by E. T. Bell
Anyone who will ponder a little on the foregoing bare outline of Dedekind’s creation will see that what he did demanded penetrating insight and a mind gifted far above the ordinary good mathematical mind in the power of abstraction. Dedekind was a mathematician after Gauss’ own heart: “At nostro quidem judicio hujusmodi veritates ex notionibus potius quam ex notationibus hauriri debeant” (But in our opinion such truths [arithmetical] should be derived from notions rather than from notations). Dedekind always relied on his head rather than on an ingenious symbolism and expert manipulations of formulas to get him forward. If ever a man put notions into mathematics, Dedekind did, and the wisdom of his preference for creative ideas over sterile symbols is now apparent although it may not have been during his lifetime. The longer mathematics lives the more abstract—and therefore, possibly, also the more practical—it becomes.
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I. If xp + yp = zp, then xp = zp−yp and resolving zp – yp, into its p factors of the first degree, we get
xp = (z-y) (z-ry) (z-r2y) . . . (z-rp-1y),
in which r is a “p th root of unity” (other than l), namely rp – 1 = 0, with r not equal to 1. The algebraic integers in the field of degree p generated by r are those which Kummer introduced into the study of Fermat’s equation, and which led him to the invention of his “ideal numbers” to restore unique factorization in the field—an integer in such a field is not uniquely the product of primes in the field for all primes p.
II. †The “infinite” in Kummer’s title is still (1936) unjustified; “many” should be put for “infinite."
III. No adequate biography of Dedekind has yet appeared. A life was to have been included in the third volume of his collected works (1932), but was not, owing to the death of the editor in chief (Robert Fricke). The account here is based on Landau’s commemorative address. Note that, following the good old Teutonic custom of some German biographers, Landau omits all mention of Dedekind’s mother. This no doubt is in accordance with the theory of the “three K’s” propounded by the late Kaiser of Germany and heartily endorsed by Adolf Hitler: “A woman’s whole duty is comprised in the three big K’s—Kissing, Kooking [cooking is spelt with a K in German], and Kids.” Still, one would like to know at least the maiden name of a great man’s mother.
CHAPTER TWENTY EIGHT
The Last Universalist
POINCARE
A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.—HENRI POINCARÉ
IN THE History of his Life and Times the astrologer William Lilly (1602-1681) records an amusing—if incredible—account of the meeting between John Napier (1550-1617), of Merchiston, the inventor of logarithms, and Henry Briggs (1561-1631) of Gresham College, London, who computed the first table of common logarithms. One John Marr, “an excellent mathematician and geometrician,” had gone “into Scotland before Mr. Briggs, purposely to be there when these two so learned persons should meet. Mr. Briggs appoints a certain day when to meet in Edinburgh; but failing thereof, the lord Napier was doubtful he would not come. It happened one day as John Marr and the lord Napier were speaking of Mr. Briggs: ‘Ah John (said Merchiston), Mr. Briggs will not now come.’ At the very moment one knocks at the gate; John Marr hastens down, and it proved Mr. Briggs to his great contentment. He brings Mr. Briggs up into my lord’s chamber, where almost one quarter of an hour was spent, each beholding other with admiration, before one word was spoke.”
Recalling this legend Sylvester tells how he himself went after Briggs’ world record for flabbergasted admiration when, in 1885, he called on the author of numerous astonishingly mature and marvellously original papers on a new branch of analysis which had been swamping the editors of mathematical journals since the early 1880’s.
“I quite entered into Briggs’ feelings at his interview with Napier,” Sylvester confesses, “when I recently paid a visit to Poincaré [18541912] in his airy perch in the Rue Gay-Lussac. . . . In the presence of that mighty reservoir of pent-up intellectual force my tongue at first refused its office, and it was not until I had taken some time (it may be two or three minutes) to peruse and absorb as it were the idea of his external youthful lineaments that I found myself in a condition to speak.”
Elsewhere Sylvester records his bewilderment when, after having toiled up the three flights of narrow stairs leading to Poincaré’s “airy perch,” he paused, mopping his magnificent bald head, in astonishment at beholding a mere boy, “so blond, so young,” as the author of the deluge of papers which had heralded the advent of a successor to Cauchy.
A second anecdote may give some idea of the respect in which Poincaré’s work is held by those in a position to appreciate its scope. Asked by some patriotic British brass hat in the rabidly nationalistic days of the World War—when it was obligatory on all academic patriots to exalt their esthetic allies and debase their boorish enemies—who was the greatest man France had produced in modern times, Bertrand Russell answered instantly, “Poincaré.” “What! That man?” his uninformed interlocutor exclaimed, believing Russell meant Raymond Poincaré, President of the French Republic. “Oh,” Russell explained when he understood the other’s dismay, “I was thinking of Raymond’s cousin, Henri Poincaré.”
Poincaré was the last man to take practically all mathematics, both pure and applied, as his province. It is generally believed that it would be impossible for any human being starting today to understand comprehensively, much less do creative work of high quality in more than two of the four main divisions of mathematics—arithmetic, algebra, geometry, analysis, to say nothing of astronomy and mathematical physics. However, even in the 1880’s, when Poincaré’s great career opened, it was commonly thought that Gauss was the last of the mathematical universalists, so it may not prove impossible for some future Poincaré once more to cover the entire field.
As mathematics evolves it both expands and contracts, somewhat like one of Lemaître’s models of the universe. At present the phase is one of explosive expansion, and it is quite impossible for any man to familiarize himself with the entire inchoate mass of mathematics that has been dumped on the world since the year 1900. But already in certain important sectors a most welcome tendency toward contraction is plainly apparent. This is so, for example, in algebra, where the wholesale introduction of postulational methods is making the subject at once more abstract, more general, and less disconnected. Unexpected similarities—in some instances amounting to disguised identity—are being disclosed by the modern attack, and it is conceivable that the next generation of algebraists will not need to know much that is now considered valuable, as many of these particular, difficult things will have been subsumed under simpler general principles of wider scope. Something of this sort happened in classical mathematical physics when relativity put the complicated mathematics of the ether on the shelf.
Another example of this contraction in the midst of expansion is the rapidly growing use of the tensor calculus in preference to that of numerous special brands of vector analysis. Such generalizations and condensations are often hard for older men to grasp at first and frequently have a severe struggle to survive, but in the end it is usually realized that general methods are essentially simpler and easier to handle than miscellaneous collections of ingenious tricks devised for special problems. When mathematicians assert that such a thing as the tensor calculus is easy—at least in comparison with some of the algorithms that preceded it—they are not trying to appear superior or mysterious but are stating a valuable truth which any student can verify for himself. This quality of inclusive generality was a distinguishing trait of Poincaré’s vast output.
If abstractness and generality have obvious advantages of the kind indicated, it is also true that they sometimes have serious drawbacks for those who must be interested in details. Of what immediate use is it to a working physicist to know that a particular differential equation occurring in his wo
rk is solvable, because some pure mathematician has proved that it is, when neither he nor the mathematician can perform the Herculean labor demanded by a numerical solution capable of application to specific problems?
To take an example from a field in which Poincaré did some of his most original work, consider a homogeneous, incompressible fluid mass held together by the gravitation of its particles and rotating about an axis. Under what conditions will the motion be stable and what will be the possible shapes of such a stably rotating fluid? Mac-Laurin, Jacobi, and others proved that certain ellipsoids will be stable; Poincaré, using more intuitive, “less arithmetical” methods than his predecessors, once thought he had determined the criteria for the stability of a pear-shaped body. But he had made a slip. His methods were not adapted to numerical computation and later workers, including G. H. Darwin, son of the famous Charles, undeterred by the horrific jungles of algebra and arithmetic that must be cleared out of the way before a definite conclusion can be reached, undertook a decisive solution.I
The man interested in the evolution of binary stars is more comfortable if the findings of the mathematicians are presented to him in a form to which he can apply a calculating machine. And since Kronecker’s fiat of “no construction, no existence,” some pure mathematicians themselves have been less enthusiastic than they were in Poincaré’s day for existence theorems which are not constructive. Poincaré’s scorn for the kind of detail that users of mathematics demand and must have before they can get on with their work was one of the most important contributory causes to his universality. Another was his extraordinarily comprehensive grasp of all the machinery of the theory of functions of a complex variable. In this he had no equal. And it may be noted that Poincaré turned his universality to magnificent use in disclosing hitherto unsuspected connections between distant branches of mathematics, for example between (continuous) groups and linear algebra.
One more characteristic of Poincaré’s outlook must be recalled for completeness before we go on to his life: few mathematicians have had the breadth of philosophical vision that Poincaré had, and none is his superior in the gift of clear exposition. Probably he had always been deeply interested in the philosophical implications of science and mathematics, but it was only in 1902, when his greatness as a technical mathematician was established beyond all cavil, that he turned as a side-interest to what may be called the popular appeal of mathematics and let himself go in a sincere enthusiasm to share with nonprofessionals the meaning and human importance of his subject. Here his liking for the general in preference to the particular aided him in telling intelligent outsiders what is of more than technical importance in mathematics without talking down to his audience. Twenty or thirty years ago workmen and shopgirls could be seen in the parks and cafés of Paris avidly reading one or other of Poincaré’s popular masterpieces in its cheap print and shabby paper cover. The same works in a richer format could also be found—well thumbed and evidently read—on the tables of the professedly cultured. These books were translated into English, German, Spanish, Hungarian, Swedish, and Japanese. Poincaré spoke the universal languages of mathematics and science to all in accents which they recognized. His style, peculiarly his own, loses much by translation.
For the literary excellence of his popular writings Poincaré was accorded the highest honor a French writer can get, membership in the literary section of the Institut. It has been somewhat spitefully said by envious novelists that Poincaré achieved this distinction, unique for a man of science, because one of the functions of the (literary) Academy is the constant compilation of a definitive dictionary of the French language, and the universal Poincaré was obviously the man to help out the poets and grammarians in their struggle to tell the world what automorphic functions are. Impartial opinion, based on a study of Poincaré’s writings, agrees that the mathematician deserved no less than he got.
Closely allied to his interest in the philosophy of mathematics was Poincaré’s preoccupation with the psychology of mathematical creation. How do mathematicians make their discoveries? Poincaré will tell us later his own observations on this mystery in one of the most interesting narratives of personal discovery that was ever written. The upshot seems to be that mathematical discoveries more or less make themselves after a long spell of hard labor on the part of the mathematician. As in literature—according to Dante Gabriel Rossetti—“a certain amount of fundamental brainwork” is necessary before a poem can mature, so in mathematics there is no discovery without preliminary drudgery, but this is by no means the whole story. All “explanations” of creativeness that fail to provide a recipe whereby a gifted human being can create are open to suspicion. Poincaré’s excursion into practical psychology, like some others in the same direction, failed to bring back the Golden Fleece, but it did at least suggest that such a thing is not wholly mythical and may some day be found when human beings grow intelligent enough to understand their own bodies.
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Poincaré’s intellectual heredity on both sides was good. We shall not go farther back than his paternal grandfather. During the Napoleonic campaign of 1814 this grandfather, at the early age of twenty, was attached to the military hospital at Saint-Quentin. On settling in 1817 at Rouen he married and had two sons: Léon Poincaré, born in 1828, who became a first-rate physician and a member of a medical faculty; and Antoine, who rose to the inspector-generalship of the department of roads and bridges. Léon’s son Henri, born on April 29, 1854, at Nancy, Lorraine, became the leading mathematician of the early twentieth century; one of Antoine’s two sons, Raymond, went in for law and rose to the presidency of the French Republic during the World War; Antoine’s other son became director of secondary education. A great-uncle who had followed Napoleon into Russia disappeared and was never heard of after the Moscow fiasco.
From this distinguished list it might be thought that Henri would have exhibited some administrative ability, but he did not, except in his early childhood when he freely invented political games for his sister and young friends to play. In these games he was always fair and scrupulously just, seeing that each of his playmates got his or her full share of officeholding. This perhaps is conclusive evidence that “the child is father to the man” and that Poincaré was constitutionally incapable of understanding the simplest principle of administration, which his cousin Raymond applied intuitively.
Poincaré’s biography was written in great detail by his fellow countryman Gaston Darboux (1842-1917), one of the leading geometers of modern times, in 1913 (the year following Poincaré’s death). Something may have escaped the present writer, but it seems that Darboux, after having stated that Poincaré’s mother “coming from a family in the Meuse district whose [the mother’s] parents lived in Arrancy, was a very good person, very active and very intelligent,” blandly omits to mention her maiden name. Can it be possible that the French took over the doctrine of “the three big K’s”—noted in connection with Dedekind—from their late instructors after the kultural drives of Germany into France in 1870 and 1914? However, it can be deduced from an anecdote told later by Darboux that the family name may have been Lannois. We learn that the mother devoted her entire attention to the education of her two young children, Henri and his younger sister (name not mentioned). The sister was to become the wife of Émile Boutroux and the mother of a mathematician (who died young).
Due partly to his mother’s constant care, Poincaré’s mental development as a child was extremely rapid. He learned to talk very early, but also very badly at first because he thought more rapidly than he could get the words out. From infancy his motor coordination was poor. When he learned to write it was discovered that he was ambidextrous and that he could write or draw as badly with his left hand as with his right. Poincaré never outgrew this physical awkwardness. As an item of some interest in this connection it may be recalled that when Poincaré was acknowledged as the foremost mathematician and leading popularizer of science of his time he submitted to the Bi
net tests and made such a disgraceful showing that, had he been judged as a child instead of as the famous mathematician he was, he would have been rated—by the tests—as an imbecile.
At the age of five Henri suffered a bad setback from diphtheria which left him for nine months with a paralyzed larynx. This misfortune made him for long delicate and timid, but it also turned him back on his own resources as he was forced to shun the rougher games of children his own age.
His principal diversion was reading, where his unusual talents first showed up. A book once read—at incredible speed—became a permanent possession, and he could always state the page and line where a particular thing occurred. He retained this powerful memory all his life. This rare faculty, which Poincaré shared with Euler who had it in a lesser degree, might be called visual or spatial memory. In temporal memory—the ability to recall with uncanny precision a sequence of events long passed—he was also unusually strong. Yet he unblushingly describes his memory as “bad.” His poor eyesight perhaps contributed to a third peculiarity of his memory. The majority of mathematicians appear to remember theorems and formulas mostly by eye; with Poincaré it was almost wholly by ear. Unable to see the board distinctly when he became a student of advanced mathematics, he sat back and listened, following and remembering perfectly without taking notes—an easy feat for him, but one incomprehensible to most mathematicians. Yet he must have had a vivid memory of the “inner eye” as well, for much of his work, like a good deal of Riemann’s, was of the kind that goes with facile space-intuition and acute visualization. His inability to use his fingers skilfully of course handicapped him in laboratory exercises, which seems a pity, as some of his own work in mathematical physics might have been closer to reality had he mastered the art of experiment. Had Poincaré been as strong in practical science as he was in theoretical he might have made a fourth with the incomparable three, Archimedes, Newton, and Gauss.