Paul Halmos

"Halmos" redirects here. For the mathematical symbol, see Tombstone (typography). For the church music composer, see László Halmos.
Paul Halmos
Born (1916-03-03)March 3, 1916
Budapest, Austria-Hungary
Died October 2, 2006(2006-10-02) (aged 90)
Los Gatos, California, U.S.
Nationality Hungarian American
Fields Mathematics
Institutions Syracuse University
University of Chicago
University of Michigan
Indiana University
Santa Clara University
Alma mater Institute for Advanced Study
University of Illinois
Doctoral advisor Joseph L. Doob
Doctoral students Errett Bishop
H. Arlen Brown
Bernard Galler
Don Hadwin
Eric Nordgren
Herman Rubin
Donald Sarason
V. S. Sunder
Notable awards Chauvenet Prize (1947)
Lester R. Ford Award (1971,1977)
Leroy P. Steele Prize (1983)

Paul Richard Halmos (Hungarian: Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-Jewish-born American mathematician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great mathematical expositor.

Early life and education

Halmos arrived in the U.S. at 13 years of age, but never lost his Hungarian accent. Halmos obtained his B.A. from the University of Illinois, majoring in mathematics, but fulfilling the requirements for both a math and philosophy degree. He took only three years to obtain the degree, and was only 19 when he graduated. He then began a Ph.D. in philosophy; but, after failing his masters' oral exams,[1] he shifted to mathematics, graduating in 1938. Joseph L. Doob supervised his dissertation, titled Invariants of Certain Stochastic Transformations: The Mathematical Theory of Gambling Systems.


Shortly after his graduation, Halmos left for the Institute for Advanced Study, lacking both job and grant money. Six months later, he was working under John 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.

Halmos taught at Syracuse University, the University of Chicago (1946–60), the University of Michigan (~1961–67), the University of California at Santa Barbara (1976–78), the University of Hawaii, and Indiana University. From his 1985 retirement from Indiana until his death, he was affiliated with the Mathematics department at Santa Clara University.


In a series of papers reprinted in his 1962 Algebraic Logic, Halmos devised polyadic algebras, an algebraic version of first-order logic differing from the better known cylindric algebras of Alfred Tarski and his students. An elementary version of polyadic algebra is described in monadic Boolean algebra.

In addition to his original contributions to mathematics, Halmos was an unusually clear and engaging expositor of university mathematics. He won the Lester R. Ford Award in 1971[2] and again in 1977 (shared with W. P. Ziemer, W. H. Wheeler, S. H. Moolgavkar, J. H. Ewing and W. H. Gustafson).[3] Halmos chaired the American Mathematical Society committee that wrote the AMS style guide for academic mathematics, published in 1973. In 1983, he received the AMS's Steele Prize for exposition.

In the American Scientist 56(4): 375–389, 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.

Paul Halmos, 1985

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 the end of a proof,[4] and this is generally agreed to be the case. The tombstone symbol ∎ (Unicode U+220E) is sometimes called a halmos.[5]

In 2005, Halmos and his wife Virginia funded the Euler Book Prize, an annual award given by the Mathematical 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 of Leonhard Euler's birth, to John Derbyshire for his book about Bernhard Riemann and the Riemann hypothesis: Prime Obsession.[6]

Books by Halmos

See also


  1. The Legend of John Von Neumann. P. R. Halmos. The American Mathematical Monthly, Vol. 80, No. 4. (Apr., 1973), pp. 382–394.
  2. Halmos, Paul R. (1970). "Finite-dimensional Hilbert spaces". Amer. Math. Monthly. 77: 457–464. doi:10.2307/2317378.
  3. Ziemer, William P.; Wheeler, William H.; Moolgavkar; Halmos, Paul R.; Ewing, John H.; Gustafson, William H. (1976). "American mathematics from 1940 to the day before yesterday". Amer. Math. Monthly. 83: 503–516.
  4. Halmos, Paul (1950). Measure Theory. New York: Van Nostrand. pp. vi. 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.
  5. "The symbol is definitely not my invention — it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like ▯, and is used to indicate an end, usually the end of a proof. It is most frequently called the 'tombstone', but at least one generous author referred to it as the 'halmos'.", Halmos (1985) p. 403.
  6. The Mathematical Association of America's Euler Book Prize, retrieved 2011-02-01.
  7. Kac, Mark (1943). "Review: Finite-dimensional vector spaces, by P. R. Halmos" (PDF). Bull. Amer. Math. Soc. 49 (5): 349–350. doi:10.1090/s0002-9904-1943-07899-8.
  8. Oxtoby, J. C. (1953). "Review: Measure theory, by P. R. Halmos" (PDF). Bull. Amer. Math. Soc. 59 (1): 89–91. doi:10.1090/s0002-9904-1953-09662-8.
  9. Lorch, E. R. (1952). "Review: Introduction to Hilbert space and the theory of spectral multiplicity, by P. R. Halmos" (PDF). Bull. Amer. Math. Soc. 58 (3): 412–415. doi:10.1090/s0002-9904-1952-09595-1.
  10. Dowker, Yael N. (1959). "Review: Lectures on ergodic theory, by P. R. Halmos" (PDF). Bull. Amer. Math. Soc. 65 (4): 253–254. doi:10.1090/s0002-9904-1959-10331-1.
  11. Zaanen, Adriaan (1979). "Review: Bounded integral operators on L² spaces, by P. R. Halmos and V. S. Sunder" (PDF). Bull. Amer. Math. Soc. (N.S.). 1 (6): 953–960. doi:10.1090/s0273-0979-1979-14699-8.


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