Paul Halmos - Accomplishments

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 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:

• 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:

What does it take to be ? 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.

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 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 Riemann and the Riemann hypothesis: Prime Obsession.