# Is Calculus a Subset of Algebra?

Or a more accurate title would be “Is the way we teach calculus a subset of algebra?” I was reading L.D. Nel’s paper “Differential Calculus Founded on an Isomorphism”, and while I have little commentary on the paper, it reminded me of a question I had while I was teaching calculus. To summarize, for differential calculus and especially, for integral calculus, it seems that the standard teaching method is to introduce the analytic definitions (e.g. the derivative in terms of limits and the integral in terms of Riemann sums) and then as quickly as possible, we abandon the analysis when we develop enough algebraic tools to do the computations that are ostensibly the main point of the subject. The analytic roots are the first thing, in my experience, that students forget. If they remember anything from those classes, it will likely be the power rule.

So how much of what we teach can be boiled down to solve algebraic problems in a certain (rather large) $D$-module? From what I can recall, there are applications of derivatives in terms of tangents, velocity, rates of growth, and optimization and applications of integrals in terms of area and volume. But in terms of mathematical content, there doesn’t seem to be much analytic content beyond their definitions and initial geometric interpretations. The only examples that come to mind are Newton’s method for approximating solutions of equations and trapezoidal and Simpson’s rules for approximating definite integrals. Of course, much of what I say doesn’t apply to sequences and series, which has a much more analytic flavor with its concerns about convergence.

But then again, to what extent is calculus part of analysis? The original texts seem more algebraic (and geometric) than its current form. The formal definitions of limits and the like were developed much later. And though it is clear that analysis was born out of calculus, whether the parent belongs to the same club as the child is harder to tell.

But then again, what makes analysis not a part of algebra? The definitions are fairly algebraic. Look at the definition of a limit. Where’s the analysis? Is it the absolute value? The inequality? Certainly, it’s not the quantifiers. I think I come back to this question because it’s not clear to me where the lines of between the mathematical subjects and what makes them so. The art of epsilon-delta is supposed to be the dividing line, but dissecting that animal doesn’t clarify the issue.