Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American

retrieved 2011-02-01. In 2009 George Csicsery featured Halmos in a documentary film also called ''I Want to Be a Mathematician''.

''How to Write Mathematics''

American Mathematical Society. *1978. (with V. S. Sunder). ''Bounded Integral Operators on L² Spaces''. Springer Verlag *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. *2009. (posthumous, with Steven Givant), ''Introduction to Boolean Algebras'', Springer.

"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

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

mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
...

and statistician who made fundamental advances in the areas of mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

, probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

, statistics
Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...

, operator theory, ergodic theory, and functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...

(in particular, Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...

s). He was also recognized as a great mathematical expositor. He has been described as one of The Martians.
Early life and education

Born inHungary
Hungary ( hu, Magyarország ) is a landlocked country in Central Europe. Spanning of the Carpathian Basin, it is bordered by Slovakia to the north, Ukraine to the northeast, Romania to the east and southeast, Serbia to the south, Croa ...

into a Jewish
Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation originating from the Israelites Israelite origins and kingdom: "The first act in the long drama of Jewish history is the age of the Israelites""The ...

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, 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''.
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 underJohn von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

, 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.
From 1967 to 1968 he was the Donegall Lecturer in Mathematics at Trinity College Dublin
, name_Latin = Collegium Sanctae et Individuae Trinitatis Reginae Elizabethae juxta Dublin
, motto = ''Perpetuis futuris temporibus duraturam'' (Latin)
, motto_lang = la
, motto_English = It will last i ...

.
Halmos taught at Syracuse University, the University of Chicago
The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the b ...

(1946–60), the University of Michigan
, mottoeng = "Arts, Knowledge, Truth"
, former_names = Catholepistemiad, or University of Michigania (1817–1821)
, budget = $10.3 billion (2021)
, endowment = $17 billion (2021)As o ...

(~1961–67), the University of Hawaii
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, th ...

(1967–68), Indiana University (1969–85), and the University of California at Santa Barbara
The University of California, Santa Barbara (UC Santa Barbara or UCSB) is a public land-grant research university in Santa Barbara, California with 23,196 undergraduates and 2,983 graduate students enrolled in 2021–2022. It is part of the U ...

(1976–78). From his 1985 retirement from Indiana until his death, he was affiliated with the Mathematics department at Santa Clara University
Santa Clara University is a private Jesuit university in Santa Clara, California. Established in 1851, Santa Clara University is the oldest operating institution of higher learning in California. The university's campus surrounds the historic M ...

(1985–2006).
Accomplishments

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 ofAlfred Tarski
Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...

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 and again in 1977 (shared with W. P. Ziemer, W. H. Wheeler, S. H. Moolgavkar, J. H. Ewing and W. H. Gustafson). 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 and , 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:
In these memoirs, Halmos claims to have invented the "iff" notation for the words "if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bico ...

" and to have been the first to use the "tombstone" notation to signify the end of a proof, and this is generally agreed to be the case. The tombstone symbol ∎ (Unicode
Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...

U+220E) is sometimes called a ''halmos''.
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
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pu ...

: Prime Obsession.The Mathematical Association of America's Euler Book Prizeretrieved 2011-02-01. In 2009 George Csicsery featured Halmos in a documentary film also called ''I Want to Be a Mathematician''.

Books by Halmos

Books by Halmos have led to so many reviews that lists have been assembled. *1942. ''Finite-DimensionalVector Spaces
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

''. Springer-Verlag.
*1950. ''Measure Theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...

''. Springer Verlag.
*1951. ''Introduction to Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...

and the Theory of Spectral Multiplicity''. Chelsea.
*1956. ''Lectures on Ergodic Theory''. Chelsea.
*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. *1978. (with V. S. Sunder). ''Bounded Integral Operators on L² Spaces''. Springer Verlag *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. *2009. (posthumous, with Steven Givant), ''Introduction to Boolean Algebras'', Springer.

See also

* Crinkled arc * Commutator subspace * Invariant subspace problem * Naive set theory * Criticism of non-standard analysis *The Martians (scientists)
"The Martians" ( hu, "A marslakók") is a term used to refer to a group of prominent Hungarian scientists (mostly, but not exclusively, physicists and mathematicians) of Jewish descent who emigrated from Europe to the United States in the early ha ...

Notes

References

* Includes a bibliography of Halmos's writings through 1991. * * *External links

*"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

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