A colleague of mine asked an interesting question reproduced below with his permission.

It is reasonable to ask whether it is to the students' advantage to learn the language of infinitesimals - even if one accepts that, all things being equal, it is an easier language to learn - when the vast majority of the mathematical and scientific community still speak in the "Weierstrassian" language. It is possible to teach them both, but this may come at a certain expense.

When one wonders whether or not to teach calculus a la Keisler, it is not because it is or isn't the right to do, but rather a matter of personal preference. A typical mathematician grew up with Weierstrass and Cauchy, feels at home with standard analysis, and is comfortable teaching it. He knows where the pitfalls are, where to appeal to intuition and where to warn students to be wary, the right examples to grab onto for support, etc.

He doesn't have the same comfort level with infinitesimals, because of how he learned to think about things and look at them, and therefore could not convey it properly to a classroom, certainly not with the same confidence and enthusiasm, that is important in these classes. Given enough time, he could probably gain that same level of comfort with infinitesimals, but it would take a lot more time and effort than he is willing to take on.

So to summarize, is it really to the students' advantage to learn the language of infinitesimals?

What is requested is a reasoned response (based on reliable sources rather than personal opinions) on (1) historical, (2) mathematical, and (3) philosophical aspects of the question.

This question was posed a while ago at MSE where some of the editors felt that ME is more appropriate.

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    $\begingroup$ As a point of information, one should point out that the poster (not the questioner, presumably) is a noted advocate for reintroducing infinitesimals. Which hopefully will encourage thoughtful responses to his colleague's question! $\endgroup$ – kcrisman Jul 3 '17 at 14:28
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    $\begingroup$ As to the question itself, I think it is quite analogous to e.g. using a new technology or "active learning" or service-learning in a course - all of which, for many, "would take a lot more time and effort" than many are able to grant it, especially with higher teaching loads. $\endgroup$ – kcrisman Jul 3 '17 at 14:30
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    $\begingroup$ Karl-Dieter, thanks for your comments. Note that the only "new technology" that one needs here is Keisler's infinitesimal microscope. I believe the latter is not patented and furthermore the cost is infinitesimal. $\endgroup$ – Mikhail Katz Jul 3 '17 at 14:37
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    $\begingroup$ Haha, yes! My point was only that this question is analogous to the discussions I have heard surrounding these other things, and so the question is quite pertinent here. $\endgroup$ – kcrisman Jul 3 '17 at 14:49
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    $\begingroup$ Thanks for this. Are they interested in teaching infinitesimals at Gordon? I could explain how this is done. $\endgroup$ – Mikhail Katz Jul 3 '17 at 14:51

This is an interesting question...

I think there is a volatile bifurcation at the very outset: certainly students who will (one way or another) be filtered/tested on the Cauchy-Weierstrass viewpoint would not immediately benefit from being made aware that there were other viable viewpoints. "Might confuse them"?!?

Yet, ironically, many physics and engineering groups (in my direct observation) say that they want "rigor", but in fact operate in the happy, optimistic world where there are infinitesimals. I think that what it amounts to is that there is filtering not on the literal mathematical content, but on ... well, obedience? Assimilation of cultural mythology?

So, mixed conclusion at this point. It is entirely plausible to me that we (collectively) could arrange that students be better able to answer operational questions (rather than questions intended to gauge their orthodoxy) by using infinitesimals. If I had to bet, I'd wager that epsilon-delta is just tooooo top-heavy to survive. That is, it was not inevitable, and is not the unique "foundationalization" of calculus... supposing that we needed a foundation (Bishop Berkeley nevertheless...)

It seems to me possible that there would be almost-completely-good reasons to introduce non-standard analysis at a graduate-math level, as a counterpoint to the usual "real analysis", and as a counterpoint to (often too-cliched) "functional analysis". Indeed, by this point, the people might be mature enough to appreciate that there truly is no mandate to "go Cauchy-Weierstrass"... even while one is "doing mathematics".

In contrast, I'd worry that 18-year-olds might not care to, or be able to, appreciate the nuances of such a discussion. Perhaps I'm too pessimistic.

A last comment: apart from orthodox math people, no one will ever object to reasoning based on infinitesimals. :) It's more persuasive than the epsilon-delta thang. Duh. :)

Edit: in regard to @DSF's and @BenCrowell's comments: yes, it is certainly possible to rigorously justify manipulation of infinitesimals, but that's not literally what people do, operationally. Rather, in my observation, they learn some "safe" manipulations, and mostly can avoid trouble by only doing the "safe" things... without necessarily knowing why they are safe.

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    $\begingroup$ Reading Cauchy's biography illuminates this discussion quite a bit - apparently the debate over how to "do this" was pretty hot at the École 200 years ago :) $\endgroup$ – kcrisman Jul 4 '17 at 2:21
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    $\begingroup$ say that they want "rigor", but in fact operate in the happy, optimistic world where there are infinitesimals Why is this a "but?" There is nothing inherently nonrigorous about infinitesimals. $\endgroup$ – Ben Crowell Jul 4 '17 at 4:08
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    $\begingroup$ @BenCrowell I think the point is they work with them non-rigorously. It seems to me much easier to slip into canceling dx's and dy's without any thought on whether this is allowed than doing the same thing with limits. But I admit being I'm biased by my education. $\endgroup$ – DRF Jul 4 '17 at 11:02
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    $\begingroup$ @DRF: Canceling infinitesimals is rigorously well justified because of the transfer principle. $\endgroup$ – Ben Crowell Jul 4 '17 at 14:32
  • $\begingroup$ in my observation, they learn some "safe" manipulations, and mostly can avoid trouble by only doing the "safe" things... without necessarily knowing why they are safe. Both historically and educationally, the quoted material could be an equally accurate observation about the plain old real number system. I've never met a college freshman who could give a coherent explanation of the completeness property of the reals. The better 50% of them know a set of safe ways of manipulating the reals, which consist of exploiting the properties that they possess in first-order logic only. $\endgroup$ – Ben Crowell Jul 4 '17 at 22:40

Yes, Karl-Dieter is referring to myself and a colleague. My current long-term project is to produce a modern full-color calculus textbook using infinitesimals, making use of my own definitions, notation, etc. I currently plan to classroom-test the first-semester portion of the book this fall, although I have already taught the infinitesimal material in all three semesters of calculus (anywhere from two to five semesters each). In regards to the original question, my plan is to make the text as easy to use as possible by instructors unfamiliar with the methods, mostly through ample comments in the margins.

In many ways, the text I am writing is similar in scope to Stewart and other popular calculus textbooks. The same topics are covered in a very similar order, with the main difference being the use of infinitesimals. I have incorporated a few other differences from a pedagogical standpoint, based upon decades of experience with algebraically-weak modern calculus students.

This is a labor of love and I have not yet spent much time seeking a publisher. As I am at a heavy teaching load university myself, it will take a few more years to finish.

  • $\begingroup$ I don't understand how this addresses the question. $\endgroup$ – Jessica B Jul 5 '17 at 6:17
  • $\begingroup$ Bryan, you seem to imply that the answer to the question is affirmative. Perhaps you could make this more explicit in your answer and also explain why this is so, as per @JessicaB 's concern. $\endgroup$ – Mikhail Katz Jul 6 '17 at 6:55

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