Bishop-Keisler controversy

The Bishop-Keisler controversy is an episode belonging to the history of mathematics, or perhaps it is the epistemology of mathematics, and definitely the polemics of mathematics. The controversy was triggered by the view of Errett Bishop that non-standard analysis is a "formal finesse" in its approach to the foundations of infinitesimal calculus, and even constitutes a "debasement" of meaning.

Context

Errett Bishop represents the constructivist school, whereas Abraham Robinson and H. Jerome Keisler represent non-standard analysis. The two fields are polar opposites of each other. It is interesting to observe that two people with very similar training in classical mathematics, can arrive at such different conclusions as to the nature of the mathematical trade.

Meta-mathematically speaking, Bishop's constructivism lies at the opposite extreme to Robinson's non-standard analysis, in the spectrum of mathematical sensibility. Bishop's criticism of the latter was expressed in a 1977 review of Keisler's textbook Elementary Calculus: an infinitesimal approach, in the Bulletin of the American Mathematical Society.

Polemical review

Bishop first provides the reader with an assortment of quotations from Keisler:

"In '60, Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."

Clearly in a disapproving fashion, Bishop quotes Keisler to the effect that

"In discussing the real line we remarked that we have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus, it is helpful to imagine a line in physical space as a hyperreal line."

Getting down to business, Bishop describes Keisler's introduction of infinitesimals in the following terms:

"The impasse is broken by forgetting that Δx is a real number, calling it something else (an infinitesimal), and telling us that it is all right to neglect it."

Bishop proceeds to refer to the theoretical underpinnings of non-standard analysis as "a supposedly consistent system of axioms". Toward the very end of the review, Bishop finally goes for the jugular:

"The real damage lies in [Keisler's] obfuscation and devitalization of those wonderful ideas."

In a final passionate appeal, Bishop notes:

"Although it seems to be futile, I always tell my calculus students that mathematics is not esoteric: It is common sense. (Even the notorious (ε, δ)-definition of limit is common sense, and moreover it is central to the important practical problems of approximation and estimation.)"

Response

As a response, Keisler published a 10-page practical guide describing the success of Elementary Calculus: an infinitesimal approach in the classroom.

Other foundational controversies

• Controversy over Cantor's theory
• Brouwer-Hilbert controversy

Notes