A true figure of merit

Abraham Robinson: The Creation of Nonstandard Analysis
September 29, 1995

Abraham Robinson, who was born in 1918 into a Jewish family in Prussia and died in 1974 in New Haven, the United States, was acknowledged as one of the greatest mathematical logicians of the century. But Robinson's life story is not simply one of outstanding professional achievement within the confines of a particular academic discipline, it is truly a "personal and mathematical odyssey", a life seemingly dedicated to the crossing of boundaries and the revision of tradition. Joseph Dauben's detailed and extensive biography manages to convey a real sense of the vast mathematical, cultural and physical distances that Robinson's life and achievements spanned.

Fleeing persecution in Nazi Germany, Robinson arrived with his mother and elder brother in Palestine in 1933. With his aptitude for mathematics already established, he studied under the logician Abraham Fraenkel at the Hebrew University in Jerusalem, writing his first paper in mathematical logic - an area in which his most profound mathematical contributions were to come, but to which it would take him some time to return once he left for Paris to take up a scholarship at the Sorbonne in January 1940. Five months later, narrowly escaping the advancing German army, he arrived in England to spend the rest of the war developing applications of mathematics to the practical problems of aeronautics at the Royal Aircraft Establishment at Farnborough. He emerged from this period a world expert on supersonic flow and wing design.

Robinson found little difficulty in pursuing his interests in what today might be rigidly separated into "pure" and "applied" mathematics. But from the time of his arrival in California, he concentrated on mathematical logic guided by the principles he had advanced in his thesis and developed in papers presented at the international conferences that he attended. These principles characterise his major contributions to logic, philosophy and pure mathematics, and bear an intriguing relationship to some of his applied work.

Robinson shared Kurt Godel's vision of mathematical logic as a tool for application in other branches of mathematics. He was instrumental in generalising logical results, obtained in pursuit of foundational goals earlier in the century, and putting them to use in mainstream mathematics. In this respect, perhaps, he maintained the attitude of an applied mathematician; only now the mathematical tools were those of logic and the subject matter to be addressed was that of pure mathematics itself.

Robinson developed ways of using properties of the formal description of a class of mathematical structures to obtain both mathematical results about that class and principles to transfer results from one class to another. These hybrid logico-algebraic methods simplified the treatment of existing results - such as his solution to Hilbert's 17th problem for real numbers - but also proved easier to generalise to more complicated situations - such as the analogue of the 17th problem for p-adic numbers. The flavour and results that Robinson brought to these investigations inspired the branch of logic known as "model theory", which today features the most extensive applications of mathematical logic within mathematics proper.

Dauben's presentation of Robinson's activities in their temporal order enables the reader to speculate on the relationship between Robinson's applied mathematics and his approach to pure mathematics. In 1951 Robinson developed an axiomatic approach to dimensional analysis - a tool much beloved of the physicist and engineer - stressing its use as a principle for transferring applied results between different practical situations. A year later he was arguing for the use of logic to achieve transfer principles between different algebraic settings.

Robinson's and John Laurmann's textbook on wing theory became a standard text "for the more mathematically inclined student" after its publication in 1956. Later, Robinson presented his interest in the boundary layer problems that arise in this applied context as the motivation for his most glamorous and philosophically intriguing creation: nonstandard analysis.

Earlier results of logicians such as Thoralf Skolem and Kurt Godel had shown how formal axiomatisations of particular mathematical ideas were often ambiguous in the sense that any one of a number of distinct nonisomorphic mathematical structures have the properties stipulated by the axiomatisation. For example, common axiomatisations of arithmetic can be properly interpreted as referring to a structure in which there are infinitely large numbers, as well as to the structure that we usually accept as arithmetic. Such unintended interpretations are called "nonstandard". Robinson saw that it was possible to work within nonstandard features, and then by inspecting the logical form of the result concluded that it is also true of the standard universe. Nonstandard analysis makes use of a nonstandard universe of real numbers, which contains among other things, objects that function as infinitesimals. The use of infinitesimals was the main casualty of developments in analysis in the late 19th century that forms the basis for our modern notion of mathematical rigour. Developed extensively by Leibniz, Newton and the early analysts, their use had been discredited due to the difficulty of reasoning with them without contradiction. Robinson's results showed us one way of making mathematical sense of these intuitions about the infinitely small. Nonstandard methods are now part of the armoury available to mathematicians in the fields of pure mathematics, applied mathematics and mathematical physics; a fitting testimony to Robinson's mathematical breadth and another realisation of his and Godel's aspirations for mathematical logic.

This broad range of activities, insights and contributions to such different mathematical fields is difficult to survey coherently at a technical level. Dauben chooses to include detailed statements of the results and mathematical issues without attempting to explain them. These passages will interest the historian of mathematics and logic, providing an absorbing read for those who have a good background in logic, algebra and elementary analysis.

The lay reader, on the other hand, will be hard pressed to appreciate the technical material. Discussion that sheds light on the workings of Robinson's mathematical mind is limited, but an adequate treatment of this would require a further book in its own right and involve even more mathematical detail of little interest to the nonspecialist.

Accounts of Robinson's activities on university committees, in education initiatives, and attending cultural events with his wife are juxtaposed with accounts of his mathematical writings and correspondence. The book goes some way to revealing the integrity, the concern for intellectual endeavour, the compassion and humour that Robinson brought to his surroundings. A picture of the man emerges from this detail, giving a sense of his values, his vision of logic as a tool in the hands of the modern mathematician, and of his dedication to education.

Lincoln Wallen is a fellow, St Catherine's College, Oxford.

Abraham Robinson: The Creation of Nonstandard Analysis: A Personal and Mathematical Odyssey

Author - Joseph W. Dauben
ISBN - 0 691 03745 0
Publisher - Princeton University Press
Price - £40.00
Pages - 559

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