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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 recently told me about a conference 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. On Gerald's suggestion, also of interest would be any educational studies comparing the two approaches (the one using infinitesimals and the one using epsilon, delta). That is, in addition to the study Sullivan, Kathleen; Mathematical Education: The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach. Amer. Math. Monthly 83 (1976), no. 5, 370–375.

Mikhail Katz
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  • Are your infinitesimals nilpotent, or invertible? Are they differential forms? Would any/all of these approaches qualify as "true infinitesimal calculus" for you? – Steven Gubkin Dec 08 '14 at 15:16
  • @StevenGubkin, Certainly infinitesimals whether nilpotent or invertible could be called TIC. Are you aware of any undergraduate courses using Lawvere/Kock/Bell? I have taught differential forms but I never thought of them as infinitesimals. I agree with you that there are similarities in notation, but if you look at Spivak's book you will see that he clearly distinguishes between the two in his historical section, and seeks to translate the infinitesimal arguments as found in Riemann and Gauss into modern techniques exploiting symmetric and antisymmetric forms. – Mikhail Katz Dec 08 '14 at 15:21
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    I don't know enough about this topic to be of any help, but you might be interested in a personal recollection I posted regarding a disagreement I witnessed (around 1980, at Duke University) between Keisler and Joseph R. Shoenfield (the logician) about the merits of Keisler's method of teaching calculus. – Dave L Renfro Dec 08 '14 at 17:00
  • @DaveLRenfro, thanks. It would be interesting to see some details on Shoenfield's critique, though I acknowledge it might be a bit late :-) One thing I do notice is that Shoenfield's criticism seems to be the diametrical opposite of Errett Bishop's. Namely, Schoenfield criticizes Keisler for watering down Robinson's mathematics too much. By contrast, Bishop thought Keisler's calculus stuff was way too hifalutin', see http://dx.doi.org/10.1007/s10699-013-9340-0 – Mikhail Katz Dec 08 '14 at 17:30
  • I don't remember enough to say much more than I did. However, I think Shoenfield's criticism was more along the lines of something like "either do it correctly (which isn't possible at this level) or stop pretending that what you're doing is correct". My impression is also that Shoenfield felt this was more of a teaching fad being promoted by the converted than something that is actually better (for teaching purposes) than existing methods. I also got the impression that other math reform battles had been previously fought by this same crowd in attendance, perhaps calculators in the classroom. – Dave L Renfro Dec 08 '14 at 18:55
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    A bit off topic, but, I wonder how many of us teach the ordinary epsilon-delta calculus "correctly". I don't know of anyone who bothers to rigorously treat the integral. Who proves half the basic continuity material in calculus I ? Not likely in this age of retention and assessment. – James S. Cook Dec 09 '14 at 04:06
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    I would hope that the answer would be that at 100% of all schools worldwide, when students learn calculus, they learn that they can think of dy and dx as very small numbers. I would also hope that 100% of calc courses covered limits. The extent to which the "very small numbers" are formalized is of course going to be a matter of taste. (Personally, I just say that dx's and dy's obey the elementary axioms of the reals.) One should also be realistic about the average freshman calc student's level of interest in foundational issues, which is zero. –  Dec 09 '14 at 04:53
  • @DaveLRenfro, thanks for the clarification. Looking at the mathforum discussion "An educational sham from 1890" that you linked, I read many of the messages but I am still not sure what the context of that mathforum discussion was. Was Shoenfield (and others) criticizing Keisler for perpetrating an alleged "education sham" or was that only a side issue? – Mikhail Katz Dec 09 '14 at 08:40
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    @JamesS.Cook, thanks for your comment. The subject of epsilon-delta and how exactly it manifests itself in Calculus 1 is certainly an interesting subject but I hesitate to get into this here as this was not the subject of this question. Perhaps elsewhere? – Mikhail Katz Dec 09 '14 at 10:23
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    @BenCrowell, thanks for your comment. I wouldn't necessarily underestimate the curiosity of calculus students. Here is a recent question I got in class after I defined continuity at a point (naturally, using Cauchy's definition as inspiration). A student asked: when we talk about continuity and continuum, things should be happening on an appreciable interval, rather than "at a point". Does it make sense to define continuity "at a point"? I was happy to tell her that Cauchy's viewpoint in 1821 was close to what she suggested. – Mikhail Katz Dec 09 '14 at 10:26
  • That Math Forum discussion was a thread I started (first post explains), and the title I used was intended to be tongue-and-cheek -- similar to words and phrases that one of the posters often uses in criticizing educational reform ideas. Someone brought up Keisler, and I happened to remember that long ago incident, so I thought it might be of interest to some readers (who likely far out-number the actual posters). So yes, it was a side issue. In fact, the entire thread was for me a side issue, although the usual degeneration by proponents of reform and those against reform began to take over. – Dave L Renfro Dec 09 '14 at 20:02
  • @Dave, thanks, I understand. You mentioned the "education scam" thing because you thought it might be relevant to Keisler's education ideas. In which direction would you have liked the discussion to develop if it were not to degenerate into a take-over by proponents of reform versus those opposing it? – Mikhail Katz Dec 10 '14 at 14:43
  • Not an answer to your question, but there is some interesting information in the recent Handbook on the History of Mathematics Education (https://books.google.com/books?id=MYy9BAAAQBAJ&q=infinitesimal#v=snippet&q=infinitesimal&f=false) around the inclusion of infinitesimal calculus in Spanish curricula in the 18th century (and, to some extent, in Italy and France). – Benjamin Dickman Dec 10 '14 at 23:42
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    In an education forum, maybe also appropriate would be to ask for published studies, where the two methods were used for instruction, comparing the outcomes. – Gerald Edgar Dec 11 '14 at 22:55
  • @Gerald, good idea. – Mikhail Katz Dec 12 '14 at 08:20
  • What is intended and what is gained by using the word "true" in the name of the course? – Dan Fox Dec 21 '14 at 15:32
  • @Dan, usually nowadays the term "infinitesimal calculus" is used as a dead metaphor for "the calculus". This means that calculus courses routinely go under the name "infinitesimal calculus" for historical reasons, whereas in point of fact no trace of an infinitesimal ever appears on the blackboard. When I refer to "true infinitesimal calculus" I mean calculus with infinitesimals, as explained in the question. – Mikhail Katz Dec 22 '14 at 09:41
  • Cross-posted on MathOverflow: http://mathoverflow.net/questions/189183/which-universities-teach-true-infinitesimal-calculus – Martin Dec 23 '14 at 10:28

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I believe St. Johns college does. They are a liberal arts school that focuses on historical texts. From their sylabus:

Junior Mathematics: Calculus and its Foundations Junior Mathematics concerns itself with questions about the continuity of motion, the infinite, and the infinitesimal, which lead to a new form of mathematics, the calculus. The initial sequence of readings (Aristotle, Galileo, and Leibniz) leads to the primary text, Newton’s Principia, which offers a sweeping vision of the mechanical motions of the universe. The year concludes with Dedekind’s Essays on the Theory of Numbers, which attempts to establish the continuity of number and prompts students to revisit questions about the nature of the infinite and the infinitesimal. Some classes continue this inquiry with a brief study of Cantor at the close of the second semester.

Also you should probably come up with a name other than "true infinitesimal calculus." It has the stench of "No true Scotsman". Perhaps reference it after the scholar that popularized it.

Greg
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This does not directly address the question about where nonstandard-analysis-based Calculus is taught, nor the request for educational studies comparing the two approaches, but it is one of the few (maybe only?) studies that take an epistemological perspective on nonstandard analysis:

Ely, Robert. (2010) Nonstandard student conceptions about infinitesimals. Journal for Research in Mathematics Education 41 (2), 117-146.

From the abstract:

This is a case study of an undergraduate calculus student’s nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson’s nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system. This provides a new perspective on students’ “struggles” with the real numbers, and adds to the discussion about the relationship between student conceptions and historical conceptions by focusing on mechanisms for maintaining cognitive and mathematical consistency.

Article available at http://www.jstor.org/stable/20720128.

mweiss
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Calculus and Differential Equations for Biology (MATH 1251 and sequels) at Northeastern University in Boston, coordinated by Dr. Samuel Blank, are taught using infinitesimals and infinite numbers rather than limits. The textbook in use, however, is a standard limits-based textbook, and is only loosely followed. The material on infinitesimals is provided in online notes.

Nick Matteo
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