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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 ?

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    $\begingroup$ @user90375 He has actually rigourised the material. $\endgroup$ – Isomorphic Dec 31 '13 at 13:59
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    $\begingroup$ 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. $\endgroup$ – Charlie Frohman Dec 31 '13 at 15:02
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    $\begingroup$ @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... $\endgroup$ – Stefan Smith Dec 31 '13 at 17:17
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    $\begingroup$ @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. $\endgroup$ – Mikhail Katz Jan 1 '14 at 12:55
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    $\begingroup$ @StefanSmith, as per your request to provide data to contradict your claims, see my answer below. $\endgroup$ – Mikhail Katz Jan 1 '14 at 12:57
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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 an 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.

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  • $\begingroup$ Corrected the URLs, now that apparently a user 'crow' at BRCC took over these pages. $\endgroup$ – MarnixKlooster ReinstateMonica Dec 20 '15 at 6:59

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