Men of Mathematics (78 page)

TO A MISSING MEMBER

OF A FAMILY OF TERMS IN AN ALGEBRAICAL FORMULA.

Lone and discarded one! divorced by fate,

From thy wished-for fellows—whither art flown?

Where lingerest thou in thy bereaved estate,

Like some lost star or buried meteor stone?

Thou mindst me much of that presumptuous one

Who loth, aught less than greatest, to be great,

From Heaven s immensity fell headlong down

To live forlorn, self-centred, desolate:

Or who, new Heraklid, hard exile bore,

Now buoyed by hope, now stretched on rack of fear,

Till throned Astraea, wafting to his ear

Words of dim portent through the Atlantic roar,

Bade him 'the sanctuary of the Muse revere

And strew with flame the dust of Isis' shore'.

Having refreshed ourselves and bathed the tips of our fingers in the Pierian spring, let us turn back for a few brief moments to a light banquet of the reason, and entertain ourselves as a sort of after-course with some general reflections arising naturally out of the previous matter of my discourse.”

If the Pierian spring was the old boy's finger bowl at this astonishing feast of reason, it is a safe bet that the faithful decanter of port was never very far from his elbow.

Sylvester's sense of the kinship of mathematics to the finer arts found frequent expression in his writings. Thus, in a paper on Newton's rule for the discovery of imaginary roots of algebraic equations, he asks in a footnote “May not Music be described as the Mathematic of sense, Mathematic as Music of the reason? Thus the musician
feels
Mathematic, the mathematician
thinks
Music—Music the dream, Mathematic the working life—each to receive its consummation from the other when the human intelligence, elevated to its perfect type,
shall shine forth glorified in some future Mozart-Dirichlet or Beethoven-Gauss—a union already not indistinctly foreshadowed in the genius and labors of a Helmholtz!”

Sylvester loved life, even when he was forced to fight it, and if ever a man got the best that is in life out of it, he did. He gloried in the fact that the great mathematicians, except for what may be classed as avoidable or accidental deaths, have been long-lived and vigorous of mind to their dying days. In his presidential address to the British Association in
1869
he called the honor roll of some of the greatest mathematicians of the past and gave their ages at death to bear out his thesis that “. . . there is no study in the world which brings into more harmonious action all the faculties of the mind than [mathematics], . . . or, like this, seems to raise them, by successive steps of initiation, to higher and higher states of conscious intellectual being. . . . The mathematician lives long and lives young; the wings of the soul do not early drop off, nor do its pores become clogged with the earthy particles blown from the dusty highways of vulgar life.”

Sylvester was a living example of his own philosophy. But even he at last began to bow to time. In
1893
—he was then seventy nine—his eyesight began to fail, and he became sad and discouraged because he could no longer lecture with his old enthusiasm. The following year he asked to be relieved of the more onerous duties of his professorship, and retired to live, lonely and dejected, in London or at Tunbridge Wells. All his brothers and sisters had long since died, and he had outlived most of his dearest friends.

But even now he was not through. His mind was still vigorous, although he himself felt that the keen edge of his inventiveness was dulled forever. Late in
1896,
in the eighty second year of his age, he found a new enthusiasm in a field which had always fascinated him, and he blazed up again over the theory of compound partitions and Goldbach's conjecture that every even number is the sum of two primes.

He had not much longer. While working at his mathematics in his London rooms early in March,
1897,
he suffered a paralytic stroke which destroyed his power of speech. He died on March
15, 1897,
at the age of eighty three. His life can be summed up in his own words, “I really love my subject.”

I
. This part of the theory was developed many years later by E. K. Wakeford (1894-1916), who lost his life in the World War. “Now thanked be God who matched us with this hour.” (Rupert Brooke.)

The theory that has had the greatest development in recent times is without any doubt the theory of functions.
—V
ITO
V
OLTERRA

Y
OUNG DOCTORS IN MATHEMATICS
, anxiously seeking positions in which their training and talents may have some play, often ask whether it is possible for a man to do elementary teaching for long and keep alive mathematically. It is. The life of Boole is a partial answer; the career of Weierstrass, the prince of analysts, “the father of modern analysis,” is conclusive.

Before considering Weierstrass in some detail, we place him chronologically with respect to those of his German contemporaries, each of whom, like him, gave at least one vast empire of mathematics a new outlook during the second half of the nineteenth century and the first three decades of the twentieth. The year 1855, which marks the death of Gauss and the breaking of the last link with the outstanding mathematicians of the preceding century, may be taken as a convenient point of reference. In 1855 Weierstrass (1815-1897) was forty; Kronecker (1823-1891), thirty two; Riemann (1826-1866), twenty nine; Dedekind (1831-1916), twenty four; while Cantor (1845-1918) was a small boy of ten. Thus German mathematics did not lack recruits to carry on the great tradition of Gauss. Weierstrass was just gaining recognition; Kronecker was well started; some of Riemann's greatest work was already behind him, and Dedekind was entering the field (the theory of numbers) in which he was to gain his greatest fame. Cantor, of course, had not yet been heard from.

We have juxtaposed these names and dates because four of the men mentioned, dissimilar and totally unrelated as much of their finest work was, came together on one of the central problems of all mathematics, that of irrational numbers: Weierstrass and Dedekind resumed the discussion of irrationals and continuity practically where
Eudoxus had left it in the fourth century
B.C
.; Kronecker, a modern echo of Zeno, made Weierstrass' last years miserable by skeptical criticism of the latter's revision of Eudoxus; while Cantor, striking out on a new road of his own, sought to compass the actual infinite itself which is implicit—according to some—in the very concept of continuity. Out of the work of Weierstrass and Dedekind developed the modern epoch of analysis, that of critical logical precision in analysis (the calculus, the theory of functions of a complex variable, and the theory of functions of real variables) in distinction to the looser intuitive methods of some of the older writers—invaluable as heuristic guides to discovery but quite worthless from the standpoint of the Pythagorean ideal of mathematical proof. As has already been noted, Gauss, Abel, and Cauchy inaugurated the first period of rigor; the movement started by Weierstrass and Dedekind was on a higher plane, suitable to the more exacting demands of analysis in the second half of the century, for which the earlier precautions were inadequate.

One discovery by Weierstrass in particular shocked the intuitive school of analysts into a decent regard for caution: he produced a continuous curve which has no tangent at any point. Gauss once called mathematics “the science of the eye”; it takes more than a good pair of eyes to “see” the curve which Weierstrass presented to the advocates of sensual intuition.

Since to every action there is an equal and opposite reaction it was but natural that all this revamped rigor should engender its own opposition. Kronecker attacked it vigorously, even viciously, and quite exasperatingly. He denied that it meant anything. Although he succeeded in hurting the venerable and kindly Weierstrass, he made but little impression on his conservative contemporaries and practically none on mathematical analysis. Kronecker was a generation ahead of his time. Not till the second decade of the twentieth century did his strictures on the currently accepted doctrines of continuity and irrational numbers receive serious consideration. Today it is true that not all mathematicians regard Kronecker's attack as merely the release of his pent-up envy of the more famous Weierstrass which some of his contemporaries imagined it to be, and it is admitted that there may be something—not much, perhaps—in his disturbing objections. Whether there is or not, Kronecker's attack was partly responsible for the
third
period of rigor in modern mathematical reasoning, that which we ourselves are attempting to enjoy. Weierstrass was not the
only fellow-mathematician whom Kronecker harried; Cantor also suffered deeply under what he considered his influential colleague's malicious persecution. All these men will speak for themselves in the proper place; here we are only attempting to indicate that their lives and work were closely interwoven in at least one corner of the gorgeous pattern.

To complete the picture we must indicate other points of contact between Weierstrass, Kronecker, and Riemann on one side and Kronecker and Dedekind on the other. Abel, we recall, died in 1829, Galois in 1832, and Jacobi in 1851. In the epoch under discussion one of the outstanding problems in mathematical analysis was the completion of the work of Abel and Jacobi on multiply periodic functions—elliptic functions, Abelian functions (see chapters 17, 18). From totally different points of view Weierstrass and Riemann accomplished what was to be done—Weierstrass indeed considered himself in some degree a successor of Abel; Kronecker opened up new vistas in elliptic functions but he did not compete with the other two in the field of Abelian functions. Kronecker was primarily an arithmetician and an algebraist; some of his best work went into the elaboration and extension of the work of Galois in the theory of equations. Thus Galois found a worthy successor not too long after his death.

Apart from his forays into the domain of continuity and irrational numbers, Dedekind's most original work was in the higher arithmetic, which he revolutionized and renovated. In this Kronecker was his able and sagacious rival, but again their whole approaches were entirely different and characteristic of the two men: Dedekind overcame his difficulties in the theory of algebraic numbers by taking refuge in the infinite (in his theory of “ideals,” as will be indicated in the proper place); Kronecker sought to solve his problems in the finite.

*  *  *

Karl Wilhelm Theodor Weierstrass, the eldest son of Wilhelm Weierstrass (1790-1869) and his wife Theodora Forst, was born on October 31, 1815, at Ostenfelde in the district of Münster, Germany. The father was then a customs officer in the pay of the French. It may be recalled that 1815 was the year of Waterloo; the French were still dominating Europe. That year also saw the birth of Bismarck, and it is interesting to observe that whereas the more famous statesman's life work was shot to pieces in the World War, if not earlier, the contributions of his comparatively obscure contemporary to science and
the advancement of civilization in general are even more highly esteemed today than they were during his lifetime.

The Weierstrass family were devout liberal Catholics all their lives; the father had been converted from Protestantism, probably at the time of his marriage. Karl had a brother, Peter (died in 1904), and two sisters, Klara (1823-1896), and Elise (1826-1898) who looked after his comfort all their lives. The mother died in 1826, shortly after Elise's birth, and the father married again the following year. Little is known of Karl's mother, except that she appears to have regarded her husband with a restrained aversion and to have looked on her marriage with moderated disgust. The stepmother was a typical German housewife; her influence on the intellectual development of her stepchildren was probably nil. The father, on the other hand, was a practical idealist, and a man of culture who at one time had been a teacher. The last ten years of his life were spent in peaceful old age in the house of his famous son in Berlin, where the two daughters also lived. None of the children ever married, although poor Peter once showed an inclination toward matrimony which was promptly squelched by his father and sisters.

One possible discord in the natural sociability of the children was the father's uncompromising righteousness, domineering authority, and Prussian pigheadedness. He nearly wrecked Peter's life with his everlasting lecturing and came perilously close to doing the same by Karl, whom he attempted to force into an uncongenial career without ascertaining where his brilliant young son's abilities lay. Old Weierstrass had the audacity to preach at his younger son and meddle in his affairs till the “boy” was nearly forty. Luckily Karl was made of more resistant stuff. As we shall see his fight against his father—although he himself was probably quite unaware that he was fighting the tyrant—took the not unusual form of making a mess of the life his father had chosen for him. It was as neat a defense as he could possibly have devised, and the best of it was that neither he nor his father ever dreamed what was happening, although a letter of Karl's when he was sixty shows that he had at last realized the cause of his early difficulties. Karl at last got his way, but it was a long, roundabout way, beset with trials and errors. Only a shaggy man like himself, huge and rugged of body and mind, could have won through to the end.

Shortly after Karl's birth the family moved to Westernkotten, Westphalia, where the father became a customs officer at the salt
works. Westernkotten, like other dismal holes in which Weierstrass spent the best years of his life, is known in Germany today only because Weierstrass once was condemned to rot there—only he did not rust; his first published work is dated as having been written in
1841
(he was then
26)
at Westernkotten. There being no school in the village, Karl was sent to the adjacent town of Münster whence, at fourteen, he entered the Catholic Gymnasium at Paderborn. Like Descartes under somewhat similar conditions, Weierstrass thoroughly enjoyed his school and made friends of his expert, civilized instructors. He traversed the set course in considerably less than the standard time, making a uniformly brilliant record in all his studies. He left in
1834
at the age of nineteen. Prizes fell his way with unfailing regularity; one year he carried off seven; he was usually first in German and in two of the three, Latin, Greek, and mathematics. By a beautiful freak of irony he never won a prize for calligraphy, although he was destined to teach penmanship to little boys but recently emancipated from their mothers' apron strings.