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I would like to learn non-standard analysis, at least the basics of it. I will make use of this book: Elementary Calculus: An Infinitesimal Approach (Dover Books on Mathematics), by H. Jerome Keisler. Before anything else, please let me take up some links that DO NOT have a fully fleshed out answer to my question, at least according to me.

Here are the links:

The content on this link 'Is non-standard analysis worth learning?' more or less discusses the use of studying non-standard analysis. The content on this link mostly refers to the link that discusses the use of studying non-standard analysis, 'Is non-standard analysis worth learning?'. This and this does not answer my question.

The problems with the answers to the question up above, is that while they may scratch on the surface and from to time take up the disadvantages of non-standard analysis, they DO NOT purely discuss the disadvantages or/and mathematical disadvantages of non-standard analysis.

'What are the disadvantages of non-standard analysis?' does not rigorously answer that which I want an answer to. For example, let me cite this answer:

I think there are a number of reasons:

Early reviews of Robinson's papers and Keisler's textbook were done by a prejudiced individual, so most mature mathematicians had a poor first impression of it. It appears to have a lot of nasty set theory and model theory in it. Start talking about nonprincipal ultrafilters and see the analysts' eyes glaze over. (This of course is silly: the construction of the hyperreals and the transfer principle is as important to NSA as construction of the reals is for real analysis, and we know how much people love that part of their first analysis course.) There is a substantial set of opinion that because NSA and standard analysis are equivalent, there's no point in learning the former. Often, the bounds created with NSA arguments are a lot weaker than standard analysis bounds. See Terry Tao's discussion here. Lots of mathematicians are still prejudiced by history and culture to instinctively think that anything infinitesimal is somewhere between false and actually sinful, and best left to engineers and physicists. As Stefan Perko mentions in the comments, there are a number of other infinitesimal approaches: smooth infinitesimals, nilpotents, synthetic differential geometry, . . . none of these is a standout candidate for replacement. It's not a widely-studied subject, so using it in papers limits the audience of your work. Most of these reasons are the usual ones about inertia: unless a radical approach to a subject is shown to have distinct advantages over the prevalent one, switching over is seen as more trouble than it's worth. And at the end of the day, mathematics has to be taught by more senior mathematicians, so they are the ones who tend to determine the curriculum.

This is a good start of an answer to the question that I am asking. What I am missing in the answer that you saw, are if there exists any mathematical problems in non-standard analysis. Does it, and if so, which?

I once read - on this forum, at a place that I really can't remember - that there exists some mathematical problems in non-standard analysis. At least some ideas or concepts that weren't, if I remember correctly, likable. The word likable points towards at least one bias. But is it a bias?

Please help me to understand if there are mathematical problems/ problems in certain concepts of non-standard analysis.

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    $\begingroup$ Feel free to state your opinions. $\endgroup$ – ಠ ಠ Feb 2 '17 at 19:00
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I'm not entirely sure what you're asking, but let me take a stab at it:

First of all, there's nothing standard analysis can do that nonstandard analysis can't. A nonstandard analyst could always decide to just study the standard hyperreals, and this would correspond to standard analysis. (The converse is also true, but nontrivially so.) So you won't find a mathematical problem in a deep sense; anytime the nonstandard approach is less useful than the standard one, a nonstandard analyst could always just use the standard approach inside nonstandard analysis.

That said, there are mathematical features of the hyperreals which are (in my opinion) less than ideal. Topologically, they are ugly: there are multiple natural topologies to put on them, and they all have odd features (see here). And I would consider the presence of lots of automorphisms to be a negative feature as well - it means that if someone asks for an example of an infinitesimal, we can't really give a satisfying answer; however, this arguably reflects my own standard bias.

I suspect there are also algebraic properties the reals have which the hyperreals lack, although at the moment I can't think of any (my previous example was incorrect and silly).

Basically, I think the bottom line is this:

  • Anything a standard analyst can do, a nonstandard analyst can also do - sometimes more easily. (Although my understanding is that that gain in ease rapidly drops off once one is comfortable with standard analysis. Some exceptions exist, however: the invariant subspaces problem was originally solved via nonstandard analysis, and I think there are some nonstandard proofs of esoteric results for which no proof via standard analysis is currently known, although we know that such a proof must exist.)

  • That said, the hyperreal field is a much less nice object than $\mathbb{R}$: the price of having a nice infinitesimal structure is that we lose good properties elsewhere. And it lacks - in my opinion - the compellingness of the structure $\mathbb{R}$. Note that this objection is completely unrelated to the question of whether nonstandard analysis and the hyperreals are useful: something need not be philosophically compelling to be a good tool. I actually think there is a really interesting philosophical phenomenon here: I find the hyperreals completely uncompelling, but the language and techniques of nonstandard analysis to be very compelling! The subject is somehow more compelling to me than its subject matter. No idea what that says about me.

  • I think there are extremely good reasons to learn standard analysis, but no good ones besides personal preference and limitations of time to not learn nonstandard analysis. I would argue that for most mathematicians, learning nonstandard analysis would not necessarily be a good use of time (a combination of unpopularity and - I suspect - a low benefit to their already-existing research interests), but the reasons for this are at least largely sociological, and not inherent to the subject.

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    $\begingroup$ I would have liked the tag 'discussion', but it did not exist. So feel free to state opinions. Particularly one of the things that you said, I found particularly interesting. Could you write a little bit more about what you mean when you say 'learning nonstandard analysis would not necessarily be a good use of time'. $\endgroup$ – ಠ ಠ Feb 2 '17 at 18:58
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    $\begingroup$ @Pantheon One of the reasons there's no discussion tag is that this site isn't really for discussions. I've added a parenthetical about my opinion there, though. $\endgroup$ – Noah Schweber Feb 2 '17 at 18:59
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    $\begingroup$ Okay, then. But feel free to state your opinions. $\endgroup$ – ಠ ಠ Feb 2 '17 at 19:02
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    $\begingroup$ Every finite dimensional algebra over a field is determined by the action of multiplication on a basis. I'm pretty sure every finite dimensional algebra over ${}^\star \mathbb{R}$ is an internal one, so we can just transfer the classification of finite-dimensional associative division algebras over from the standard case. $\endgroup$ – Hurkyl Feb 2 '17 at 19:47
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    $\begingroup$ I'm curious, why the downvote? $\endgroup$ – Noah Schweber Mar 6 '17 at 15:42
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About the likeability issue raised in the question: indeed, A. Robinson's original version of non-standard analysis (which is not quite necessary to follow J. Keisler's treatment of calculus!) does require understanding a bit of model theory. In contrast, E. Nelson's "IST" version is a much more user-friendly version (see A. Robert's very nice little introductory book, and/or Nelson's essay), that has managed to package most of Robinson's stuff in a fashion that does not require frequent interaction with (or understanding of) the model theory. (Nevertheless, I gather that there are some possibilities in Robinson's version that are not fully represented in Nelson's).

I have not quite used non-standard analysis "in research", instead finding that L. Schwartz' notion of "distribution/generalized function", as expanded-on by A. Grothendieck's early work, and Gelfand-Graev-et al, is adequate (so far) for my purposes (and in my context, obviously). Nevertheless, non-standard analysis is interesting to me for at least two reasons. First, it approximately shows (in a revisionist and anachronistic way, of course) that L. Euler's and A. Cauchy's use of "infinitesimals" can be turned into a completely legitimate argument (in contrast to various naive dismissals that often claim that epsilon-delta arguments are the only legitimate way to do analysis). Second, non-standard analysis does seem to better capture certain intuitions about "orders of magnitude" that are a bit clumsy to formulate epsilon-delta-wise... Not that it is impossible to capture them, but that it requires a priori an understanding that might only be achieved by thinking in non-standard terms. Not that I've made anything of this myself, only that I have a vague feeling in this direction. Again, A. Robert's book gives the delightful example of canards.

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  • $\begingroup$ What is a canard? $\endgroup$ – Akiva Weinberger Mar 6 '17 at 13:35
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    $\begingroup$ @AkivaWeinberger, a canard is according to merriam-webster.com/dictionary/canard a false or ungrounded story. Used in the context of Robinson's framework, the term may also contain an allusion to solutions of a particular type of PDE that were first obtained using infinitesimals. $\endgroup$ – Mikhail Katz Mar 6 '17 at 13:43
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To fill in some historical background, such concerns about Abraham Robinson's framework stem from two main sources: (1) Errett Bishop and (2) Alain Connes. Most of the negative comments you hear about this at MSE arguably emanate ultimately from these two.

The Bishop-Connes critique of Robinson's framework has in my opinion been definitively refuted by Sam Sanders in his latest article in the leading philosophy journal Synthese.

For some details on Connes's critique see this answer.

A further canard, to the effect that anything proved by the tools provided in Robinson's framework, can also be proved without infinitesimals has been refuted in the leading logic journal Journal of Symbolic Logic, long ago by Henson and Keisler.

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