#### Paul Halmos, 90, Mathematician Known for Simplifying Concepts, Dies

Paul R. Halmos, a mathematician known for exploring the implications of probability theory and helping simplify the expression of mathematical concepts in writing and speech, died on Oct. 2 in San Jose, Calif. He was 90.

The cause was pneumonia, said a spokesman for Santa Clara University in California, where Dr. Halmos taught from 1985 to 1995.

Dr. Halmos, whose work included algebraic logic, measure theory and naïve set theory, once described mathematics as "almost like being in touch with God."

"God is keeping secrets from us," he continued, "and it's fun to try to learn some of the secrets."

Dr. Halmos worked on probability theory, the study of randomness under differing conditions, and contributed to the field of operator theory, a branch of higher analysis related to calculus.

In synthesizing concepts within operator theory, he "made the theory more coherent and a vibrant area of research," said John H. Ewing, a mathematician and the executive director of the American Mathematical Society. With F. W. Gehring, Dr. Ewing edited a review of Dr. Halmos's work, "Paul Halmos: Celebrating 50 Years of Mathematics" (1991).

Whether speaking about probability theory or ergodic theory, which is connected to chaos theory and concerned with changes to systems over time, Dr. Halmos was recognized for his ability to express complex ideas concisely. In the 50's, he helped simplify written mathematics and became an early advocate for using the tombstone symbol to signify the end of a proof. The symbol, now sometimes called a halmos, acts as a punctuation mark.

Paul Richard Halmos was born in Budapest. He earned an undergraduate degree and a doctorate from the University of Illinois, where he wrote a dissertation on theories behind gambling systems in 1938.

In 1950, he published "Measure Theory" on measuring volume, length and area in general spaces, that Dr. Ewing said "remains an outstanding work on the subject." Dr. Halmos also wrote "Algebraic Logic" (1962), "A Hilbert Space Problem Book" (1967) and an autobiography, "I Want to Be a Mathematician" (1985).

He taught at the University of Chicago, the University of Michigan, Syracuse University, Indiana University and the University of Hawaii before joining Santa Clara. In the 80's, Dr. Halmos was editor of The American Mathematical Monthly.

Surviving is his wife of 61 years, Virginia, of Los Gatos, Calif.

By JEREMY PEARCE, October 20, 2006 © The New York Times Company