Is this elementary, nilpotent-free approach to automatic differentiation strong enough for real analysis? How similar is it to Newton's system?

Question

This is a sequel to this question: Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?

The ring of "dual numbers" is of the form $a+b\,dx$, where $dx$ is an "infinitesimal" quantity that squares to 0; they can be constructed pretty simply as the quotient of the polynomial ring $\Bbb R[dx]/dx^2$.

In general, the useful thing about the dual numbers is in automatic differentiation: for any analytic function $f$, we have

$f(x+dx) = f(x) + f'(x) dx$

These are interesting because they form a minimalist structure that is sufficient for a good portion of calculus, and have even been used to teach high school calculus. They also share some superficial similarity with Newton's approach in neglecting the square of an infinitesimal. The main issue people mentioned about them in my original thread is that they don't form a field. This can cause some deep issues, as also pointed out by Terry Tao in the above blog post. As a result, things like nonstandard analysis or smooth infinitesimal analysis can be better behaved in some circumstances.

The strange thing is, however, these issues also seem to vanish if we just totally drop the notion that $dx^2 = 0$, and just let it be another real variable with an odd name. Or, if we want, we can think of this as something like the ring of formal power series $\Bbb R[[dx]]$, or the field of formal Laurent series $\Bbb R((dx))$ if a field is desired. Later edit: or something like the Levi-Civita field would probably be best.

Either way, for any analytic function, we get an expansion of derivatives of all orders:

$\displaystyle f(x+dx) = f(x) + f'(x) dx + \frac{f''(x)}{2!} dx^2 + \frac{f'''(x)}{3!} dx^3 + ...$

which can be calculated symbolically to any desired degree, and of which the "dual number" version is a truncation. Both expansions are easy to see by starting with polynomials and extending to analytic functions from there. This can also be thought of as analogous to Newton's approach, where we are simply using the "coefficient extraction" operator to get the coefficient of the $dx$ term, ignoring the higher-order terms.

It is fairly easy to define other approaches to differentiation in terms of this, since we have:

$\displaystyle f(x+dx) - f(x) = f'(x) dx + \frac{f''(x)}{2!} dx^2 + \frac{f'''(x)}{3!} dx^3 + ...$

All of these different approaches are just different ways to pluck off that $dx$ term:

• Quotienting the formal power series ring by $dx^2$, canceling out the rest of the $dx^n$ terms (the dual number approach)
• Treating $dx$ as a real variable, rather than a formal quantity, dividing by it, and taking the limit as $dx \to 0$, canceling out the rest of the $dx^n$ terms (Cauchy's approach)
• Treating $dx$ as an element in the formal Laurent series, dividing by it, and using the "coefficient extraction" operator on the formal power series to get the constant term (i.e. the non-standard analysis "standard part" approach, basically)
• Just noting formally that the $dx$ term is the one you want, and ignoring $dx^2$ and larger (Newton's approach?)

The strange thing is, though, although we have presented this in the context of automatic differentiation, this is literally the most elementary theorem possible - we've just rediscovered Taylor's theorem with a change of variables, evaluating the function $f(x+dx)$ symbolically at some $x$ while leaving $dx$ indeterminate. We could even treat $dx$ as a simple real variable rather than a new formal quantity, with the only caveat being that the above may only hold for "sufficiently small" $dx$, due to convergence issues. So in some sense we have gone in a huge circle, but gotten a better theory than that of the dual numbers, but which has no nilpotents or infinitesimals at all - just indeterminate real variables.

My questions:

1. This seems to be so basic that it's almost surprising that it works so well. Is there some drawback I'm not seeing?
2. Thinking of this as the field of formal Laurent series $\Bbb R((dx))$ (or the Levi-Civita field) is algebraically interesting, but is there any technical difference between doing this and just treating $dx$ as a real indeterminate and getting this result as Taylor's theorem?
3. Is this in any way similar to how Newton used "fluxions," historically? Or similar to Leibniz's infinitesimal approach?
4. Is there some benefit to going with non-standard analysis, or smooth infinitesimal analysis, etc, rather than this elementary approach?

EDIT: In the original question I asked about Laurent series, although I am noting that perhaps the Levi-Civita field would have been better to ask about.

Answer 1

I think you need to think more about what this is meant to achieve. Also, you are considering two different rings, and it is important to avoid equivocating between them - formal power series have some nice properties that formal Laurent series do not have, like being closed under $\exp.$ Anyway, here is a partial answer to 1 or 4.

Formal power series and formal Laurent series are both useful in certain contexts. They faithfully model germs of analytic (respectively, meromorphic) functions.

In nonstandard analysis $\epsilon$ is a real number. In $\mathbb R((dx)),$ you don't have $|dx|,$ or $\sqrt{|dx|},$ or $\exp(dx^{-1}).$

Smooth infinitesimal analysis provides a system of logic for doing differential geometry. A benefit is that you can reason naively about function spaces, such as defining a tangent bundle as the space of functions from a infinitesimally-thickened point. It says that every function has a Taylor series, but that is a small part of it.