Weierstrass' method for handling limits with the epsilon and delta symbols is very useful for rigorous analysis of math but it is terrible in terms of any intuitive approach to limits. There are are other ways to teach limits. For instance with infinitesimals yet I don't know if that has been simplified. Why does the 'epsilon- delta' method seem to dominate many calculus courses? Are there Calculus Courses that don't use this 'epsilon-delta' method?
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6$\begingroup$ Not sure why you apologize for spelling Weierstrass correctly. And roughly how many calculus courses have you looked at? Seems to me that either you use an actual definition, which is epsilon-delta, or you handwave it. (The idea of using nonstandard analysis when teaching calculus is unrealistic.) $\endgroup$– KCdMar 25, 2015 at 2:37
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4$\begingroup$ @KCd the name is spelled Weierstraß (in German), and Weierstrass is a transliteration. $\endgroup$– quid ♦Mar 25, 2015 at 12:08
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1$\begingroup$ Over the years I've looked at dozens of Calculus Textbooks. Has the epsilon-delta approach been improved in any way since Weierstrass wrote it ? $\endgroup$– 201044Mar 26, 2015 at 1:02
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$\begingroup$ @quid, the question was written in English, not German, and the name was spelled correctly in English. Apologizing in that context makes as much sense as apologizing for writing Chebotarev in English instead of Чеботарёв, i.e., it really is nothing to apologize for at all. $\endgroup$– KCdMar 26, 2015 at 2:03
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2$\begingroup$ @201044 David Bressoud gives a spirited take in Radical Approach to Real Analysis. He tells an engaging historical story with a zoo of examples exposing the subtleties before casting limit as "winning the epsilon-delta game" jstor.org/stable/20454048?seq=1#page_scan_tab_contents $\endgroup$– ConifoldMar 26, 2015 at 17:24
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2 Answers
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Teaching calculus using infinitesimals from nonstandard analysis has been tried, Keisler's text is considered a staple in this regard. But I am curious as to what is so "terrible" about epsilon-delta approach? Using infinitesimals invites us to envision a hidden world of numbers beyond the real numbers, into which all elementary functions miraculously extend, and their properties there are then mysteriously reflected back into the world of real numbers.
The intuitive idea of a limit, as taught to non-majors, is "$f(x)$ approaches $L$ as $x$ approaches $a$", epsilon and delta simply quantify what "approaches" means for $f(x)$ and $x$ respectively. Admittedly, epsilon and delta arguments are technical, but it does not mean that they are unintuitive. We know from the early history of calculus that the simplicity of intuitions about limits is misleading because those intuitions are self-contradictory. Resolving these contradictions inevitably leads to subtleties and technicalities, and refinement of intuition, under any systematic approach. At least, epsilon and delta does it without invoking extra phantom entities, that would have to be either explained or glossed over in addition to real numbers, so why is it worse than infinitesimals?
This is not to say that calculus with epsilon and delta can not be taught terribly, and that unfortunately it often is. But I am not sure that this is the fault of the approach rather than of the fact that there is a fundamental difficulty in the concept itself, as its 2000 year long genesis suggests. Instructors are often unable and/or unwilling to dedicate the time and the effort required to teach it properly. Especially since the algorithmic aspect of calculus is so much easier to teach, and can be taught while ignoring limits almost completely.
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6$\begingroup$ One could already skeptically claim that "the real numbers" is a hidden world beyond constructible numbers, and that the miraculous extension of our usual functions from rationals to reals already burdens the imagination... Competent intuitions about limits (e.g., Euler's) are not innately self-contradictory, insofar as they "got the right answer", even if the narrative was "not portable". The phantom entities in the delta-epsilon game are the quantifiers, not to mention the construction of the reals, either by cuts or Cauchy sequences. Mass of tradition is not really persuasive, per se. $\endgroup$ Mar 25, 2015 at 22:59
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$\begingroup$ I assumed a typical school background, with changes made in the calculus course only. Remaking the curriculum comprehensively based on infinitesimals is a different issue. It would be difficult in practical terms, especially because instructors familiar with the new setting will have to be produced en masse somehow. Of course, students are not really taught real numbers, they are simply made comfortable with the idea of decimal fractions extending indefinitely, and lines having no holes. But some similar "therapy" will have to be worked out and implemented for infinitesimals, starting early. $\endgroup$– ConifoldMar 26, 2015 at 1:12
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$\begingroup$ The case where | f(x) - f(a) | < e whenever | x-a| < d for any 'small e' ( e for epsilon ,d for delta) ; this might be replaced with '| f(x) - f(a)| < e whenever | x- a| < e ,for any small e. Would this work? $\endgroup$– 201044Aug 15, 2015 at 10:57
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1$\begingroup$ @201044: no, this would not work, not by a long shot, but comment are for commenting, not for discussions, so I will not detail here. $\endgroup$ Apr 21, 2017 at 7:44
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The idea that historical infinitesimals were self-contradictory is prevalent among historians (see e.g., the accepted answer above) and also many mathematicians, but it has been challenged in the recent literature; see e.g., this article in Erkenntnis as well as its MathSciNet review. Furthermore this recent study presents the results of a questionnaire that indicate that students themselves overwhelmingly prefer infinitesimal definitions a la Cauchy of basic calculus concepts like continuity and derivative. The reason epsilon-delta paraphrases still dominate in teaching are due to a history of prejudice.