Paul Halmos

1. Early life and education

Paul Richard Halmos was born on March 3, 1916, in Budapest, Kingdom of Hungary, into a Jewish family. At the age of 13, he immigrated to the United States, though he retained a Hungarian accent throughout his life. He pursued his undergraduate studies at the University of Illinois, where he majored in mathematics while simultaneously fulfilling the requirements for a degree in philosophy. He completed his Bachelor of Arts degree in just three years, graduating at the age of 19.

Following his undergraduate studies, Halmos initially enrolled in a Ph.D. program in philosophy at the same Champaign-Urbana campus. However, after failing his master's oral examinations, he decided to switch his focus to mathematics. He successfully completed his doctoral degree in mathematics in 1938. His dissertation, titled Invariants of Certain Stochastic Transformations: The Mathematical Theory of Gambling Systems, was supervised by Joseph L. Doob.

2. Career

Shortly after earning his doctorate, Halmos moved to the Institute for Advanced Study in Princeton, New Jersey, without a job or grant funding. Six months later, he began working under the guidance of John von Neumann, an experience that proved to be profoundly influential and decisive for his career. While at the Institute, Halmos authored his first book, Finite-Dimensional Vector Spaces, which quickly established his reputation as an outstanding expositor of mathematics.

Halmos had an extensive teaching career at several prestigious universities across the United States. He taught at Syracuse University, the University of Chicago from 1946 to 1960, the University of Michigan from approximately 1961 to 1967, the University of Hawaii from 1967 to 1968, and Indiana University from 1969 to 1985. Additionally, he served as the Donegall Lecturer in Mathematics at Trinity College Dublin from 1967 to 1968 and taught at the University of California, Santa Barbara from 1976 to 1978. After his retirement from Indiana University in 1985, he remained affiliated with the Mathematics department at Santa Clara University until his death in 2006.

3. Mathematical Contributions

Paul Halmos made fundamental contributions across various branches of mathematics, including mathematical logic, probability theory, operator theory, ergodic theory, and functional analysis, with a particular focus on Hilbert spaces.

In a series of papers that were later reprinted in his 1962 book Algebraic Logic, Halmos developed polyadic algebras. This was an algebraic formulation of first-order logic, distinct from the more widely known cylindric algebras developed by Alfred Tarski and his students. An elementary version of polyadic algebra is described in the context of monadic Boolean algebra. His work significantly advanced the understanding of these abstract algebraic structures and their relationship to logical systems.

4. Publications and Expository Skills

Paul Halmos was celebrated not only for his original mathematical research but also for his exceptional ability to write and explain complex mathematical concepts with remarkable clarity and engaging style. He authored numerous influential books and articles that have become standard texts in their respective fields.

5. Mathematical Philosophy and Practice

Halmos held unique and influential perspectives on the nature of mathematics and its study. In a 1968 article in American Scientist, he argued that mathematics is fundamentally a creative art, asserting that mathematicians should be viewed as artists rather than mere "number crunchers." He discussed a conceptual division of the field into "mathology" and "mathophysics," further contending that the thought processes and work methods of mathematicians and painters are remarkably similar.

His pedagogical philosophy emphasized active engagement and discovery. In his "automathography," I Want to Be a Mathematician, he famously advised:
: "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?"
This quote encapsulates his belief that true mathematical understanding comes from deep interaction with the material, rather than passive absorption. He also shared his view on what it takes to be a mathematician, stating: "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."

Halmos is also credited with inventing several widely adopted mathematical notations. He claimed to have coined the "iff" notation as an abbreviation for "if and only if", which has become standard in mathematical texts. Furthermore, he is generally recognized as the first to use the "tombstone" symbol (∎, Unicode U+220E) to signify the end of a proof. This symbol is sometimes affectionately referred to as a "halmos" in his honor.

6. Personal Life and Philanthropy

Paul Halmos's personal life, particularly his dedication to mathematics, is reflected in his 1985 book, I Want to Be a Mathematician. He referred to this work as an "automathography" rather than an autobiography, indicating its primary focus on his life as a mathematician rather than his private affairs.

Halmos was also involved in significant philanthropic efforts aimed at promoting mathematics to a broader audience. In 2005, he and his wife, Virginia Halmos, provided funding to establish the Euler Book Prize. This annual award, administered by the Mathematical Association of America, is given to a book that is expected to enhance the public's appreciation and understanding of mathematics. The inaugural prize was awarded in 2007, coinciding with the 300th anniversary of Leonhard Euler's birth, to John Derbyshire for his book Prime Obsession, which explores the lives of Bernhard Riemann and the Riemann hypothesis.

7. Death

Paul Richard Halmos passed away on October 2, 2006.

8. Legacy and Honors

Paul Halmos's legacy endures through his profound mathematical contributions, his influential textbooks, and his innovative approaches to mathematical exposition and pedagogy. His impact on how mathematics is taught and communicated continues to be felt by generations of students and educators. In 2009, filmmaker George Csicsery featured Halmos in a documentary film also titled I Want to Be a Mathematician, further cementing his place in the history of mathematics.

9. See also

• Crinkled arc
• Commutator subspace
• Invariant subspace problem
• Naive set theory
• Criticism of non-standard analysis
• The Martians (scientists)