Paul Halmos
Encyclopedia
Paul Richard Halmos was a Hungarian
-born American
mathematician
who made fundamental advances in the areas of probability theory
, statistics
, operator theory
, ergodic theory
, and functional analysis
(in particular, Hilbert space
s). He was also recognized as a great mathematical expositor.
, majoring in philosophy and minoring in mathematics. 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, shifted to mathematics, graduating in 1938. Joseph Doob supervised his dissertation, titled Invariants of Certain Stochastic Transformations: The Mathematical Theory of Gambling Systems. Shortly thereafter, 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
, the University of California at Santa Barbara (about 1977), 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
.
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. This was so even though Halmos arrived in the USA at 13 years of age and never lost his Hungarian accent. He 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. Some of his classics were:
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:
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
, and this is generally agreed to be the case. The tombstone symbol (Unicode
U+220E) is sometimes called a halmos.
In 2005, Halmos and his wife Viriginia 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.
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...
-born American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who made fundamental advances in the areas of probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
, ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
, and functional analysis
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
(in particular, Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
s). He was also recognized as a great mathematical expositor.
Career
Halmos obtained his B.A. from the University of IllinoisUniversity of Illinois at Urbana-Champaign
The University of Illinois at Urbana–Champaign is a large public research-intensive university in the state of Illinois, United States. It is the flagship campus of the University of Illinois system...
, majoring in philosophy and minoring in mathematics. 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, shifted to mathematics, graduating in 1938. Joseph Doob supervised his dissertation, titled Invariants of Certain Stochastic Transformations: The Mathematical Theory of Gambling Systems. Shortly thereafter, Halmos left for the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
, lacking both job and grant money. Six months later, he was working under John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
, 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
Syracuse University
Syracuse University is a private research university located in Syracuse, New York, United States. Its roots can be traced back to Genesee Wesleyan Seminary, founded by the Methodist Episcopal Church in 1832, which also later founded Genesee College...
, the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
(1946–60), the University of Michigan
University of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...
, the University of California at Santa Barbara (about 1977), the University of Hawaii
University of Hawaii
The University of Hawaii System, formally the University of Hawaii and popularly known as UH, is a public, co-educational college and university system that confers associate, bachelor, master, and doctoral degrees through three university campuses, seven community college campuses, an employment...
, and Indiana University
Indiana University
Indiana University is a multi-campus public university system in the state of Indiana, United States. Indiana University has a combined student body of more than 100,000 students, including approximately 42,000 students enrolled at the Indiana University Bloomington campus and approximately 37,000...
. From his 1985 retirement from Indiana until his death, he was affiliated with the Mathematics department at Santa Clara University
Santa Clara University
Santa Clara University is a private, not-for-profit, Jesuit-affiliated university located in Santa Clara, California, United States. Chartered by the state of California and accredited by the Western Association of Schools and Colleges, it operates in collaboration with the Society of Jesus , whose...
.
Accomplishments
In a series of papers reprinted in his 1962 Algebraic Logic, Halmos devised polyadic algebraPolyadic algebra
Polyadic algebras are algebraic structures introduced by Paul Halmos. They are related to first-order logic in a way analogous to the relationship between Boolean algebras and propositional logic .There are other ways to relate first-order logic to algebra, including Tarski's cylindric algebras...
s, an algebraic version of first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
differing from the better known cylindric algebra
Cylindric algebra
The notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of first-order logic with equality. This is comparable to the role Boolean algebras play for propositional logic. Indeed, cylindric algebras are Boolean algebras equipped with additional...
s of Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
and his students. An elementary version of polyadic algebra is described in monadic Boolean algebra
Monadic Boolean algebra
In abstract algebra, a monadic Boolean algebra is an algebraic structure with signaturewhere ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra.The prefixed unary operator ∃ denotes the existential quantifier, which satisfies the identities:...
.
In addition to his original contributions to mathematics, Halmos was an unusually clear and engaging expositor of university mathematics. This was so even though Halmos arrived in the USA at 13 years of age and never lost his Hungarian accent. He chaired the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
committee that wrote the AMS style guide for academic mathematics, published in 1973. In 1983, he received the AMS's Steele Prize for exposition. Some of his classics were:
- How to read mathematics
- How to write mathematics
- How to speak mathematics.
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:
In these memoirs, Halmos claims to have invented the "iff" notation for the words "if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
" and to have been the first to use the “tombstone”
Tombstone (typography)
The tombstone, halmos, or end of proof mark "" is used in mathematics to denote the end of a proof, in place of the traditional abbreviation "QED" for the Latin phrase "quod erat demonstrandum" ....
notation to signify the end of a proof
Q.E.D.
Q.E.D. is an initialism of the Latin phrase , which translates as "which was to be demonstrated". The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when what was specified in the enunciation — and in the setting-out —...
, and this is generally agreed to be the case. The tombstone symbol (Unicode
Unicode
Unicode is a computing industry standard for the consistent encoding, representation and handling of text expressed in most of the world's writing systems...
U+220E) is sometimes called a halmos.
In 2005, Halmos and his wife Viriginia funded the Euler Book Prize
Euler Book Prize
The Euler Book Prize is an award named after Leonhard Euler and given annually at the Joint Mathematics Meetings by the Mathematical Association of America to an outstanding book in mathematics that is likely to improve the public view of the field....
, an annual award given by the Mathematical Association of America
Mathematical Association of America
The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
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
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
's birth.
Books by Halmos
- 1942. Finite-Dimensional Vector SpacesVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
. Springer-Verlag. - 1950. Measure Theory. Springer Verlag.
- 1951. Introduction to Hilbert SpaceHilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
and the Theory of Spectral Multiplicity. Chelsea. - 1956. Lectures on Ergodic TheoryErgodic theoryErgodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
. Chelsea. - 1960. Naive Set TheoryNaive Set Theory (book)Naive Set Theory is a mathematics textbook by Paul Halmos originally published in 1960. This book is an undergraduate introduction to not-very-naive set theory. It is still considered by many to be the best introduction to set theory for beginners...
. Springer Verlag. - 1962. Algebraic Logic. Chelsea.
- 1963. Lectures on Boolean Algebras. Van Nostrand.
- 1967. A Hilbert SpaceHilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
Problem Book. Springer-Verlag. - 1978 (with V. 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 AmericaMathematical Association of AmericaThe Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
. - 1991. Problems for Mathematicians, Young and Old, Dolciani Mathematical Expositions, Mathematical Association of AmericaMathematical Association of AmericaThe Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
. - 1996. Linear Algebra Problem Book, Dolciani Mathematical Expositions, Mathematical Association of AmericaMathematical Association of AmericaThe Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
. - 1998 (with Steven Givant). Logic as Algebra, Dolciani Mathematical Expositions No. 21, Mathematical Association of AmericaMathematical Association of AmericaThe Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
.
External links
- "Paul Halmos: A Life in Mathematics", Mathematical Association of AmericaMathematical Association of AmericaThe Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
(MAA)