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I'm not interested in an in-depth answer. Here are some specific questions for which I couldn't find an answer:

  • With non-standard analysis, can we solve problems that can't be solved using standard analysis?
  • Do we have some results that differ from standard and non-standard fields? (E.g.: is there a function which derivative is different if calculated with the standard and the non-standard definition of derivative?)
  • More importantly: is non-standard analysis just a formalism, or is it needed for practical reasons?

Simple yes/no answers are fine. Short examples are a bonus.

user16538
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    (1) Yes and no. If there is a non-standard solution, there is also a standard solution, but it may be harder to find. (2) No, as long as you’re talking about standard objects. (3) Depends on your definition of practical. For at least some people it provides an easier, more intuitive way to think about at least some parts of analysis. – Brian M. Scott Mar 16 '15 at 04:14

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Look at the answers to Is non-standard analysis worth learning? Do they answer your question?

If I understood your question correctly, it seems the answer is that non-standard analysis (NSA) is not technically needed for the kind of practical reasons you mention. in particular, this answer to the previously-cited question mentions a few results that were first found via non-standard analysis, but also says these results were all found later by means of standard analysis. Indeed it seems that theoretically, the interesting thing about NSA is how well its theorems correspond to statements that are also true (and provable) in standard analysis.

On the other hand, consider this answer concerning the teaching of calculus. It cites evidence that students of calculus learn the concepts better when things are presented first in terms of infinitesimals and the epsilon-delta formalisms are introduced later. Not everyone agrees with this conclusion. (And that's an understatement.) But I think this might qualify as some kind of practical use of NSA if the conclusion is true.

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David K
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