[Bahamut]

I do have an issue with this line "Ultimately, infinitesimals were discredited and discarded by mathematicians (though they continued to be mentioned in some physics books many decades later)"

Infinitesimals have been made rigorous with modern mathematics.

[ot]

Indeed, many mathematicians think in terms of non-standard analysis, and then translate their proofs into standard arguments, even if the non-standard ones can be made just as rigorous as the standard ones.

Terry Tao has a wonderful series of posts about hard and soft analysis, ultrafilters, and non-standard analysis. He writes

    I feel that one of the reasons that non-standard analysis is
    not embraced more widely is because the transfer principle,
    and the ultrafilter that powers it, is often regarded as some
    sort of “black box” which mysteriously bestows some
    certificate of rigour on non-standard arguments used to prove
    standard theorems, while conveying no information whatsoever
    on what the quantitative bounds for such theorems should
    be. Without a proper understanding of this black box, a
    mathematician may then feel uncomfortable with any
    non-standard argument, no matter how impressive and powerful
    the result.

and

    The main drawbacks to use of non-standard notation (apart
    from the fact that it tends to scare away some of your
    audience) is that a certain amount of notational setup is
    required at the beginning, and that the bounds one obtains at
    the end are rather ineffective (though, of course, one can
    always, after painful effort, translate a non-standard
    argument back into a messy but quantitative standard argument
    if one desires)

(from http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonsta...)

[igravious]

I agree. It would be truer to say that infinitesimals are studiously ignored by modern mainstream mathematicians because they feel that Dedekind and co. have put the calculus on a firm footing way back when.

Anybody with a small bit of curiosity or a dashing of non-conformity will be suspicious of this narrative.

If anything, infinitesimals in their various guises carry a certain explanatory heft, and are quite beguiling little creatures if you take the time to get to know them. I'd be happy to elaborate or leave a few links here if anybody is interested.

[protonfish]

I loathed limit-based calculus in High School and College. Later I read Elementary Calculus: An Infinitesimal Approach http://www.math.wisc.edu/~keisler/calc.html and it all came clear in a fraction of the pages. It's infuriating that most math curricula won't drop those old, bloated, overly formal calculus tomes to improve the clarity and effectiveness of the instruction method.

[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.

[Patient0]

For me it was completely the other way around: I was "taught" calculus using the infinitesimal approach but without any rigour. Statements like "As dx gets really really small x+dx/x becomes 1" drove me crazy! Why was it sometimes ok to replace dx with 0!? The idea of an "infinitely" small number to me was always vague and suspect. So while I could do the calculations I never trusted the results.

This meant that maths stopped having the same appeal to me as computer programming.

It was only years later when I revisited the epsilon delta arguments that it finally made sense. It was a revelation to me that you could explain all of calculus without ever talking about "infinite".

I wish it had been taught to me rigorously the first time around: I would have been much better off.

[jfarmer]

Conversely, I find infinitessimals vague and woo-woo, especially the way physicists and statisticians often use them. Once the epsilon-delta style "clicked" for me, it felt like second nature.

How can you tell whether it's standard analysis that's confusing per se or you just had poor math teachers?

[baddox]

That's funny. I despised my college calculus courses because they were so informal. Much like the author of this post said, my calculus education focused entirely on boring rote computation and not at all on proofs, the,a tater of which is. The only part I really consider to be mathematics.

[acjohnson55]

I'm interested! I think Dedekind cuts are reasonably understandable, but infinitesimals are on the surface of much of our calculus syntax, so I'd be glad to understand where they become so tricky formally.

[igravious]

I wrote a short paper on the topic once upon a time[1] which you may find interesting. It's part history of math, part philosophy of math.

It's not a great paper and most of the insights in it come from others but here is some of the arithmetic of nilpotent[1] infinitesimals as shown in the appendix.

Imagine an entity which is not equal to zero but that when raised to the power of 2 or higher is equal to zero! Sounds odd, doesn't it, but it works! (ϵ is an infinitesimal)

ϵ != 0 but ϵ^n = 0 | n>1

ok? so we get:

(ϵ + 1)^n = 1 + nϵ thus: (ϵ + 1)^−1 = 1 − ϵ

e^ϵ = 1+ϵ

(ϵ + 1)(ϵ−1) = −1, or alternately (1 + ϵ)(1 − ϵ) = −1

and finally (for calculus): ϵf′(x) = f(x + ϵ)−f(x)

1: http://leto.electropoiesis.org/propaganda/The_Analyst_Revisi...

2: https://en.wikipedia.org/wiki/Nilpotent

edit: clarity, line breaks!

[JadeNB]

How do these differ from the [dual numbers](https://en.wikipedia.org/wiki/Dual_number)?

[ColinWright]

You appear to have an error. You write:

  (ϵ + 1)(ϵ−1) = −1, or alternately (1 + ϵ)(1 − ϵ) = −1

That alternative should surely be:

    (1 + ϵ)(1 − ϵ) = 1

Not least, in a commutative system (1+x)(1-x) = 1-x^2. Thus

    (1 + ϵ)(1 − ϵ) = 1 - ϵ^2 = 1

[igravious]

Thanks, well caught :)

[cokernel]

To be fair to the author, he does discuss Robinson's approach to infinitesimals (but not Skolem's) in the very next paragraph.

[jonsterling]

The article actually does discuss Robinson's non-standard analysis, but it's a shame it does not deal with smooth infinitesimal analysis and intuitionism.