What are the disadvantages of non-standard analysis?

Question

Most students are first taught standard analysis, and the might learn about NSA later on. However, what has kept NSA from becoming the standard? At least from what I've been told, it is much more intuitive than the standard. So why has it not been adopted as the mainstream analysis, especially for lower-level students?

Answer 1

I think there are a number of reasons:

1. 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.
2. 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.)
3. There is a substantial set of opinion that because NSA and standard analysis are equivalent, there's no point in learning the former.
4. Often, the bounds created with NSA arguments are a lot weaker than standard analysis bounds. See Terry Tao's discussion here.
5. 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.
6. 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.
7. 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.

Answer 2

So far (55 years from Robinson's book) it has not caught on. Working mathematicians have not seen much benefit to learning it. Mathematical research relies on communication with others: so although I know NSA I normally to not use it to talk to others, since they don't. I believe NSA does have some important uses among logicians.

Nevertheless, amateur mathematicians keep posting here about non-standard analysis

Answer 3

Robinson's framework today is a flourishing field, with its own journal: Journal of Logic and Analysis, and applications to other fields like differential geometry.

The way Robinson originally presented his theory made it appear as if one needs to learn a substantial amount of mathematical logic in order to use infinitesimals. This perception lingers on combined with the feeling, reinforced by the choice of the term nonstandard, that one requires a brave new world of novel axioms in order to do mathematics with infinitesimals.

The fact that Robinson's, as well as Ed Nelson's, frameworks are conservative with respect to the traditional Zermelo-Fraenkel (ZFC) foundations did not "trickle down to the poor" as it should have.

In undergraduate teaching, it is insufficiently realized that just as one doesn't construct the real numbers in freshman calculus, there is no need to introduce the maximal ideals, either. Emphasis on rigorous procedures (rather than set-theoretic foundations) needs to be clarified further.

The proven effectiveness of the infinitesimal approach in the classroom parallels its increasing use around the world, including the US, Belgium, Israel, Switzerland, and Italy.

Answer 4

NSA is an interesting intellectual game in its own right, but it is not helping the student to a better understanding of multivariate analysis: volume elements, $ds$ versus $dx$, etcetera. The difficulties there reside largely in the geometric intuition, and not in the $\epsilon/\delta$-procedures reformulated in terms of NSA.

We are still awaiting a "new analysis" reconciling the handling of calculus using the notation of engineers (and mathematicians as well, when they are alone) with the sound concepts of "modern analysis".

And while I'm at it: Why should we introduce more orders of infinity than there are atoms in the universe in order to better understand $\int_\gamma \nabla f\cdot dx=f\bigl(\gamma(b)\bigr)-f\bigl(\gamma(a)\bigr)\>$?