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Also, I've heard that initially calculus was based upon infinitesimals but then it's foundations were changed from that which led to the foundations of modern theory of analysis. Does it corresponds to the lack of rigour in infinitesimal, if any, or something?

What is non-standard analysis? Is it related to the theme question?

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    $\begingroup$ Initially how infinitesimals were used was lacking in rigour. Non-standard analysis, however, is a rigorous foundation that includes the notion of infinitesimals. $\endgroup$ – Eff Oct 15 '16 at 13:58
  • $\begingroup$ @ankit infinitesimals were negligible compared to the main variables however not their ratios, the newly found derivatives.The relation of these ratios to variables were recognized as growth or change descriptors. $\endgroup$ – Narasimham Oct 15 '16 at 14:21
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The procedures of Leibnizian calculus with infinitesimals find faithful proxies in the procedures of Robinson's framework for calculus and analysis with infinitesimals, putting to rest centuries of anti-infinitesimal vitriol from Bishop Berkeley to Georg Cantor. Berkeley and Cantor thought infinitesimals were contradictory. Both published articles attempting to "prove" this. Both were wrong, as demonstrated in recent literature. See this site for a wide variety of published articles on the topic by a group of over 25 authors.

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Yes, Leibniz used something one could call static infinitesimals, in modern terms one can replicate some of that with (truncated) power series arithmetic. Newton used a more dynamic idea of infinitesimals, which led to criticisms about ghost of vanishing quantities...

The use of infinitesimals in that time was more an art than a science, leading to many mistakes and misunderstandings.

Non-standard analysis tries to capture some of the ease and directness of infinitesimals, which is demonstrated by the multiple approaches (Keiser, Diener/Diener, Laugwitz) for introductory analysis courses, i.e., real analysis, metric topology. But for more advanced topics like functional analysis NSA is quite cumbersome to formulate, as one has at every step to distinguish the standard and non-standard entities.

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  • $\begingroup$ Do you understand Robinson's infinitesimals? $\endgroup$ – Mikhail Katz Oct 16 '16 at 9:41
  • $\begingroup$ The basics, yes. I prefer Nelson's IST, much more direct, no secondary starred objects,... $\Bbb R$ remains Archimedean, in contrast to, as claimed, $\Bbb R^*$,... $\endgroup$ – LutzL Oct 16 '16 at 9:48
  • $\begingroup$ I really don't know why you think this is cumbersome and moreover I have evidence quite to the contrary as we have been teaching calculus with infinitesimals for several years now to hundreds of students. This approach is both (1) rigorous (2) accessible and (3) preferable to the epsilon-delta approach as far as the students are concerned. I can send you some material with additional details. $\endgroup$ – Mikhail Katz Oct 16 '16 at 9:52
  • $\begingroup$ Yes, real analysis is the topic where all the approaches, Laugwitz, Robinson and Nelson, demonstrate that NSA is more intuitive than epsilontics. Now reformulate functional analysis in terms of NSA. Nelson gave a remarkable example in Radically elementary probability theory, but it is only concerned with integrability, not general Banach spaces. $\endgroup$ – LutzL Oct 16 '16 at 9:58
  • $\begingroup$ What this comment overlooks is that Robinson's framework (unlike Laugwitz's framework) is a conservative extension of the Weierstrassian framework. All the old tools are there still, nothing is missing. If you don't want to use the new tools you don't have to. In some cases they shed new light that is not provided by other methods, as argued by Terry Tao. You should probably make it clear to the OP (who is likely an undergraduate) that your concern is limited to research topics that he probably did not have in mind. $\endgroup$ – Mikhail Katz Oct 16 '16 at 10:23

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