[ColinWright]

Added in edit to emphasise a point:

    If all you want to do is differentiate and integrate,
    then non-standard analysis is probably, for most people,
    a faster way to be able to do just that.

Now read on ...

Non-standard analysis has been put on a firm, formal footing. Theorems have been proven showing that (largely) it's equivalent to the regular form of analysis. Some things are easier to prove in standard analysis, some things are easier to prove in non-standard analysis, etc, etc.

However, this is only really of use if all you want to do is calculus. If you want to go beyond calculus, almost everything (in this and related areas) is about sequences, limits, limiting processes, functions, and transformations. There, non-standard analysis tends not to help, and unless you've done calculus the standard way, you have to learn all this stuff in an unfamiliar and difficult-to-visualize, abstract area.

One of the main reasons for continuing to learn calculus in the epsilon-delta limiting process manner is exactly because it's not only formally sound, it's also giving you tools for moving beyond the rather limited world of differential calculus.

Speculating wildly from limited experience, it might also be the case that starting people with the non-standard approach in calculus is actually just as confusing. You may find that you really only got the insights you did because you had already struggled with the standard approach, and then were given something that made it all fall into place. Perhaps some people they think the non-standard approach is easier, but in fact it's only because they've actually got the foundations from the other. Just a thought.

[pash]

> If you want to go beyond calculus, almost everything (in this and related areas) is about sequences, limits, limiting processes, functions, and transformations. There, non-standard analysis tends not to help ...

Why do you say this? I ask because I've found internal set theory, Edward Nelson's axiomatic version of nonstandard analysis, to be a lovely tool for doing typical sorts of things in analysis.

You have to learn to wield the "standard" predicate [0], which is too dark an art for some mathematicians, I suppose. But, in my opinion, nonstandard characterizations of notions like convergence and continuity are delightfully simple and direct.

It also turns out that when you have nonstandard numbers at hand, infinity is an over-powerful abstraction for some purposes. Nelson came up with a new formalism for probability theory [1], for example, that makes finite spaces powerful enough to capture what's interesting for most purposes. Similarly, finite but unlimited sequences often are "long enough" to incorporate all the interesting behavior of infinite sequences.

0. Alain Robert's Nonstandard Analysis is a good starting point.

1. See his short book Radically Elementary Probability Theory. I love this book, and didn't much like probability theory before reading it.

[protonfish]

I disagree. The vast majority of students take math classes for the practical applications - science and engineering - not to continue theoretical pure math study. Therefore the focus should be on effective teaching of applied math. I am sure that if a student wishes to explore their studies in pure mathematics they will be clever enough to learn whatever they need in specialized classes.

[ColinWright]

Actually, you are agreeing with me. You are saying that doing calculus was, for you, much easier using the infinitesimal approach. I'm not disagreeing with you. In fact, you'll find that advanced mathematicians think in that way, although they can drop back to epsilon-delta work if they need to (which they often do).

So we are in agreement. My point is that if you teach calculus that way you have immediately ham-strung anyone who might go on and do anything other than engineering or physics. In fact, there are deep theoretical arguments in physics where you need to use the standard approach, and the non-standard approaches are much more difficult.

My point is that if all you want is calculus then it's very likely that the non-standard approach is fine. I'm also arguing that this is limited thinking. Clearly you were never going to go further in these sorts of subjects - does that mean that everyone else should also be taught in a similarly limited way?

I also observe that limiting arguments are essential in anything other than the most direct and practical versions of engineering, so again, the point isn't in the calculus, the point is learning about limits.

Many people don't need any math at all beyond arithmetic, and I know a lot of people who proudly announce that they can't even do that. And to some extent it's true - most people don't need any math at all. Why were you bothering to take calculus? I'm sure you've never needed it.

But let me add that if all you want to do is arithmetic, why bother? Just use a calculator. If all you want to be able to do is differentiate, why bother? Feed it to Wolfram Alpha. If all you want to do is program, why bother? Hire someone to do it.

But yes, if all you want to do is high-school calculus, there are easier ways to learn the processes to jump through the hoops, pass the exam, and get the piece of paper. For most people that's all they care about. We probably agree on that.