Infinitesimal calculus

Building calculus on infinitely small numbers rather than on limits. The corpus holds one account, keisler-elementary-calculus, which teaches the entire standard sequence this way and is the reason this page exists.

The three-hundred-year gap

The history is the argument, and keisler-elementary-calculus tells it in four dates:

• 1670s. Leibniz and Newton develop the calculus using infinitesimals. The intuition — a derivative is a ratio of infinitely small changes — is the one that produced the subject.
• Two centuries of successful use, alongside sustained doubt about whether an infinitely small quantity means anything.
• 1870s. Weierstrass supplies the first rigorous treatment, using the ε-δ definition of limit. Infinitesimals are banished from the curriculum for rigour, and the standard course has begun with limits ever since.
• 1960. Abraham Robinson gives infinitesimals a precise treatment, closing the gap from the other direction and making the original intuition legitimate.

Keisler’s summary of what the intervening century cost: “Students have had to learn the subject without the original intuition.”

The mechanism

The real line is extended to the hyperreal line, which contains the reals plus infinitesimals (non-zero numbers smaller in magnitude than every positive real) and their reciprocals, the infinite numbers. Every finite hyperreal sits infinitely close to exactly one real, its standard part.

That single operation replaces the limit in the definitions. A derivative becomes the standard part of a ratio of infinitesimal increments; a definite integral, the standard part of an infinite sum of infinitesimal pieces. Both are computations rather than limiting processes, which is the pedagogical claim in one line.

Keisler builds this in chapter 1 and has derivative, continuity and the integral by the end of chapter 4. Limits appear in chapter 5, motivated by approximation, and the traditional definition is taught rather than skipped.

What it buys, on the source’s own account

Three claimed advantages: proximity to the founders’ intuition, easier access to derivative and integral, and the student ending up with both tools instead of one. There is also at least one place where the approach reaches further rather than merely arriving sooner — §14.4 uses infinitesimals for a simple existence proof for y′ = f(t,y) with continuous f, which Keisler says is out of reach for a traditional elementary course.

The restraint matters as much as the claim. Robinson’s construction used mathematical logic; Keisler’s textbook does not, and the logical machinery is exiled to a companion instructor volume. The hyperreals arrive as a stated extension of the reals with rules, not as an ultrapower.

The fork this creates in the corpus

This is not a competing set of theorems. It is a competing starting point, and the results come out the same — which is precisely why the disagreement is about teaching rather than truth. corral-vector-calculus assumes the standard limit-based Calculus I and II as its prerequisite; keisler-elementary-calculus rebuilds exactly that prerequisite on a different foundation and then covers Corral’s material again in its chapters 11–13. Two sources, one subject, two answers to what a student should meet first. See synthesis and vector-calculus.

Related

keisler-elementary-calculus · vector-calculus · mathematics · h-jerome-keisler · synthesis