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| '''Errett Albert Bishop''' (July 14, 1928 – April 14, 1983)<ref>[http://libraries.ucsd.edu/historyofucsd/newsreleases/1983/19830422.html UCSD Obituary]</ref> was an [[United States|American]] [[mathematician]] known for his work on analysis. He is the father of [[constructive analysis]], because of his 1967 ''Foundations of Constructive Analysis'', where he [[Mathematical proof|proved]] most of the important [[theorem]]s in [[real analysis]] by [[Constructivism (mathematics)|constructive]] methods.
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| ==Life==
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| 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 year 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.
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| 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''.
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| Bishop taught at the [[University of California]], 1954–65. He spent the 1964–65 academic year at the [[Miller Institute|Miller Institute for Basic Research]] in [[Berkeley, California|Berkeley]]. He was a visiting scholar at the [[Institute for Advanced Study]] in 1961–62.<ref>[http://www.ias.edu/people/cos/ Institute for Advanced Study: A Community of Scholars]</ref> From 1965 until his death, he was professor at the [[University of California at San Diego]].
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| ==Work== | | == do not inquire about Nima chaos.' I am sorry to say sin == |
| Bishop's wide-ranging work falls into five categories:
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| # 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.
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| # The general theory of [[function algebra]]s. Here Bishop worked on [[uniform algebra]]s (commutative [[Banach algebra]]s with unit whose norms are the [[spectral norm]]s) proving results such as antisymmetric decomposition of a uniform algebra, the [[Bishop-DeLeeuw theorem]], and the proof of existence of [[Jensen measure]]s. Bishop wrote a 1965 survey "Uniform algebras," examining the interaction between the theory of uniform algebras and that of several complex variables.
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| # [[Banach space]]s and [[operator theory]], the subject of his thesis. He introduced what is now called the [[Bishop condition]], useful in the theory of [[decomposable operator]]s.
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| # 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 <math>\mathbb{C}^n</math>, and a new proof of [[Reinhold Remmert|Remmert]]'s [[proper mapping theorem]].
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| # [[Constructive mathematics]]. Bishop became interested in foundational issues while at the Miller Institute. His now-famous ''Foundations of Constructive Analysis'' (1967)<ref>{{cite journal|author=Stolzenberg, Gabriel|title=Review: Errett Bishop, ''Foundations of Constructive Analysis''|journal=Bull. Amer. Math. Soc.|year=1970|volume=76|issue=2|pages=301–323|url=http://projecteuclid.org/euclid.bams/1183531480}}</ref> aimed to show that a constructive treatment of analysis is feasible, something about which [[Hermann Weyl|Weyl]] had been pessimistic. A 1985 revision, called ''Constructive Analysis'', was completed with the assistance of Douglas Bridges.
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| 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 mathematics on constructive mathematics. His talk was titled "The Constructivisation of Abstract Analysis." The American mathematical society 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." A. 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." {{harv|Warschawski|1985}} Robinson wrote in his review of Bishop's book that Bishop's historical commentary is "more vigorous than accurate".
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| ==Quotes==
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| *(A) "Mathematics is common sense";
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| *(B) "Do not ask whether a statement is true until you know what it means";
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| *(C) "A proof is any completely convincing argument";
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| *(D) "Meaningful distinctions deserve to be preserved".
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| :(Items A through D are principles of constructivism from his {{cite book|title=Schizophrenia in Contemporary Mathematics|url=http://books.google.com/books?id=fyUJGwAACAAJ|year=1973|publisher=[[American Mathematical Society]]}}, reprinted in Rosenblatt 1985)
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| *"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)
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| *"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)
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| *"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)
| | </ul> |
| *"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)
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| *"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)
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| *"Very possibly classical mathematics will cease to exist as an independent discipline" (Bishop, 1970, p. 54)
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| *"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) | |
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| ==Bishopian constructivism and attitude toward classical mathematics==
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| Bishop described what he perceived to be a lack of "meaning" in classical mathematics, a condition he described both as "[[schizophrenia]]" and a "debasement of meaning", and expressed the sentiment in 1968 that its demise is "very possible".<ref>{{citation
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| | last1 = Katz | first1 = Karin Usadi
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| | last2 = Katz | first2 = Mikhail G.
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| | arxiv = 1110.5456
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| | doi =
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| | issue = 2
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| | journal = [[Intellectica]]
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| | pages = 223–302
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| | title = Meaning in Classical Mathematics: Is it at Odds with Intuitionism?
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| | volume = 56
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| | year = 2011}}</ref>
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| ==See also==
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| *[[Constructive analysis]]
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| *[[Constructivism (mathematics)]]
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| *[[Criticism of non-standard analysis]]
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| ==Re-notes==
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| {{reflist}}
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| ==References==
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| *Bishop, Errett 1967. ''Foundations of Constructive Analysis'', New York: Academic Press. ISBN 4-87187-714-0
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| *Bishop, Errett and Douglas Bridges, 1985. ''Constructive Analysis''. New York: Springer. ISBN 0-387-15066-8.
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| *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.
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| *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.
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| *Bridges, Douglas, "Constructive Mathematics", The Stanford Encyclopedia of Philosophy (Winter 2004 Edition), Edward N. Zalta (ed.), [http://plato.stanford.edu/archives/win2004/entries/mathematics-constructive/] - Online article by Douglas Bridges, a collaborator of Bishop.
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| *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.
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| *{{citation|editor1-last=Rosenblatt|editor1-first=M.|last1=Warschawski|first1=S.|chapter=Errett Bishop - In Memoriam|title=Errett Bishop: Reflections on him and his research|volume= 39|series=Contemporary Mathematics|publisher=American Mathematical Society}}
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| *Schechter, Eric 1997. ''Handbook of Analysis and its Foundations''. New York: Academic Press. ISBN 0-12-622760-8 — Constructive ideas in analysis, cites Bishop.
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| == External links == | |
| *{{MacTutor Biography|id=Bishop}}
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| *{{MathGenealogy|id=6445|title=Errett Bishop}}
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| {{Authority control|VIAF=29612057}}
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME =Bishop, Errett
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = American mathematician
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| | DATE OF BIRTH = July 10, 1928
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| | PLACE OF BIRTH =
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| | DATE OF DEATH = April 14, 1983
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| | PLACE OF DEATH =
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| }}
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| {{DEFAULTSORT:Bishop, Errett}}
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| <!-- no such cat yet [[Category:University of California, San Diego faculty]] --> | |
| [[Category:1928 births]]
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| [[Category:1983 deaths]]
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| [[Category:20th-century mathematicians]]
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| [[Category:American mathematicians]]
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| [[Category:Mathematical analysts]]
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| [[Category:University of Chicago alumni]]
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| [[Category:University of California, Berkeley faculty]]
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| [[Category:University of California, San Diego faculty]]
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| [[Category:Institute for Advanced Study visiting scholars]]
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| [[Category:Functional analysts]]
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| [[Category:United States Army personnel]]
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