James B. Ax was a mathematician well-known for his work in algebra and number theory. He also worked in mathematical physics and was an active developer of financial algorithms.

James Ax attended Peter Stuyvesant High School in New York City, and Brooklyn Polytechnic University. He received a Ph.D. in Mathematics from U.C. Berkeley in 1961. He held a one-year instructorship position at Stanford, and then accepted an assistant professorship at Cornell. In 1965-66, Ax was a Guggenheim Fellow at Harvard University.

In 1967, James Ax, together with Simon Kochen, was awarded the AMS Frank Nelson Cole Prize in Number Theory for their proof of the "Ax-Kochen" theorem about polynomials over the p-adic numbers. The proof of the Ax-Kochen theorem is an unusual blend of tools that apply mathematical logic, especially ultraproducts and model theory, to prove an elementary equivalence between Laurent series and p-adic numbers. This work appeared in a series of three papers between 1965 and 1966 in the American Journal of Mathematics and the Annals of Mathematics. In subsequent work, Ax proved the "Ax Theorem" about polynomials over p-adics and the decidability of the p-adics, which has applications in diverse fields including information theory.

After winning the Cole Prize, Ax received offers from a number of universities including Berkeley and Yale, but decided to remain at Cornell. He became Associate Professor in 1967 and was promoted to Full Professor in 1969, as the youngest full professor in the history of Cornell at that time. However, endless wooing by the SUNY Stony Brook mathematics department chair, James Simons, and the prospect of starting in with a young and upcoming university, attracted him to Stony Brook in 1969. At Stony Brook, Ax began working on the mathematical foundations of physics, and he published papers on the axiomatic foundations for spacetime and on the group theoretic foundations of the axioms of quantum mechanics. He remained at Stony Brook as a Full Professor until retiring from academics in 1977.

James Ax founded a quantitative finance firm called Axcom in the 1980's. Axcom was later acquired by James Simons' Renaissance Technologies Corporation and renamed the "Medallion Fund" in honor of the Cole Prize that Ax won and the Veblen prize that Simons won. The Medallion Fund has been considered to be one of the most consistently successful hedge funds. The Axcom and Medallion Funds pioneered the use of sophisticated mathematical modeling for financial trading.

In the early 1990's, he retired and moved to San Diego, California. He remained an active researcher and continued to work in mathematical physics on foundational aspects of quantum mechanics with his long-time collaborator Simon Kochen of Princeton University. He also taught himself to play golf. In 2004 he attended playwriting and screenwriting classes at the UCSD Extension school. In 2005 he finished work on a scientific thriller screenplay, "Bots".