# D D Kosambi on Einstein and G D Birkhoff

We present below versions of two short essays by D D Kosambi, the first on Einstein and the second on G D Birkhoff. Einstein: The Passionate Adventurer first appeared in The Times of India in 1959 while G D Birkhoff - A Tribute first appeared in Maths Student XII (3-4) (1944).

"That's our professor Einstein", said the American taxi driver with tenderness and pride rare in his notoriously disrespectful élan. The recipient of this accolade was clumping vigorously along the Princeton sidewalk one gloomy December afternoon in 1948. The famous mane of hair, now white and, thinner, was still unprotected by a hat. Shirt and necktie had been replaced by a sailor's knit jersey, socks dispensed with altogether.

This was an unforgettable first glimpse of the man whose theories had evoked the complete spectrum of comment from derision to adulation. The simplicity of his dress was matched by his bearing at all times, even in scientific discussion. His life, however, had not been as tranquil as his aspect. This pioneer citizen of a new universe bad had to change his earthly citizenship several times. Germany, the country of his birth and education, denied him full effectiveness because he was a Jew. Zionism, which he tried so hard to serve, did not gain because he took the advice of people without vision. His personal judgment of scientific merit came to be regarded as worthless, for anyone could - prey upon his abundant kindness to obtain a superlative testimonial.

Popularly labelled the world's greatest mathematician, he could never be compared to the really great mathematicians of his day such as Dedekind, Poincaré, Hilbert. The imputation that he was the father of the atom bomb led him to say bitterly towards the end of his life that if it were to do over again, he would prefer to be a plumber or a tramp rather than a scientist. He did write the letter that led president Roosevelt to allocate funds for the immediate development of nuclear chain reactions. Not he but other Princeton colleagues brought their talents to bear upon the technical development of the A-bomb, heedless of disaster to humanity. It is stated that his equation $E = mc^{2}$ led inevitably to the horrors of atomic warfare. This is true to about the same extent that the Sermon on the Mount led inevitably to the sack of Constantinople in the Fourth Crusade.

Einstein's great achievement was a completely new way of looking at the material universe. The amount of matter in space - at any time affects the properties of space itself, and also the measurement of time. Other radically new scientific ideas that characterise the first half of our century crystallised rapidly about the theory of relativity.

The fine mechanical system developed by Newton and his successors had begun to show small but clear and unmistakable flaws by the end of the 19th century. The planet Mercury did not move with the proper clockwork accuracy. Both electricity and magnetism obeyed Newton's inverse square law, just like gravitation; but what was the connection between them and gravity? Ultimate particles of matter carried electromagnetic charges. Why did they send out electromagnetic waves-light, as gravitation did not? Why was the velocity of that light completely unaffected by the earth's rapid movement through space? If mass and energy were indestructible, how did the Curies' new element Radium constantly shoot off particles as well as the x-rays discovered by Roentgen? Man's search for new sources of power and energy was being blocked by outworn notions of matter.

Einstein helped solve more than one of these problems, but his main work developed out of the question: why does light travel with a speed independent of its source? He turned the question about, and said that the constant velocity of light is a fundamental property of space. Two observers at a distance could compare their watches and yardsticks only by flashing light signals whose velocity remained the same for both, no matter how they moved. This leads to entirely new concepts of measurement and simultaneity. It also relates mass and energy, which become two interchangeable aspects of the same thing.

Philosophers and theologians dragged in the Bible, Karl Marx, immorality of the soul and God's knowledge of mathematics to attack Einstein with a viciousness not yet forgotten. He alone saw beyond mere verbal controversy. Some new, powerful tool was needed for the analysis of time and space. Not only had all major known facts to be explained, but it was essential to predict phenomena not as yet observed. This tool was discovered by him in the work of two Italian mathematicians Ricci and Levi-Civita. Its use had to be mastered painfully. Then came the "passionate adventure into the unknown" upon which he looked back as filling the finest years of his life: the precise mathematical formulation of the unity of space, time, and matter. The sublime exaltation of such discovery has to be experienced. It cannot be explained to those who seek it in mescaline, or the ascent of impossible mountain peaks. He took good care to associate competent mathematicians with his work. The insight, however, was his alone. So many of us produced beautiful and intricate formulae without knowing what to do with them, while he thought his way slowly to Nature's secrets.

The first magnificent results were published during the early years of that, senseless slaughter - the First World War. The new theory explained not only Newton's gravitational law, but also the curious behaviour of Mercury. There was a spectacular prediction, that a ray of light passing close to the sun would be bent slightly. As light has no mass, Newton's theory could not explain this; even if the rays did have weight, the deflection by Newton's theory would be only half that given by Einstein. Special observations made during solar eclipses confirmed the Einstein law. The theory passed thereafter as current coin into the common treasury of man's knowledge.

It was not in astronomy but in the opposite direction that the influence of relativity was indispensable. What happened inside the atom received a better explanation. Einstein went on to combine gravitation with electro-magnetism in a succession of unified field theories, over the years 1929-1949. When it appeared that virtually no solutions existed of his final equations, he had the courage to face possible ruin of twenty years' hard work: "Perhaps, my dear colleague, Nature does not obey differential equations after all". The solutions were found later by Hlavaty. However, Nature still has the last word. The inexhaustible properties of matter continually revealed by experiments in nuclear physics have outstripped all theories.

It seems to me that we now stand close to the threshold of a new life, as far above any pre-atomic utopia as that was above the early Stone Age. If we really cross the threshold into a new age, it will be by renouncing war and controlling greed for individual profit. Then indeed may our descendants abandon this little cinder of a planet for really brave new worlds in unbounded space. Busy with the creation of real history, they might no longer be conscious of their historical past. But one of the individuals who led us nearer to the threshold was the passionate adventurer - Einstein.

2. G D Birkhoff: A Tribute

By last airmail came unexpected news that the leading American mathematician Prof G D Birkhoff (not to be confused with his distinguished son and colleague Garrett Birkhoff) of Harvard had suddenly passed away. It is unfortunate that this note, which could have served as tardy appreciation on the occasion of his sixtieth birthday (March 21, 1944) should have to be turned into a rather slight obituary.

Birkhoff's principal significance in contemporary science was that with him American mathematics came of age. He first reversed the general trend of a pilgrimage to Europe for deeper mathematical studies, which was considered essential for every serious American student before him. When continental mathematicians first referred to his work, it was (as for example Wilhelm Blaschke) with the emphatic qualification "der Amerikaner Birkhoff" and that citation, I feel certain, must have been regarded as a greater triumph by Birkhoff himself than the innumerable honorary degrees, fellowships of academies and learned societies (led by the Academie des Sciences at Paris), prizes (none of greater distinction than that of the newly opened pontifical academy) showered upon him throughout his abruptly terminated scientific life. Not that he was not an internationalist, for he did his very best to save Göttingen as a mathematical centre during the depression, before Hitler destroyed all such hopes. After that he was instrumental in bringing to the USA the finest of the exiled talent, though he could never have been mistaken for a pro-Semite by anyone who spoke with him for more than thirty seconds on the subject. Nevertheless, he was and always remained an American, at times aggressively so, with forthright direct action and oversimplified thinking - except in his mathematics. When Indians like T Vijayaraghavan and S Chandrasekhar received the courtesy that he always lavished upon all whom he believed to possess any degree of scientific competence, Birkhoff would manage to persuade himself that he was doing it for the good of America and not because of his own genuinely kind and hospitable nature.

To appreciate what this means, without letting it go as a grudging compliment to the memory of great scholar, one would have to study the mathematical atmosphere of Birkhoff's younger days when the very claim to be an American scholar exposed the claimant to ridicule at home and polite contempt abroad. The true scholars of his formative period were either imported, like Sylvester and Bolza, or were isolated and frozen out for lack of appreciation. Josiah Willard Gibbs had to be resurrected by the German chemist Ostwald before the USA realised that it had actually produced a great scientist. Eliakim Hastings Moore, Birkhoff's beloved and most influential teacher, found his 'general analysis' totally neglected till the fame of his distinguished pupil directed the attention of others there-to. By that time it was much too late for the method to be effective or Moore's notation to be adopted, for Hilbert's followers had skimmed all the cream off that particular jug. This was inevitable in the undeveloped state of American mathematics, though Moore himself was the first in any country to realise the full connotation of Hilbert's work on integral equations which has had so profound an influence upon current scientific activity through modern quantum theory.

Birkhoff studied under Moore at Chicago in 1902, came to Harvard (the only other American mathematical centre then existing) next year with Moore's approval, worked rather unenthusiastically for two years, and returned to Chicago for another two years of real inspiration. The stage at Harvard was then fully occupied by the imposing presence of B O Peirce who made up for a total lack of mathematical intelligence by his preternatural solemnity and an impressive beard (which earned him the soubriquet of "Santa Claus" from Birkhoff's generation of students). Peirce's ostentation overshadowed the real mathematical ability of his son Charles, whose unconventional life and thought (like that of G W Hill) led to complete intellectual obscurity and a miserable academic career with the doubtful compensation of posthumous honour.

Far better as a teacher was William Fogg Osgood, a much younger member of the Harvard mathematical department which he succeeded in raising to pre-eminence in latter days. Osgood had an excellent mathematical training under Felix Klein, was instrumental in bringing the modern analysis of Weierstrass with its battery of epsilons and deltas to bear upon the American student, knew what the Erlanger programme was, possessed the adjuncts considered necessary for a good teacher in his day including beard, pompous mannerisms (plus the affectionate student-bestowed title of "Foggy Bill") and lack of imagination. He failed to inspire Birkhoff, though he did succeed in bringing him as a professor to Harvard from Princeton in later days, and Birkhoff treated him with all the formal respect due to a teacher. But their mutual regard went no further. Osgood privately characterizsed Birkhoff's presentation as "sloppy", which it certainly was, and hence by innuendo his thinking too, which was certainly never sloppy. It is to be noted that these two leaders of scientific endeavour managed to keep their personal views about each other from interfering with their work and close cooperation in the same University department; this is a lesson that has yet to be learned by academicians in our own country.

The last of Birkhoff's teachers worth mentioning was Maxime Bocher whose crystal clear lectures derived from his crystal clear thinking. The very perfection of these lectures, however, left the budding young research worker cold with unfortunate after effects; for, Birkhoff's own lectures left very much to be desired even twenty-five years later. Those of his lectures that the advanced students did understand were invariably considered by them the most inspiring that they had ever heard, but usually he lost himself, the subject, and the audience in his own latest brainwave which might have developed that very morning between the breakfast table and the lecture room.

The rest of Birkhoff's story is precisely that of the growth of modern mathematics in the USA, of the substitution of a few isolated individual workers by the general development of indigenous talent and, a powerful school took the lead over all others soon after the First World War. The best view of this development as regards persons and performance is to be had from Birkhoff's own semi centennial address (1938) to the American Mathematical Society, "Fifty Years of American Mathematics". In this his thorough grasp of many branches of the subject shows itself, along with his breadth of vision, though he consistently minimises his own contributions and the reader will not realise that a goodly number of those he praises with justice were made by his own brilliant students, often with his direct inspiration.

He was early associated with Vandiver in work on Fermat's last theorem, and his pleasure was unbounded when he saw in the advance copy of Landau's book on number theory that a whole section had been devoted to the work of his early collaborator, gaining him long-delayed recognition. What first made Birkhoff himself internationally famous was his simple proof of Poincaré's geometric theorem on the existence of invariant points for a ring transformed into itself with the two boundaries advanced in opposite direction. Poincaré died without proving the theorem and a proof supposed to have been discovered by Phragmen actually came to nothing, leaving Birkhoff in sole possession of the field and the eminence that was his due. Before this, his work on existence theorems for difference equations, showing their close parallel behaviour with differential equations had given him substantial reputation.

He followed this success up with further important work on dynamics, a continuation of Poincaré's ideas, which approached the dynamical existence problems of stable motions, of periodic orbits, etc. by methods of topology and differential geometry im Grossen. This was, perhaps, his most important and successful work. His attack on the four-colour map problem was not successful in spite of intensive study and effort but it inspired others like P Franklin to make their own important advances. In mathematical physics his book on relativity and modern physics deserves to be far better known than it is; though it came rather late in the relativistic day, his contributions there have been regarded as important by no less a geometer than Levi-Civita. If he did not follow up the enticing field of tensor analysis, it must undoubtedly have been because the Princeton school (which he himself helped found with Veblen) had plunged into the heart of the subject with greater vigour and success than could have been expected at that time by a solitary research worker whose main interest lay in other branches of mathematics.

Birkhoff's proof in 1931 of the ergodic theorem was due to his having gone first to the main point of the topic towards which such capable workers as J von Neumann were also converging. The actual flawless proof was supplied a year later by the Soviet mathematician Khinchin in a definitive paper which is at the same time a model of graceful tact; but the basic ideas derived admittedly from Birkhoff who showed the fundamental position of the concept of measure and integration, particularly of Lebesgue integration, in such problems. On the other hand, I find it impossible to take Birkhoff's "Aesthetic Measure" seriously.

Authoritative and complete obituaries will rewritten by those who knew Birkhoff much better than the present writer, and whose mathematical stature could be said to equal his in some directions at least. One can only point to Birkhoff here as an important subject for study in the development of science in a country which changes over rapidly to an advanced mode and greater concentration of production. The precise function of the individual can only be studied against the background of the changing system, and this is generally not done in obituaries of the type to which we are accustomed. India must some day take the lead in a similar way - though a tremendous expansion of business is now going on in this land without the corresponding scientific advance one should naturally expect, for the simple reason that our intellectual life is still disoriented and based (even debased!) upon concepts originally derived from the scanty overflow of a colonial economy.

If an answer is wanted some day as to why a real school of Indian mathematicians (did; or) did not develop after the second world war, the historian of science could do worse than study the development of Birkhoff as a facet of the development of his country. He used to mention in moments of confidence (expressing amusement mingled with slight but noticeable regret) that a Dutch relative of his had offered him an active share in a cable company, and had the laugh of him ever since by making a fortune out of the enterprise to which Birkhoff had preferred the obscurity of a simple professorship! One wonders what had happened to the Dutch cousin and the cable business by the end of 1940. But that must have been a year of disappointment for Birkhoff too, for the international mathematical congress scheduled for Harvard that year had to be cancelled because of the war. It would have been an occasion to round out his whole career, an occasion upon which he could see his students carry his country to the forefront of the science to which he had devoted his entire life, giving up all the financial opportunities that were open to any American of his mental powers in the period of expanding bourgeois capitalist economy.

One wonders whether the war had changed Birkhoff's naive views about the importance of race and the essential glory of Nordics, which he held very strongly at least till 1934. Did he realise that "pure Anglo-Saxon" was even more meaningless than "American scholar" had been before him, and that "American" was not a race but a mentality? It shows the fundamental nobility of his character that he never allowed such views to interfere with his scientific judgment, nor to prejudice him in the slightest in the matter of adjudicating research fellowships and prizes. He had no hesitation in recommending for important posts people with whom he was not and did not want to be on visiting terms. It was noticeable that when he expressed his views on politics or sociology in any sort of mixed company, anyone could (and someone often did) contradict him flatly without offence on either side. But the moment he began to talk about mathematics, the others (no matter what their specialty) quietly stopped their own chatter to listen. It was impossible not to love such a teacher.

In the lifetime of any scientist who has not the good fortune to die young, (like Abel, Galois, Riemann, Ramanujan) there must inevitably come a moment when he begins to sense his own limitations, to feel the joints of his mind stiffening no less than those of his body, to shrink a little from the thought of testing his ability against totally new problems. Far too many scientists of first rank then go the way of popular acclaim, newspaper reputations and public performances, to drown the insistent voice from within which begins to ask "the end is near, you are going down the intellectual slope; how much of your work will survive? What is your share of immortality?" If Birkhoff had any inkling of the end, one feels that he would have faced it without a qualm, for his work was safe (in spite of wartime regression) in the hands of the American school. His great contemporaries greater in absolutely effective total scientific achievement as for example, Hilbert, Levi-Civita, Lebesgue could not have had this assurance of continuity dying as they did in the midst of universal distress and the collapse of all they had been brought up to honour. In this, Birkhoff was fortunate above the rest, and none deserved such fortune better.

Last Updated March 2022