by E. T. Bell
g11x1′2 + g12x1′2x2′ + g22x2′2,
expressing (as before) the square of the distance between neighboring points on a given surface (determined when the functions g11, g12, g22 are given), to calculate the measure of curvature of any point of the surface wholly in terms of the given functions g11, g12, g22. Now, in ordinary language, to speak of the “curvature” of a space of more than two dimensions is to make a meaningless noise. Nevertheless Riemann, generalizing Gauss, proceeded in the same mathematical way to build up an expression involving all the g’s in the general case of an n-dimensional space, which is of the same kind mathematically as the Gaussian expression for the curvature of a surface, and this generalized expression is what he called the measure of curvature of the space. It is possible to exhibit visual representations of a curved space of more than two dimensions, but such aids to perception are about as useful as a pair of broken crutches to a man with no feet, for they add nothing to the understanding and they are mathematically useless.
Why did Riemann do all this and what has come out of it? Not attempting to answer the first, except to suggest that Riemann did what he did because his daemon drove him, we may briefly enumerate some of the gains that have accrued from Riemann’s revolution in geometrical thought. First, it put the creation of “spaces” and “geometries” in unlimited number for specific purposes—use in dynamics, or in pure geometry, or in physical science—within the capabilities of professional geometers, and it baled together huge masses of important geometrical theorems into compact bundles that could be handled easily as wholes. Second, it clarified our conception of space, at least so far as mathematicians deal in “space,” and stripped that mystic nonentity Space of its last shred of mystery. Riemann’s achievement has taught mathematicians to disbelieve in any geometry, or in any space, as a necessary mode of human perception. It was the last nail in the coffin of absolute space, and the first in that of the “absolutes” of nineteenth century physics.
Finally, the curvature which Riemann defined, the processes which he devised for the investigation of quadratic differential forms (those giving the formula for the square of the distance between neighboring points in a space of any number of dimensions), and his recognition of the fact that the curvature is an invariant (in the technical sense explained in previous chapters), all found their physical interpretations in the theory of relativity. Whether the latter is in its final form or not is beside the point; since relativity our outlook on physical science is not what it was before. Without the work of Riemann this revolution in scientific thought would have been impossible—unless some later man had created the concepts and the mathematical methods that Riemann created.
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I. If z = x 4- iy, and w = u -f iv, is an analytic function of z, Riemann’s equations are
These equations had been given much earlier by Cauchy, and even Cauchy was not the first, as D’Alembert had stated the equations in the eighteenth century.
CHAPTER TWENTY SEVEN
Arithmetic the Second
KUMMER AND DEDEKIND
We see therefore that ideal prime factors reveal the essence of complex numbers, make them transparent, as it were, and disclose their inner crystalline structure.
—E. E. KUMMER
The majority of my readers will be greatly disappointed to learn that by this commonplace observation the secret of continuity is to be revealed.—R. DEDEKIND
IT IS A CURIOUS FACT that although arithmetic—the theory of numbers—has been the fertile mother of more profound problems and powerful methods than any other discipline of mathematics, it is usually regarded as standing rather to one side of the main progress as a more or less cold-blooded spectator of the flashier achievements of geometry and analysis, particularly in their services to physical science, and comparatively few of the great mathematicians of the past two thousand years have expended their more serious efforts on the advancement of the science of “pure number.”
Many causes have determined this strange neglect of what, after all, is mathematics par excellence. Among these we need note only the following: arithmetic at present is on a higher plane of intrinsic difficulty than the other great fields of mathematics; the immediate applications of the theory of numbers to science are few and not readily perceptible to the ordinary run of creative mathematicians, although some of the greatest have felt that the proper mathematics of nature will be found ultimately in the behavior of the common whole numbers; and, finally, it is only human for mathematicians—at least for some, even the great—to court reputation and popularity in their own generation by reaping the easier harvests of a spectacular success in analysis, geometry, or applied mathematics. Even Gauss succumbed, to his keen regret in middle life.
Modern arithmetic—after Gauss—began with Kummer. The origin of Kummer’s theory in his attempt to prove Fermat’s Last Theorem has already been noted (Chapter 25). Something of the man’s long life may be told before we pass to Dedekind. Kummer was a typical German of the old school with all the blunt simplicity, good nature, and racy humor, which characterized that fast-vanishing species at its best. Museum specimens, aged in the wood, could be found behind the bar in any San Francisco German beer garden a generation ago.
Although Ernst Eduard Kummer (January 29, 1810-May 14, 1893) was born only five years before the deflation of Napoleon, the glorious Emperor of the French played an important if unwitting part in Kummer’s life. The son of a physician of Sorau (then in the principality of Brandenburg), Germany, Kummer at the age of three lost his father: the lousy remnant of Napoleon’s Grand Army, filtering back through Germany to France, brought with it the characteristically Russian gift of typhus, which it shared freely with the well-washed Germans. The overworked physician caught the disease, died of it, and left Ernst and an elder brother to the care of his widow. Young Kummer grew up in cramping poverty, but his struggling mother contrived somehow or another to see her sons through the local Gymnasium. The arrogance and exactions of the Napoleonic French, no less than the memory of his father, which the mother kept alive, made young Kummer an extremely practical patriot, and it was with real gusto that he devoted much of his superb scientific talent in later life to training German army officers in ballistics at the war college of Berlin. Many of his students gave good accounts of themselves in the Franco-Prussian War.
At the age of eighteen (in 1828) Kummer was sent by his mother to the University of Halle to study theology and otherwise fit himself for a career in the church. Owing to his poverty Kummer did not reside at the University, but tramped back and forth every day from Sorau to Halle with his food and books in a knapsack on his back. Regarding his theological studies Kummer makes the interesting observation that it is more or less a matter of accident or environment whether a mind with a gift for abstract speculation turns to philosophy or to mathematics. The accident in his own case was the presence at Halle of Heinrich Ferdinand Scherk (1798-1885) as professor of mathematics. Scherk was rather old fashioned, but he had an enthusiasm for algebra and the theory of numbers which he imparted to young Kummer. Under Scherk’s guidance Kummer soon abandoned his moral and theological studies in favor of mathematics. Echoing Descartes, Kummer said he preferred mathematics to philosophy because “mere errors and false views cannot enter mathematics.” Had Kummer lived till today he might have modified his statement, for he was a broadminded man, and the present philosophical tendencies in mathematics are sometimes curiously reminiscent of medieval theology. In his third year at the University Kummer solved a prize problem in mathematics and was awarded his Ph.D. degree (September 10, 1831) at the age of twenty one. No university position being open at the time, Kummer began his career as a teacher in his old Gymnasium.
In 1832 he moved to Liegnitz, where he taught for ten years in the Gymnasium. It was there that he started Kronecker off on his revolutionary career. Fortunately Kummer was not so hard up as Weierstrass under similar circumstances and was able to afford postage for scientific corr
espondence. The eminent mathematicians (including Jacobi) with whom Kummer shared his mathematical discoveries saw to it that the young genius of a schoolteacher was lifted into a more suitable position at the earliest opportunity, and in 1842 Kummer was appointed Professor of Mathematics at the University of Breslau. He taught there till 1855, when the death of Gauss caused extensive revisions in the mathematical map of Europe.
It had been assumed that Dirichlet was contented at Berlin, then the mathematical capital of the world. But when Gauss died, Dirichlet could not resist the temptation of succeeding the Prince of Mathematicians and his own former master as professor at Göttingen. Even today the glory of being a “successor of Gauss” has an almost irresistible attraction for mathematicians who might easily earn more money in other positions, and until quite recently Göttingen could choose whom it would. The high esteem in which Kummer was held by his fellow mathematicians can be judged by the fact that he was the unanimous choice to succeed Dirichlet at Berlin. Since the age of twenty nine he had been a corresponding member of the Royal Berlin Academy. He now (1855) succeeded Dirichlet in both the University and the Academy, and was also appointed professor at the Berlin War College.
Kummer was one of those rarest of all scientific geniuses who are first class in the most abstract mathematics, the applications of mathematics to practical affairs, including war, which is the most unblushingly practical of all human idiocies, and finally in the ability to do experimental physics of a high degree of excellence. His finest work was in the theory of numbers where his profound originality led him to inventions of the very first order of importance, but in other fields—analysis, geometry, and applied physics—he also did outstanding work. Although Kummer’s advance in the higher arithmetic was of the pioneering sort that justifies comparing him with the creators of non-Euclidean geometry, we somehow get the impression on reviewing his life of eighty three years, that splendid as his achievement was, he did not accomplish all that he must have had in him. Possibly his lack of personal ambition (an instance is given presently), his easygoing geniality, and his broad sense of humor prevented him from winding himself in an attempt to beat the record.
The nature of what Kummer did in the theory of numbers has been described in the chapter on Kronecker: he restored the fundamental theorem of arithmetic to those algebraic number fields which arise in the attempt to prove Fermat’s hast Theorem and in the Gaussian theory of cyclotomy, and he effected this restoration by the creation of an entirely new species of numbers, his so-called “ideal numbers.” He also carried on the work of Gauss on the law of biquadratic reciprocity and sought the laws of reciprocity for degrees higher than the fourth.
As has already been mentioned in preceding chapters, Kummer’s “ideal numbers” are now largely displaced by Dedekind’s “ideals,” which will be described when we come to them, so it is not necessary to attempt here the almost impossible feat of explaining in untechnical language what Kummer’s “numbers” are. But what he accomplished by means of them can be stated with sufficient accuracy for an account like the present: Kummer proved that xp + yp = zp, where p is a prime, is impossible in integers x, y, z, all different from zero, for a whole very extensive class of primes p. He did not succeed in proving Fermat’s theorem for all primes; certain slippery “exceptional primes” eluded Kummer’s net—and still do. Nevertheless the step ahead which he took so far surpassed everything that all his predecessors had done that Kummer became famous almost in spite of himself. He was awarded a prize for which he had not competed.
The report in full of the French Academy of Sciences on the competition for its “Grand Prize” in 1857 ran as follows. “Report on the competition for the grand prize in mathematical sciences. Already set in the competition for 1853 and prorogued to 1856. The committee, having found no work which seemed to it worthy of the prize among those submitted to it in competition, proposed to the Academy to award it to M. Kummer, for his beautiful researches on complex numbers composed of roots of unityI and integers. The Academy adopted this proposal.”
Kummer’s earliest work on Fermat’s Last Theorem is dated October, 1835. This was followed by further papers in 1844-47, the last of which was entitled Proof of Fermat’s Theorem on the Impossibility of xp + yp = zp for an InfiniteII Number of Primes p. He continued to add improvements to his theory, including its application to the laws of higher reciprocity, till 1874, when he was sixty four years old.
Although these highly abstract researches were the field of his greatest interest, and although he said of himself, “To describe my personal scientific attitude more exactly, I may conveniently designate it as theoretical . . .; I have particularly striven for that mathematical knowledge which finds its proper sphere in mathematics without reference to applications,” Kummer was no narrow specialist. Somewhat like Gauss, he appeared to take equal pleasure in both pure and applied science. Gauss indeed, through his works, was Kummer’s real teacher, and the apt pupil proved his mettle by extending his master’s work on the hypergeometric series, adding to what Gauss had done substantial developments which today are of great use in the theory of those differential equations which recur most frequently in mathematical physics.
Again, the magnificent work of Hamilton on systems of rays (in optics) inspired Kummer to one of his own most beautiful inventions, that of the surface of the fourth degree which is known by his name and which plays a fundamental part in the geometry of Euclidean space when that space is four-dimensional (instead of three-dimensional, as we ordinarily imagine it), as happens when straight lines instead of points are taken as the irreducible elements out of which the space is constructed. This surface (and its generalizations to higher spaces) occupied the center of the stage in a whole department of nineteenth century geometry;it was found (by Cayley) to be representable (parametrically—see the chapter on Gauss) by means of the quadruply periodic functions to which Jacobi and Hermite devoted some of their best efforts.
Quite recently (since 1934) it has been observed by Sir Arthur Eddington that Kummer’s surface is a sort of cousin to Dirac’s wave equation in quantum mechanics (both have the same finite group; Kummer’s surface is the wave surface in space of four dimensions).
To complete the circle, Kummer was led back by his study of systems of rays to physics, and he made important contributions to the theory of atmospheric refraction. In his work at the War College he astonished the scientific world by proving himself a first-rate experimenter in his work on ballistics. With characteristic humor Kummer excused himself for this bad fall from mathematical grace: “When I attack a problem experimentally,” he told a young friend, “it is a proof that the problem is mathematically impregnable.”
Remembering his own struggles to get an education and his mother’s sacrifices, Kummer was not only a father to his students but something of a brother to their parents. Thousands of grateful young men who had been helped on their way by Kummer at the University of Berlin or the War College remembered him all their lives as a great teacher and a great friend. Once a needy young mathematician about to come up for his doctor’s examination was stricken with smallpox and had to return to his home in Posen near the Russian border. No word came from him, but it was known that he was desperately poor. When Kummer heard that the young man was probably unable to afford proper care, he sought out a friend of the student, gave him the requisite money and sent him off to Posen to see that what was necessary was done. In his teaching Kummer was famous for his homely similes and philosophical asides. Thus, to drive home the importance of a particular factor in a certain expression, he observed that “If you neglect this factor you will be like a man who in eating a plum swallows the pit and spits out the pulp.”
The last nine years of Kummer’s life were spent in complete retirement. “Nothing will be found in my posthumous papers,” he said, thinking of the mass of work which Gauss left to be edited after his death. Surrounded by his family (nine children survived him), Kummer gave up mathematics for
good when he retired, and except for occasional trips to the scenes of his boyhood lived in the strictest seclusion. He died after a short attack of influenza on May 14, 1893, aged eighty three.
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Kummer’s successor in arithmetic was Julius Wilhelm Richard Dedekind (he dropped the first two names when he grew up), one of the greatest mathematicians and one of the most original Germany—or any other country—has produced. Like Kummer, Dedekind had a long life (October 6, 1831-February 12, 1916), and he remained mathematically active to within a short time of his death. When he died in 1916 Dedekind had been a mathematical classic for well over a generation. As Edmund Landau (himself a friend and follower of Dedekind in some of his work) said in his commemorative address to the Royal Society of Göttingen in 1917: “Richard Dedekind was not only a great mathematician, but one of the wholly great in the history of mathematics, now and in the past, the last hero of a great epoch, the last pupil of Gauss, for four decades himself a classic, from whose works not only we, but our teachers and the teachers of our teachers, have drawn.”
Richard Dedekind, the youngest of the four children of Julius Levin Ulrich Dedekind, a professor of law, was born in Brunswick, the natal place of GaussIII. From the age of seven to sixteen Richard studied at the Gymnasium in his home town. He gave no early evidence of unmistakable mathematical genius; in fact his first loves were physics and chemistry, and he looked upon mathematics as the handmaiden—or scullery slut—of the sciences. But he did not wander long in darkness. By the age of seventeen he had smelt numerous rats in the alleged reasoning of physics and had turned to mathematics for less objectionable logic. In 1848 he entered the Caroline College—the same institution that gave the youthful Gauss an opportunity for self-instruction in mathematics. At the college Dedekind mastered the elements of analytic geometry, “advanced” algebra, the calculus, and “higher” mechanics. Thus he was well prepared to begin serious work when he entered the University of Göttingen in 1850 at the age of nineteen. His principal instructors were Moritz Abraham Stern (1807-1894), who wrote extensively on the theory of numbers, Gauss, and Wilhelm Weber the physicist. From these three men Dedekind got a thorough grounding in the calculus, the elements of the higher arithmetic, least squares, higher geodesy, and experimental physics.