Errett Albert Bishop (July 14, 1928 – April 14, 1983) was an
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, pe ...
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
On ...
known for his work on analysis. He expanded
constructive analysis
In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics.
This contrasts with ''classical analysis'', which (in this context) simply means analysis done according to the (more comm ...
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 th ...
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 converg ...
by
constructive
Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has ...
methods.
Life
Errett Bishop's father, Albert T. Bishop, graduated from the
United 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 groun ...
at
West Point
The United States Military Academy (USMA), also known Metonymy, metonymically as West Point or simply as Army, is a United States service academies, United States service academy in West Point, New York. It was originally established as a f ...
, 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 b ...
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, cla ...
, 1950–52, doing mathematical research at the
National Bureau of Standards
The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
. 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, operator ...
; 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 Francisco, ...
, 1954–65. He spent the 1964–65 academic year at the
Miller Institute for Basic Research in
Berkeley
Berkeley most often refers to:
*Berkeley, California, a city in the United States
**University of California, Berkeley, a public university in Berkeley, California
* George Berkeley (1685–1753), Anglo-Irish philosopher
Berkeley may also refer ...
. 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
The University of California, San Diego (UC San Diego or colloquially, UCSD) is a public land-grant research university in San Diego, California. Established in 1960 near the pre-existing Scripps Institution of Oceanography, UC San Diego is t ...
.
Work
Bishop's wide-ranging work falls into five categories:
# Polynomial and rational approximation. Examples are extensions of
Mergelyan's approximation theorem Mergelyan's theorem is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951.
Statement
:Let ''K'' be a compact subset of the complex plane C such that C∖''K'' is conne ...
and the theorem of
Frigyes 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
Marcel Riesz ( hu, Riesz Marcell ; 16 November 1886 – 4 September 1969) was a Hungarian mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations ...
concerning measures on the unit circle orthogonal to polynomials.
# The general theory of
function algebra In functional analysis, a Banach function algebra on a compact Hausdorff space ''X'' is unital subalgebra, ''A'', of the commutative C*-algebra ''C(X)'' of all continuous, complex-valued functions from ''X'', together with a norm on ''A'' that m ...
s. Here Bishop worked on
uniform algebra
In functional analysis, a uniform algebra ''A'' on a compact Hausdorff topological space ''X'' is a closed (with respect to the uniform norm) subalgebra of the C*-algebra ''C(X)'' (the continuous complex-valued functions on ''X'') with the followi ...
s (commutative
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach spa ...
s with unit whose norms are the
spectral norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Preliminaries
Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m ro ...
s) 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 vector ...
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 operat ...
, the subject of his thesis. He introduced what is now called the
Bishop condition, useful in the theory of
decomposable operator In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced i ...
s.
# The theory of functions of
several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
. An example is his 1962 "Analyticity in certain Banach spaces." He proved important results in this area such as the
biholomorphic embedding theorem
In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic.
Formal definit ...
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 Stei ...
as a closed
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...
in
, and a new proof of
Remmert's
proper mapping theorem
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
.
#
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 th ...
. 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
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is ass ...
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
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorpo ...
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
In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics.
This contrasts with ''classical analysis'', which (in this context) simply means analysis done according to the (more comm ...
*
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 analysis i ...
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
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{{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