Men of Mathematics (92 page)

Kronecker entered this beautifully difficult field of algebraic numbers in
1845
at the age of twenty two with his famous dissertation
De Unitatibus Complexis (On Complex Units).
The particular units he discussed were those in algebraic number fields arising from the
Gaussian problem of the division of the circumference of a circle into
n
equal arcs. For this work he got his Ph.D.

The German universities used to have—and may still have—a laudable custom in connection with the taking of a Ph.D.: the successful candidate was in honor bound to fling a party—usually a prolonged beer bust with all the trimmings—for his examiners. At such festivities a mock examination consisting of ridiculous questions and more ridiculous answers was sometimes part of the fun. Kronecker invited practically the whole faculty, including the Dean, and the memory of that undignified feast in celebration of his degree was, he declared in later years, the happiest of his life.

In at least one respect Kronecker and his scientific enemy Weierstrass were much alike: they were both very great gentlemen, as even those who did not particularly care for either admitted. But in nearly everything else they were almost comically different. The climax of Kronecker's career was his prolonged mathematical war against Weierstrass, in which quarter was neither given nor asked. One was a born algebraist, the other almost made a religion of analysis. Weierstrass was large and rambling, Kronecker a compact, diminutive man, not over five feet tall, but perfectly proportioned and sturdy. After his student days Weierstrass gave up his fencing; Kronecker was always an expert gymnast and swimmer and in later life a good mountaineer.

Eyewitnesses of the battles between this curiously mismatched pair tell how the big fellow, annoyed by the persistence of the little fellow, would stand shaking himself like a good-natured St. Bernard dog trying to rid himself of a determined fly, only to excite his persecutor to more ingenious attacks, till Weierstrass, giving up in despair, would amble off, Kronecker at his heels still talking maddeningly. But for all their scientific differences the two were good friends, and both were great mathematicians without a particle of the “great man” complex that too often inflates the shirts of the would-be mighty.

Kronecker was blessed with a rich uncle in the banking business. The uncle also controlled extensive farming enterprises. All this fell into young Kronecker's hands for administration on the death of the uncle, shortly after the budding mathematician had taken his degree at the age of twenty two. The eight years from
1845
to
1853
were spent in managing the estate and running the business, which
Kronecker did with great thoroughness and financial success. To manage the landed property efficiently he even mastered the principles of agriculture.

In
1848,
at the age of twenty five, the energetic young business man very prudently fell in love with his cousin, Fanny Prausnitzer, daughter of the defunct wealthy uncle, married her, and settled down to raise a family. They had six children, four of whom survived their parents. Kronecker's married life was ideally happy, and he and his wife—a gifted, pleasant woman—brought up their children with the greatest devotion. The death of Kronecker's wife a few months before his own last illness was the blow which broke him.

During his eight years in business Kronecker produced no mathematics. But that he did not stagnate mathematically is shown by his publication in
1853
of a fundamental memoir on the algebraic solution of equations. All through his activity as a man of affairs Kronecker had maintained a lively scientific correspondence with his former master, Kummer, and on escaping from business in
1853
he visited Paris, where he made the acquaintance of Hermite and other leading French mathematicians. Thus he did not sever communications with the scientific world when circumstances forced him into business, but kept his soul alive by making mathematics rather than whist, pinochle, or checkers his hobby.

In
1853,
when Kronecker's memoir on the algebraic solvability of equations (the nature of the problem was discussed in the chapters on Abel and Galois) was published, the Galois theory of equations was understood by very few. Kronecker's attack was characteristic of much of his finest work. Kronecker had mastered the Galois theory, indeed he was probably the only mathematician of the time (the late
1840
's) who had penetrated deeply into Galois' ideas; Liouville had contented himself with a sufficient insight into the theory to enable him to edit some of Galois' remains intelligently.

A distinguishing feature of Kronecker's attack was its comprehensive thoroughness. In this, as in other investigations in algebra and the theory of numbers, Kronecker took the refined gold of his predecessors, toiled over it like an inspired jeweler, added gems of his own, and made from the precious raw material a flawless work of art with the unmistakable impress of his artistic individuality upon it. He delighted in perfect things; a few of his pages will often exhibit a complete development of one isolated result with all its implications
immanent but not loading the unique theme with expressed detail. Consequently even the shortest of his papers has suggested important developments to his successors, and his longer works are inexhaustible mines of beautiful things.

Kronecker was what is called an “algorist” in most of his works. He aimed to make concise, expressive formulas tell the story and automatically reveal the action from one step to the next so that, when the climax was reached, it was possible to glance back over the whole development and see the apparent inevitability of the conclusion from the premises. Details and accessory aids were ruthlessly pruned away until only the main trunk of the argument stood forth in naked strength and simplicity. In short, Kronecker was an artist who used mathematical formulas as his medium.

After Kronecker's works on the Galois theory the subject passed from the private ownership of a few into the common property of all algebraists, and Kronecker had wrought so artistically that the next phase of the theory of equations—the current postulational formulation of the theory and its extensions—can be traced back to him. His aim in algebra, like that of Weierstrass in analysis, was to find the “natural” way—a matter of intuition and taste rather than scientific definition—to the heart of his problems.

The same artistry and tendency to unification appeared in another of his most celebrated papers, which occupies only a couple of pages in his collected works,
On the Solution of the General Equation of the Fifth Degree,
first published in 1858. Hermite, we recall, had given the first solution, by means of elliptic (modular) functions in the same year. Kronecker attains Hermite's solution—or what is practically the same—by applying the ideas of Galois to the problem, thereby making the miracle appear more “natural.” In another paper, also short, over which he has spent most of his time for five years, he returns to the subject in 1861, and seeks the reason
why
the general equation of the fifth degree is solvable in the manner in which it is, thus taking a step beyond Abel who settled the question of solvability “by radicals.”

Much of Kronecker's work has a distinct arithmetical tinge, either of rational arithmetic or of the broader arithmetic of algebraic numbers. Indeed, if his mathematical activity had any guiding clue, it may be said to have been his desire, perhaps subconscious, to
arithmetize
all mathematics, from algebra to analysis. “God made the integers,” he said, “all the rest is the work of man.” Kronecker's demand that analysis
be replaced by finite arithmetic was the root of his disagreement with Weierstrass. Universal arithmetization may be too narrow an ideal for the luxuriance of modern mathematics, but at least it has the merit of greater clarity than is to be found in some others.

Geometry never seriously attracted Kronecker. The period of specialization was already well advanced when Kronecker did most of his work, and it would probably have been impossible for any man to have done the profoundly perfect sort of work that Kronecker did as an algebraist and in his own peculiar type of analysis and at the same time have accomplished anything of significance in other fields. Specialization is frequently damned, but it has its virtues.

A distinguishing feature of many of Kronecker's technical discoveries was the intimate way in which he wove together the three strands of his greatest interests—the theory of numbers, the theory of equations, and elliptic functions—into one beautiful pattern in which unforeseen symmetries were revealed as the design developed and many details were unexpectedly imaged in others far away. Each of the tools with which he worked seemed to have been designed by fate for the more efficient functioning of the others. Not content to accept this mysterious unity as a mere mystery, Kronecker sought and found its underlying structure in Gauss' theory of binary quadratic forms, in which the main problem is to investigate the solutions in integers of indeterminate equations of the second degree in two unknowns.

Kronecker's great work in the theory of algebraic numbers was not part of this pattern. In another direction he also departed occasionally from his principal interests when, according to the fashion of his times, he occupied himself with the purely mathematical aspects of certain problems (in the theory of attraction as in Newton's gravitation) of mathematical physics. His contributions in this field were of mathematical rather than physical interest.

*  *  *

Up till the last decade of his life Kronecker was a free man with obligations to no employer. Nevertheless he voluntarily assumed scientific duties, for which he received no remuneration, when he availed himself of his privilege as a member of the Berlin Academy to lecture at the University of Berlin. From
1861
to
1883
he conducted regular courses at the university, principally on his personal researches, after the necessary introductions. In
1883
Kummer, then at Berlin, retired, and Kronecker succeeded his old master as ordinary
professor. At this period of his life he travelled extensively and was a frequent and welcome participant in scientific meetings in Great Britain, France, and Scandinavia.

Throughout his career as a mathematical lecturer Kronecker competed with Weierstrass and other celebrities whose subjects were more popular than his own. Algebra and the theory of numbers have never appealed to so wide an audience as have geometry and analysis, possibly because the connections of the latter with physical science are more apparent.

Kronecker took his aristocratic isolation good-naturedly and even with a certain satisfaction. His beautifully clear introductions deluded his auditors into a belief that the subsequent course of lectures would be easy to follow. This belief evaporated rapidly as the course progressed, until after three sessions all but a faithful and obstinate few had silently stolen away—many of them to listen to Weierstrass. Kronecker rejoiced. A curtain could now be drawn across the room behind the first few rows of chairs, he joked, to bring lecturer and auditors into cosier intimacy. The few disciples he retained followed him devotedly, walking home with him to continue the discussions of the lecture room and frequently affording the crowded sidewalks of Berlin the diverting spectacle of an excited little man talking with his whole body—especially his hands—to a spellbound group of students blocking the traffic. His house was always open to his pupils, for Kronecker really liked people, and his generous hospitality was one of the greatest satisfactions of his life. Several of his students became eminent mathematicians, but his “school” was the whole world and he made no effort to acquire an artifically large following.

The last is characteristic of Kronecker's own most startlingly independent work. In an atmosphere of confident belief in the soundness of analysis Kronecker assumed the unpopular rôle of the philosophical doubter. Not many of the great mathematicians have taken philosophy seriously; in fact the majority seem to have regarded philosophical speculations with repugnance, and any epistemological doubt affecting the soundness of their work has usually been ignored or impatiently brushed aside.

With Kronecker it was different. The most original part of his work, in which he was a true pioneer, was a natural outgrowth of his philosophical inclinations. His father, Werner, Kummer, and his own wide reading in philosophical literature had influenced him in the direction
of a critical outlook on all human knowledge, and when he contemplated mathematics from this questioning point of view he did not spare it because it happened to be the field of his own particular interest, but infused it with an acid, beneficial skepticism. Although but little of this found its way into print it annoyed some of his contemporaries intensely and it has survived. The doubter did not address himself to the living but, as he said, “to those who shall come after me.” Today these followers have arrived, and due to their united efforts—although they often succeed only in contradicting one another—we are beginning to get a clearer insight into the nature and meaning of mathematics.

Weierstrass (Chapter 22) would have constructed mathematical analysis on his conception of irrationals as defined by infinite sequences of rationals. Kronecker not only disputes Weierstrass; he would nullify Eudoxus. For him as for Pythagoras only the God-given integers 1, 2, 3, . . . ,” “exist”; all the rest is a futile attempt of mankind to improve on the creator. Weierstrass on the other hand believed that he had at last made the square root of 2 as comprehensible and as safe to handle as 2 itself; Kronecker denied that the square root of 2 “exists,” and he asserted that it is impossible to reason consistently with or about the Weierstrassian construction for this root or for any other irrational. Neither his older colleagues nor the young to whom Kronecker addressed himself gave his revolutionary idea a very enthusiastic welcome.