by E. T. Bell
The whole of Weierstrass’ work in analysis can be regarded as a grand attack on his main problem. Isolated results, special developments and even extensive theories—for example that of irrational numbers as developed by him—all originated in some phase or another of the central problem. He early became convinced that for a clear understanding of what he was attempting to do a radical revision of the fundamental concepts of mathematical analysis was necessary, and from this conviction he passed to another, of more significance today perhaps than the central problem itself: analysis must be founded on the common whole numbers 1, 2, 3, . . .. The irrationals which give us the concepts of limits and continuity, from which analysis springs, must be referred back by irrefrangible reasoning to the integers; shoddy proofs must be discarded or reworked, gaps must be filled up, and obscure “axioms” must be dragged out into the light of critical inquiry till all are understood and all are stated in comprehensible language in terms of the integers. This in a sense is the Pythagorean dream of basing all mathematics on the integers, but Weierstrass gave the program constructive definiteness and made it work.
Thus originated the nineteenth century movement known as the arithmetization of analysis—something quite different from Kronecker’s arithmetical program, at which we shall glance in a later chapter; indeed the two approaches were mutually antagonistic.
In passing it may be pointed out that Weierstrass’ plan for his life work and his magnificent accomplishment of most of what he set himself as a young man to do, is a good illustration of the value of the advice Felix Klein once gave a perplexed student who had asked him the secret of mathematical discovery. “You must have a problem,” Klein replied. “Choose one definite objective and drive ahead toward it. You may never reach your goal, but you will find something of interest on the way.”
From Deutsch-Krone Weierstrass moved to Braunsberg, where he taught in the Royal Catholic Gymnasium for six years, beginning in 1848. The school “program” for 1848-49 contains a paper by Weierstrass which must have astonished the natives: Contributions to the Theory of Abelian Integrals. If this work had chanced to fall under the eyes of any of the professional mathematicians of Germany, Weierstrass would have been made. But, as his Swedish biographer, Mittag-Leffler, dryly remarks, one does not look for epochal papers on pure mathematics in secondary-school programs. Weierstrass might as well have used his paper to light his pipe.
His next effort fared better. The summer vacation of 1853 (Weierstrass was then 38) was passed in his father’s house at Westernkotten. Weierstrass spent the vacation writing up a memoir on Abelian functions. When it was completed he sent it to Crelle’s great Journal. It was accepted and appeared in volume 47 (1854).
This may have been the paper whose composition was responsible for an amusing incident in Weierstrass’ career as a schoolteacher at Braunsberg. Early one morning the director of the school was startled by a terrific uproar proceeding from the classroom where Weierstrass was supposed to be holding forth. On investigation he discovered that Weierstrass had not shown up. He hurried over to Weierstrass’ dwelling, and on knocking was bidden to enter. There sat Weierstrass pondering by the glimmering light of a lamp, the curtains of the room still drawn. He had worked the whole night through and had not noticed the approach of dawn. The director called his attention to the fact that it was broad daylight and told him of the uproar in his classroom. Weierstrass replied that he was on the trail of an important discovery which would rouse great interest in the scientific world and he could not possibly interrupt his work.
The memoir on Abelian functions published in Crelle’s Journal in 1854 created a sensation. Here was a masterpiece from the pen of an unknown schoolmaster in an obscure village nobody in Berlin had ever heard of. This in itself was sufficiently astonishing. But what surprised those who could appreciate the magnitude of the work even more was the almost unprecedented fact that the solitary worker had published no preliminary bulletins announcing his progress from time to time, but with admirable restraint had held back everything till the work was completed.
Writing to a friend some ten years later, Weierstrass gives his modest version of his scientific reticence: “. . . the infinite emptiness and boredom of those years [as a schoolteacher] would have been unendurable without the hard work that made me a recluse—even if I was rated rather a good fellow by the circle of my friends among the junkers, lawyers, and young officers of the community. . . . The present offered nothing worth mentioning, and it was not my custom to speak of the future.”
Recognition was immediate. At the University of Königsberg, where Jacobi had made his great discoveries in the field which Weierstrass had now entered with a masterpiece of surpassing excellence, Richelot, himself a worthy successor of Jacobi in the theory of multiply periodic functions, was Professor of Mathematics. His expert eyes saw at once what Weierstrass had done. He forthwith persuaded his university to confer the degree of doctor, honoris causa, on Weierstrass and himself journeyed to Braunsberg to present the diploma.
At the dinner organized by the director of the Gymnasium in Weierstrass’ honor Richelot asserted that “we have all found our master in Mr. Weierstrass.” The Ministry of Education immediately promoted him and granted him a year’s leave to prosecute his scientific work. Borchardt, the editor of Crelle’s Journal at the time, hurried to Braunsberg to congratulate the greatest analyst in the world, thus starting a warm friendship which lasted till Borchardt’s death a quarter of a century later.
None of this went to Weierstrass’ head. Although he was deeply moved and profoundly grateful for all the generous recognition so promptly accorded him, he could not refrain from casting a backward glance over his career. Years later, thinking of the happiness of the occasion and of what that occasion had opened up for him when he was forty years of age, he remarked sadly that “everything in life comes too late.”
* * *
Weierstrass did not return to Braunsberg. No really suitable position being open at the time, the leading German mathematicians did what they could to tide over the emergency and got Weierstrass appointed Professor of Mathematics at the Royal Polytechnic School in Berlin. This appointment dated from July 1, 1856; in the autumn of the same year he was made Assistant Professor (in addition to the other post) at the University of Berlin and was elected to the Berlin Academy.
The excitement of novel working conditions and the strain of too much lecturing presently brought on a nervous breakdown. Weierstrass had also been overworking at his researches. In the summer of 1859 he was forced to abandon his course and take a rest cure. Returning in the fall he continued his work, apparently refreshed, but in the following March was suddenly attacked by spells of vertigo, and he collapsed in the middle of a lecture.
All the rest of his life he was bothered with the same trouble off and on, and after resuming his work—as full professor, with a considerably lightened load—never trusted himself to write his own formulas on the board. His custom was to sit where he could see the class and the blackboard, and dictate to some student delegated from the class what was to be written. One of these “mouthpieces” of the master developed a rash propensity to try to improve on what he had been told to write. Weierstrass would reach up and rub out the amateur’s efforts and make him write what he had been told. Occasionally the battle between the professor and the obstinate student would go to several rounds, but in the end Weierstrass always won. He had seen little boys misbehaving before.
As the fame of his work spread over Europe (and later to America), Weierstrass’ classes began to grow rather unwieldy and he would sometimes regret that the quality of his auditors lagged far behind their rapidly mounting quantity. Nevertheless he gathered about him an extremely able band of young mathematicians who were absolutely devoted to him and who did much to propagate his ideas, for Weierstrass was always slow about publication, and without the broadcasting of his lectures which his disciples took upon themselves his influence on the mathematical thought of
the nineteenth century would have been considerably retarded.
Weierstrass was always accessible to his students and sincerely interested in their problems, whether mathematical or human. There was nothing of the “great man” complex about him, and he would as gladly walk home with any of the students—and there were many—who cared to join him as with the most famous of his colleagues, perhaps more gladly when the colleague happened to be Kronecker. He was happiest when, sitting at a table over a glass of wine with a few of his devoted disciples, he became a jolly student again himself and insisted on paying the bill for the crowd.
An anecdote (about Mittag-Leffler) may suggest that the Europe of the present century has partly lost something it had in the 1870’s. The Franco-Prussian war (1870-71) had left France pretty sore at Germany. But it had not befogged the minds of mathematicians regarding one another’s merits irrespective of their nationalities. The like holds for the Napoleonic wars and the mutual esteem of the French and British mathematicians. In 1873 Mittag-Leffler arrived in Paris from Stockholm all set and full of enthusiasm to study analysis under Hermite. “You have made a mistake, sir,” Hermite told him: “you should follow Weierstrass’ course at Berlin. He is the master of all of us.”
Mittag-Leffler took the sound advice of the magnanimous Frenchman and not so long afterward made a capital discovery of his own which is to be found today in all books on the theory of functions. “Hermite was a Frenchman and a patriot,” Mittag-Leffler remarks; “I learned at the same time in what degree he was also a mathematician.”
* * *
The years (1864-97) of Weierstrass’ career at Berlin as Professor of Mathematics were full of scientific and human interests for the man who was acknowledged as the leading analyst in the world. One phase of these interests demands more than the passing reference that might suffice in a purely scientific biography of Weierstrass: his friendship with his favorite pupil, Sonja (or Sophie) Kowalewski.
Madame Kowalewski’s maiden name was Sonja Corvin-Kroukow-sky; she was born at Moscow, Russia, on January 15, 1850, and died at Stockholm, Sweden, on February 10, 1891, six years before the death of Weierstrass.
At fifteen Sonja began the study of mathematics. By eighteen she had made such rapid progress that she was ready for advanced work and was enamored of the subject. As she came of an aristocratic and prosperous family, she was enabled to gratify her ambition for foreign study and matriculated at the University of Heidelberg.
This highly gifted girl became not only the leading woman mathematician of modern times, but also made a reputation as a leader in the movement for the emancipation of women, particularly as regarded their age-old disabilities in the field of higher education.
In addition to all this she was a brilliant writer. As a young girl she hesitated long between mathematics and literature as a career. After the composition of her most important mathematical work (the prize memoir noted later), she turned to literature as a relaxation and wrote the reminiscences of her childhood in Russia in the form of a novel (published first in Swedish and Danish). Of this work it is reported that “the literary critics of Russia and Scandinavia were unanimous in declaring that Sonja Kowalewski had equalled the best writers of Russian literature in style and thought.” Unfortunately this promising start was blocked by her premature death, and only fragments of other literary works survive. Her one novel was translated into many languages.
Although Weierstrass never married he was no panicky bachelor who took to his heels every time he saw a pretty woman coming. Sonja, according to competent judges who knew her, was extremely good-looking. We must first tell how she and Weierstrass met.
Weierstrass used to enjoy his summer vacations in a thoroughly human manner. The Franco-Prussian war caused him to forego his usual summer trip in 1870, and he stayed in Berlin, lecturing on elliptic functions. Owing to the war his class had dwindled to only twenty instead of the fifty who heard the lectures two years before. Since the autumn of 1869 Sonja Kowalewski, then a dazzling young woman of nineteen, had been studying elliptic functions under Leo Königsberger (born 1837) at the University of Heidelberg, where she had also followed the lectures on physics by Kirchhoff and Helmholtz and had met Bunsen the famous chemist under rather amusing circumstances—to be related presently. Königsberger, one of Weierstrass’ first pupils, was a first-rate publicity agent for his master. Sonja caught her teacher’s enthusiasm and resolved to go directly to the master himself for inspiration and enlightenment.
The status of unmarried women students in the 1870’s was somewhat anomalous. To forestall gossip, Sonja at the age of eighteen contracted what was to have been a nominal marriage, left her husband in Russia, and set out for Germany. Her one indiscretion in her dealings with Weierstrass was her neglect to inform him at the beginning that she was married.
Having decided to learn from the master himself, Sonja took her courage in her hands and called on Weierstrass in Berlin. She was twenty, very earnest, very eager, and very determined; he was fifty five, vividly grateful for the lift Gudermann had given him toward becoming a mathematician by taking him on as a pupil, and sympathetically understanding of the ambitions of young people. To hide her trepidation Sonja wore a large and floppy hat, “so that Weierstrass saw nothing of those marvelous eyes whose eloquence, when she wished it, none could resist.”
Some two or three years later, on a visit to Heidelberg, Weierstrass learned from Bunsen—a crabbed bachelor—that Sonja was “a dangerous woman.” Weierstrass enjoyed his friend’s terror hugely, as Bunsen at the time was unaware that Sonja had been receiving frequent private lessons from Weierstrass for over two years.
Poor Bunsen based his estimate of Sonja on bitter personal experience. He had proclaimed for years that no woman, and especially no Russian woman, would ever be permitted to profane the masculine sanctity of his laboratory. One of Sonja’s Russian girl friends, desiring ardently to study chemistry in Bunsen’s laboratory, and having been thrown out herself, prevailed upon Sonja to try her powers of persuasion on the crusty chemist. Leaving her hat at home, Sonja interviewed Bunsen. He was only too charmed to accept Sonja’s friend as a student in his laboratory. After she left he woke up to what she had done to him. “And now that woman has made me eat my own words,” he lamented to Weierstrass.
Sonja’s evident earnestness on her first visit impressed Weierstrass favorably and he wrote to Königsberger inquiring about her mathematical aptitudes. He asked also whether “the lady’s personality offers the necessary guarantees.” On receiving an enthusiastic reply, Weierstrass tried to get the university senate to admit Sonja to his mathematical lectures. Being brusquely refused he took care of her himself on his own time. Every Sunday afternoon was devoted to teaching Sonja at his house, and once a week Weierstrass returned her visit. After the first few lessons Sonja lost her hat. The lessons began in the autumn of 1870 and continued with slight interruptions due to vacations or illnesses till the autumn of 1874. When for any reason the friends were unable to meet they corresponded. After Sonja’s death in 1891 Weierstrass burnt all her letters to him, together with much of his other correspondence and probably more than one mathematical paper.
The correspondence between Weierstrass and his charming young friend is warmly human, even when most of a letter is given over to mathematics. Much of the correspondence was undoubtedly of considerable scientific importance, but unfortunately Sonja was a very untidy woman when it came to papers, and most of what she left behind was fragmentary or in hopeless confusion.
Weierstrass himself was no paragon in this respect. Without keeping records he loaned his unpublished manuscripts right and left to students who did not always return what they borrowed. Some even brazenly rehashed parts of their teacher’s work, spoiled it, and published the results as their own. Although Weierstrass complains about this outrageous practice in letters to Sonja his chagrin is not over the petty pilfering of his ideas but of their bungling in incompetent hands and the consequent damage to ma
thematics. Sonja of course never descended to anything of this sort, but in another respect she was not entirely blameless. Weierstrass sent her one of his unpublished works by which he set great store, and that was the last he ever saw of it. Apparently she lost it, for she discreetly avoids the topic—to judge from his letters—whenever he brings it up.
To compensate for this lapse Sonja tried her best to get Weierstrass to exercise a little reasonable caution in regard to the rest of his unpublished work. It was his custom to carry about with him on his frequent travels a large white wooden box in which he kept all his working notes and the various versions of papers which he had not yet perfected. His habit was to rework a theory many times until he found the best, the “natural” way in which it should be developed. Consequently he published slowly and put out a work under his own name only when he had exhausted the topic from some coherent point of view. Several of his rough-hewn projects are said to have been confided to the mysterious box. In 1880, while Weierstrass was on a vacation trip, the box was lost in the baggage. It has never been heard of since.
After taking her degree in absentia from Göttingen in 1874, Sonja returned to Russia for a rest as she was worn out by excitement and overwork. Her fame had preceded her and she “rested” by plunging into the hectic futilities of a crowded social season in St. Petersburg while Weierstrass, back in Berlin, pulled wires all over Europe trying to get his favorite pupil a position worthy of her talents. His fruitless efforts disgusted him with the narrowness of the orthodox academic mind.