Robinson also lectured on nonstandard analysis at the Castelnuovo Institute at the University of Rome, which resulted in two papers in the Proceedings of the Accademia dei Lincei, both devoted to Dedekind domains in the theory of algebraic numbers. Back at UCLA, the administrative turmoil and constant department meetings began to take their toll, as did the continuing financial woes of the entire university. Enrollments at UCLA had increased to nearly 30, students.

Financially the university was facing severe cutbacks, and faculty salaries were no longer competitive on a national scale. This resulted in a month-long course on the fundamentals of model theory, which included applications to both algebra and nonstandard analysis. When he returned to UCLA for the fall term, Robinson taught a course on set theory, another on lattice theory and Boolean algebras, and a three-.

## W. A. J. Luxemburg & Abraham Robinson, Contributions to Non-Standard Analysis - PhilPapers

Luxemburg as the culmination of a semester-long seminar on nonstandard analysis at Caltech. This meeting brought together for the first time a large number of mathematicians, all of whom were making use of nonstandard analysis, and the list of participants was impressive. This was actually the fourteenth in a series of annual summer research institutes sponsored by the American Mathematical Society and the Association for Symbolic Logic, with financial support from the National Science Foundation. While at UCLA Robinson had a diverse group of graduate students, and of those who wrote their dissertations with him, the ones who obtained the most significant results were Allen R.

Subsequently, Robinson worked with Larry E. A smaller private university offered him close contact with all members of a distinguished senior faculty in mathematics, a regular flow of talented postdoctoral researchers and graduate students, and the opportunity for easy contact with scholars in fields other than his own. No doubt his awareness of tradition lent appeal to the case of a university nearly twice as old as the oldest of his previous affiliations Seligman, , pp.

Michael Rabin had prepared the way for Robinson by teaching a course the previous semester on model theory. With Robinson now at Yale he served as a magnet for other logicians, and soon he had established a strong group of postdoctoral students, including Jon Barwise, Paul Eklof, Manuel Lerman, James H. It was not long after his move to Yale that Robinson was elected to a two-year term as president of the Association for Symbolic Logic — One of the major responsibilities of the Association was overseeing publication of its Journal of Symbolic Logic.

Robinson was also interested in ways the Association could serve to promote logic in parts of the world where it had not as yet been established. Among the efforts Robinson championed during his presidency was development of mathematical logic in Japan and Latin America, where the Association, with support from the National Science Foundation, sponsored a number of special logic seminars and summer schools. Thus both standard and nonstandard versions of arithmetic and analysis were possible, which served to reinforce the reasonableness of a formalist foundation for all of mathematics.

The following summer, , Robinson was back in Heidelberg, to work again with Peter Roquette on nonstandard number theory. Mostow, quoted in Dauben, , p.

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In the spring of while on leave from Yale, Robinson was invited to give three Shearman lectures back at his alma mater, the University of London. This was related to work he had been doing with his colleague at Yale, Jon Barwise. Somewhat later Jerome Keisler summarized the significance of all this as follows:. Robinson showed that the generic models constructed by his forcing are closely related to model completions. If a theory has a model completion, then it must be the set of sentences true in all generic models of the theory.

Whether the theory has a model completion or not, all generic models are existentially closed in the theory. These results created new interest in model completeness and suggested many questions in particular areas of algebra. As a result there has been a substantial increase recently in research activity in the whole area of model-theoretic algebra Keisler, , p. From Oslo Robinson went on to Chile, where he was a major figure in the first Latin American meeting on logic.

This had been organized by W. Luxemburg and Detlef Laugwitz, and brought together. When Robinson returned to New Haven in the fall of , he had been made a Sterling Professor, a prestigious named professorship at Yale. Among the applications of nonstandard analysis that had begun to interest Robinson in the early s were results he and his colleague at Yale Donald J. Among those whom Robinson invited to Yale as a visiting professor was Gabriel Sabbagh, who spent the fall of in New Haven. Angus Macintyre had also accepted a position at Yale that same year as an associate professor, which meant that Yale was fast becoming one of the most stimulating centers for mathematical logic in the world.

Robinson was away for a part of the year at the Institute for Advanced Study. In my opinion Nonstandard Analysis perhaps in some non-conservative version will become increasingly important in the future development of Analysis and Number Theory. While at the Institute, Robinson was working on a paper dedicated to Andrzej Mostowski for his sixtieth birthday.

He also wrote another commemorative article that he dedicated to his colleague A. Brouwer Medal by the Dutch Mathematical Society. In his Brouwer lecture Robinson also articulated. Indeed there seems to be no natural way to give preference to just one among them. This contrasts with the classical approach to the real numbers, which are supposed to constitute a unique or, more precisely, categorical totality.

However, as I have stated elsewhere, I belong to those who consider that it is in the realm of possibility that at some stage even the established number systems will, perhaps under the influence of developments in set theory, bifurcate so that, for example, future generations will be faced with several coequal systems of real numbers in place of just one. Robinson spent part of the summer of back in Heidelberg.

This also allowed Robinson and Roquette to continue their collaboration on nonstandard number theory.

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Together they were working on nonstandard approaches to diophantine equations, in particular C. Kurt Mahler had generalized the theorem, allowing for certain rational as well as integer solutions. By exploiting the idea of enlargements, specifically of an algebraic number field in a nonstandard setting, Robinson and Roquette hoped that nonstandard methods would help them to go beyond the results Siegel and Mahler had obtained.

His own response was basically a formalist one:. I expect that future work on formalism may well include general epistemo-logical and even ontological considerations. Indeed, I think that there is a real need, in formalism and elsewhere, to link our understanding of mathematics with our understanding of the physical world.

The notions of objectivity, existence, infinity, are all relevant to the latter as they are to the former although this again may be contested by a logical positivist and a discussion of these notions in a purely mathematical context is, for that reason, incomplete. The collection of all number systems is not a finished totality whose discovery was complete around , or , or , but that it has been and still. When Robinson returned to Yale in the fall of , he had been experiencing stomach pains and finally underwent a series of tests at the end of November.

## Non-standard analysis

Robinson began to cancel commitments, lectures he had agreed to give, and meetings he had hoped to attend, but he continued to meet with his students. The class was removed to his modest office, where a dozen or so hearers crowded in. Robinson was not able to withstand the progressive advance of the cancer, and in April he was forced to cancel his one class and return to the Yale Infirmary. Shortly thereafter he died quietly in hospital on April 11, A few days earlier he had just been elected a member of the National Academy of Sciences.

He was also happy to remind people that his own. Robinson had indeed produced excellent work at the beginning of his own career, but his best-known and most often discussed work was done well after he was It can even be said that Robinson was only just beginning to develop the potential of nonstandard analysis and model theory when he died so prematurely at the age of Robinson was in many respects a universal mathematician, at home in many fields and thus able to exploit the power of model theory in many different areas.

As one sympathetic to the work of applied mathematicians as well as the most theoretical, he was also interested in finding applications of nonstandard analysis in a host of disciplines, from quantum physics to economics.

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And yet as the work he did at Yale clearly shows, he was not only aware of its powerful applications in certain contexts, but he appreciated the fact that it was historically revolutionary as well. As a tool, however, it required an experienced hand, and he was among the few who knew other parts of mathematics well enough to know where nonstandard analysis might be most helpful, or even essential.

In the course of his 55 years Robinson accomplished more than most can claim to have accomplished in far longer lifetimes. Indeed, he was a man who made mathematics a thing of beauty, and equally important, he had the remarkable ability to reveal that beauty to all who wished to learn from his example.

Brown, D. A limit theorem on the cores of large standard exchange economies. Proceedings of the National Academy of Sciences U. Dauben, J. Abraham Robinson.

The Creation of Nonstandard Analysis. A Personal and Mathematical Odyssey. Princeton, N. Engeler, E. Halmos, P.

I Want to Be a Mathematician. An Automathography. New York: Springer. Hirschfeld, J. Forcing, Arithmetic, and Division Rings. Springer Lecture Notes in Mathematics, vol. Berlin: Springer Verlag. Keisler, H. Studies in model theory. Luxemburg, and A. Young, eds.

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Selected Papers of Abraham Robinson, vol. New Haven, Conn. Kochen, S. The pure mathematician. Bulletin of the London Mathematical Society — In Selected Papers of Abraham Robinson , vol. Keisler, S. Young, pp. Introduction to papers on philosophy. Lutz, R. Nonstandard Analysis.

A Practical Guide with Applications. Springer Lecture Notes in Mathematics no.