Paul Halmos | |
|---|---|
 | |
| Born | Paul Richard Halmos (1916-03-03)3 March 1916 |
| Died | 2 October 2006(2006-10-02) (aged 90) Los Gatos, California, U.S. |
| Nationality | Hungarian American |
| Alma mater | University of Illinois |
| Awards | Chauvenet Prize(1947) Lester R. Ford Award (1971,1977) Leroy P. Steele Prize(1983) |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Syracuse University University of Chicago University of Michigan University of Hawaiʻi Indiana University Santa Clara University |
| Doctoral advisor | Joseph L. Doob |
| Doctoral students | Errett Bishop Bernard Galler Donald Sarason V. S. Sunder Peter Rosenthal |
Paul Richard Halmos (Hungarian:Halmos Pál; 3 March 3 1916 – 2 October 2006) was aHungarian-bornAmericanmathematician and probabilist who made fundamental advances in the areas ofmathematical logic,probability theory,operator theory,ergodic theory, andfunctional analysis (in particular,Hilbert spaces). He was also recognized as a great mathematical expositor. He has been described as one ofThe Martians.[1]
Born in theKingdom of Hungary into aJewish family, Halmos immigrated to the United States at age 13. He obtained his B.A. from theUniversity of Illinois, majoring in mathematics while also fulfilling the requirements for a degree in philosophy. He obtained the degree after only three years, and was 19 years old when he graduated. He then began a Ph.D. in philosophy, still at theChampaign–Urbana campus. However, after failing his masters' oral exams,[2] he shifted to mathematics and graduated in 1938.Joseph L. Doob supervised his dissertation, titledInvariants of Certain Stochastic Transformations: The Mathematical Theory of Gambling Systems.[3]
Shortly after his graduation, Halmos left for theInstitute for Advanced Study, lacking both job and grant money. Six months later, he was working underJohn von Neumann, which proved a decisive experience. While at the Institute, Halmos wrote his first book,Finite Dimensional Vector Spaces, which immediately established his reputation as a fine expositor of mathematics.[4]
From 1967 to 1968 he was theDonegall Lecturer in Mathematics atTrinity College Dublin.
Halmos taught atSyracuse University, theUniversity of Chicago (1946–60), theUniversity of Michigan (~1961–67), theUniversity of Hawaii (1967–68),Indiana University (1969–85), and theUniversity of California at Santa Barbara (1976–78). From his 1985 retirement from Indiana until his death, he was affiliated with the Mathematics department atSanta Clara University (1985–2006).
In a series of papers reprinted in his 1962Algebraic Logic, Halmos devisedpolyadic algebras, an algebraic version offirst-order logic differing from the better knowncylindric algebras ofAlfred Tarski and his students. An elementary version of polyadic algebra is described inmonadic Boolean algebra.
In addition to his original contributions to mathematics, Halmos was an unusually clear and engaging expositor of university mathematics. He won theLester R. Ford Award in 1971[5] and again in 1977 (shared with W. P. Ziemer, W. H. Wheeler, S. H. Moolgavkar, J. H. Ewing and W. H. Gustafson).[6] Halmos chaired theAmerican Mathematical Society committee that wrote the AMS style guide for academic mathematics, published in 1973. In 1983, he received the AMS'sLeroy P. Steele Prize for exposition.
In theAmerican Scientist 56(4): 375–389 (Winter 1968), Halmos argued that mathematics is a creative art, and that mathematicians should be seen as artists, not number crunchers. He discussed the division of the field into mathology and mathophysics, further arguing that mathematicians and painters think and work in related ways.
Halmos's 1985 "automathography"I Want to Be a Mathematician is an account of what it was like to be an academic mathematician in 20th century America. He called the book "automathography" rather than "autobiography", because its focus is almost entirely on his life as a mathematician, not his personal life. The book contains the following quote on Halmos' view of what doing mathematics means:
Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
What does it take to be [a mathematician]? I think I know the answer: you have to be born right, you must continually strive to become perfect, you must love mathematics more than anything else, you must work at it hard and without stop, and you must never give up.
In these memoirs, Halmos claims to have invented the "iff" notation for the words "if and only if" and to have been the first to use the"tombstone" notation to signify theend of a proof,[7] and this is generally agreed to be the case. The tombstone symbol ∎ (Unicode U+220E) is sometimes called ahalmos.[8]
In 1994, Halmos received theDeborah and Franklin Haimo Award for Distinguished College or University Teaching of Mathematics.[9]
In 2005, Halmos and his wifeVirginia Halmos funded theEuler Book Prize, an annual award given by theMathematical Association of America for a book that is likely to improve the view of mathematics among the public. The first prize was given in 2007, the 300th anniversary ofLeonhard Euler's birth, toJohn Derbyshire for his book aboutBernhard Riemann and theRiemann hypothesis:Prime Obsession.[10]
In 2009George Csicsery featured Halmos in a documentary film also calledI Want to Be a Mathematician.[11]
Books by Halmos have led to so many reviews that lists have been assembled.[12][13]
The symbol ∎ is used throughout the entire book in place of such phrases as "Q.E.D." or "This completes the proof of the theorem" to signal the end of a proof.
