, but no one will understand what you did.

If we just do what our ancestors did because its what everyone already understand, we'd never improve.

The true telos of a mathematics education is gaining a genuine understanding of the principles of mathematics. Pragmatic concerns of the nature you describe should be treated as secondary concerns in an educational environment.

We learnt some non-standard in a logic class (as a capstone application of FOL and model theory).

Perhaps there’s a perceived notion that to understand infinitesimals and such you need a logic background, but logic classes are generally an upper-half class that doesn’t have a full track for undergrads. This would be in addition to the fact that most professors are familiar with ε,δ-proofs and AP calculus classes cover the limit approach (though not rigorously).

The entrenched pipeline of mathematics students and professors would have to face a period of getting flushed out and refitted, and this is probably not attractive to administrators.