Men of Mathematics (85 page)

The tremendous power and fluent ease of the method can be readily appreciated by seeing what it does in any work on symbolic logic. But, as already emphasized, the importance of this “symbolic reasoning” is in its applicability to subtle questions regarding the foundations of all mathematics which, were it not for this precise method of fixing meanings of “words” or other “symbols” once for all, would probably be unapproachable by ordinary mortals.

Like nearly all novelties, symbolic logic was neglected for many years after its invention. As late as 1910 we find eminent mathematicians scorning it as a “philosophical” curiosity without mathematical significance. The work of Whitehead and Russell in
Principia Mathematica
(1910-1913) was the first to convince any considerable body of professional mathematicians that symbolic logic might be worth their serious attention. One staunch hater of symbolic logic may be mentioned—Cantor, whose work on the infinite will be noticed in the concluding chapter. By one of those little ironies which make mathematical history such amusing reading for the open-minded, symbolic logic was to play an important part in the drastic criticism of Cantor's work that caused its author to lose faith in himself and his theory.

Boole did not long survive the production of his masterpiece. The year after its publication, still subconsciously striving for the social respectability that he once thought a knowledge of Greek could confer, he married Mary Everest, niece of the Professor of Greek in Queen's College. His wife became his devoted disciple. After her husband's death, Mary Boole applied some of the ideas which she had acquired from him to rationalizing and humanizing the education of young children. In her pamphlet,
Boole's Psychology,
Mary Boole records an interesting speculation of Boole's which readers of
The Laws of Thought
will recognize as in keeping with the unexpressed but implied personal philosophy in certain sections. Boole told his wife that in 1832, when he was about seventeen, it “flashed upon” him as he was walking across a field that besides the knowledge gained from
direct observation, man derives knowledge from some source un-definable and invisible—which Mary Boole calls “the unconscious.” It will be interesting (in a later chapter) to hear Poincaré expressing a similar opinion regarding the genesis of mathematical “inspirations” in the “subconscious mind.” Anyhow, Boole was inspired, if ever a mortal was, when he wrote
The Laws of Thought.

Boole died, honored and with a fast-growing fame, on December 8, 1864, in the fiftieth year of his age. His premature death was due to pneumonia contracted after faithfully keeping a lecture engagement when he was soaked to the skin. He fully realized that he had done great work.

Talk with M. Hermite: he never evokes a concrete image; yet you soon perceive that the most abstract entities are for him like living creatures.
—H
ENRI
P
OINCARé

O
UTSTANDING UNSOLVED PROBLEMS
demand new methods for their solution, while powerful new methods beget new problems to be solved. But, as Poincaré observed, it is the man, not the method, that solves a problem.

Of old problems responsible for new methods in mathematics that of motion and all it implies for mechanics, terrestrial and celestial, may be recalled as one of the principal instigators of the calculus and present attempts to put reasoning about the infinite on a firm basis. An example of new problems suggested by powerful new methods is the swarm which the tensor calculus, popularized to geometers by its successes in relativity, let loose in geometry. And finally, as an illustration of Poincaré's remark, it was Einstein, and not the method of tensors, that solved the problem of giving a coherent mathematical account of gravitation. All three theses are sustained in the life of Charles Hermite, the leading French mathematician of the second half of the nineteenth century—if we except Hermite's pupil Poincaré, who belonged partly to our own century.

Charles Hermite, born at Dieuze, Lorraine, France, on December 24, 1822, could hardly have chosen a more propitious era for his birth than the third decade of the nineteenth century. His was just the rare combination of creative genius and the ability to master the best in the work of other men which was demanded in the middle of the century to coordinate the arithmetical creations of Gauss with the discoveries of Abel and Jacobi in elliptic functions, the striking advances of Jacobi in Abelian functions, and the vast theory of algebraic invariants in process of rapid development by the English mathematicians Boole, Cayley, and Sylvester.

Hermite almost lost his life in the French Revolution—although the last head had fallen nearly a quarter of a century before he was born. His paternal grandfather was ruined by the Commune and died in prison; his grandfather's brother went to the guillotine. Hermite's father escaped owing to his youth.

If Hermite's mathematical ability was inherited, it probably came from the side of the father, who had studied engineering. Finding engineering uncongenial, Hermite senior gave it up, and after an equally distasteful start in the salt industry, finally settled down in business as a cloth merchant. This resting place was no doubt chosen by the rolling stone because he had married his employer's daughter, Madeleine Lallemand, a domineering woman who wore the breeches in her family and ran everything from the business to her husband. She succeeded in building both up to a state of solid bourgeois prosperity. Charles was the sixth of seven children—five sons and two daughters. He was born with a deformity of the right leg which rendered him lame for life—possibly a disguised blessing, as it effectively barred him from any career even remotely connected with the army—and he had to get about with a cane. His deformity never affected the uniform sweetness of his disposition.

Hermite's earliest education was received from his parents. As the business continued to prosper, the family moved from Dieuze to Nancy when Hermite was six. Presently the growing demands of the business absorbed all the time of the parents and Hermite was sent as a boarder to the
lycée
at Nancy. This school proving unsatisfactory the prosperous parents decided to give Charles the best and packed him off to Paris. There he studied for a short time at the Lycée Henri IV, moving on at the age of eighteen (1840) to the more famous (or infamous) Louis-le-Grand—the “Alma” Mater of the wretched Galois—to prepare for the Polytechnique.

For a while it looked as if Hermite was to repeat the disaster of his untamable predecessor at Louis-le-Grand. He had the same dislike for rhetoric and the same indifference to the elementary mathematics of the classroom. But the competent lectures on physics fascinated him and won his cordial cooperation in the bilateral process of acquiring an education. Later on, unpestered by pedants, Hermite became a good classicist and the master of a beautifully clear prose.

Those who hate examinations will love Hermite. There is something in the careers of these two most famous alumni of Louis-le-Grand,
Galois and Hermite, which might well cause the advocates of examinations as a reliable yardstick for arranging human beings in order of intellectual merit to ask themselves whether they have used their heads or their feet in arriving at their conclusions. It was only by the grace of God and the diplomatic persistence of the devoted and intelligent Professor Richard, who had done his unavailing best fifteen years before to save Galois for science, that Hermite was not tossed out by stupid examiners to rot on the rubbish heap of failure. While still a student at the
lycée,
Hermite, following in the steps of Galois, supplemented and neglected his elementary lessons by private reading at the library of Sainte-Geneviève, where he found and mastered the memoir of Lagrange on the solution of numerical equations. Saving up his pennies, he bought the French translation of the
Disquisitiones Arithmeticae
of Gauss and, what is more, mastered it as few before or since have mastered it. By the time he had followed what Gauss had done Hermite was ready to
go on.
“It was in these two books,” he loved to say in later life, “that I learned Algebra.” Euler and Laplace also instructed him through their works. And yet Hermite's performance in examinations was, to say the most flattering thing possible of it, mediocre. Mathematical nonentities beat him out of sight.

Mindful of the tragic end of Galois, Richard tried his best to steer Hermite away from original investigation to the less exciting though muddier waters of the competitive examinations for entrance to the École Polytechnique—the filthy ditch in which Galois had drowned himself. Nevertheless the good Richard could not refrain from telling Hermite's father that Charles was “a young Lagrange.”

The
Nouvelles Annales de Mathématiques,
a journal devoted to the interests of students in the higher schools, was founded in 1842. The first volume contains two papers composed by Hermite while he was still a student at Louis-le-Grand. The first is a simple exercise in the analytic geometry of conic sections and betrays no originality. The second, which fills only six and a half pages in Hermite's collected works, is a horse of quite a different color. Its unassuming title is
Considerations on the algebraic solution of the equation of the fifth degree
(translation).

“It is known,” the modest mathematician of twenty begins, “that Lagrange made the algebraic solution of the general equation of the fifth degree depend on the determination of a root of a
particular
equation of the sixth degree, which he calls a
reduced equation
£ today, a
'resolvent']. . . . So that, if this resolvent were decomposable into rational factors of the second or third degrees, we should have the solution of the equation of the fifth degree. I shall try to show that such a decomposition is impossible.” Hermite not only succeeded in his attempt—by a beautifully simple argument—but showed also in doing so that he was an algebraist. With but a few slight changes this short paper will do all that is required.

It may seem strange that a young man capable of genuine mathematical reasoning of the caliber shown by Hermite in his paper on the general quintic should find elementary mathematics difficult. But it is not necessary to understand—or even to have heard of—much of classical mathematics as it has evolved in the course of its long history in order to be able to follow or work creatively in the mathematics that has been developed since 1800 and is still of living interest to mathematicians. The geometrical treatment (synthetic) of conic sections of the Greeks, for instance, need not be mastered today by anyone who wishes to follow modern geometry; nor need any geometry at all be learned by one whose tastes are algebraic or arithmetical. To a lesser degree the same is true for analysis, where such geometrical language as is used is of the simplest and is neither necessary nor desirable if up-to-date proofs are the object. As a last example, descriptive geometry, of great use to designing engineers, is of practically no use whatever to a working mathematician. Some quite difficult subjects that are still mathematically alive require only a school education in algebra and a clear head for their comprehension. Such are the theory of finite groups, the mathematical theory of the infinite, and parts of the theory of probabilities and the higher arithmetic. So it is not astonishing that large tracts of what a candidate is required to know for entrance to a technical or scientific school, or even for graduation from the same, are less than worthless for a mathematical career. This accounts for Hermite's spectacular success as a budding mathematician and his narrow escape from complete disaster as an examinee.

Late in 1842, at the age of twenty, Hermite sat for the entrance examinations to the École Polytechnique. He passed, but only as sixty eighth in order of merit. Already he was a vastly better mathematician than some of the men who examined him were, or were ever to become. The humiliating outcome of this test made an impression on the young master which all the triumphs of his manhood never effaced.

Hermite stayed only one year at the Polytechnique. It was not his head that disqualified him but his lame foot which, according to a ruling of the authorities, unfitted him for any of the positions open to successful students of the school. Perhaps it is as well that Hermite was thrown out; he was an ardent patriot and might easily have been embroiled in one or other of the political or military rows so precious to the effervescent French temperament. However, the year was by no means wasted. Instead of slaving over descriptive geometry, which he hated, Hermite spent his time on Abelian functions, then (1842) perhaps the topic of outstanding interest and importance to the great mathematicians of Europe. He had also made the acquaintance of Joseph Liouville (1809-1882), a first-rate mathematician and editor of the
Journal des Mathématiques.

Liouville recognized genius when he saw it. In passing it may be amusing to recall that Liouville inspired William Thomson, Lord Kelvin, the famous Scotch physicist, to one of the most satisfying definitions of a mathematician that has ever been given. “Do you know what a mathematician is?” Kelvin once asked a class. He stepped to the board and wrote