Timeline for Which universities teach true infinitesimal calculus? [closed]
Current License: CC BY-SA 3.0
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| Dec 31, 2014 at 10:42 | history | closed |
Andy Putman Asaf Karagila♦ Alexandre Eremenko R W Neil Strickland |
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| Dec 30, 2014 at 16:10 | review | Close votes | |||
| Dec 31, 2014 at 10:42 | |||||
| Dec 30, 2014 at 16:05 | comment | added | Dima Pasechnik | @Mikhail, this looks way too general for the purpose of "smoothing out" real algebraic and semialgebraic sets. In the computable constructions there one anyway uses only $\epsilon^{p/q}$ for bounded $|p/q|$. | |
| Dec 30, 2014 at 15:35 | history | notice removed | François G. Dorais | ||
| Dec 30, 2014 at 15:20 | comment | added | Mikhail Katz | @Dima, here is what I found at wiki: "His foundational work on algebraic geometry is at a higher level of abstraction than all prior versions. He adapted the use of non-closed generic points, which led to the theory of schemes. He also pioneered the systematic use of nilpotents. As 'functions' these can take only the value 0, but they carry infinitesimal information, in purely algebraic settings." | |
| Dec 30, 2014 at 14:59 | comment | added | Dima Pasechnik | @Mikhail, it could be enough for our purposes, although I never heard about nilpotent infinitesimals. References? | |
| Dec 30, 2014 at 13:11 | comment | added | Mikhail Katz | @Dima, it has been my impression that in the context of algebraic geometry one doesn't need hyperreal infinitesimals and Grothendieck-style nilpotent infinitesimals are sufficient. Is this correct in the context of real algebraic geometry? | |
| Dec 30, 2014 at 12:57 | comment | added | Dima Pasechnik | @Mikhail, it's a common place in real algebraic geometry to work over the field of Puiseux series in an infinitesimal $\epsilon$ over $\mathbb{R}$ (or other real closed fields). Cf. e.g. perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html | |
| Dec 30, 2014 at 5:41 | comment | added | Mikhail Katz | @DimaPasechnik, I am not familiar with the literature using a hyperreal framework to do real algebraic geometry. What are you referring to exactly? | |
| Dec 29, 2014 at 18:40 | comment | added | Dima Pasechnik | @Mikhail : I always found it very tricky to apply the transfer principle in the setting of real algebraic geometry - IMHO it's easier in the setting of univariate calculus... Well, I am no Euler, and I very often got lost. | |
| Dec 29, 2014 at 17:48 | comment | added | Mikhail Katz | @Dima, that's precisely the point. When it comes to interpreting the procedures found in Euler, modern Weierstrassian frameworks are less suitable than modern infinitesimal frameworks, be it Robinson's, Lawvere's, or Bell's. Mostly scholars have been trying to understand Euler from the viewpoint of Weierstrassian frameworks, hence the routine claims of "non-rigorous" and even "dismal" as J. Gray put it. The source of the disdain for Euler lies in the inappropriateness of conceptual frameworks being applied to interpret his procedures. | |
| Dec 29, 2014 at 17:40 | comment | added | Dima Pasechnik | @Mikhail : don't you need some NSA to make proper sense of Euler's derivation of the infinite product formula for $\sin x$ ? | |
| Dec 29, 2014 at 16:26 | comment | added | Mikhail Katz | @Dima, Euler's computations are far more rigorous than they are reputed to be. See for example this text: ams.org/notices/201408/rnoti-p848.pdf | |
| Dec 29, 2014 at 9:34 | comment | added | Dima Pasechnik | @Mikhail: cf. en.wikipedia.org/wiki/Infinitesimal --- as well, I don't think that by "infinitesimal calculus" most people mean non-rigorous Euler-style computations. Call it "Transfer principle-based infinitesimal calculus" if you must. | |
| Dec 29, 2014 at 9:08 | comment | added | Mikhail Katz | @DanFox, to respond to your comment about "non-standard analysis", I would like to share with you the reaction of philosopher Salanskis to the term "non-standard" from 1988: S'il a donc choisi d'emblée le nom d'analyse non standard pour le jeu qu'il définissait, c'est en référence à la notion de modèle non standard plutôt que pour suggérer le caractère divergent, subversif ou extravagant de ce jeu. De nombreux esprits, même les meilleurs, s'y sont pourtant trompés, de manière regrettable." What Salanskis is pointing out is that the term "nonstandard" is misleading. | |
| Dec 29, 2014 at 9:05 | comment | added | Mikhail Katz | @Dima, the term "rigorous infinitesimal calculus" sounds precisely like epsilon-delta calculus, in other words the opposite of what is intended here. Similarly, "axiomatic infinitesimal calculus" sounds like you axiomatize the real numbers instead of building them, and then again do epsilon-delta. | |
| Dec 29, 2014 at 8:56 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 29, 2014 at 8:50 | history | notice added | Mikhail Katz | Draw attention | |
| S Dec 29, 2014 at 8:49 | history | bounty ended | Mikhail Katz | ||
| S Dec 29, 2014 at 8:49 | history | notice removed | Mikhail Katz | ||
| Dec 29, 2014 at 8:49 | vote | accept | Mikhail Katz | ||
| Dec 28, 2014 at 23:05 | answer | added | Bjørn Kjos-Hanssen | timeline score: 3 | |
| Dec 28, 2014 at 17:58 | comment | added | Dima Pasechnik | perhaps "rigorous" or "axiomatic" would be better word than "true". | |
| Dec 28, 2014 at 15:53 | comment | added | Dan Fox | @katz: Thanks for the clarification. Probably it is usually not a good practice to use "true" as part of mathematical terminology except in very particular contexts, for example expressly discussing the mathematical concept of truth. While one understands what is meant by "true infinitesimal", this terminology is not totally neutral. With a bit of effort one could surely find synonymous terminology less likely to provoke emotional reactions (perhaps "nonstandard analysis" was chosen partly with this sort of thinking in mind?) that only serve to confuse the discussion of the math. | |
| Dec 28, 2014 at 13:43 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 28, 2014 at 13:39 | comment | added | Pietro Majer | ( books.google.it/books/about/… ) | |
| Dec 28, 2014 at 13:36 | comment | added | Pietro Majer | Dear Katz, let me gather more information -I have no teaching experience about NSA, and no, I do not know Goldoni's book. I recall there has been a meeting on NSA in Pisa, around 2000, which included teaching experiences of it at high-school level. | |
| Dec 28, 2014 at 13:26 | comment | added | Mikhail Katz | @PietroMajer, since I see you are on the faculty at Pisa in Italy, could you comment on the recent interest in teaching calculus in highschool using infinitesimals? Are you familiar with Giorgio Goldoni's book? | |
| Dec 28, 2014 at 13:17 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 28, 2014 at 13:14 | comment | added | Mikhail Katz | @PietroMajer, nice to hear from you. Why don't you post this as an answer? | |
| Dec 28, 2014 at 13:13 | comment | added | Pietro Majer | @katz: Vieri Benci tought calculus for freshmen based on Non Standard Analysis in the spirit of Keisler book for several years (I guess 15 or so) at the Engeneering Faculty, Pisa. You may like to hear him for a feed-back. | |
| Dec 28, 2014 at 13:08 | comment | added | Mikhail Katz | @Dan, I added a note to the question to explain the use of the term. This does not mean that the approach limited purely to epsilon-delta and an Archimedean continuum is "false"; rather, it means that it does not use true infinitesimals. | |
| Dec 28, 2014 at 13:05 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 28, 2014 at 13:01 | comment | added | Dan Fox | @katz: the use of the word "true" suggests an implicit evaluation of relative merits. The question "Which schools, colleges, or universities teach calculus with infinitesimals?" - unadorned with the evaluative qualifier "true" - might receive a better response. | |
| Dec 28, 2014 at 12:20 | vote | accept | Mikhail Katz | ||
| Dec 28, 2014 at 12:20 | |||||
| Dec 28, 2014 at 12:20 | answer | added | Mikhail Katz | timeline score: 1 | |
| Dec 27, 2014 at 23:17 | comment | added | fedja | @katz Neither was it my intention (the main purpose of my comment was to prevent this thread from being closed), so let us stop that side discussion here (though I would gladly talk about it somewhere else) :-). | |
| Dec 27, 2014 at 19:48 | comment | added | Mikhail Katz | @fedja, it wasn't my intention to discuss the relative merits of "epsilon-delta language" as you put it versus the infinitesimal method, but since you brought it up I would mention that we do teach epsilon-delta once the students have understood the basic concepts of the calculus like derivative and continuity via infinitesimals. Once they understand the concepts, they already know what the epsilon-delta definitions are trying to say, and it becomes much easier to absorb the intimidating alternating quantifiers. In fact I taught epsilon-delta last week, both continuity and uniform continuity. | |
| Dec 26, 2014 at 0:20 | comment | added | fedja | 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. | |
| Dec 23, 2014 at 17:22 | comment | added | mme | There is also a(n unanswered) version on MSE. | |
| Dec 23, 2014 at 9:19 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 22, 2014 at 9:44 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 21, 2014 at 12:40 | comment | added | Gerald Edgar | @Willie: the question now has a bounty, which prevents it from begin closed! | |
| S Dec 21, 2014 at 10:12 | history | bounty started | Mikhail Katz | ||
| S Dec 21, 2014 at 10:12 | history | notice added | Mikhail Katz | Draw attention | |
| Dec 21, 2014 at 10:08 | history | reopened |
Mikhail Katz Andrey Rekalo Ramiro de la Vega Dan Petersen Fernando Muro |
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| Dec 21, 2014 at 9:53 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 20, 2014 at 20:09 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 18, 2014 at 16:49 | comment | added | Joseph O'Rourke | @katz: Yes, I understand. I was just notifying those who suggested it post on MESE that it already was posted there. | |
| Dec 18, 2014 at 16:40 | history | edited | Mikhail Katz |
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| Dec 18, 2014 at 16:06 | comment | added | Mikhail Katz | @Joseph, the exposure here is much broader and therefore there is a better chance of getting an answer. | |
| Dec 18, 2014 at 10:47 | comment | added | Joseph O'Rourke | A version of this question was posted at MESE on Dec 8th: link. Comments but no answers. | |
| Dec 18, 2014 at 10:03 | review | Reopen votes | |||
| Dec 18, 2014 at 18:24 | |||||
| Dec 18, 2014 at 9:45 | comment | added | Mikhail Katz | @Willie, this question is similar to other questions under the "teaching" tag, and should be within the scope. | |
| Dec 18, 2014 at 9:44 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 8, 2014 at 14:01 | history | closed |
abx Willie Wong Stefan Waldmann Stefan Kohl♦ paul garrett |
Not suitable for this site | |
| Dec 8, 2014 at 12:37 | comment | added | Willie Wong | I've voted to close as off-topic as this question belongs better at matheducators.stackexchange.com | |
| Dec 8, 2014 at 12:05 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Dec 8, 2014 at 11:28 | review | Close votes | |||
| Dec 8, 2014 at 14:02 | |||||
| Dec 8, 2014 at 11:22 | comment | added | Neil Strickland | This would be better at matheducators.stackexchange.com | |
| Dec 8, 2014 at 10:18 | history | asked | Mikhail Katz | CC BY-SA 3.0 |