Errett Albert Bishop (July 14, 1928 – April 14, 1983) was an

- 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. — Constructive ideas in analysis, cites Bishop.

American
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the "United States" or "America"
** Americans, citizens and nationals of the United States of America
** American ancestry, p ...

mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
...

known for his work on analysis. He expanded constructive analysis in his 1967 ''Foundations of Constructive Analysis'', where he proved most of the important theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the ...

s in real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...

by constructive methods.
Life

Errett Bishop's father, Albert T. Bishop, graduated from theUnited States Military Academy
The United States Military Academy (USMA), also known metonymically as West Point or simply as Army, is a United States service academy in West Point, New York. It was originally established as a fort, since it sits on strategic high gro ...

at West Point
The United States Military Academy (USMA), also known metonymically as West Point or simply as Army, is a United States service academy in West Point, New York. It was originally established as a fort, since it sits on strategic high grou ...

, ending his career as professor of mathematics at Wichita State University
Wichita State University (WSU) is a public research university in Wichita, Kansas, United States. It is governed by the Kansas Board of Regents. The university offers more than 60 undergraduate degree programs in more than 200 areas of study in ...

in Kansas. Although he died when Errett was less than 4 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
Newton is a city in and the county seat of Harvey County, Kansas, United States. As of the 2020 census, the population of the city was 18,602. Newton is located north of Wichita. The city of North Newton is located immediately north and e ...

.
Errett and his sister were apparent math prodigies.
Bishop entered the University of Chicago
The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the ...

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
The United States Army (USA) is the land service branch of the United States Armed Forces. It is one of the eight U.S. uniformed services, and is designated as the Army of the United States in the U.S. Constitution.Article II, section 2, c ...

, 1950–52, doing mathematical research at the National Bureau of Standards. He completed his Ph.D. in 1954 under Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operato ...

; his thesis was titled ''Spectral Theory for Operations on Banach Spaces''.
Bishop taught at the University of California
The University of California (UC) is a public land-grant research university system in the U.S. state of California. The system is composed of the campuses at Berkeley, Davis, Irvine, Los Angeles, Merced, Riverside, San Diego, San Franc ...

, 1954–65. He spent the 1964–65 academic year at the Miller Institute for Basic Research in Berkeley. He was a visiting scholar at the Institute for Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

in 1961–62. 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 ofFrigyes Riesz
Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathema ...

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 space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vecto ...

s and operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...

, 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 In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a S ...

as a closed submanifold in $\backslash mathbb^n$, and a new proof of Remmert's proper mapping theorem.
# Constructive mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove t ...

. Bishop became 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''. In the later part of his life Bishop was seen as the leading mathematician in the area of constructive mathematics. In 1966 he was invited to speak at the International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the Nevanlinna Prize (to be rename ...

on constructive mathematics. His talk was titled "The Constructivisation of Abstract Mathematical Analysis." The American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

invited him to give four hour-long lectures as part of the Colloquium Lectures series. The title of his lectures was "Schizophrenia of Contemporary Mathematics." Abraham Robinson wrote of his work in constructive mathematics: "Even those who are not willing to accept Bishop's basic philosophy must be impressed with the great analytical power displayed in his work." Robinson wrote in his review of Bishop's book that Bishop's historical commentary is "more vigorous than accurate".
Quotes

*(A) "Mathematics is common sense"; *(B) "Do not ask whether a statement is true until you know what it means"; *(C) "A proof is any completely convincing argument"; *(D) "Meaningful distinctions deserve to be preserved". :(Items A through D are principles of constructivism from his (Reprinted in Rosenblatt 1985.) *"The primary concern of mathematics is number, and this means the positive integers. . . . In the words of Kronecker, the positive integers were created by God. Kronecker would have expressed it even better if he had said that the positive integers were created by God for the benefit of man (and other finite beings). Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. 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, Chapter 1, A Constructivist Manifesto, page 2) *"We are not contending that idealistic mathematics is worthless from the constructive point of view. This would be as silly as contending that unrigorous mathematics is worthless from the classical point of view. Every theorem proved with idealistic methods presents a challenge: to find a constructive version, and to give it a constructive proof." (Bishop 1967, Preface, page x) *"Theorem 1 is the famous theorem of Cantor, that the real numbers are uncountable. The proof is essentially Cantor's 'diagonal' proof. Both Cantor's theorem and his method of proof are of great importance." (Bishop 1967, Chapter 2, Calculus and the Real Numbers, page 25) *"The real numbers, for certain purposes, are too thin. Many beautiful phenomena become fully visible only when the complex numbers are brought to the fore." (Bishop 1967, Chapter 5, Complex Analysis, page 113) *"It is clear that many of the results in this book could be programmed for a computer, by some such procedure as that indicated above. In particular, it is likely that most of the results of Chaps. 2, 4, 5, 9, 10, and 11 could be presented as computer programs. As an example, a complete separable metric space ''X'' can be described by a sequence of real numbers, and therefore by a sequence of integers, simply by listing the distances between each pair of elements of a given countable dense set. . . . As written, this book is person-oriented rather than computer-oriented. It would be of great interest to have a computer-oriented version." (Bishop 1967, Appendix B, Aspects of Constructive Truth, pages 356 and 357) *"Very possibly classical mathematics will cease to exist as an independent discipline" (Bishop, 1970, p. 54) *"Brouwer's criticisms of classical mathematics were concerned with what I shall refer to as 'the debasement of meaning (Bishop in Rosenblatt, 1985, p. 1)See also

* Bishop set * Constructive analysis *Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove t ...

* Criticism of non-standard analysis Nonstandard analysis and its offshoot, nonstandard calculus, have been criticized by several authors, notably Errett Bishop, Paul Halmos, and Alain Connes. These criticisms are analyzed below.
Introduction
The evaluation of nonstandard analysi ...

Notes

References

*Bishop, Errett 1967. ''Foundations of Constructive Analysis'', New York: Academic Press. *Bishop, Errett and Douglas Bridges, 1985. ''Constructive Analysis''. New York: Springer. . *Bishop, Errett (1970) Mathematics as a numerical language. 1970 Intuitionism and Proof Theory (Proc. Conf., Buffalo, N.Y., 1968) pp. 53–71. North-Holland, Amsterdam. *Bishop, E. (1985) Schizophrenia in contemporary mathematics. In Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), 1–32, Contemp. Math. 39, Amer. Math. Soc., Providence, RI. *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. — Constructive ideas in analysis, cites Bishop.

External links

* * {{DEFAULTSORT:Bishop, Errett 1928 births 1983 deaths 20th-century American mathematicians Complex analysts Mathematical analysts University of Chicago alumni University of California, Berkeley College of Letters and Science faculty University of California, San Diego faculty Institute for Advanced Study visiting scholars Functional analysts United States Army personnel