Men of Mathematics

by E. T. Bell

  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.

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  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 w
ith 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.

  Weierstrass himself seems to have felt uneasy; certainly he was hurt. His strong emotion is released mostly in one tremendous German sentenceII like a fugue, which it is almost impossible to preserve in English. “But the worst of it is,” he complains, “that Kronecker uses his authority to proclaim that all those who up to now have labored to establish the theory of functions are sinners before the Lord. When a whimsical eccentric like Christoffel [the man whose somewhat neglected work was to become, years after his death, an important tool in differential geometry as it is cultivated today in the mathematics of relativity] says that in twenty or thirty years the present theory of functions will be buried and that the whole of analysis will be referred to the theory of forms, we reply with a shrug. But when Kronecker delivers himself of the following verdict which I repeat word for word: ‘If time and strength are granted me, I myself will show the mathematical world that not only geometry, but also arithmetic can point the way to analysis, and certainly a more rigorous way. If I cannot do it myself those who come after me will . . . and they will recognize the incorrectness of all those conclusions with which so-called analysis works at present’—such a verdict from a man whose eminent talent and distinguished performance in mathematical research I admire as sincerely and with as much pleasure as all his colleagues, is not only humiliating for those whom he adjures to acknowledge as an error and to forswear the substance of what has constituted the object of their thought and unremitting labor, but it is a direct appeal to the younger generation to desert their present leaders and rally around him as the disciple of a new system which must be founded. Truly it is sad, and it fills me with a bitter grief, to see a man, whose glory is without flaw, let himself be driven by the well justified feeling of his own worth to utterances whose injurious effect upon others he seems not to perceive.

  “But enough of these things, on which I have touched only to explain to you the reason why I can no longer take the same joy that I used to take in my teaching, even if my health were to permit me to continue it a few years longer. But you must not speak of it; I should not like others, who do not know me as well as you, to see in what I say the expression of a sentiment which is in fact foreign to me.”

  Weierstrass was seventy and in poor health when he wrote this. Could he have lived till today he would have seen his own great system still flourishing like the proverbial green bay tree. Kronecker’s doubts have done much to instigate a critical re-examination of the foundations of all mathematics, but they have not yet destroyed analysis. They go deeper, and if anything of far-reaching significance is to be replaced by something firmer but as yet unknown, it seems likely that a good part of Kronecker’s own work will go too, for the critical attack which he foresaw has uncovered weaknesses where he suspected nothing. Time makes fools of us all. Our only comfort is that greater shall come after us.

  Kronecker’s “revolution,” as his contemporaries called his subversive assault on analysis, would banish all but the positive integers from mathematics. Geometry since Descartes has been largely an affair of analysis applied to ordered pairs, triples, . . . of real numbers (the “numbers” which correspond to the distances measured on a given straight line from a fixed point on the line); hence it too would come under the sway of Kronecker’s program. So familiar a concept as that of a negative integer, −2 for instance, would not appear in the mathematics Kronecker prophesied, nor would common fractions.

  Irrationals, as Weierstrass points out, roused Kronecker’s special displeasure. To speak of x2— 2 = 0 having a root would be meaningless. All of these dislikes and objections are of course themselves meaningless unless they can be backed by a definite program to replace what is rejected.

  Kronecker actually did this, at least in outline, and indicated how the whole of algebra and the theory of numbers, including algebraic numbers, can be reconstructed in accordance with his demand. To get rid of for example, we need only put a letter for it temporarily, say i, and consider polynomials containing i and other letters, say x, y, z, . . . . Then we manipulate these polynomials as in elementary algebra, treating i like any of the other letters, till the last step, when every polynomial containing i is divided by i2 + 1 and everything but the remainder obtained from this division is discarded. Anyone who remembers a little elementary algebra may readily convince himself that this leads to all the familiar properties of the mysteriously misnamed “imaginary” numbers of the text books. In a similar manner negatives and fractions and all algebraic numbers (other than the positive rational integers) are eliminated from mathematics—if desired—and only the blessed positive integers remain. The inspiration about discarding goes back to Cauchy in 1847. This was the germ of Kronecker’s program.

  Those who dislike Kronecker’s “revolution” call it a Putsch, which is more like a drunken brawl than an
orderly revolution. Nevertheless it has led in recent years to two constructively critical movements in the whole of mathematics: the demand that a construction in a finite number of steps be given or proved to be possible for any “number” or other mathematical “entity” whose “existence” is indicated, and the banishment from mathematics of all definitions that cannot be stated explicitly in a finite number of words. Insistence upon these demands has already done much to clarify our conception of the nature of mathematics, but a vast amount remains to be done. As this work is still in progress we shall defer further consideration of it until we come to Cantor, when it will be possible to exhibit examples.

  Kronecker’s disagreement with Weierstrass should not leave an unpleasant impression, as it may do if we ignore the rest of Kronecker’s generous life. Kronecker had no intention of wounding his kindly old senior; he merely let his tongue run away with him in the heat of a purely mathematical argument, and Weierstrass, when he was in good spirits, laughed the whole attack off, as he should have done, knowing well that just as he had improved on Eudoxus, so his successors would probably improve upon him. Possibly if Kronecker had been six or seven inches taller than he was he would not have felt constrained to overemphasize his objections to analysis so vociferously. Much of the whole wordy dispute sounds suspiciously like the overcorrection of an unjustified inferiority complex.

  The reaction of many mathematicians to Kronecker’s “revolution” was summed up by Poincaré when he said that Kronecker had been enabled to do so much fine mathematics because he frequently forgot his own mathematical philosophy. Like not a few epigrams this one is just untrue enough to be witty.