Errett Bishop

Errett Albert Bishop (July 14, 1928 – April 14, 1983)[1] was an American mathematician known for his work on analysis. He expanded constructive analysis in 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. Based on personal conversations with Bishop, D. Hill recounted that Bishop's rejection of what he viewed as the fundamentalist nature of classical mathematics was closely related in Bishop's mind with his rejection of what he viewed as his fundamentalist Protestant upbringing.[2] Errett 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. He was a visiting scholar at the Institute for Advanced Study in 1961–62.[3] 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:

  1. 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.
  2. 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.
  3. 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.
  4. 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 \mathbb{C}^n, and a new proof of Remmert's proper mapping theorem.
  5. Constructive mathematics. Bishop became interested in foundational issues while at the Miller Institute. His now-famous Foundations of Constructive Analysis (1967)[4] 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 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." (Warschawski 1985) Robinson wrote in his review of Bishop's book that Bishop's historical commentary is "more vigorous than accurate".

Quotes

(Items A through D are principles of constructivism from his Schizophrenia in Contemporary Mathematics. American Mathematical Society. 1973. , reprinted in Rosenblatt 1985)

Bishopian constructivism and attitude toward classical mathematics

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".[5]

See also

Re-notes

  1. ↑ UCSD Obituary
  2. ↑ Kanovei, Vladimir; Katz, Karin U.; Katz, Mikhail G.; Schaps, Mary, "Proofs and Retributions, Or: Why Sarah Can't Take Limits", Foundations of Science, doi:10.1007/s10699-013-9340-0.
  3. ↑ Institute for Advanced Study: A Community of Scholars
  4. ↑ Stolzenberg, Gabriel (1970). "Review: Errett Bishop, Foundations of Constructive Analysis". Bull. Amer. Math. Soc. 76 (2): 301–323. doi:10.1090/s0002-9904-1970-12455-7.
  5. ↑ Katz, Karin Usadi; Katz, Mikhail G. (2011), "Meaning in Classical Mathematics: Is it at Odds with Intuitionism?", Intellectica 56 (2): 223–302, arXiv:1110.5456

References

External links

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