Philip J. Davis (1923–2018): The Humanistic Mathematician
Philip J. Davis was a polymathic figure in 20th-century mathematics. While many mathematicians confine themselves to the abstract rigors of their discipline, Davis spent his long career bridging the gap between high-level numerical analysis and the philosophical, historical, and social dimensions of mathematics. A National Book Award winner and a pioneer in numerical integration, Davis is remembered as much for his technical brilliance as for his insistence that mathematics is a deeply human activity.
1. Biography: Early Life and Career Trajectory
Philip J. Davis was born on January 2, 1923, in Lawrence, Massachusetts. His path to mathematics was paved during the golden age of American science. He attended Harvard University, where he earned his B.S. in 1943. His education was interrupted by World War II, during which he served in the U.S. Army Air Corps, working on aerodynamics at the National Advisory Committee for Aeronautics (NACA), the precursor to NASA.
Returning to Harvard after the war, he completed his Ph.D. in 1950 under the supervision of Ralph Boas Jr., focusing on complex analysis. Davis’s career took him to the National Bureau of Standards (now NIST) in Washington, D.C., where he served as the Chief of the Numerical Analysis Section from 1952 to 1963. This period was pivotal, as it coincided with the birth of electronic computing; Davis was among the first to grapple with how machines could handle complex mathematical proofs and calculations.
In 1963, he joined the Division of Applied Mathematics at Brown University. He remained at Brown for the rest of his career, becoming Professor Emeritus and a beloved figure in the department until his death on March 13, 2018, at the age of 95.
2. Major Contributions: Technical and Philosophical
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Numerical Analysis and Approximation Theory
Davis was a master of "approximation theory"—the art of finding simpler functions to represent complex ones. He made significant contributions to numerical quadrature (the calculation of definite integrals) and the study of the Gamma function. His work helped establish the theoretical framework for how computers calculate areas under curves and solve differential equations.
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The Philosophy of Mathematics (Social Constructivism)
Later in his career, Davis became a leading voice in the "humanist" school of mathematics. He argued against the Platonist view (that math exists in an abstract, eternal realm) and instead proposed that mathematics is a social construct—a product of human culture, language, and historical consensus.
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Circulant Matrices
He wrote the definitive text on circulant matrices, which are essential in signal processing, image encoding, and the study of cyclic structures in data.
3. Notable Publications
Davis was a prolific writer, known for a prose style that was witty, erudite, and accessible.
- Interpolation and Approximation (1963): A classic textbook that remains a staple for graduate students in numerical analysis.
- Methods of Numerical Integration (1967, with Philip Rabinowitz): This remains one of the most cited and used references in the field of computational mathematics.
- The Mathematical Experience (1981, with Reuben Hersh): This is his most famous work. It won the National Book Award and is credited with changing how the public (and many mathematicians) view the field. It explores the history, philosophy, and psychology of mathematics.
- Descartes' Dream: The World According to Mathematics (1986, with Reuben Hersh): A critical look at the "mathematization" of the world and the ethical implications of relying on mathematical models for social policy.
- The Thread: A Mathematical Yarn (1983): A quirky, semi-fictional exploration of the history of the "Tschebyscheff" name, demonstrating his love for mathematical history and storytelling.
4. Awards and Recognition
Davis’s ability to communicate complex ideas earned him some of the highest honors in the scientific and literary worlds:
- National Book Award (1983): For The Mathematical Experience (Science category).
- Chauvenet Prize (1963): Awarded by the Mathematical Association of America (MAA) for outstanding expository writing.
- Lester R. Ford Award (1982): For his contributions to mathematical exposition.
- Guggenheim Fellowship: Awarded for his research in mathematics.
- Honorary Doctorates: Received from several institutions, including the University of Roskilde in Denmark, in recognition of his influence on the philosophy of science.
5. Impact and Legacy
Davis’s legacy is twofold. In the technical realm, his algorithms and theorems regarding numerical integration are embedded in the software libraries that modern scientists use every day (such as those in MATLAB or Mathematica).
In the intellectual realm, he was a pioneer of the "Humanistic Mathematics" movement. He challenged the "coldness" of the field, arguing that mathematicians must be aware of their social responsibilities. His work influenced a generation of educators to teach mathematics not as a list of static rules, but as a dynamic, evolving human conversation. He was one of the first to warn about the dangers of over-relying on algorithmic "objectivity," a topic that is now central to modern debates about Artificial Intelligence.
6. Collaborations
- Reuben Hersh: Perhaps his most significant collaborator, Hersh was a philosopher of mathematics. Together, they wrote the books that defined the social-constructivist view of math.
- Philip Rabinowitz: A long-time collaborator at the Weizmann Institute of Science, with whom Davis wrote the definitive texts on numerical integration.
- Ralph Boas Jr.: His mentor at Harvard, who influenced Davis’s rigorous yet clear approach to complex analysis.
- The Brown University Faculty: Davis was a cornerstone of the Brown Applied Math department, collaborating with figures like David Mumford on the intersection of mathematics and culture.
7. Lesser-Known Facts
- A Writer of Fiction: Beyond his technical papers, Davis wrote several works of fiction and "mathematical whimsy," including Thomas Gray: Philosopher Cat, a book that blends 18th-century history with the life of a cat.
- The SEAC Computer: While at the National Bureau of Standards, Davis worked with the SEAC (Standards Eastern Automatic Computer), one of the very first stored-program electronic computers in the United States. He was one of the first humans to see a computer solve a problem that was previously thought to be uncomputable by hand.
- Historical Sleuth: Davis was obsessed with the 18th-century mathematician Leonhard Euler and spent years tracking down obscure biographical details of mathematicians, often treating mathematical history like a detective novel.
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Critic of "Certainty": Despite being a master of a "precise" science, Davis was deeply skeptical of absolute certainty. He famously said:
"The price of mathematical purity is logical impossibility."