Raphael Mitchel Robinson

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2 November 1911
National City, California, USA
27 January 1995
Berkeley, California, USA

Raphael Robinson was an American mathematician who was interested in a wide variety of problems including number theory and tilings.


Raphael Robinson's mother was Bessie Stevenson and his father was Bertram H Robinson. Bertram was a lawyer who travelled from place to place. He gave his sons romantic names, Raphael being the name he gave to the youngest of his four children, which was in keeping with his nature and his love of poetry. However his desire to move around eventually saw him move on and leave Bessie to bring up the family on her own. Bessie was a school teacher who taught in an elementary school and had to work very hard to give her children a good education.

Robinson entered the University of California at Berkeley from where he graduated with a B.A. in 1932 and an M.A. in the following year. He undertook research in complex analysis supervised by John McDonald and he was awarded a Ph.D. in December 1934 for his thesis Some results in the theory of Schlicht functions.

The Great Depression began in 1929 while Robinson was an undergraduate and by 1932, when he graduated with a B.A., one quarter of the workers in the United States were unemployed. The Depression lasted for around ten years so when Robinson began to look for a post in 1935 there was still a great scarcity of college positions and those which did exist paid very low wages. He was offered a half-time position at Brown University as an instructor which he accepted despite the fact that it really did not pay enough for him to survive. Indeed he suffered great hardship for two years and as a result of the poverty he suffered from tuberculosis. By 1937 employment opportunities were improving and Robinson was offered a full-time instructorship at Berkeley which he gladly accepted.

In 1939 Robinson taught a course in number theory and one of his students was Julia Bowman. Raphael and Julia began going for walks together; on these he would teach her more mathematics which she found very exciting. When Bowman's job applications failed, Neyman found a small amount of money to allow her to stay on at Berkeley as his assistant and in 1941 she was awarded her M.A. By this time Raphael and Julia planned to marry so Julia turned down a civil service job to remain at Berkeley as a teaching assistant. Raphael married Julia on 22 December 1941 but after this she was no longer allowed to teach in the mathematics department since Raphael was on the mathematics staff. Many years later Julia Robinson spoke about her husband:-
He taught me and has continued to teach me, has encouraged me, and has supported me in many ways.
Robinson was steadily promoted, becoming a full professor in 1949. He remained on the Faculty at Berkeley until he retired in 1973.

We record details of his character and interests as given in an obituary written by John Addison, David Gale, Leon Henkin, and Constance Reid:-
At the age of 61, when "early retirement" was not yet a popular option, Raphael chose to retire - at considerable financial sacrifice - so that he could devote more time to mathematics. Even in retirement Robinson owned no casual clothes. His pleasures were sedentary. He enjoyed challenging table games, novels as well as non-fiction, old movies, and the verse of Ogden Nash (on occasion turning out efforts of his own in that genre). He was a generous donor to many causes and a thorough reader of the Chronicle, the New Yorker, and the Nation as well as Martin Gardner's columns and selected comic strips. He was also a faithful contributor to the Problems Section of the American Mathematical Monthly. What the section editor described as "a beautiful short paper" of his was accepted for publication just days before his death.
Julia Robinson died in July 1985 and, in the following year, Raphael established the Julia Bowman Robinson Fund for fellowships for graduate students in mathematics at Berkeley. On 4 December 1994 Robinson suffered a stroke from which he never recovered, dying eight weeks later.

Robinson worked on a wide variety of mathematical topics. His doctoral dissertation was on complex analysis, but he also worked on logic, set theory, geometry, number theory, and combinatorics. In 1939 he published On numerical bounds in Schottky's theorem in the Bulletin of the American Mathematical Society, and in the following year published On the mean values of an analytic function in the same journal.

As an example of another of his early papers let us say a little about The approximation of irrational numbers by fractions with odd or even terms which he published in the Duke Mathematical Journal in 1940. The paper looks at a problem first studied by Hurwitz in 1891, namely to approximate an irrational number xx by rational numbers AB\large\frac{A}{B}\normalsize subject to the condition of xAB<1mB2| x - \large\frac{A}{B}\normalsize | < \large\frac{1}{mB^{2}\normalsize} for various values of mm. Robinson obtains best possible results using methods involving continued fractions, their convergents and their secondary convergents.

A typical paper on logic was Finite sequences of classes which appeared in 1945. He made a major contribution to the study of the foundations of mathematics, in particular the study of undecidable theories. In a series of papers Robinson showed that a number of mathematical theories are undecidable. He also examined the concept of 'essentially undecidable' introduced by Tarski, and answered an important open question by constructing a theory with a finite number of axioms that is essentially undecidable. In 1953 Tarski, together with Robinson and Mostowski, published Undecidable theories. G Kreisel writes:-
The book gives an introductory account of the methods introduced by Tarski for establishing the undecidability of several fairly simple branches of mathematics (group theory, lattices, abstract projective geometry, closure algebras and others). The methods and aims of this work are probably more easily intelligible and more interesting to the 'ordinary' mathematician than those of any other branch of mathematical logic.
As we mentioned above, Robinson worked in number theory and he used the earliest computers to obtain results. He coded the Lucas test for primality and tested whether 2n12^{n} - 1 was prime for all primes n<2304n < 2304 on the SWAC computer. He gave his results in Mersenne and Fermat numbers published in the Proceedings of the American Mathematical Society in 1954. These showed that these Mersenne numbers were all composite except for the seventeen values: nn = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, for which 2n12^{n} - 1 is a prime. At the time that Robinson wrote this paper the last five of these primes were larger than any that had previously been found.

A number theory colleague wrote the following about Robinson's number theory papers:-
In an age where most of our journals are filled with papers which (even if good) exploit theories for their own sake ... it is refreshing and stimulating to encounter one of Robinson's papers. In each of them he takes a problem, old or new, which can be stated in simple and intelligible terms, and either solves it, or at least adds much that is new. His scholarship is impeccable; it is plain that he never writes until he has thought deeply, and until he has sought out every relevant piece of existing knowledge.
Another major interest was tilings of the plane. In a major paper Undecidability and nonperiodicity for tilings of the plane published in 1971, Robinson continued to study problems of a kind that he had examined for a long while. D A Klarner writes in a review:-
This paper not only makes a considerable contribution in simplifying a tangled body of theory; it is wonderfully clear in exposition. The general mathematical reader will take pleasure in reading this paper; it is a remarkable piece of work.
In fact Robinson had already made a substantial contribution to problems of this type in earlier papers. We give a description of the type of problems Robinson was considering:-
Imagine the plane cut with two sets of parallel lines into an infinite grid of unit squares called cells. These cells are to be filled with translates of unit squares called tiles. A tile is a unit square cut with its diagonals into four triangles that are coloured; furthermore, a tile has an orientation in the plane so that rotations and reflections of the tile may not be allowed. Finally, there is a rule about adjacent tiles: their abutting edges must be the same colour. Given a finite set of types of tiles, the question is raised whether translates of copies of tiles in this set may be used to fill every cell in the plane subject to the rule that abutting edges be the same colour. If this is possible, the set of tiles is said to tile the plane. H Wang (1961) asked if there exists a general decision method for deciding every question of this kind. Also, he conjectured that if a set of tiles tiles the plane, then the set may be used to tile the plane periodically. If this conjecture were true (it has been shown to be false), then a general decision method would exist; namely, we systematically tile larger and larger square arrays of cells in every possible way with the given set of tiles. If the set tiles the plane periodically, this procedure will eventually turn up a period of a tiling. If the set does not pack the plane, then it follows from the König infinity lemma that there is a square array that cannot be tiled at all. Of course, this decision method is not effective if the set tiles the plane, but there is no way to tile the plane with the set periodically. This is the problem considered by [Robinson]: to construct a set of tiles that tiles the plane but does not admit a periodic tiling. Actually, such a set containing over twenty thousand tiles had already been found by R Berger (1966) who needed the result in the course of his proof that no general decision method exists for Wang's tiling problems. [Robinson] has found a set of 52 tiles that tile the plane, but do not admit a periodic tiling. There are variations on the rules about adjacent tiles, and for each rule the decidability question and the periodicity question have been settled.
In the 1971 paper mentioned above, Robinson asks a question about undecidability and nonperiodicity results for tilings of the hyperbolic plane. He partially answered his own question in Undecidable tiling problems in the hyperbolic plane which was published in 1978. Undecidability involves the halting problem for Turing machines and in 1991, when Robinson was aged 80, he published Minsky's small universal Turing machine which describes a universal Turing machine with 4 symbols and 7 states. In 1994 Robinson (now aged 83!) published Two figures in the hyperbolic plane which presents some properties of tilings of the hyperbolic plane by equilateral triangles having angles of size 2πn\Large\frac{2\pi}{n}, where n=7n = 7 or n=9n = 9.

References (show)

  1. J Brillhart, Raphael M Robinson, Bull. Inst. Combin. Appl. 16 (1996), 15-18.
  2. L Henkin, In memoriam : Raphael Mitchell Robinson, Bull. Symbolic Logic 1 (3) (1995), 340-343.
  3. In memoriam : Raphael Mitchell Robinson (1911-1995), Modern Logic 5 (3) (1995), 329.

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Written by J J O'Connor and E F Robertson
Last Update November 2004