Errett Bishop

Errett A. Bishop
Born	(1928-07-14)July 14, 1928 Newton, Kansas
Died	April 14, 1983(1983-04-14)(aged 54) San Diego, California
Nationality	American
Alma mater	University of Chicago
Known for	Bishop set,Constructive analysis
Scientific career
Fields	Mathematics
Institutions	University of California at San Diego
Doctoral advisor	Paul Halmos

Errett Albert Bishop(July 14, 1928 – April 14, 1983)was anAmericanmathematicianknown for his work on analysis. He is best known for developingconstructive analysisin his 1967Foundations of Constructive Analysis,where heprovedmost of the importanttheoremsinreal analysisusing "constructivist"methods.

Life

Errett Bishop's father, Albert T. Bishop, graduated from theUnited States Military AcademyatWest Point,ending his career as professor of mathematics atWichita State Universityin 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 inNewton, Kansas. Errett and his sister were apparent math prodigies.

Bishop entered theUniversity of Chicagoin 1944, obtaining both the BS and MS in 1947. The doctoral studies he began in that year were interrupted by two years in theUS Army,1950–52, doing mathematical research at theNational Bureau of Standards.He completed his Ph.D. in 1954 underPaul Halmos;his thesis was titledSpectral Theory for Operations on Banach Spaces.

Bishop taught at theUniversity of California,1954–65. He spent the 1964–65 academic year at theMiller Institute for Basic ResearchinBerkeley.He was a visiting scholar at theInstitute for Advanced Studyin 1961–62.From 1965 until his death, he was professor at theUniversity of California at San Diego.

Work

Bishop's work falls into five categories:

1. Polynomial and rational approximation. Examples are extensions ofMergelyan's approximation theoremand the theorem ofFrigyes RieszandMarcel Rieszconcerning measures on the unit circle orthogonal to polynomials.
2. The general theory offunction algebras.Here Bishop worked onuniform algebras(commutativeBanach algebraswith unit whose norms are thespectral norms) proving results such as antisymmetric decomposition of a uniform algebra, theBishop–DeLeeuw theorem,and the proof of existence ofJensen measures.Bishop wrote a 1965 survey "Uniform algebras," examining the interaction between the theory of uniform algebras and that of several complex variables.
3. Banach spacesandoperator theory,the subject of his thesis. He introduced what is now called theBishop condition,useful in the theory ofdecomposable operators.
4. The theory of functions ofseveral complex variables.An example is his 1962 "Analyticity in certain Banach spaces." He proved important results in this area such as thebiholomorphic embedding theoremfor aStein manifoldas a closedsubmanifoldin$\mathbb {\mathbb {C} } ^{n}$,and a new proof ofRemmert'sproper mapping theorem.
5. Constructive mathematics.Bishop became interested in foundational issues while at the Miller Institute. His now-famousFoundations of Constructive Analysis(1967)aimed to show that a constructive treatment of analysis is feasible, something about whichWeylhad been pessimistic. A 1985 revision, calledConstructive Analysis,was completed with the assistance of Douglas Bridges.

In 1972, Bishop (with Henry Cheng) publishedConstructive Measure Theory.

In the later part of his life, Bishop was seen as the leading mathematician in the area of constructivist mathematics. In 1966, he was invited to speak at theInternational Congress of Mathematicianson that theme. His talk was titled "The Constructivisation of Abstract Mathematical Analysis."TheAmerican Mathematical Societyinvited 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 Robinsonwrote of Bishop's work in constructivist 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, however, 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 hisSchizophrenia in Contemporary Mathematics.American Mathematical Society.1973.(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 thecomplex numbersare 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 spaceXcan 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, page 1)

See also

• Bishop set
• Constructive analysis
• Constructivism (mathematics)
• Criticism of non-standard analysis

References

• Bishop, Errett 1967.Foundations of Constructive Analysis,New York: Academic Press.ISBN4-87187-714-0
• Bishop, Errett and Douglas Bridges, 1985.Constructive Analysis.New York: Springer.ISBN0-387-15066-8.
• Bishop, Errett (1970) Mathematics as a numerical language. 1970 Intuitionism and Proof Theory (Proc. Conf., Buffalo, New York 1968) pages 53–71. North-Holland, Amsterdam.
• Bishop, E. (1985) Schizophrenia in contemporary mathematics. In Errett Bishop: reflections on him and his research (San Diego, California, 1983), 1–32, Contemp. Math. 39, American Math. Society, Providence, Rhode Island.
• Bridges, Douglas, "Constructive Mathematics", The Stanford Encyclopedia of Philosophy (Winter 2004 Edition), Edward N. Zalta (ed.),[1]- 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.
• Warschawski, S. (1985), "Errett Bishop - In Memoriam", in Rosenblatt, M. (ed.),Errett Bishop: Reflections on him and his research,Contemporary Mathematics, vol. 39, American Mathematical Society
• Schechter, Eric 1997.Handbook of Analysis and its Foundations.New York: Academic Press.ISBN0-12-622760-8— Constructive ideas in analysis, cites Bishop.

External links

• O'Connor, John J.;Robertson, Edmund F.,"Errett Bishop",MacTutor History of Mathematics Archive,University of St Andrews
• Errett Bishopat theMathematics Genealogy Project