Abraham Robinson

Polish-born mathematician

• Born: October 6, 1918
• Birthplace: Waldenburg, Germany (now Wałbrzych, Poland)
• Died: April 11, 1974
• Place of death: New Haven, Connecticut

Robinson is known for his contributions to the study of calculus and of more advanced branches of mathematics, using sets of objects that satisfy the axioms for the real numbers but that are demonstrably different from them.

Early Life

Abraham Robinson (AY-bruh-ham RAH-bihn-suhn) was born during the difficult final days of World War I. His father, Abraham Robinsohn (a spelling his son was to alter after he became an adult), was active in the Zionist movement, but he died unexpectedly before the birth of his son. Robinson’s mother, Hedwig Bähr, moved in with her parents, bringing Robinson and his older brother. They moved to Breslau, which had an active Jewish community, where Robinson’s mother would work for Keren Hayesod, the agency raising money for the Jewish state in Palestine. Robinson was successful in the local school, but the entire family was uprooted when the Nazis came to power in 1933. While living in Palestine had always been part of the family’s dream, political events made it a necessity.

On arriving in Palestine, Robinson finished his secondary schooling and became a student at the Hebrew University in Jerusalem. There he studied under some of the distinguished mathematicians who had left Europe after making their reputations, especially Abraham Fraenkel. Fraenkel had worked in set theory during his years in the German academic world, and Robinson was influenced by Fraenkel’s choice of topics and his philosophical approach. Robinson joined the Haganah, the Jewish militia designed to protect the Jewish population from attacks by Arab groups. He also developed interests in art and in music, which he would pursue for the rest of his life.

Robinson’s work in mathematics was successful enough for him to receive a scholarship to the Sorbonne. After six months in Paris, the German invasion led to his hasty departure, and he took refuge across the English Channel. He was interned briefly on his arrival, but he was able to benefit from friendships with members of the English mathematical community. His contribution to the war effort involved service at the Royal Aircraft Establishment, and he rapidly became an expert in aeronautics. In particular, after the war, he was on the faculty of the Royal College of Aeronautics, although he also pursued his graduate studies in mathematics at Birkbeck College, part of the University of London. That afforded him the chance to return to the study of mathematical logic, and his thesis dealt with the way algebraic questions could be settled by logical means. In 1944, he married Renée Kopel, who came from a wealthy Viennese family and who had worked as an actor and a fashion designer. They had no children.

Life’s Work

After getting his degree at London, Robinson was invited to the University of Toronto. His strength in applied mathematics appealed to the mathematical community there, and he continued to publish and to advise students in that area. In the meantime, he devoted increasing time to logic. He then accepted an invitation from the Hebrew University to return to Israel to take over the position that Fraenkel had vacated. He found the atmosphere in Israel after the 1956 Sinai campaign calmer than when he had lived there as a student, and he had some distinguished students. A sabbatical at Princeton University enabled him to arrange for a permanent position at the University of California, Los Angeles (UCLA). UCLA gave him a joint appointment in mathematics and in philosophy (ideal for a logician), although he was sorry to leave Israel once again. He remained at UCLA for five years, and then he took a position at Yale, where he found the culture reminiscent of what he had enjoyed about British academia. He found himself on the liberal side of some of the political turmoil during those years, but he remained active, working on his research and with students. He died in 1974, shortly after being diagnosed with pancreatic cancer.

In the midst of all these travels, Robinson developed the branch of mathematical logic known as model theory. If one looks at the axioms, or fundamental principles, for real numbers, one would not be surprised to find that the real numbers satisfy them. The surprising feature of model theory was the discovery of nonstandard models for sets of axioms.

Robinson’s work affected model theory in general, but he is best known for his discovery of nonstandard analysis. This is the study of calculus and more advanced branches of mathematics, using sets of objects that satisfy the axioms for the real numbers but that are demonstrably different from them. In particular, Robinson’s nonstandard real numbers included infinitesimal numbers, which had been used by Gottfried Wilhelm Leibniz in the seventeenth century to serve as a basis for calculus. They had been abandoned as indefensible in an age of increasing rigor, but Robinson demonstrated that they had a firm foundation. He even demonstrated how nonstandard analysis could solve problems whose proofs by standard means were more cumbersome. It was his simplified proof of one of the conjectures of David Hilbert that put nonstandard analysis on the map to stay. Robinson also drew some philosophical conclusions from the viability of nonstandard analysis, although there is continued debate over how convincing his arguments were.

Significance

Robinson took the relationship between algebra and logic and made clear that logic had important methods and results to offer. In fact, those results spread across the rest of mathematics. Textbooks on calculus using Robinson’s nonstandard real numbers as a foundation ensured that even freshmen could look at mathematics from a Robinsonian perspective. His foundation for the real numbers offers an approach for tackling problems that have otherwise proved difficult to understand.

Bibliography

Dauben, Joseph Warren. Abraham Robinson: The Creation of Nonstandard Analysis. Princeton, N.J.: Princeton University Press, 1995. A full chronicle of Robinson’s life against its varied backgrounds.

Keisler, H. Jerome. Elementary Calculus: An Infinitesimal Approach. Boston: Prindle, Weber, Schmidt, 1986. The textbook that brought Robinson’s work to engineers.

Odifreddi, Piergiorgio. The Mathematical Century. Princeton, N.J.: Princeton University Press, 2004. Reflects the importance of Robinson’s work by indicating the number of branches of mathematics that it has affected.

Saracino, D. H., and V. B. Weispfenning, eds. Model Theory and Algebra: A Memorial Tribute to Abraham Robinson. New York: Springer Verlag, 1975. Collection of papers in the immediate aftermath of Robinson’s death describing his personal as well as his mathematical influence.

Yandell, Ben H. The Honors Class. Natick, Mass.: A. K. Peters, 2002. Evaluation of Robinson’s role in contributing to the solution of Hilbert’s problems from 1900.