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My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by Keisler https://www.math.wisc.edu/~keisler/calc.html. Two of my colleagues in Belgium are similarly teaching TIC at two universities there. I am also aware of such teaching going on in France in the Strasbourg area, based on Edward Nelson's approach, though I don't have any details on that.

Which schools, colleges, or universities teach true infinitesimal calculus?

A colleague in Italy has told me about a conference a few months ago on using infinitesimals in teaching in Italian highschools. This NSA (nonstandard analysis) conference was apparently well attended (over 100 teachers showed up).

In Geneva, there are two highschools that have been teaching calculus using ultrasmall numbers for the past 10 years.

Anybody with more information about this (who to contact, what the current status of the proposal is, etc.) is hereby requested to provide such information here.

Note 1 in response to Dan's comment: usually nowadays the term infinitesimal calculus is used as a dead metaphor for "the calculus". Thus, calculus courses routinely go under the name "infinitesimal calculus" for historical reasons, whereas in point of fact no infinitesimal ever appears on the blackboard. When I refer to "true infinitesimal calculus" I mean calculus with infinitesimals (as explained above) as opposed to ordinary "infinitesimal calculus" as found in Thomas-Finney and other textbooks.

Note 2 in response to Pietro's comment: I should clarify that calculus using infinitesimals is not limited to Keisler's book. There are several books of this sort, including some that are unrelated to Robinson's theory, such as those by A. Kock and J. Bell.

Note 3: thanks to Bjørn Kjos-Hanssen for the update on the history of TIC teaching at the University of Hawaii.

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    $\begingroup$ This would be better at matheducators.stackexchange.com $\endgroup$
    – Neil Strickland
    Commented Dec 8, 2014 at 11:22
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    $\begingroup$ I've voted to close as off-topic as this question belongs better at matheducators.stackexchange.com $\endgroup$
    – Willie Wong
    Commented Dec 8, 2014 at 12:37
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    $\begingroup$ @Willie, this question is similar to other questions under the "teaching" tag, and should be within the scope. $\endgroup$
    – Mikhail Katz
    Commented Dec 18, 2014 at 9:45
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    $\begingroup$ A version of this question was posted at MESE on Dec 8th: link. Comments but no answers. $\endgroup$
    – Joseph O'Rourke
    Commented Dec 18, 2014 at 10:47
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    $\begingroup$ The obvious observation is: "Very few places do", so, definitely, it makes more sense to ask this question on the professional mathematician forum than on the general teaching forum (just to throw my weight to keep it open here). I would never do it in the undergraduate curriculum myself (unless the standard epsilon-delta language is taught in parallel) for the simple reason that very few texts in analysis are written in this language, so, alas, I don't know much as far as the main question is concerned. $\endgroup$
    – fedja
    Commented Dec 26, 2014 at 0:20

2 Answers 2

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Since you mentioned University of Hawaii --

David Ross regularly teaches nonstandard analysis and infinitesimal calculus at University of Hawaii in various forms, for instance

  • MATH 649K (a regular grad course), Spring 2008

  • MATH 699 (reading course), Spring 2011

We also usually cover the topic in our senior-level course on mathematical logic (MATH 455), although I suppose a lot of universities do that.

But infinitesimal calculus is not part of our standard calculus syllabus.

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So far I have been able to find out that true infinitesimal calculus was taught at the following universities:

University of Hawaii;

University of Illinois at Urbana-Champaign;

University of Iowa;

University of Wisconsin at Madison.

Anyone who has either taught or studied in such courses there is requested to share his or her experiences.

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  • $\begingroup$ news.slashdot.org/comments.pl?sid=99232&cid=8500301 $\endgroup$
    – user21349
    Commented Dec 29, 2014 at 2:13
  • $\begingroup$ Thanks, @BenCrowell. I found the following comment there: "I find it disheartening that the book was allowed to go out of print, and that there are now no (as far as I'm aware) current popular calculus texts using the infinitesimal-based approach. I, like the original poster, and like most students learning today, was always confused by what you could and couldn't do with dy and dx. How I wish I'd had this book 20 years ago." It is nice to see such enthusiasm about Keisler's book but one correction: the book is currently in print at Dover. Furthermore, there are today numerous TIC textbooks. $\endgroup$
    – Mikhail Katz
    Commented Dec 29, 2014 at 8:54

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