Limits vs infinitesimals

Question

Many years ago in college when I went from Calculus to Differential Equations, there was a distinct difference in the classes. All three semesters of Calculus had spent considerable time on justifying the mathematics. In Differential Equations, they didn't bother. They started doing all of these algebraic operations with dy and dx quantities without ever justifying why the operations were valid.

From what I'd learned in Calculus, I always imagined that there was an unstated "limit as dx approaches 0" around the equations. This made sense to me, although some of the operations still seemed to need additional justification.

There is also a theory of non-standard analysis involving infinitesimal quantities that I'm not too familiar with, but as I understand it, it dispenses with the notion of a limit and instead treats dx as a positive number smaller than any positive real number. I don't have any problem with the idea itself; if it can be shown to be consistent and to get to the right solutions, I'm fine with it, but what I don't get is the purpose. Does this give some power that limit theory does not? Are there limits to limit theory that I didn't notice? Is this just another formulation of the same thing as limit theory?

Answer 1

The situation you describe with the transition from calculus to ODE is indeed somewhat comical, as noted by leading educators such as David Tall. This may be less incoherent than it sounds because, once the students master the epsilon-delta technique, in principle they should be able to convert a manipulation of $dx$'s and $dy$'s such as the one occurring in the technique called the separation of variables for ODE, into a rigorous argument. A typical example would be solving $\frac{dy}{dx}=y$ by replacing this by $\frac{dy}{y}=dx$, integrating, etc. Furthermore, the technique can be justified in terms of differential 1-forms, so long as one is careful not to keep the $dx$ in the denominator (i.e., not to divide by $dx$).

As you mention, another possibility would be to work in infinitesimal analysis where the derivative $\frac{dy}{dx}$ can be viewed as a quotient of infinitesimals (rather than as a formal symbol). This involves some work with taking the standard part and is slightly more involved than what Leibniz originally envisioned. Note that limits are used in both approaches. For example, the limit of $f(x)$ as $x$ tends to zero can be evaluated as the standard part of $f(h)$ where $h$ is infinitesimal. Thus the true dichotomy is not between limits and infinitesimals, but rather between the following possibilities:

(A) working with an Archimedean number system and making use of epsilon-delta definitions (of the concept of limit and other concepts), and

(B) working with a number system that incorporates infinitesimals, and defining concepts such as derivative and integral (as well as limit) in terms of taking standard part.

Which brings us to your question: What is the purpose? The question can be answered at several levels. At the research level, the article "On the Strength of Nonstandard Analysis" from 1986 by C. WARD HENSON AND H. JEROME KEISLER showed, roughly, that using second-order arithmetic in the context of a number system of type (B) is equivalent to using third-order arithmetic in the context of a number system of type (A). A similar shift occurs for higher orders. This suggests that, while in principle it may be possible to translate any proof using infinitesimals into a (more complex) proof without them, in practice such a translation may make a proof incomprehensible given a sufficiently complex starting point. In the current literature, there are results that currently have only a proof via infinitesimals and no proof of type (A), such as Renling Jin's results on sumsets in additive combinatorics.

At a more basic level of calculus pedagogy, infinitesimals enable one to give rigorous - as well as lucid - definitions of basic concepts such as continuity. A (standard) function $f(x)$ is continuous at a (standard) real $c$ if and only if an infinitesimal increment $\alpha$ always produces an infinitesimal change in the function: $f(c+\alpha)-f(c)$. This happens to be Cauchy's definition of continuity dating from 1821. In my own teaching experience, students find this definition more accessible than the epsilon-delta alternating-quantifier one.

At an intermediate level of difficulty, approach (B) enables one to represent, for example, the Dirac delta function at a point $c$ by an actual (internal) function, so that integrating a standard function against it would literally produce the value of the function at $c$. Of course there are many other applications, such as the theory of Loeb measures.