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Errett Bishop

Errett Albert Bishop (1928–1983) was an American mathematician known for his work on analysis. He is the father of constructivist analysis, by virtue of his 1967 Foundations of Constructive Analysis, where he proved most of the important theorems in real analysis by constructive methods.

Life

Errett Bishop's father, Albert T Bishop, graduated from the United States Military Academy at West Point, ending his career as professor of mathematics at Wichita State University in Kansas. Although he died when Errett was only 5 years old, he influenced Errett's eventual career by the math texts he left behind, which is how Errett discovered mathematics. Errett grew up in Newton, Kansas. He and his sister were apparent math prodigies.

Bishop entered the University of Chicago in 1944, obtaining both the BS and MS in 1947. The doctoral studies he began in that year were interrupted by two years in the US Army, 1950–52, doing mathematical research at the National Bureau of Standards. He completed his Ph.D. in 1954 under Paul Halmos; his thesis was titled Spectral Theory for Operations on Banach Spaces.

Bishop taught at the University of California, 1954–65. He spent the 1964–65 academic year at the Miller Institute for Basic Research in Berkeley. From 1965 until his death, he was professor at the University of California at San Diego.

Work

Bishop's wide-ranging work falls into five categories:

Polynomial and rational approximation. Examples are extensions of Mergelyan's approximation theorem and the theorem of Frigyes Riesz and Marcel Riesz concerning measures on the unit circle orthogonal to polynomials.
The general theory of function algebras. Here Bishop worked on uniform algebras (commutative Banach algebras with unit whose norms are the spectral norms) proving results such as antisymmetric decomposition of a uniform algebra, the Bishop-DeLeeuw theorem, and the proof of existence of Jensen measures. Bishop wrote a 1965 survey "Uniform algebras," examining the interaction between the theory of uniform algebras and that of several complex variables.
Banach spaces and operator theory, the subject of his thesis. He introduced what is now called the Bishop condition, useful in the theory of decomposable operators.
The theory of functions of several complex variables. An example is his 1962 "Analyticity in certain Banach spaces." He proved important results in this area such as the biholomorphic embedding theorem for a Stein manifold as a closed submanifold in Cn, and a new proof of Remmert's proper mapping theorem.
Constructive mathematics. Bishop become interested in foundational issues while at the Miller Institute. His now-famous Foundations of constructive analysis (1967) aimed to show that a constructive treatment of analysis is feasible, something about which Weyl had been pessimistic. A 1985 revision, called Constructive analysis, was completed with the assistance of Douglas Bridges.

In 1972, Bishop (with Henry Cheng) published Constructive measure theory.

Bishop vs Keisler

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

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.)"

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

Quotes

"Classical mathematics concerns itself with operations that can be carried out by God.. Mathematics belongs to man, not to God... When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself." (Bishop 1967)

From his 1973 Schizophrenia in contemporary mathematics, reprinted in Rosenblatt 1985:

*"Mathematics is common sense."

*"Do not ask whether a statement is true until you know what it means."

*"A proof is any completely convincing argument."

*"Meaningful distinctions deserve to be preserved."

References

Bishop, Errett 1967. Foundations of Constructive Analysis, New York: Academic Press.
Bishop, Errett and Douglas Bridges, 1985. Constructive Analysis. New York: Springer. ISBN 0-387-15066-8.
Bridges, Douglas, "Constructive Mathematics", The Stanford Encyclopedia of Philosophy (Winter 2004 Edition), Edward N. Zalta (ed.), - Online article by Douglas Bridges, a collaborator of Bishop.
Rosenblatt, M., ed., 1985. Errett Bishop: Reflections on him and his research. Proceedings of the memorial meeting for Errett Bishop held at the University of California-San Diego, September 24, 1983. Contemporary Mathematics 39. AMS.
Schechter, Eric 1997. Handbook of Analysis and its Foundations. New York: Academic Press. ISBN 0-12-622760-8 — Constructive ideas in analysis, cites Bishop.