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I have been reading some non-standard analysis from Keisler's book and I think it is logically consistent till now but there are criticisms against it and why isn't non-standard analysis accepted more widely, the whole book is almost similar to the concept of limit. I have read Apostol's books earlier . Then why aren't we taught only the infinitesimal calculus only and what is special about the $\epsilon - \delta$ approach ?

Mikhail Katz
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Isomorphic
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  • In my high school, the teachers were already experimenting these methods (called infinitesimal 'tiny numbers'), some of my schoolmates met analysis first time like that. – Berci Dec 31 '13 at 11:48
  • There are textbooks that teach it that way. The same theorems are true, but having learned the infinitesmal approach means that later analysis which is conceptualized in the $\delta-\epsilon$ format will be harder to follow. The goal of a teacher is to enculturate the students. It would be sort of unfair to mislead them by teaching the material from an obscure viewpoint. – Charlie Frohman Dec 31 '13 at 13:34
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    @user90375 He has actually rigourised the material. – Isomorphic Dec 31 '13 at 13:59
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    It's been known to be rigorous for at least 40 years. Twice I have taught it that way. First in the early 80's at a small college that was using it, and once in the 90's at a large state university. After deep introspection, I rejected it as bad pedagogy, as it is unfair to beginners to make their road to learning unnecessarily longer. It's fine if someone chooses that for themselves. The intuition the viewpoint encompasses is good. It doesn't belong in the classroom, at this time. – Charlie Frohman Dec 31 '13 at 15:02
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    @Iota : I cannot back this up with data but I heard that many experiments to teach beginning calculus students using infinitesimals were unsuccessful. I'd appreciate it if anyone can back this up or contradict it. Personally I don't find the idea of an infinitesimal number very intuitive and I hope no one ever expects me to teach undergraduates, many of whom cannot find the square root of $1/4$, how to work with them. From what I have seen, working with infinitesimals rigorously requires a fair knowledge of set theory and logic, whereas the traditional epsilon - delta approach... – Stefan Smith Dec 31 '13 at 17:17
  • ...only requires some use of quantifiers and "common-sense" type logic (as opposed to hard-core mathematical logic). Consider that it took mathematicians an additional century or so to make the use of infinitesimals rigorous, after they'd succeeded in doing so with the traditional approach. I'm not saying that Keisler isn't a genius or that his book is bad, but the disadvantages of trying to teach calculus using infinitesimals seem to outweigh the advantages. I agree with user90375's first comment expecially. – Stefan Smith Dec 31 '13 at 17:24
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    @user90375, your "deep introspection" as you put it led you to certain conclusions and opinions which you are certainly entitled to. However, educational studies paint a different picture; see my answer below. – Mikhail Katz Jan 01 '14 at 12:55
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    @StefanSmith, as per your request to provide data to contradict your claims, see my answer below. – Mikhail Katz Jan 01 '14 at 12:57
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    Very interesting to see this topic discussed in here! I wrote about Leibniz and the invention of the differential calculus with a particular focus on understanding the notation and mathematical approach given by Leibniz. Infinitesimals do have a closer connection to the geometric spirit of the mathematics of the 17th century. So if one accepts the rules without proving them, they provide a great insight to the geometry involved. – String Jan 01 '14 at 14:12
  • @String, that's interesting. Where did you write about this? – Mikhail Katz Jan 01 '14 at 14:13
  • @user72694: It was my bachelor's degree project as we call it in Denmark, a semi-large project ending the first 3 years of study at the university here. I wrote about differentials of higher orders too and compared them geometrically to the modern approach taught in our upper secondary school (gymnasium). – String Jan 01 '14 at 14:18
  • @user72694: And actually we even teach it very vaguely here. The epsilon-delta approach is not encountered until the first year at the university. What I found most amusing was to learn how deeply the geometrical way of thinking was built into the concepts and workings by Leibniz and his early followers! – String Jan 01 '14 at 14:21
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    @String, fascinating. Is your work on Leibniz online? You may be interested in http://dx.doi.org/10.1007/s10670-012-9370-y and http://www.ams.org/notices/201211/ and more generally recent literature on infinitesimals – Mikhail Katz Jan 01 '14 at 14:24
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    @user72694: My project argued that Leibniz and his contemporaries were still so entangled with the inherited division from the ancient between numbers and geometry as being distinct fields of study so they were bound to think of geometrical concepts as curves in a much more geometrical manner than we often times do nowadays. When we talk about functions 'spitting out' numbers in return for numbers, they were thinking of curves with line segments $x$ and $y$ mutually connected in a geometrical fashion. – String Jan 01 '14 at 14:26
  • @user72694: Thank you! Very interesting, indeed! No, my work is not online. Sadly it is all in Danish so it only has the potential of a very limited audience until now. The summary is in English as it has to be, though :) – String Jan 01 '14 at 14:31
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    And just as a fun observation I have about teaching mathematics: sometimes when I teach in the upper secondary school, one student sitting there with a calculator goes "I really don't understand this!". Then I go there and we find an error typed on the calculator. Having corrected this, my student replies: "Ah, now I totally get it!". But all that really happened there was merely a correction of a typo. So how on earth could that feel like such a great leap in understanding? I doubt it did :) – String Jan 01 '14 at 14:37
  • But then again, confirming with others that we agree on how something should be done and what the result should be, we get a feeling that we are learning the subject. We get to believe more in ourselves that way. So the question could be: Which part of learning comes first? The conceptual understanding and factual knowledge or the confidence that you are able to work with the subject correctly which then in turn leads to a conceptual understanding and factual knowledge? – String Jan 01 '14 at 14:41
  • @String, incidentally who was your advisor on the Leibniz project? – Mikhail Katz Jan 01 '14 at 15:03
  • @user72694: It was Kirsti Andersen who retired shortly after that. I also spoke to her husband Henk J.M. Bos, who wrote the noteworthy and very impressive Ph.D. about 'Higher Order Differentials' back in 1974. Do you happen know these people? – String Jan 01 '14 at 15:20
  • @String, congratulations upon such distinguished mentors. We cite both of them in our recent work, and discuss Professor Bos's analysis of Leibniz in detail in our Erkenntnis text here – Mikhail Katz Jan 01 '14 at 15:28
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    Seeing the active discussion here, can you please help me with this problem too. http://math.stackexchange.com/questions/618509/differentials-definition . – Isomorphic Jan 01 '14 at 15:29
  • @user72694: Actually I pointed out a very petite error in the Ph.D. stating on p. 34 that the quotient $\frac{\Delta^2 y}{h_1\cdot h_2}$ has no limit for $h_1,h_2\rightarrow 0$. This is only true if we forget that $h_1$ and $h_2$ must conform to the nature of the second order derivative of $x$, namely that $\Delta^2 x=h_2-h_1$ converges to zero much faster than does $h_1$ and $h_2$. If so, the limit certainly exists. Having said that, this does not affect the main point Bos makes, that choosing $x$ i arithmetic progression is essential to the modern rigorous approach! – String Jan 01 '14 at 15:34
  • @user72694: I want that book of yours! It sure does look interesting. Actually this inspires me to re-consider writing my master thesis on historical calculus as well. Hopefully next autumn. Maybe your recent book could be of relevance to that :) The sad thing is that I cannot find nearly as distinguished mentors at our university any more to the best of my knowledge :( – String Jan 01 '14 at 15:40
  • @String, notice that the concept of "chosing $x$ in arithmetic progression", while familiar to the historians of mathematics, is incomprehensible to a traditionally (namely, sans infinitesimals) trained mathematician. In a hyperreal context, this acquires a precise meaning in terms of hyperfinite grids. – Mikhail Katz Jan 01 '14 at 15:42
  • @String, wait a minute, I have not written a book yet :-) What you can find at my page is an assortment of 21 recent articles. – Mikhail Katz Jan 01 '14 at 15:43
  • @user72694: Yes, of course! What I meant was the other way around. Leaving the progression undetermined was a symptom of not defining curves in terms of functions. BTW did Bos coin the term 'progression of the variables' in that thesis for the first time? – String Jan 01 '14 at 15:55
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    @user72694: Regarding Leibniz' dispute with Nieuwentijt: I think geometry was so pervading in the mind set of the 17th century that the 'intuitive' connection between first order differentials and tangents bestowed meaning on the former. In order to 'vindicate' the higher order differentials, Leibniz should equivalently have found geometrical properties (similar to 'being tangent') that higher order differentials 'intuitively' possessed thus bestowing meaning on them. But geometrical concepts like 'radius of curvature' were internal to the subject of differential calculus, thus not convincing. – String Jan 01 '14 at 16:08
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    @user72694: So from my perspective, Leibniz could possibly have 'vindicated' his own differentials of higher orders, had there only been a geometrical concept known prior to the differential calculus connected 'intuitively' to them. Of course the modern function-approach offers them a place as tangent slopes to differentiated functions, but then they loose their close connection as being part of the infinitesimal structure of the original curve. Such a wonderful imaginative world is lost in the differentiation of functions :) But then again, I am a novice to that subject! – String Jan 01 '14 at 16:14
  • @user72694: Sorry for spamming the comment, but actually nowadays we are still left with the intuitive geometrical connection between secants and tangents through limits. This is still something we have to believe. So intuition still plays a role though everything else have been made so apparently rigorous. – String Jan 01 '14 at 16:17
  • @String, the "vindication" as you put it of higher-order differentials and infinitesimals can be argued via centers of curvature and the like, but perhaps more convincingly via Leibniz's law of continuity. The ultimate foundation of his disagreement with Nieuwentijt was Leibniz's intuition that inassignable quantities should obey the "same" laws as the assignable ones. Mathematicians were able to assign a precise meaning to this in the 20th century, showing how far ahead Leibniz was of Nieuwentijt. – Mikhail Katz Jan 01 '14 at 18:05
  • @user72694 Of course you teach in an educational system that is fundamentally incommensurable with the American system. My guess is the world can survive another day with both our opinions intact, and correct given where we are and who we are. – Charlie Frohman Jan 01 '14 at 19:01
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    @user90375, as I mentioned in my answer, the compatibility of infinitesimals with "the American approach to mathematics education" was conclusively demonstrated in Kathleen Sullivan's study in the Chicago area. Your own comment's "incommensurability" with civilized speech may well end you up in a suspension in the not too distant future. – Mikhail Katz Jan 01 '14 at 19:16
  • http://math.stackexchange.com/questions/852394/failure-of-differential-notation –  Feb 14 '16 at 07:36
  • PUN: Application of Infinitesimal Calculus: Wee did it. –  Dec 10 '18 at 16:41

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As far as teaching the calculus is concerned, infinitesimals are useful in explaining concepts such as derivative, integral, and even limit. That's why Kathleen Sullivan's controlled study of infinitesimal and epsilontic methodologies in the 1970s revealed that students taught using infinitesimals possess better conceptual understanding of the fundamental concepts of the calculus; see here and here. (A PDF copy can be found here, through a related page Calculus with Infinitesimals that the OP may be interested in.)

Once the students have mastered the key concepts, one can explain the epsilon, delta definitions in an accessible way (the students already understand what the definition is trying to tell us). But one can't do away with $(\epsilon, \delta$)-type definitions altogether. For example, Keisler's proof of the ratio test on page 524 exploits the $\epsilon, N$ definition.

So we still need these definitions, even in the context of teaching infinitesimal calculus. Furthermore, they are needed when developing the foundations of analysis, for example to define Cauchy sequences or rationals, etc.

As far as your subsidiary question "why isn't non-standard analysis accepted more widely", this is a larger problem that should perhaps be treated in a separate question if you are still interested in the issue.

Mikhail Katz
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