Editing Paul Halmos

{{Short description|Hungarian-American mathematician (1916–2006)}}
{{redirect|Halmos|the mathematical symbol|Tombstone (typography)|the church music composer|László Halmos}}
{{Hungarian name|Halmos Pál}}
{{Use dmy dates|date=April 2022}}
{{Infobox scientist
| name              = Paul Halmos
| image             = Paul Halmos.jpeg
| image_size        = 225px
| caption           = 
| birth_name        = Paul Richard Halmos
| birth_date        = {{Birth date|1916|03|03|df=y}}
| birth_place       = [[Budapest]], [[Austria-Hungary]]
| death_date        = {{Death date and age|2006|10|02|1916|03|03|df=y}}
| death_place       = [[Los Gatos, California]], U.S.
| nationality       = Hungarian<br />American
| fields            = [[Mathematics]]
| workplaces        = [[Syracuse University]]<br />[[University of Chicago]]<br />[[University of Michigan]] <br />[[University of Hawaiʻi]] <br />[[Indiana University]]<br />[[Santa Clara University]]
| alma_mater        = [[University of Illinois at Urbana-Champaign|University of Illinois]]
| doctoral_advisor  = [[Joseph L. Doob]]
| doctoral_students = [[Errett Bishop]]<br />[[Bernard Galler]]<br />[[Donald Sarason]]<br />[[V. S. Sunder]]<br />[[Peter Rosenthal]]
| known_for         = 
| awards            = [[Chauvenet Prize]] <small>(1947)</small><br />[[Lester R. Ford Award]]<small> (1971,1977)</small><br />[[Leroy P. Steele Prize]] <small>(1983)</small>
}}

'''Paul Richard Halmos''' ({{langx|hu|Halmos Pál}}; 3 March 1916 – 2 October 2006) was a [[Kingdom of Hungary|Hungarian]]-born [[United States|American]] [[mathematician]] and probabilist who made fundamental advances in the areas of [[mathematical logic]], [[probability theory]], [[operator theory]], [[ergodic theory]], and [[functional analysis]] (in particular, [[Hilbert space]]s). He was also recognized as a great mathematical expositor. He has been described as one of [[The Martians (scientists)|The Martians]].<ref>[http://fizikaiszemle.hu/archivum/fsz9703/marsl.html A marslakók legendája] - [[György Marx]]</ref>

==Early life and education==
Born in the [[Kingdom of Hungary]] into a [[History of the Jews in Hungary|Jewish]] family, Halmos immigrated to the United States at age 13. He obtained his B.A. from the [[University of Illinois at Urbana-Champaign|University 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 the [[Champaign–Urbana]] campus. However, after failing his masters' oral exams,<ref>The Legend of John Von Neumann. P. R. Halmos. The American Mathematical Monthly, Vol. 80, No. 4. (Apr., 1973), pp.&nbsp;382–394.</ref> he shifted to mathematics and graduated in 1938. [[Joseph L. Doob]] supervised his dissertation, titled ''Invariants of Certain Stochastic Transformations: The Mathematical Theory of Gambling Systems''.<ref>Halmos, Paul R. [http://projecteuclid.org/euclid.dmj/1077491241 "Invariants of certain stochastic transformations: The mathematical theory of gambling systems."] Duke Mathematical Journal 5, no. 2 (1939): 461–478.</ref>

==Career==
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.<ref>{{cite journal | title=Paul Halmos: Maverick Mathologist | first=Donald J. | last=Albers | journal=[[Two-Year College Mathematics Journal]] | volume=13 | number=4 | year=1982 | pages=226–242 | publisher=[[Mathematical Association of America]] | jstor=3027125 | doi=10.2307/3027125 }}</ref>

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).

==Accomplishments==
In a series of papers reprinted in his 1962 ''Algebraic Logic'', Halmos devised [[polyadic algebra]]s, an algebraic version of [[first-order logic]] differing from the better known [[cylindric algebra]]s 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<ref>{{cite journal|author=Halmos, Paul R.|title=Finite-dimensional Hilbert spaces|journal=Amer. Math. Monthly|volume=77|year=1970|issue=5|pages=457–464|doi=10.2307/2317378|jstor=2317378}}</ref> and again in 1977 (shared with W. P. Ziemer, W. H. Wheeler, S. H. Moolgavkar, J. H. Ewing and W. H. Gustafson).<ref>{{cite journal|author=Ziemer, William P.|author2=Wheeler, William H.|author3=Moolgavkar|author4=Halmos, Paul R.|author5=Ewing, John H.|author6=Gustafson, William H.|title=American mathematics from 1940 to the day before yesterday|journal=Amer. Math. Monthly|volume=83|year=1976|issue=7|pages=503–516|doi=10.2307/2319347|jstor=2319347}}</ref> 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 (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 {{not a typo|mathology}} and {{not a typo|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:

{{blockquote|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?}}

{{quote box
|align=right
|width=33%
|quote=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.
|source= — 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 (typography)|"tombstone"]] notation to signify the [[Q.E.D.|end of a proof]],<ref>{{Cite book
 | last = Halmos
 | first = Paul
 | year = 1950
 | title = Measure Theory
 | publisher = Van Nostrand
 | location = New York
 | quote = 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.
 | pages = vi
}}</ref> and this is generally agreed to be the case.  The tombstone symbol ∎ ([[Unicode]] U+220E) is sometimes called a ''halmos''.<ref>"The symbol is definitely not my invention&nbsp;— 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.</ref>

In 1994, Halmos received the [[Deborah and Franklin Haimo Awards for Distinguished College or University Teaching of Mathematics|Deborah and Franklin Haimo Award for Distinguished College or University Teaching of Mathematics]].<ref name=web>{{Cite web|url=https://maa.org/member-communities/maa-awards/teaching-awards/section-award|title=Recipients of the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics; Mathematical Association of America|website=www.maa.org|archive-url=https://web.archive.org/web/20240608175318/https://maa.org/member-communities/maa-awards/teaching-awards/section-award |archive-date=8 June 2024 }}</ref>

In 2005, Halmos and his wife [[Virginia Halmos]] 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]].<ref name="maa">{{cite web |url-status=dead |url=http://www.maa.org/awards/eulerbook.html |title=The Mathematical Association of America's Euler Book Prize |archive-url=https://web.archive.org/web/20130127024145/http://www.maa.org/awards/eulerbook.html |archive-date=27 January 2013 |access-date=2011-02-01 |website=Mathematical Association of America }}</ref>

In 2009 [[George Csicsery]] featured Halmos in a documentary film also called ''I Want to Be a Mathematician''.<ref>"[https://www.imdb.com/title/tt1417077/ I Want to Be a Mathematician (Video 2009)]" on IMdB.</ref>

==Books==
Books by Halmos have led to so many reviews that lists have been assembled.<ref>{{cite web|title=Reviews of Paul Halmos' books Part 1 |date=August 2016 |website=MacTutor|url=https://mathshistory.st-andrews.ac.uk/Extras/Halmos_books_1/ |url-status=live |archive-url=https://web.archive.org/web/20230903153502/https://mathshistory.st-andrews.ac.uk/Extras/Halmos_books_1/ |archive-date= Sep 3, 2023 }}</ref><ref>{{cite web|title=Reviews of Paul Halmos's books Part 2 |date=August 2016 |website=MacTutor|url=https://mathshistory.st-andrews.ac.uk/Extras/Halmos_books_2/ |url-status=live |archive-url=https://web.archive.org/web/20230903153429/https://mathshistory.st-andrews.ac.uk/Extras/Halmos_books_2/ |archive-date= Sep 3, 2023 }}</ref>
*1942. ''Finite-Dimensional [[Vector space|Vector Spaces]]''. Springer-Verlag.<ref>{{cite journal|author=Kac, Mark|author-link=Mark Kac|title=Review: ''Finite-dimensional vector spaces'', by P. R. Halmos|journal=Bull. Amer. Math. Soc.|year=1943|volume=49|issue=5|pages=349–350|url=https://www.ams.org/journals/bull/1943-49-05/S0002-9904-1943-07899-8/S0002-9904-1943-07899-8.pdf|doi=10.1090/s0002-9904-1943-07899-8|doi-access=free |url-status=live |archive-url=https://web.archive.org/web/20240218001403/https://www.ams.org/journals/bull/1943-49-05/S0002-9904-1943-07899-8/S0002-9904-1943-07899-8.pdf |archive-date= Feb 18, 2024 }}</ref>
*1950. ''[[Measure Theory]]''. Springer Verlag.<ref>{{cite journal|author=Oxtoby, J. C.|title= Review: ''Measure theory'', by P. R. Halmos|journal=Bull. Amer. Math. Soc.|year=1953|volume=59|issue=1|pages=89–91|url=https://www.ams.org/journals/bull/1953-59-01/S0002-9904-1953-09662-8/S0002-9904-1953-09662-8.pdf|doi=10.1090/s0002-9904-1953-09662-8|doi-access=free |url-status=live |archive-url= https://web.archive.org/web/20230903153422/https://www.ams.org/journals/bull/1953-59-01/S0002-9904-1953-09662-8/S0002-9904-1953-09662-8.pdf |archive-date= Sep 3, 2023 }}</ref>
*1951. ''Introduction to [[Hilbert space|Hilbert Space]] and the Theory of [[spectral multiplicity|Spectral Multiplicity]]''. Chelsea.<ref>{{cite journal|author=Lorch, E. R.|author-link=Edgar Lorch|title=Review: ''Introduction to Hilbert space and the theory of spectral multiplicity'', by P. R. Halmos|journal=Bull. Amer. Math. Soc.|year=1952|volume=58|issue=3|pages=412–415|url=https://www.ams.org/journals/bull/1952-58-03/S0002-9904-1952-09595-1/S0002-9904-1952-09595-1.pdf|doi=10.1090/s0002-9904-1952-09595-1|doi-access=free |url-status=live |archive-url= https://web.archive.org/web/20240218001321/https://www.ams.org/journals/bull/1952-58-03/S0002-9904-1952-09595-1/S0002-9904-1952-09595-1.pdf |archive-date= Feb 18, 2024 }}</ref>
*1956. ''Lectures on [[ergodic theory|Ergodic Theory]]''. Chelsea.<ref>{{cite journal|author=Dowker, Yael N.|author-link=Yael Dowker|title= Review: ''Lectures on ergodic theory'', by P. R. Halmos|journal=Bull. Amer. Math. Soc.|year=1959|volume=65|issue=4|pages=253–254|url=https://www.ams.org/journals/bull/1959-65-04/S0002-9904-1959-10331-1/S0002-9904-1959-10331-1.pdf|doi=10.1090/s0002-9904-1959-10331-1|doi-access=free |url-status=live |archive-url=https://web.archive.org/web/20230903153423/https://www.ams.org/journals/bull/1959-65-04/S0002-9904-1959-10331-1/S0002-9904-1959-10331-1.pdf |archive-date= Sep 3, 2023 }}</ref>
*1960. ''[[Naive Set Theory (book)|Naive Set Theory]]''. Springer Verlag.
*1962. ''Algebraic Logic''. Chelsea.
*1963. ''Lectures on [[Boolean algebra (structure)|Boolean Algebras]]''. Van Nostrand.
*1967. ''A Hilbert Space Problem Book''. Springer-Verlag.
*1973. (with [[Norman Steenrod|Norman E. Steenrod]], [[Menahem Max Schiffer|Menahem M. Schiffer]], and [[Jean A. Dieudonné|Jean A. Dieudonne]]). [https://bookstore.ams.org/hwm ''How to Write Mathematics'']. American Mathematical Society. {{ISBN|978-0-8218-0055-3}}
*1978. (with [[V. S. Sunder]]). ''Bounded [[Integral transform|Integral Operators]] on L² Spaces''. Springer Verlag {{ISBN|978-3540088943}} <ref>{{cite journal|author=Zaanen, Adriaan|author-link=Adriaan Cornelis Zaanen|title= Review: ''Bounded integral operators on L² spaces'', by P. R. Halmos and V. S. Sunder|journal=Bull. Amer. Math. Soc. (N.S.)|year=1979|volume=1|issue=6|pages=953–960|url=https://www.ams.org/journals/bull/1979-01-06/S0273-0979-1979-14699-8/S0273-0979-1979-14699-8.pdf|doi=10.1090/s0273-0979-1979-14699-8|doi-access=free}}</ref>
*1985. ''I Want to Be a Mathematician''. Springer-Verlag. {{ISBN|978-0387960784}}
*1987. ''I Have a Photographic Memory''. [[Mathematical Association of America]]. {{ISBN|978-0821819395}}
*1991. ''Problems for Mathematicians, Young and Old'', Dolciani Mathematical Expositions, Mathematical Association of America. {{ISBN|978-0883853207}}
*1996. ''Linear Algebra Problem Book'', Dolciani Mathematical Expositions, Mathematical Association of America. {{ISBN|978-0883853221}}
*1998. (with Steven Givant). ''Logic as Algebra'', Dolciani Mathematical Expositions No. 21, Mathematical Association of America.{{ISBN|978-1470451134}}  <ref>{{cite web|author=Johnson, Mark|date=February 11, 1999|title=Review of ''Logic as Algebra'' by Paul Halmos and Steven Givant|website=MAA Reviews, Mathematical Association of America|url=https://old.maa.org/press/maa-reviews/logic-as-algebra}}</ref>
*2009. (posthumous, with Steven Givant), ''Introduction to Boolean Algebras'',<ref>{{Cite book|isbn = 978-0387402932|title = Introduction to Boolean Algebras|last1 = Givant|first1 = Steven|last2 = Halmos|first2 = Paul|date = 2 December 2008| publisher=Springer }}</ref> Springer.

==See also==
*[[Crinkled arc]]
*[[Commutator subspace]]
*[[Invariant subspace problem]]
*[[Naive set theory]]
*[[Criticism of non-standard analysis]]
*[[The Martians (scientists)]]

==Notes==
{{reflist|30em}}

==References==
* {{cite book | author1=J. H. Ewing | author2=F. W. Gehring | author-link2=Frederick Gehring | year=1991 | title=Paul Halmos: Celebrating 50 Years of Mathematics | publisher=Springer-Verlag | isbn=0-387-97509-8 | oclc=22859036 | url=https://books.google.com/books?id=Xnv5fXu1vFoC}}  Includes a bibliography of Halmos's writings through 1991.
* {{cite journal | author=John Ewing |date=October 2007 | title = Paul Halmos: In His Own Words | journal = [[Notices of the American Mathematical Society]] | volume = 54 | issue = 9 | pages =1136–1144 | url = https://www.ams.org/notices/200709/tx070901136p.pdf | access-date = 2008-01-15 }}
* {{cite book | author=Paul Halmos | title=I want to be a Mathematician: An Automathography | publisher=[[Springer-Verlag]] | year=1985 | isbn=0-387-96470-3 | oclc=230812318 }}
* {{cite journal | url=https://entropiesschool.sciencesconf.org/data/How_to_Write_Mathematics.pdf | author=Paul R. Halmos | title=How to Write Mathematics | journal=[[L'Enseignement mathématique]] | volume=16 | number=2 | pages=123&ndash;152 | year=1970 }}

==External links==
{{wikiquote}}
* {{MacTutor Biography|id=Halmos}}
* [https://web.archive.org/web/20121230180325/http://www.maa.org/news/100306halmos.html "Paul Halmos: A Life in Mathematics"], [[Mathematical Association of America]] (MAA)
* [https://archive.org/details/HalmosP.R.FiniteDimensionalVectorSpaces.SpringerVerlag205s_201703 Finite-Dimensional Vector Spaces]
* [https://digitalcollections.briscoecenter.org/collection/897 "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]
{{Chauvenet Prize recipients}}

{{Authority control}}

{{DEFAULTSORT:Halmos, Paul}}
[[Category:1916 births]]
[[Category:2006 deaths]]
[[Category:20th-century Hungarian mathematicians]]
[[Category:20th-century American mathematicians]]
[[Category:Algebraists]]
[[Category:American logicians]]
[[Category:American people of Hungarian-Jewish descent]]
[[Category:American statisticians]]
[[Category:Donegall Lecturers of Mathematics at Trinity College Dublin]]
[[Category:Functional analysts]]
[[Category:Hungarian emigrants to the United States]]
[[Category:Hungarian Jews]]
[[Category:Indiana University faculty]]
[[Category:Jewish American scientists]]
[[Category:Mathematical analysts]]
[[Category:Measure theorists]]
[[Category:Operator theorists]]
[[Category:Probability theorists]]
[[Category:Set theorists]]
[[Category:The American Mathematical Monthly editors]]
[[Category:University of Chicago faculty]]
[[Category:University of Illinois Urbana-Champaign alumni]]
[[Category:University of Michigan faculty]]