Men of Mathematics

by E. T. Bell

  Kronecker died of a bronchial illness in Berlin on December 29, 1891, in his sixty ninth year.

  * * *

  I. One problem in this subject: an algebraic curve may have loops on it, or places where the curve crosses its tangents; given the degree of the curve, how many such points are there? Or if we cannot answer that, what equations connecting the number of these and other exceptional points must hold? Similarly for surfaces.

  II. In a letter to Sonja Kowalewski, 1885.

  CHAPTER TWENTY SIX

  Anima Candida

  RIEMANN

  A geometer like Riemann might almost have foreseen the more important features of the actual world.—A. S. EDDINGTON

  IT HAS BEEN SAID OF coleridge that he wrote but little poetry of the highest order of excellence, but that that little should be bound in gold. The like has been said of Bernhard Riemann, the mathematical fruits of whose all too brief summer fill only one octavo volume. It may also be truly said of Riemann that he touched nothing that he did not in some measure revolutionize. One of the most original mathematicians of modern times, Riemann unfortunately inherited a poor constitution, and he died before he had reaped a tithe of the golden harvests in his fertile mind. Had he been born a century later than he was, medical science could probably have leased him twenty or thirty more years of life, and mathematics would not now be waiting for his successor.

  Georg Friedrich Bernhard Riemann, the son of a Lutheran pastor, and the second of six children (two boys, four girls), was born in the little village of Breselenz, in Hanover, Germany, on September 17, 1826. His father had fought in the Napoleonic wars, and on settling down to a less barbarous mode of living had married Charlotte Ebell, daughter of a court councillor. Hanover in 1826 was not exactly prosperous, and the circumstances of an obscure country parson with a wife and six children to feed and clothe were far from affluent. It is claimed by some biographers, apparently with justice, that the frail health and early deaths of most of the Riemann children were the result of undernourishment in their youth and were not due to poor stamina. The mother also died before her children were grown.

  In spite of poverty the home life was happy, and Riemann always retained the warmest affection—and homesickness, when he was absent—for all his lovable family. From his earliest years he was a timid, diffident soul with a horror of speaking in public or attracting attention to himself. In later life this chronic shyness proved a very serious handicap and occasioned him much agonized misery till he overcame it by diligent preparation for every public utterance he was likely to make. The engaging bashfulness of Riemann’s boyhood and early manhood, which endeared him to all who met him, was in strange contrast to the ruthless boldness of his matured scientific thought. Supreme in the world of his own creation, he realized his transcendent powers and shrank from nobody, real or imaginary.

  While Riemann was still an infant his father was transferred to the pastorate of Quickborn. There young Riemann received his first instruction, from his father, who appears to have been an excellent teacher. From the very first lessons Bernhard showed an unquenchable thirst for learning. His earliest interests were historical, particularly in the romantic and tragic history of Poland. As a boy of five Bernhard gave his father no peace about unhappy Poland, but demanded to be told over and over again the legend of that heroic country’s gallant (and at times slightly fatuous) struggles for liberty and, in the late Woodrow Wilson’s rich, fruity phrase, “self-determination.”

  Arithmetic, begun at about six, offered something less harrowing for the sensitive young boy to dwell on. His inborn mathematical genius now asserted itself. Bernhard not only solved all the problems shoved at him, but invented more difficult teasers to exasperate his brother and sisters. Already the creative impulse in mathematics dominated the boy’s mind. At the age of ten he received instruction in more advanced arithmetic and geometry from a professional teacher, one Schulz, a fairly good pedagogue. Schulz soon found himself following his pupil, who often had better solutions than he.

  At fourteen Riemann went to stay with his grandmother at Hanover, where he entered his first Gymnasium, in the upper third class. Here he endured his first overwhelming loneliness. His shyness made him the butt of his schoolfellows and drove him in upon his own resources. After a temporary setback his schoolwork was uniformly excellent, but it gave him no comfort, and his only solace was the joy of buying such inconsiderable presents as his pocket money would permit, to send home to his parents and brother and sisters on their birthdays. One present for his parents he invented and made himself, an original perpetual calendar, much to the astonishment of his incredulous schoolfellows. On the death of his grandmother two years later, Riemann was transferred to the Gymnasium at Lüneburg, where he studied till he was prepared, at the age of nineteen, to enter the University of Göttingen. At Lüneburg Riemann was within walking distance of home. He took full advantage of his opportunities to escape to the warmth of his own fireside. These years of his secondary education, while his health was still fair, were the happiest of his life. The tramps back and forth between the Gymnasium and Quickborn taxed his strength, but in spite of his mother’s anxiety that he might wear himself out, Riemann continued to over-exert himself in order that he might be with his family as often as possible.

  While still at the Gymnasium Riemann suffered from the itch for finality and perfection which was later to slow up his scientific publication. This defect—if such it was—caused him great difficulty in his written language exercises and at first made it doubtful whether he would “pass.” But this same trait was responsible later for the finished form of two of his masterpieces, one of which even Gauss declared to be perfect. Things improved when Seyffer, the teacher of Hebrew, took young Riemann into his own house as a boarder and ironed him out.

  The two studied Hebrew together, Riemann frequently giving more than he took, as the future mathematician at that time was all set to gratify his father’s wishes and become a great preacher—as if Riemann, with his tongue-tied bashfulness, could ever have thumped hell and damnation or redemption and paradise out of any pulpit. Riemann himself was enamored of the pious prospect, and although he never got as far as a probationary sermon, he did employ his mathematical talents in an attempted demonstration, in the manner of Spinoza, of the truth of Genesis. Undaunted by his failure, young Riemann persevered in his faith and remained a sincere Christian all his life. As his biographer (Dedekind) states, “He reverently avoided disturbing the faith of others; for him the main thing in religion was daily self-examination.” By the end of his Gymnasium course it was plain even to Riemann that Great Headquarters could have but little use for him as a router of the devil, but might be able to employ him profitably in the conquest of nature. Thus once again, as in the cases of Boole and Kummer, a brand was plucked from the burning, ad majoram Dei gloriam.

  The director of the Gymnasium, Schmalfuss, having observed Riemann’s talent for mathematics, had given the boy the run of his private library and had excused him from attending mathematical classes. In this way Riemann discovered his inborn aptitude for mathematics, but his failure to realize immediately the extent of his ability is so characteristic of his almost pathological modesty as to be ludicrous.

  Schmalfuss had suggested that Riemann borrow some mathematical book for private study. Riemann said that would be nice, provided the book was not too easy, and at the suggestion of Schmalfuss carried off Legendre’s Théorie des Nombres (Theory of Numbers). This is a mere trifle of 859 large quarto pages, many of them crabbed with very close reasoning indeed. Six days later Riemann returned the book. “How far did you read?” Schmalfuss asked. Without replying directly, Riemann expressed his appreciation of Legendre’s classic. “That is certainly a wonderful book. I have mastered it.” And in fact he had. Some time later when he was examined he answered perfectly, although he had not seen the book for months.

  No doubt this is the origin of Riemann’s interest in the riddle of prime numbers
. Legendre has an empirical formula estimating the approximate number of primes less than any preassigned number; one of Riemann’s profoundest and most suggestive works (only eight pages long) was to be in the same general field. In fact “Riemann’s hypothesis,” originating in his attempt to improve on Legendre, is today one of the outstanding challenges, if not the outstanding challenge, to pure mathematicians.

  To anticipate slightly, we may state here what this hypothesis is. It occurs in the famous memoir Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse (On the number of prime numbers under a given magnitude), printed in the monthly notices of the Berlin Academy for November, 1859, when Riemann was thirty three. The problem concerned is to give a formula which will state how many primes there are less than any given number n. In attempting to solve this Riemann was driven to an investigation of the infinite series

  in which s is a complex number, say where u and v are real numbers, so chosen that the series converges. With this proviso the infinite series is a definite function of s, say fζ(5) (the Greek zeta, f, is always used to denote this function, which is called “Riemann’s zeta function”); and as s varies, ζ (s) continuously takes on different values. For what values of s will ζ (s) be zero? Riemann conjectured that all such values of s for which u lies between 0 and 1 are of the form ½ + iv, namely, all have their real part equal to ½.

  This is the famous hypothesis. Whoever proves or disproves it will cover himself with glory and incidentally dispose of many extremely difficult questions in the theory of prime numbers, other parts of the higher arithmetic, and in some fields of analysis. Expert opinion favors the truth of the hypothesis. In 1914 the English mathematician G. H. Hardy proved that an infinity of values of s satisfy the hypothesis, but an infinity is not necessarily all. A decision one way or the other disposing of Riemann’s conjecture would probably be of greater interest to mathematicians than a proof or disproof of Fermat’s Last Theorem. Riemann’s hypothesis is not the sort of problem that can be attacked by elementary methods. It has already give rise to an extensive and thorny literature.

  Legendre was not the only great mathematician whose works Riemann absorbed by himself—always with amazing speed—at the Gymnasium; he became familiar with the calculus and its ramifications through the study of Euler. It is rather surprising that from such an antiquated start in analysis (Euler’s approach was out of date by the middle 1840’s owing to the work of Gauss, Abel, and Cauchy), Riemann later became the acute analyst that he did. But from Euler he may have picked up something which also has its place in creative mathematical work, an appreciation of symmetrical formulas and manipulative ingenuity. Although Riemann depended chiefly on what may be called deep philosophical ideas—those which get at the heart of a theory—for his greater inspirations, his work nevertheless is not wholly lacking in the “mere ingenuity” of which Euler was the peerless master and which it is now quite the fashion to despise. The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so also can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular. Riemann’s instinctive mathematical tact preserved him from the bad taste of either extreme.

  In 1846, at the age of nineteen, Riemann matriculated as a student of philology and theology at the University of Göttingen. His desire to please his father and possibly help financially by securing a paying position as quickly as possible dictated the choice of theology. But he could not keep away from the mathematical lectures of Stern on the theory of equations and on definite integrals, those of Gauss on the method of least squares, and Goldschmidt’s on terrestrial magnetism. Confessing all to his indulgent father, Riemann prayed for permission to alter his course. His father’s ungrudging consent that Bernhard follow mathematics as a career made the young man supremely happy—also profoundly grateful.

  After a year at Göttingen, where the instruction was decidedly antiquated, Riemann migrated to Berlin to receive from Jacobi, Dirichlet, Steiner, and Eisenstein his initiation into new and vital mathematics. From all of these masters he learned much—advanced mechanics and higher algebra from Jacobi, the theory of numbers and analysis from Dirichlet, modern geometry from Steiner, while from Eisenstein, three years older than himself, he learned not only elliptic functions but self-confidence, for he and the young master had a radical and most energizing difference of opinion as to how the theory should be developed. Eisenstein insisted on beautiful formulas, somewhat in the manner of a modernized Euler; Riemann wanted to introduce the complex variable and derive the entire theory, with a minimum of calculation, from a few simple, general principles. Thus, no doubt, originated at least the germs of one of Riemann’s greatest contributions to pure mathematics. As the origin of Riemann’s work in the theory of functions of a complex variable is of considerable importance in his own history and in that of modern mathematics, we shall glance at what is known about it.

  Briefly, nothing definite. The definition of an analytic function of a complex variable, discussed in connection with Gauss’ anticipation of Cauchy’s fundamental theorem, was essentially that of Riemann. When expressed analytically instead of geometrically that definition leads to the pair of partial differential equations II which Riemann took as his point of departure for a theory of functions of a complex variable. According to Dedekind, “Riemann recognized in these partial differential equations the essential definition of an [analytic] function of a complex variable. Probably these ideas, of the highest importance for his future career, were worked out by him in the fall vacation of 1847 [Riemann was then twenty one] for the first time.”

  Another version of the origin of Riemann’s inspiration is due to Sylvester, who tells the following story, which is interesting even if possibly untrue. In 1896, the year before his death, Sylvester recalls staying at “a hotel on the river at Nuremberg, where I conversed outside with a Berlin bookseller, bound, like myself, for Prague. . . . He told me he was formerly a fellow pupil of Riemann, at the University, and that, one day, after receipt of some numbers of the Comptes rendus from Paris, the latter shut himself up for some weeks, and when he returned to the society of his friends, said (referring to the newly published papers of Cauchy), ’This is a new mathematic.’ ”

  Riemann spent two years at the University of Berlin. During the political upheaval of 1848 he served with the loyal student corps and had one weary spell of sixteen hours’ guard duty protecting the jittery if sacred person of the king in the royal palace. In 1849 he returned to Göttingen to complete his mathematical training for the doctorate. His interests were unusually broad for the pure mathematician he is commonly rated to be, and in fact he devoted as much of his time to physical science as he did to mathematics.

  From this distance it seems as though Riemann’s real interest was in mathematical physics, and it is quite possible that had he been granted twenty or thirty more years of life he would have become the Newton or Einstein of the nineteenth century. His physical ideas were bold in the extreme for his time. Not till Einstein realized Riemann’s dream of a geometrized (macroscopic) physics did the physics which Riemann foreshadowed—somewhat obscurely, it may be—appear reasonable to physicists. In this direction his only understanding follower till our own century was the English mathematician William Kingdon Clifford (1845-1879), who also died long before his time.

  During his last three semesters at Göttingen Riemann attended lectures on philosophy and followed the course of Wilhelm Weber in experimental physics with the greatest interest. The philosophical and psychological fragments left by Riemann at his death show that as a philosophical thinker he was as original as he was in mathematics and science. Weber recognized Riemann’s scientific genius and became his warm friend and helpful counsellor. To a far higher degree than the majority of great mathematicians who have written on physical science, Riemann had a feeling for what is important—or likely to be so—in physics, and this feeling is no doubt
due to his work in the laboratory and his contact with men who were primarily physicists and not mathematicians. The contributions of even great pure mathematicians to physical science have usually been characterized by a singular irrelevance so far as the universe observed by scientists is concerned. Riemann, as a physical mathematician, was in the same class as Newton, Gauss, and Einstein in his instinct for what is likely to be of scientific use in mathematics.

  As a sequel to his philosophical studies with Johann Friedrich Herbart (1776-1841), Riemann came to the conclusion in 1850 (he was then twenty four) that “a complete, well-rounded mathematical theory can be established, which progresses from the elementary laws for individual points to the processes given to us in the plenum (’continuously filled space’) of reality, without distinction between gravitation, electricity, magnetism, or thermostatics.” This is probably to be interpreted as Riemann’s rejection of all “action at a distance” theories in physical science in favor of field theories. In the latter the physical properties of the “space” surrounding a “charged particle,” say, are the object of mathematical investigation. Riemann at this stage of his career seems to have believed in a space-filling “ether,” a conception now abandoned. But as will appear from his epochal work on the foundations of geometry, he later sought the description and correlation of physical phenomena in the geometry of the “space” of human experience. This is in the current fashion, which rejects an existent, unobservable ether as a cumbersome superfluity.

  Fascinated by his work in physics, Riemann let his pure mathematics slide for a while and in the fall of 1850 joined the seminar in mathematical physics which had just been founded by Weber, Ulrich, Stern, and Listing. Physical experiments in this seminar consumed the time that scholarly prudence would have reserved for the doctoral dissertation in mathematics, which Riemann did not submit till he was twenty five.