Infinitesimals have been made rigorous with modern mathematics.
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.
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)
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.
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.
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.
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.
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.
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.
How can you tell whether it's standard analysis that's confusing per se or you just had poor math teachers?
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!
(ϵ + 1)(ϵ−1) = −1, or alternately (1 + ϵ)(1 − ϵ) = −1
(1 + ϵ)(1 − ϵ) = 1
(1 + ϵ)(1 − ϵ) = 1 - ϵ^2 = 1