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Biographical Memoirs: Volume 82

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Biographical Memoirs: Volume 82 ABRAHAM ROBINSON October 6, 1918–April 11, 1974 BY JOSEPH W.DAUBENPlayfulness is an important element in the makeup of a good mathematician. —Abraham Robinson ABRAHAM ROBINSON WAS BORN on October 6, 1918, in the Prussian mining town of Waldenburg (now Walbrzych), Poland.1 His father, Abraham Robinsohn (1878–1918), after a traditional Jewish Talmudic education as a boy went on to study philosophy and literature in Switzerland, where he earned his Ph.D. from the University of Bern in 1909. Following an early career as a journalist and with growing Zionist sympathies, Robinsohn accepted a position in 1912 as secretary to David Wolfson, former president and a leading figure of the World Zionist Organization. When Wolfson died in 1915, Robinsohn became responsible for both the Herzl and Wolfson archives. He also had become increasingly involved with the affairs of the Jewish National Fund. In 1916 he married Hedwig Charlotte (Lotte) Bähr (1888– 1949), daughter of a Jewish teacher and herself a teacher. 1   Born Abraham Robinsohn, he later changed the spelling of his name to Robinson shortly after his arrival in London at the beginning of World War II. This spelling of his name is used throughout to distinguish Abby Robinson the mathematician from his father of the same name, the senior Robinsohn. Abby’s older brother, Shaul, always used Robinsohn, the traditional form of the family name.

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Biographical Memoirs: Volume 82 In 1916 their first son, Saul Benjamin, was born in Cologne, Germany. Robinsohn had been appointed the first director of the Jewish National Library in Jerusalem, but the family’s plans to emigrate to Palestine were unexpectedly precluded when Robinsohn prematurely died of a heart attack in Berlin on May 3, 1918. Five months later their second son, Abraham (Abby), was born in Waldenburg, Lower Silesia, where Lotte Robinsohn had moved to live with her parents. In 1925 the family moved to Breslau, capital of Silesia, where Lotte Robinsohn worked for the Keren Hajessod, a Zionist organization devoted to the emigration of Jews to Palestine. Both Saul and Abby were educated at a private Jewish school in Breslau, where Abby was very soon identified as “a genius.” He liked to hike, and enjoyed writing short stories, poems, plays, and even a five-act comedy, “Aus einer Tierchronik” (From a Chronicle of Animals). Both brothers attended the Jewish High School in Breslau and looked forward to spending their summers in Vienna with their uncle Isak Robinsohn, a prominent radiologist. In 1933, however, as Hitler and the National Socialists came to power in Germany, Lotte Robinsohn decided it was time to realize her lifelong dream of settling in Palestine. The family left Berlin by train on April 1, 1933, as Jewish businesses were being boycotted throughout the country. The trip south through Austria to Italy afforded the family a chance to see Rome, where Abby was greatly impressed by the Coliseum and found that he especially liked Italian pastries and espresso. In Naples they boarded the Volcania, a ship that sailed for Palestine via Greece. From Piraeus the Robinsohns were able to spend a day in Athens, and the Acropolis naturally made a lasting impression. A day later, when their ship docked in Haifa, as Abby recalled in his diary, everyone on board was singing the Ha-tikvah:

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Biographical Memoirs: Volume 82 “Our hope is not yet lost, the age-old hope, to return to the land of our fathers….” But under the British Mandate refugees could not establish legal residence, and so the Robinsohns arrived in Palestine as “tourists,” with ongoing tickets to Trieste, which they never used. PALESTINE (1933–1939) To support her family Lotte Robinsohn ran a small pension in Tel Aviv, but when Saul Robinsohn went to Jerusalem in 1934 to attend Hebrew University, within a year she and Abby also moved to Jerusalem, where Abby finished high school before going on to the university as well. To help meet family expenses he began tutoring students in various subjects, including Hebrew. The first evidence of his mathematical interests also dates from this time: a set of notes in German on the properties of conics (Seligman, 1979, p. xii). Jewish immigration to Palestine grew dramatically in the late 1930s; simultaneously the Arab population increasingly rebelled against the mandate and Zionism. The Jewish response was the creation of an illegal organization for the defense of Palestine, the Haganah. Robinson joined the Haganah and often assumed night watches. From time to time there were also paramilitary exercises in the mountains near Jerusalem that would keep him away from his studies, sometimes for weeks at a time. When six students at Hebrew University were killed on Mt. Scopus in 1936, the immediacy of the danger was apparent. It was not long before Robinson was made a junior officer of the Haganah. When Robinson entered Hebrew University in 1936, the Mathematics Department (added to the faculty in 1927) was barely a decade old. But, given the exodus of Jews from Europe, a number of impressive mathematicians had settled in Palestine, including Abraham Fraenkel, Michael Fekete,

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Biographical Memoirs: Volume 82 Jacob Levitzki—and Robinson studied with them all. The library was built around the collection of Felix Klein, whose books the university had obtained in 1926. Edmund Landau taught briefly in the newly founded Einstein Institute of Mathematics, and was followed by the appointment of Fraenkel. Robinson was among Fraenkel’s first students, but within two years Fraenkel said that he had already taught Robinson, his brightest student, all that he could (Seligman, 1979, p. xv). In addition to mathematics Robinson also took a number of courses in theoretical physics, an introductory course in Greek, as well as readings in ancient philosophy, especially the pre-Socratics and Plato. He also took a course devoted specifically to Leibniz. One of his fellow students Ernst Straus recalls, “When we did not understand something, we would ask him to explain it to us later” (quoted in Seligman, 1979, p. xvii). Robinson was also active in the university’s mathematics club, which he had helped organize and to which he once gave a lecture on the zeta function. Robinson’s first publication appeared in 1939 in the Journal of Symbolic Logic. This showed that the axiom of definiteness (the axiom of extensionality, or the axiom that establishes the character of equality within the system) was independent of the axioms of Zermelo-Fraenkel set theory. The paper was reviewed by Paul Bernays, who recommended it with various revisions to Alonzo Church for publication. Another paper accepted in 1939 for publication in Compositio Mathematica offered a simple proof of the theorem that for rings with minimal conditions for right ideals, every right nil ideal is nilpotent. This work drew on ideas inspired by courses Robinson had taken with Jacob Levitzky, but when World War II broke out, Compositio Mathematica ceased publication, and Robinson’s paper, though corrected

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Biographical Memoirs: Volume 82 in page proof, did not appear (although it is included in the edition of Robinson’s Selected Papers). At the end of 1939 Robinson was awarded a special French government scholarship. In his application he explained that he needed to broaden his mathematical horizons, especially with respect to “mathematical methodology” and that in France he hoped to be able to read the vast literature on the subject not available in Palestine. And so, despite the war that had already begun in Eastern Europe in 1939, Robinson set off in January of 1940 by ship from Beirut to Marseilles and then went on by train to Paris. PARIS: JANUARY-JUNE 1940 In Paris Robinson lived in a small pension in the Quartier Latin, not far from the Sorbonne. There is no record of what Robinson may have done with respect to his mathematical studies in Paris, apart from an enthusiastic letter of introduction Fraenkel wrote on Robinson’s behalf to the philosopher of mathematics Leon Brunschvicg. While in Paris, Robinson’s diary records visits to the museums and galleries, public concerts, the opera, cinemas, and the theatre. He noted in particular a play he saw by Jean Giraudoux, Ondine, based on a German novella but presented in Paris with typical French “esprit” and “clarté” as Robinson said (Dauben, 1995, p. 65). He also commented specifically on an exhibition he had seen by the Belgian artist Frans Masereel, a member of the Association of Revolutionary Artists and Writers, an antifascist group. Masereel had been sympathetic to the Republican cause in Spain and stressed socially progressive and radical themes in much of his artistic work, which Robinson regarded as better known in Palestine than in France. When the Germans invaded Holland, Belgium, and Luxemburg in May of 1940, Robinson first thought about

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Biographical Memoirs: Volume 82 trying to make his way back to Palestine. But, when Mussolini sided with Germany on June 10, declaring war on France and England, any easy route back to Palestine was effectively blocked. The next day Robinson left Paris, having learned that the German army was only some 30 kilometers northwest of the city. Relying on a combination of suburban trains, trucks, and often making his way on foot, he headed for Bordeaux. Fortunately, traveling on a British passport, Robinson was able to secure a place on a coal tender, one of the last to carry refugees across the channel from France to England. The trip took four days, slowed by intermittent shelling from a German ship and occasional strafings by enemy planes overhead. Everyone slept on the open deck until the boat reached Falmouth in Cornwall. From there Robinson was taken to a holding facility in London for processing along with thousands of other refugees. LONDON (1940–1946) Thanks to the Jewish National League in London, Robinson eventually found a place to stay in Brixton, “a quarter of ill repute,” as his diary put it. Soon the Germans were bombing London, and the Battle of Britain was underway. For weeks on end the blitzkrieg was relentless. One morning, returning from a night’s refuge in one of the underground stations, Robinson found his quarters destroyed by a bomb, and for nearly two weeks he was homeless. By November of 1940, however, he had enlisted in the Free French Forces under the command of General Charles de Gaulle. Despite the war Robinson did his best to keep his mathematics alive and even wrote a short paper on a generalized distributive law for commutative fields. M.H.Etherington, in reviewing the paper for the Proceedings of the Royal Society of Edinburgh, described the results as “entirely new,”

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Biographical Memoirs: Volume 82 “interesting,” and “could not be of any assistance to the enemy,” whereupon the article, “On a Certain Variation of the Distributive Law for a Commutative Algebraic Field,” was published in 1941. Meanwhile, Robinson had by a fortuitous set of circumstances been asked to help in writing a report on aircraft design for the Ministry of Aircraft Production. The results were sufficiently impressive that the British government requested Robinson’s transfer from the Free French to the Ministry of Aircraft Production at the Royal Aircraft Establishment in Farnborough, just southwest of London. Immediately Robinson began to study aerodynamics in earnest and soon passed a special examination administered by the Royal Aeronautical Society, whereby in June of 1942 he was made an “associate fellow.” In January of 1943 Robinson was visiting friends in London when he met Renée Kopel, a refugee from Vienna who was working in London as an actress and fashion designer. The two soon found that they both enjoyed the theatre, art galleries, nature, walking, and above all, music. Exactly a year after they met, Renée and Abby were married at Temple Fortune in Golders’ Green. At first they lived in West Byfleet, nearly equidistant between Farnborough and London, and later Surbiton, somewhat closer to London. In the meantime, Robinson had joined the Home Guard, to have a more active, physical involvement with the war. At Farnborough Robinson’s research was devoted in part to a study of the merits of single- versus double-engine designs for planes on aircraft carriers, but soon he was transferred to the aerodynamics department, where he began research on supersonic aerodynamics. One of the last of Robinson’s projects at Farnborough was the reconstruction of a German V-2 rocket from bits and pieces of debris the

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Biographical Memoirs: Volume 82 Royal Air Force had managed to collect from test firings from Peenemünde that landed in Sweden and Poland. Once Allied forces had made their beachhead at Normandy in June of 1944, it was another two months before Paris was liberated, on August 25. Within months Mussolini was dead in Italy, Hitler had committed suicide in Berlin, and Churchill finally proclaimed an end to the war in Europe, V-E Day, May 8, 1945. Abby donned his Royal Air Force uniform and went to London, where he and Renée listened to Churchill address the nation from Whitehall, after which they joined the crowds celebrating in central London. With the war in Europe finally at an end Robinson was assigned to an intelligence reconnaissance task force sent to Germany to debrief scientists in hopes of learning what they had accomplished in aerodynamical research. While in Frankfurt, he also made a special side trip to Breselenz to see the house where Riemann was born. LONDON (1946–1951) One of the first things Robinson did after the war was to make a brief return visit to Jerusalem, in part to take his examinations for his diploma from Hebrew University, and to see his mother, brother, and friends whom he had not seen for six years. Robinson was subsequently awarded his M.S. degree, with minors in physics and philosophy. He also used the month he was there to work on a paper with his former instructor Theodore Motzkin, the result of which was “Characterization of Algebraic Plane Curves,” published in the Duke Mathematical Journal the following year. Robinson returned to London, having accepted a position as senior lecturer at the newly founded College of Aeronautics at Cranfield (northwest of London), where he taught mathematics in the Department of Aerodynamics. At

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Biographical Memoirs: Volume 82 Cranfield Robinson spent considerable amounts of time conducting experiments in the wind tunnels and also learned to fly in order to gain the practical experience many theoreticians never acquired. Among Robinson’s continuing research interests at this time was the design of delta wings for supersonic travel. Wanting to further his own mathematical studies, Robinson enrolled as a graduate student at Birkbeck College, University of London, where he studied with Richard Cooke and Paul Dienes. He originally thought to devote a doctoral thesis to the syntax of algebra, but this eventually became “On the Metamathematics of Algebra.” He reported some of the early results of his thesis in a brief abstract he sent to the Journal of Symbolic Logic in 1949: “Analysis and Development of Algebra by the Methods of Symbolic Logic.” Even from this very concise note it was clear that his interests were much more mathematical in a strict sense than were those of other pioneers of the subject like Alfred Tarski and Leon Henkin. Robinson’s major interest was algebra, and he regarded logic as a means of obtaining new and more general results—not as an end in itself. Robinson first came to the attention of a worldwide audience at the International Congress of Mathematicians held in 1950 in Cambridge, Massachusetts. Based on the strength of a proposal he had submitted, Robinson was invited to give a lecture, “On the Applications of Symbolic Logic to Algebra,” which presented further results from his just completed Ph.D. thesis (in 1949). Here he was in excellent company; the other invited lecturers in the section on logic, in addition to Tarski, were Stephen Kleene and Thoralf Skolem. Both Robinson’s congress lecture and his thesis dealt with models and algebras of axioms, in which his introduction of diagrams and transfer principles was especially in-

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Biographical Memoirs: Volume 82 novative—and where he established a variety of results concerning algebraically closed fields. Philosophically, at this point Robinson was committed as he said to a “fairly robust philosophical realism,” meaning that he accepted the full “reality” of any given mathematical structure. The formal languages he drew upon were simply constructs to describe structures, and these he took for granted. His methods above all made it possible to establish results “whose proof by conventional means is not apparent” (Dauben, 1995, p. 175). Later, as a mature mathematician he would adopt a more formalist position with respect to the foundations of mathematics. Robinson’s thesis from Birkbeck College was published by North-Holland in 1951 as On the Metamathematics of Algebra. Robinson was also made deputy head of the Department of Aeronautics at Cranfield. However, in February 1951 he received an invitation to accept a position as an associate professor at the University of Toronto. There he would replace Leopold Infeld, the Polish physicist whose presumed Communist sympathies had created certain difficulties that eventually persuaded him to leave Canada and return to his native Poland. UNIVERSITY OF TORONTO (1951–1957) Robinson always tried to write at a constant pace, “three good pages” a day (Dauben, 1995, p. 185). As Abby and Renée sailed from Liverpool to Montreal on a Cunard liner in August of 1951, they not only went first class but in the course of the trip Robinson also completed the 25-page manuscript “On the Foundations of Dimensional Analysis.” (The original manuscript dated “RMS Franconia, August/ September 1951” is preserved among Robinson’s papers in the Yale University archives.) Dimensional analysis, as he explained, was an especially useful tool for engineers and

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Biographical Memoirs: Volume 82 analysis and its potential applications in other parts of mathematics. He had suggested in fact that Robinson come to the Institute for an extended period of time, and even hoped that Robinson might one day be his successor (Robinson to Gödel, April 14, 1971; Gödel papers #011957, Princeton University archives; cited in Dauben, 1995, p. 458). On the subject of nonstandard analysis Gödel had the following to say: In my opinion Nonstandard Analysis (perhaps in some non-conservative version) will become increasingly important in the future development of Analysis and Number Theory. The same seems likely to me, with regard to all of mathematics, for the idea of constructing “complete models” in various senses, depending on the nature of the problem under discussion (Gödel to Robinson, December 29, 1972; Gödel papers #011962, Princeton University archives; cited in Dauben, 1995, p. 459). While at the Institute, Robinson was working on a paper dedicated to Andrzej Mostowski for his sixtieth birthday. Of the “metamathematical problems” Robinson had raised in his Association for Symbolic Logic presidential address, he now turned to answer some of the questions he had posed on the “emerging field” of topological model theory. He also wrote another commemorative article that he dedicated to his colleague A.I.Mal’cev, “On Bounds in the Theory of Polynomial Ideals.” The most significant honor of Robinson’s entire career was conferred in April of 1973 when he was awarded the L. E.J.Brouwer Medal by the Dutch Mathematical Society. The first recipient, three years earlier, had been the French pioneer of catastrophe theory, René Thom. Robinson’s Brouwer lecture was devoted to “Standard and Nonstandard Number Systems” and constituted a mathematical tour of the major highlights of his best-known results, showing the power of nonstandard approaches to mathematics in general. In his Brouwer lecture Robinson also articulated

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Biographical Memoirs: Volume 82 further his views on the foundations of mathematics, which now reflected his experience with nonstandard number systems in particular: [Nonstandard analysis] does not present us with a single number system which extends the real numbers, but with many related systems. 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 1973 back in Heidelberg. Gert Müller had invited him to spend a week with the model theoreticians in Müller’s seminar, but this also gave Robinson a chance to meet with Roquette’s group, which took advantage of his visit as well. This also allowed Robinson and Roquette to continue their collaboration on nonstandard number theory. Together they were working on nonstandard approaches to diophantine equations, in particular C.I.Siegel’s theorem on integer points on curves. 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. As Roquette later explained, “These ideas of Abraham Robinson are of far-reaching importance, providing us with a new viewpoint and guideline towards our understanding of diophantine problems” (Roquette and Robinson, 1975, p. 424). From Heidelberg Robinson flew to Bristol for a European meeting of the Association for Symbolic Logic, where

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Biographical Memoirs: Volume 82 he discussed problems of foundations. Despite the great advances made in contemporary mathematics, where technical developments in particular had been “spectacular,” he was concerned about how little the essential nature of mathematics itself had been illuminated with respect to the problem of infinite totalities ontologically. 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. Robinson ended the summer of 1973 back in Princeton with a brief visit, again to see Gödel. When the fall term began at Yale, Robinson offered a course with the philosopher Stephan Körner on the “Philosophical Foundations of Mathematics.” He later confided to Körner that he doubted if the students were enjoying the seminar, “because we are enjoying it too much” (Stephan Körner in a letter to Renée Robinson, cited in Dauben, 1995, p. 471). In the end Robinson expressed his philosophy of mathematics in his usual light-hearted way in an account he wrote for the Yale Scientific Magazine (47, 1973), “Numbers—What Are They and What Are They Good For?” Number systems, like hair styles, go in and out of fashion—its what’s underneath that counts. He explained this in part as follows: The collection of all number systems is not a finished totality whose discovery was complete around 1600, or 1700, or 1800, but that it has been and still

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Biographical Memoirs: Volume 82 is a growing and changing area, sometimes absorbing new systems and sometimes discarding old ones, or relegating them to the attic. Nonstandard analysis and Robinson’s nonstandard real numbers were just another step in the continuing evolution of mathematics, which served to broaden and deepen the number systems available to mathematicians and logicians alike. When Robinson returned to Yale in the fall of 1973, he had been experiencing stomach pains and finally underwent a series of tests at the end of November. “These gave abundant grounds for suspecting cancer of the pancreas, and an exploratory operation revealed that the disease was beyond surgical remedy” (Seligman, 1979, p. xxxi). 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. The disease and the drugs forced him to struggle to concentrate, but his wit still could flash out, and his listeners’ laughter would then fill the narrow corridor outside his office (Seligman, 1979, p. xxxi). 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, 1974. A few days earlier he had just been elected a member of the National Academy of Sciences. CONCLUSION Robinson once said that “playfulness is an important element in the makeup of a good mathematician,” and he was certainly a mathematician who enjoyed his work to the fullest. He was also happy to remind people that his own

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Biographical Memoirs: Volume 82 career was the perfect counter-example to the old myth that mathematicians do their best work before they are 30, at the beginning of their careers. 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 40. 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 55. 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. 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. As he once told Greg Cherlin, “At first it was easy to get results—now you have to do more” (Gregory Cherlin, cited in Dauben, 1995, p. 492). 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.

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Biographical Memoirs: Volume 82 REFERENCES Brown, D.J., and A.Robinson. 1972. A limit theorem on the cores of large standard exchange economies. Proceedings of the National Academy of Sciences U. S. A. 69:1258–60. Dauben, J.W. 1995. Abraham Robinson. The Creation of Nonstandard Analysis. A Personal and Mathematical Odyssey. Princeton, N.J.: Princeton University Press. Engeler, E. 1964. Review of Robinson’s Introduction to Model Theory and to the Metamathematics of Algebra (1963). Math. Rev. 27: no. 3533. Halmos, P.R. 1985. I Want to Be a Mathematician. An Automathography. New York: Springer. Hirschfeld, J.M., and W.H.Wheeler. 1975. Forcing, Arithmetic, and Division Rings. Springer Lecture Notes in Mathematics, vol. 454. Berlin: Springer Verlag. Keisler, H.J. 1973. Studies in model theory.” Stud. Math. 8:96–133. Keisler, H.J., S.Körner, W.A.J. Luxemburg, and A.D.Young, eds. 1979. Selected Papers of Abraham Robinson, vol. 1: Model Theory and Algebra, vol. 2: Nonstandard Analysis and Philosophy, vol. 3: Aeronautics. New Haven, Conn.: Yale University Press. Kochen, S. 1976. The pure mathematician. On Abraham Robinson’s work in mathematical logic. Bulletin of the London Mathematical Society 8:312–15. Kochen, S. 1979. Introduction. In Selected Papers of Abraham Robinson, vol. 1, eds. H.J.Keisler, S.Körner, W.A.J.Luxemburg, and A. D.Young, pp. xxxiii-xxxvii. New Haven, Conn.: Yale University Press. Körner, S. 1979. Introduction to papers on philosophy. In Selected Papers of Abraham Robinson, vol. 2, eds. H.J.Keisler, S.Körner, W.A.J.Luxemburg, and A.D.Young, pp. xii-xiv. New Haven, Conn.: Yale University Press. Lutz, R., and M.Goze. 1981. Nonstandard Analysis. A Practical Guide with Applications. Springer Lecture Notes in Mathematics no. 881. Berlin: Springer-Verlag. Seligman, G. 1979. Biography of Abraham Robinson. In Selected Papers of Abraham Robinson, vol. 1, eds. H.J.Keisler, S.Körner, W. A.J.Luxemburg, and A.D.Young, pp. xiii-xxxii. New Haven, Conn.: Yale University Press.

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Biographical Memoirs: Volume 82 Young, A.D. 1979. Introduction. In Selected Papers of Abraham Robinson, vol. 3, eds. H.J.Keisler, S.Körner, W.A.J.Luxemburg, and A.D.Young, pp. xxix-xxxii. New Haven, Conn.: Yale University Press.

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Biographical Memoirs: Volume 82 SELECTED BIBLIOGRAPHY 1951 On the application of symbolic logic to algebra. In Proceedings of the International Congress of Mathematicians. Cambridge, Massachusetts, 1950 , vol. 1, pp. 686–94. Providence, R.I.: American Mathematical Society. On the Metamathematics of Algebra. Amsterdam: North-Holland. 1954 On predicates and algebraically closed fields. J. Symb. Logic 19:103– 14. 1955 Théorie métamathématique des idéaux. Paris: Gauthier-Villars. On ordered fields and definite functions. Math. Ann. 130:257–71. 1956 Completeness and persistence in the theory of models. Z. Math. Logik Grundlagen Math. 2:15–26. Complete Theories. Amsterdam: North-Holland. With J.A.Laurmann. Wing Theory. Cambridge: Cambridge University Press. 1958 Relative model-completeness and the elimination of quantifiers. Published in typescript as part of Summaries of Talks, Summer Institute for Symbolic Logic. Cornell University, 1957. Princeton, N.J.: Institute for Defense Analysis, Communications Research Division. Relative model-completeness and the elimination of quantifiers. Dialectica 12:394–407. On the concept of a differentially closed field. Bull. Res. Counc. Isr. 8:113–28. 1959 On the concept of a differentially closed field. Bull. Res. Counc. Isr. 8F:113–28.

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Biographical Memoirs: Volume 82 1961 Model theory and non-standard arithmetic. In Infinitistic Methods. Proceedings of the Symposium on Foundations of Mathematics, Warsaw, September 2–9, 1959, pp. 265–302. Oxford: Pergamon Press. Non-standard analysis. K. Ned. Akad. Wet. Proc. 64; Indag. Math. 23:432–40. 1962 Recent developments in model theory. In Logic, Methodology and Philosophy of Science. Proceedings of the 1960 International Congress for Logic, Methodology and Philosophy of Science, eds. E.Nagel, P.Suppes, and A.Tarski, pp. 60–79. Stanford, Calif.: Stanford University Press. Complex Function Theory over Non-Archimedean Fields. Doc. no. 282416. Arlington, Va.: Armed Services Technical Information Agency. 1963 Introduction to Model Theory and to the Metamathematics of Algebra. Amsterdam: North-Holland. 1964 Formalism 64. In Proceedings of the International Congress for Logic, Methodology and Philosophy of Science, Jerusalem, 1964, pp. 228–523. Amsterdam: North-Holland. 1965 Numbers and Ideals. An Introduction to Some Basic Concept of Algebra and Number Theory. San Francisco: Holden-Day. On the theory of normal families. In Studia logico-mathematica et philosophica, in honorem Rolf Nevanlinna die natali eius septuagesimo 22.X.1965. Acta Philos. Fenn. 18:159–84. 1966 Nonstandard Analysis. Amsterdam: North-Holland. On some applications of model theory to algebra and analysis. Rend. Mat. Appl. 25:562–92. With A.R.Bernstein. Solution of an invariant subspace problem of K.T.Smith and P.R.Halmos. Pac. J. Math. 16:421–31.

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Biographical Memoirs: Volume 82 1967 Nonstandard arithmetic. Bull. Am. Math. Soc. 73:818–43. 1968 Some thoughts on the history of mathematics. Compos. Math. 20:188– 93. 1971 Infinite forcing in model theory. In Proceedings of the Second Scandinavian Logic Symposium, Oslo, 1970, pp. 317–40. Amsterdam: North Holland.. Forcing in model theory. In Symposia Mathematica, no. 5, pp. 69– 82. New York: Academic Press. 1974 With D.J.Brown. The cores of large standard exchange economies. J. Econ. Theory 9:245–54. 1975 With P.Roquette. On the finiteness theorem of Siegel and Mahler concerning diophantine equations. J. Number Theory 7:121–76. 1988 On a relatively effective procedure for getting all quasi-integer solutions of diophantine equations with positive genus. Ann. Jap. Assoc. Philos. Sci. 7:111–15.

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