# Which universities teach true infinitesimal calculus?

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.

• 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. Dec 8, 2014 at 17:00
• 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. Dec 9, 2014 at 4:06
• 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.
– user507
Dec 9, 2014 at 4:53
• @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. Dec 9, 2014 at 10:26
• 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. Dec 11, 2014 at 22:55

## 3 Answers

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.

• Thanks @Greg and I tried to reach somebody at St. Johns but was unable to. Dec 7, 2015 at 14:05

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.

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.

• Thanks, @kundor and I would be interested in getting in touch with Blank. Dec 7, 2015 at 14:04