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.

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    $\begingroup$ This might be more appropriate on mathematics educators SE. $\endgroup$ Feb 10 '16 at 9:43
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    $\begingroup$ I support the comment of @BrandonThomasVanOver $\endgroup$
    – user65203
    Feb 10 '16 at 9:52
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    $\begingroup$ While this post is interesting and erudite, the subject matter is manifestly pedagogical. The community at Mathematics Educators SE has nine Questions about infinitesimals, all answered (and most with Accepted Answers). $\endgroup$
    – hardmath
    Feb 10 '16 at 14:49
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    $\begingroup$ @user72694: As Math.SE seeks to curate excellent content in the form of questions with answers based on reasoned mathematical argument, I apply the criteria of Good Subjective, Bad Subjective in voting "primarily opinion-based" in relation to Math.SE. This is entirely consistent with another SE community having "the field of teaching mathematics" base their criteria differently. $\endgroup$
    – hardmath
    Feb 10 '16 at 15:11
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    $\begingroup$ Reopening appeals do not belong in the question. You can post in the meta thread (but wait to give the review queue some time first). $\endgroup$ Feb 20 '16 at 19:57

Yes, it is to the students' advantage to learn about infinitesimals. But Keisler is not the royal road. Nelson's approach in his Radically Elementary Probability Theory is much better.

  • $\begingroup$ Thanks for this. Would you elaborate as to my colleague's concerns? $\endgroup$ Feb 10 '16 at 15:16
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    $\begingroup$ Just as an aside: I think "royal road" has the right connotations in English here and not "king's road" (which to any British English speaker is a street name). $\endgroup$
    – Rob Arthan
    Feb 20 '16 at 21:51

My colleague's thoughtful question is a stimulating examination entertaining the possibility that the infinitesimal approach might be better at least in principle. I would like to respond separately to its historical, mathematical, and philosophical components.

(1) The historical component leaves to be desired. In fact it is a good gauge of received views, because it lumps Cauchy with Weierstrass. Such a view implies that they had similar attitudes toward foundations. This is incorrect since Cauchy routinely used infinitesimals in his work, both in textbooks and research articles. The common view that Cauchy gave an epsilon-delta definition of continuity is a misconception that has been analyzed in the literature; see e.g., this article.

(2) Mathematically speaking, one needs to acknowledge that there is a grain of truth in the comment that this approach constitutes a departure from hallowed tradition. However, there is also a conflation of two separate issues with regard to using infinitesimals: (a) a social problem with regard to the students' potential future colleagues, and (b) a stigma involved.

As far as (a) is concerned, I would answer that teaching students an additional technique cannot possibly place them at a disadvantage and can possibly place them at an advantage. Evidence for this is provided by the fact that often mathematicians get further than their colleagues through intuitions they developed while learning physics, which bestows an advantage in math research.

A shining example of this is Seiberg-Witten theory. This article discusses a short proof of Donaldson's famous diagonalisation theorem that he proved using difficult gauge theory. This is one of the results Donaldson got his Fields medal for. The result turned out to be almost trivial using Witten's approach that was motivated by the physics Witten was doing.

As far as issue (b) is concerned, it is true that some mathematicians tend to react with puzzlement when they hear about Robinson's framework. However, I feel this way of objecting to infinitesimals is several years behind the times and is essentially untenable today now that Terry Tao published a book exploiting ultraproducts to prove Hilbert's fifth problem and other theorems.

(3) Philosophically speaking, at least some of the opposition appears to stem from a (stated or unstated) commitment to the Cantor-Dedekind postulate. The Cantor-Dedekind postulate identifies the line in physical space with the real line. This is a philosophical presupposition that can and has been challenged. Thus, Keisler has written that there is no more reason to assume an identification of the line in physical space with the real line than to assume its identification with the hyperreal line. In both cases we don't, and can't in principle have, any concrete evidence to support such a postulate.

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    $\begingroup$ You have now answered your own Question. Although the points you make are interesting to someone who saw nonstandard analysis qua Robinson et al as an undergraduate, it confirms my impression that you are looking to debate rather than to learn. $\endgroup$
    – hardmath
    Feb 18 '16 at 11:53
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    $\begingroup$ @hardmath, accusing an editor of "debating" is not a very charitable interpretation of a question. My colleague's question is a different type of criticism from what I have heard before and I was hoping to elicit some thoughtful reactions from fellow editors. Notice that I organized my thoughts on the matter five days after posting the question. There was one additional reaction but it was brief. I hope other editors join in to clarify the picture. $\endgroup$ Feb 18 '16 at 16:25
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    $\begingroup$ My attention was brought to this question, which I think is a good question, and maybe a bit too technical for Math Educators SE, ... by chance on an afternoon when I'm writing up a note about "point-scatterers of infinitesimal range" as providing precedent/intuition (1931 quantum mechanics) for some subtle possibilities in automorphic forms and number theory... I did vote to re-open. $\endgroup$ Feb 20 '16 at 20:00

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