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I have heard of an infinitesimal approach to calculus. Is it better than the normal approach or is it the other way around.

alkabary
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Abhijith S. Raj
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    What does it mean to be better? – davidlowryduda May 31 '15 at 06:36
  • easier persay ? – alkabary May 31 '15 at 06:37
  • I mean which is better for studying. There are these books which follow an infinitesimal approach to calc. – Abhijith S. Raj May 31 '15 at 06:37
  • Such as (elementary calculus an infinitesimal approach.) – Abhijith S. Raj May 31 '15 at 06:38
  • The infinitesimal means something very small, and so that approach kinda involves more details when studying calculus. – alkabary May 31 '15 at 06:39
  • Is it more interesting than the normal approach. – Abhijith S. Raj May 31 '15 at 06:40
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    Based on your background, you would do well to study calculus using the standard approach (epsilon-delta definition of limits, continuity, differentiability, ...) and then study Big-O notation which can often make things much easier or cleaner. The order is important, because using Big-O notation can make you very prone to logical errors if you don't have a solid grasp of first-order logic. – user21820 May 31 '15 at 06:40
  • I have studied big o in this course by prof. Robert ghrist. – Abhijith S. Raj May 31 '15 at 06:41
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    @alkabary: It's precisely the opposite. People who use infinitesimals often use ridiculously hand-wavy arguments that sometimes cannot even be made rigorous because the important details are missing. – user21820 May 31 '15 at 06:42
  • Ok does that mean you recommend the standard approach. – Abhijith S. Raj May 31 '15 at 06:43
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    @AbhijithS.Raj: Big-O notation is prone to error unless you are absolutely confident with manipulation of formulas in first-order logic. So yes, study the standard approach first. – user21820 May 31 '15 at 06:45
  • I have spend quite some time with the standard approach. – Abhijith S. Raj May 31 '15 at 06:50
  • Maybe i was thinking of shifting to the infinitesimal approach based on what Ittay said. – Abhijith S. Raj May 31 '15 at 06:50
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    @AbhijithS.Raj: Also, the reason I suggest studying Big-O notation after that instead of non-standard analysis (using infinitesimals) is because non-standard analysis is not as convenient or useful as some people think, even after the non-trivial hurdle that often goes through rather advanced theorems about first-order logic. Big-O notation in contrast can be easily phrased in first-order logic and simply reduces the quantifier depth, and so is completely compatible with the standard form. – user21820 May 31 '15 at 06:50
  • @user21820 I understand... – Abhijith S. Raj May 31 '15 at 06:51
  • I have seen the power of Big O. – Abhijith S. Raj May 31 '15 at 06:54
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    @AbhijithS.Raj: Did you try using it to re-derive all the basic theorems of calculus? See http://math.stackexchange.com/a/1298332/21820 for a brief example. I personally found that it is often as easy or easier than even non-standard analysis. Anyway if you seriously want to learn about non-standard analysis rigorously you would probably need to learn some logic, at least up to the theorem that a first-order theory has a countable model if it is consistent. That is one way of building the non-standard reals. – user21820 May 31 '15 at 07:03

1 Answers1

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The approach with infinitesimals is equivalent to the one without infinitesimals. Anything you can prove using one formalism, you can prove in the other formalism. The difference is thus cosmetic, and a matter of taste and preference. That is largely influenced by what is historically more common-place, which of course is the infinitesimal-free approach. Whether or not this will remain the situation is hard to predict (if forced, I will bet against infinitesimals).

There is a trade-off. The infinitesimal-free approach does not require any sophisticated machinery to get going. Taking any definition of the reals one wishes, one can start doing analysis immediately with the $\epsilon - \delta$ definition. However, the arguments take some time to get used to, and they are not particularly a direct translation of one's geometric intuitions into a formal system. The infinitesimal approach (no matter which one) requires some considerable amount of preparatory work (either in the form of logic, or some detailed discussion of axioms which are unfamiliar). There are numerous texts that attempt to ease the reader into the world of infinitesimals. As far as I know, they all present an initial hurdle that is not trivial to get around before one can start doing analysis, though once one is past that hurdle, the arguments are nicer and closer to one's intuition.

So, if you want to learn analysis, it's probably a good idea to do it the infinitesimal-free way. This is by far the standard language analysts use. If you want to broaden your horizons and learn some more formalisms and deepen your foundations of the subject, spending some time with infinitesimals is a good idea. Also, if you want to impress people in the pub knowing your infinitesimals is a good idea (and also to quickly point at silly fallacies people at the pub say about infinitesimals, you might wish to properly learn the basics).

Ittay Weiss
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