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**Paul Richard Halmos** (Hungarian: *Halmos Pál*; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician 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. He has been described as one of The Martians.^{[1]}

Paul Halmos | |
---|---|

Born | Paul Richard Halmos March 3, 1916 |

Died | October 2, 2006 Los Gatos, California, U.S. | (aged 90)

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 Indiana University Santa Clara University |

Doctoral advisor | Joseph L. Doob |

Doctoral students | Errett Bishop Bernard Galler Donald Sarason V. S. Sunder Peter Rosenthal |

Born in Hungary into a Jewish family, Halmos arrived in the U.S. at 13 years of age. He 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, still at the Champaign–Urbana campus; but, after failing his masters' oral exams,^{[2]} 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*.^{[3]}

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.^{[4]}

From 1967 to 1968 he was the Donegall Lecturer in Mathematics at Trinity College Dublin.

Halmos taught at Syracuse University, the University of Chicago (1946–60), the University of Michigan (~1961–67), the University of Hawaii (1967–68), Indiana University (1969–85), and the University of California at Santa Barbara (1976–78). From his 1985 retirement from Indiana until his death, he was affiliated with the Mathematics department at Santa Clara University (1985–2006).

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^{[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 the American Mathematical Society committee that wrote the AMS style guide for academic mathematics, published in 1973. In 1983, he received the AMS's Leroy P. 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,^{[7]} and this is generally agreed to be the case. The tombstone symbol ∎ (Unicode U+220E) is sometimes called a *halmos*.^{[8]}

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.^{[9]}

In 2009 George Csicsery featured Halmos in a documentary film also called *I Want to Be a Mathematician*.^{[10]}

Books by Halmos have led to so many reviews that lists have been assembled.^{[11]}^{[12]}

- 1942.
*Finite-Dimensional Vector Spaces*. Springer-Verlag.^{[13]} - 1950.
*Measure Theory*. Springer Verlag.^{[14]} - 1951.
*Introduction to Hilbert Space and the Theory of Spectral Multiplicity*. Chelsea.^{[15]} - 1956.
*Lectures on Ergodic Theory*. Chelsea.^{[16]} - 1960.
*Naive Set Theory*. Springer Verlag. - 1962.
*Algebraic Logic*. Chelsea. - 1963.
*Lectures on Boolean Algebras*. Van Nostrand. - 1967.
*A Hilbert Space Problem Book*. Springer-Verlag. - 1973. (with Norman E. Steenrod, Menahem M. Schiffer, and Jean A. Dieudonne).
*How to Write Mathematics*. American Mathematical Society. ISBN 978-0-8218-0055-3 - 1978. (with V. S. Sunder).
*Bounded Integral Operators on L² Spaces*. Springer Verlag^{[17]} - 1985.
*I Want to Be a Mathematician*. Springer-Verlag. - 1987.
*I Have a Photographic Memory*. Mathematical Association of America. - 1991.
*Problems for Mathematicians, Young and Old*, Dolciani Mathematical Expositions, Mathematical Association of America. - 1996.
*Linear Algebra Problem Book*, Dolciani Mathematical Expositions, Mathematical Association of America. - 1998. (with Steven Givant).
*Logic as Algebra*, Dolciani Mathematical Expositions No. 21, Mathematical Association of America.^{[18]} - 2009. (posthumous, with Steven Givant),
*Introduction to Boolean Algebras*,^{[19]}Springer.

**^**A marslakók legendája - György Marx**^**The Legend of John Von Neumann. P. R. Halmos. The American Mathematical Monthly, Vol. 80, No. 4. (Apr., 1973), pp. 382–394.**^**Halmos, Paul R. "Invariants of certain stochastic transformations: The mathematical theory of gambling systems." Duke Mathematical Journal 5, no. 2 (1939): 461–478.**^**Albers, Donald J. (1982). "Paul Halmos: Maverick Mathologist".*Two-Year College Mathematics Journal*.**13**(4). Mathematical Association of America: 226–242. doi:10.2307/3027125. JSTOR 3027125.**^**Halmos, Paul R. (1970). "Finite-dimensional Hilbert spaces".*Amer. Math. Monthly*.**77**(5): 457–464. doi:10.2307/2317378. JSTOR 2317378.**^**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**(7): 503–516. doi:10.2307/2319347. JSTOR 2319347.**^**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.

**^**"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.**^**"The Mathematical Association of America's Euler Book Prize".*Mathematical Association of America*. Archived from the original on 27 January 2013. Retrieved 1 February 2011.**^**"I Want to Be a Mathematician (Video 2009)" on IMdB.**^**"Reviews of Paul Halmos' books Part 1".*MacTutor*. August 2016. Archived from the original on 3 September 2023.**^**"Reviews of Paul Halmos's books Part 2".*MacTutor*. August 2016. Archived from the original on 3 September 2023.**^**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. Archived (PDF) from the original on 18 February 2024.**^**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. Archived (PDF) from the original on 3 September 2023.**^**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. Archived (PDF) from the original on 18 February 2024.**^**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. Archived (PDF) from the original on 3 September 2023.**^**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.**^**Johnson, Mark (11 February 1999). "Review of*Logic as Algebra*by Paul Halmos and Steven Givant".*MAA Reviews, Mathematical Association of America*.**^**Givant, Steven; Halmos, Paul (2 December 2008).*Introduction to Boolean Algebras*. Springer. ISBN 978-0387402932.

- J. H. Ewing; F. W. Gehring (1991).
*Paul Halmos: Celebrating 50 Years of Mathematics*. Springer-Verlag. ISBN 0-387-97509-8. OCLC 22859036. Includes a bibliography of Halmos's writings through 1991. - John Ewing (October 2007). "Paul Halmos: In His Own Words" (PDF).
*Notices of the American Mathematical Society*.**54**(9): 1136–1144. Retrieved 15 January 2008. - Paul Halmos (1985).
*I want to be a Mathematician: An Automathography*. Springer-Verlag. ISBN 0-387-96470-3. OCLC 230812318. - Paul R. Halmos (1970). "How to Write Mathematics" (PDF).
*L'Enseignement mathématique*.**16**(2): 123–152.

Wikiquote has quotations related to **Paul Halmos**.

- O'Connor, John J.; Robertson, Edmund F., "Paul Halmos",
*MacTutor History of Mathematics Archive*, University of St Andrews - "Paul Halmos: A Life in Mathematics", Mathematical Association of America (MAA)
- Finite-Dimensional Vector Spaces
- "Examples of Operators" a series of video lectures on operators in Hilbert Space given by Paul Halmos during his 2-week stay in Australia, Briscoe Center Digital Collections