VI.95 Abraham Robinson

b. Waldenburg (Lower Silesia; now Walbrzych, Poland), 1918; d. New Haven, Connecticut, 1974

Applied mathematics; logic; model theory; nonstandard analysis

Robinson was educated at a private Rabbinical school and then at the Jewish High School in Breslau until 1933, when he emigrated with his family to Palestine. There Robinson finished high school, going on to study mathematics at the Hebrew University under Abraham Fraenkel. He spent the spring of 1940 at the Sorbonne, but when the Germans invaded France Robinson made his way to England. There he spent the war as a refugee, in the service of the Free French Forces. Robinson’s mathematical talents were soon recognized, and he was assigned to the Royal Aircraft Establishment in Farnborough, where he was part of a team designing supersonic delta wings and reconstructing German V2 rockets to determine how they worked. After the war, Robinson received his M.Sc. degree in mathematics from the Hebrew University, with minors in physics and philosophy. Several years later, he completed his Ph.D. in mathematics at Birkbeck College, London. His thesis, “On the metamathematics of algebra,” was published in 1951.

Meanwhile, Robinson had been teaching at the Royal College of Aeronautics since its founding in Cranfield in October of 1946. Although promoted to Deputy Head of the Department of Aeronautics in 1950, in the following year Robinson accepted a position, at the rank of associate professor, at the University of Toronto in the Department of Applied Mathematics. While at Toronto, most of his publications were devoted to applied mathematics, including papers on supersonic airfoil design and a book he coauthored with his former student from Cranfield, J. A. Laurmann, on Wing Theory.

His years at Toronto (1951–57) proved to be a transitional period in Robinson’s career, as his interests turned increasingly toward mathematical logic, beginning with studies of algebraically closed fields of characteristic zero. In 1955 he published a book in French summarizing much of his early work in mathematical logic and MODEL THEORY [IV.23], Théorie Métamathématique des Ideaux Robinson was a pioneering contributor to model theory, which at its simplest uses mathematical logic to analyze mathematical structures (like groups, fields, or even set theory itself). Given an axiomatic system, a model is a structure that satisfies the axioms. One of his early impressive results was a model-theoretic proof, which he published in 1955 in Mathematische Annalen, of Hilbert’s seventeenth problem, namely that a positive-definite rational function over the reals can be expressed as a sum of squares of rational functions. This was soon followed by another book, Complete Theories (1956), which further extended ideas he had explored earlier in his thesis on model-theoretic algebra. Here Robinson introduced such important concepts as model completeness, model completion, and the “prime model test,” along with proofs of the completeness of REAL-CLOSED FIELDS [IV.23 §5] and the uniqueness of the model completion of a model-complete theory.

In the fall of 1957 Robinson returned to the Hebrew University, where he assumed the chair formerly held by his teacher Abraham Fraenkel in the Einstein Institute of Mathematics. While at the Hebrew University, Robinson worked on aspects of local differential algebra, differentially closed fields, and in logic on SKOLEM’s [VI.81] results dealing with nonstandard models of arithmetic. These provide models of ordinary PEANO ARITHMETIC [III.67], the usual arithmetic of the integers (0, 1, 2, 3, . . .), but ones that include “nonstandard” elements, “numbers” that extend the scope of the standard model to models that are larger but nevertheless satisfy the axioms of the standard structure. A nonstandard model of arithmetic may include, for example, infinite integers. As Haim Gaifman puts it succinctly, “A nonstandard model is one that constitutes an interpretation of a formal system that is admittedly different from the intended one.”

Robinson spent the year 1960–61 in the United States, at Princeton, replacing CHURCH [VI.89], who was on sabbatical leave. It was there that Robinson was inspired to make his most revolutionary contribution to mathematics, nonstandard analysis, using model theory to allow the rigorous introduction of infinitesimals. In fact, this extended the usual, standard model of the real numbers to a nonstandard model that included both infinite and infinitesimal elements. He first published on this topic in 1961 in the Proceedings of the Netherlands Royal Academy of Sciences. This paper was soon followed by a book, Introduction to Model Theory and to the Metamathematics of Algebra (1963), a thorough revision of his earlier book of 1951, including a new section on nonstandard analysis.

Meanwhile, Robinson had left Jerusalem for Los Angeles, where he was appointed as Carnap’s chair at UCLA in mathematics and philosophy. In addition to writing an introductory text, Numbers and Ideals. An Introduction to Some Basic Concepts of Algebra and Number Theory (1965), he also published his definitive introduction to Nonstandard Analysis (1966). Among the important results he obtained while at UCLA (1962–67) was his proof of the invariant subspace theorem in Hilbert space for the case of polynomially compact operators, published with his graduate student Allen Bernstein. (The case for compact operators had been established by Aronszajn and Smith in 1954; what Bernstein and Robinson did was extend this to the case of an operator T such that some nonzero polynomial of T is compact.)

In 1967 Robinson moved to Yale University (1967–74), where he was eventually given a Sterling Professorship in 1971. Among Robinson’s most important mathematical achievements during this period were his extension of Paul Cohen’s method of FORCING [IV.22 §5.2] in set theory to model theory, and applications of nonstandard analysis in economics and quantum physics. He also applied nonstandard analysis to achieve an outstanding result in number theory, namely a simplification of Carl Ludwig Siegel’s theorem regarding integer points on curves (1929), as generalized by Kurt Mahler for rational as well as integer solutions (1934). This was work that Robinson did jointly with Peter Roquette; together they extended the Siegel–Mahler theorem by considering nonstandard integer points and nonstandard prime divisors. After Robinson’s death from pancreatic cancer in 1974, Roquette published this work in the Journal of Number Theory in 1975.

Further Reading

Dauben, J. W. 1995. Abraham Robinson. The Creation of Nonstandard Analysis. A Personal and Mathematical Odyssey. Princeton, NJ: Princeton University Press.

——. 2002. Abraham Robinson. 1918–1974. Biographical Memoirs of the National Academy of Sciences 82:1–44.

Davis, M., and R. Hersh. 1972. Nonstandard analysis. Scientific American 226:78–86.

Gaifman, H. 2003. Non-standard models in a broader perspective. In Nonstandard Models of Arithmetic and Set Theory, edited by A. Enayat and R. Kossak, pp. 1–22. Providence, RI: American Mathematical Society.

Joseph W. Dauben