# Infinitesimal calculus

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 ?

• @user90375 He has actually rigourised the material. Dec 31, 2013 at 13:59
• 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. Dec 31, 2013 at 15:02
• @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... Dec 31, 2013 at 17:17
• @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. Jan 1, 2014 at 12:55
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.