Timeline for Which universities teach true infinitesimal calculus?
Current License: CC BY-SA 3.0
32 events
| when toggle format | what | by | license | comment | |
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| Apr 20, 2017 at 13:33 | history | edited | Mikhail Katz | edited tags | |
| Nov 17, 2015 at 23:39 | answer | added | Nick Matteo | timeline score: 2 | |
| Nov 4, 2015 at 19:40 | answer | added | mweiss | timeline score: 4 | |
| Nov 4, 2015 at 18:17 | answer | added | Greg | timeline score: 5 | |
| Dec 23, 2014 at 10:28 | comment | added | Martin | Cross-posted on MathOverflow: mathoverflow.net/questions/189183/… | |
| Dec 23, 2014 at 9:20 | history | edited | Mikhail Katz | CC BY-SA 3.0 | pruning |
| Dec 22, 2014 at 9:41 | comment | added | Mikhail Katz | @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. | |
| Dec 21, 2014 at 15:32 | comment | added | Dan Fox | What is intended and what is gained by using the word "true" in the name of the course? | |
| S Dec 18, 2014 at 16:19 | history | bounty ended | CommunityBot | ||
| S Dec 18, 2014 at 16:19 | history | notice removed | CommunityBot | ||
| Dec 17, 2014 at 17:57 | history | edited | Mikhail Katz | CC BY-SA 3.0 | added 391 characters in body |
| Dec 14, 2014 at 16:14 | history | edited | Mikhail Katz | CC BY-SA 3.0 | added 46 characters in body |
| Dec 12, 2014 at 8:22 | history | edited | Mikhail Katz | CC BY-SA 3.0 | added 179 characters in body |
| Dec 12, 2014 at 8:20 | comment | added | Mikhail Katz | @Gerald, good idea. | |
| Dec 11, 2014 at 22:55 | comment | added | Gerald Edgar | 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 10, 2014 at 23:42 | comment | added | Benjamin Dickman | Not an answer to your question, but there is some interesting information in the recent Handbook on the History of Mathematics Education (books.google.com/…) around the inclusion of infinitesimal calculus in Spanish curricula in the 18th century (and, to some extent, in Italy and France). | |
| Dec 10, 2014 at 15:13 | history | edited | Mikhail Katz | edited tags | |
| S Dec 10, 2014 at 15:12 | history | bounty started | Mikhail Katz | ||
| S Dec 10, 2014 at 15:12 | history | notice added | Mikhail Katz | Draw attention | |
| Dec 10, 2014 at 14:43 | comment | added | Mikhail Katz | @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? | |
| Dec 9, 2014 at 20:02 | comment | added | Dave L Renfro | 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. | |
| Dec 9, 2014 at 10:26 | comment | added | Mikhail Katz | @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 10:23 | comment | added | Mikhail Katz | @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? | |
| Dec 9, 2014 at 8:40 | comment | added | Mikhail Katz | @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? | |
| Dec 9, 2014 at 4:53 | comment | added | user507 | 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 9, 2014 at 4:06 | comment | added | James S. Cook | 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 8, 2014 at 18:55 | comment | added | Dave L Renfro | 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. | |
| Dec 8, 2014 at 17:30 | comment | added | Mikhail Katz | @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 dx.doi.org/10.1007/s10699-013-9340-0 | |
| Dec 8, 2014 at 17:00 | comment | added | Dave L Renfro | 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 15:21 | comment | added | Mikhail Katz | @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. | |
| Dec 8, 2014 at 15:16 | comment | added | Steven Gubkin | Are your infinitesimals nilpotent, or invertible? Are they differential forms? Would any/all of these approaches qualify as "true infinitesimal calculus" for you? | |
| Dec 8, 2014 at 15:01 | history | asked | Mikhail Katz | CC BY-SA 3.0 |