Men of Mathematics (93 page)

Weierstrass himself seems to have felt uneasy; certainly he was hurt. His strong emotion is released mostly in one tremendous German sentence
^II
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
x
²
—
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
i
²
+ 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.

Kronecker died of a bronchial illness in Berlin on December 29, 1891, in his sixty ninth year.

I
. One problem in this subject: an algebraic curve may have loops on it, or places where the curve crosses its tangents; given the degree of the curve, how many such points are there? Or if we cannot answer that, what equations connecting the number of these and other exceptional points must hold? Similarly for surfaces.

II
. In a letter to Sonja Kowalewski, 1885.

A geometer like Riemann might almost have foreseen the more important features of the actual world.
—A. S. E
DDINGTON

I
T HAS BEEN SAID OF
coleridge that he wrote but little poetry of the highest order of excellence, but that that little should be bound in gold. The like has been said of Bernhard Riemann, the mathematical fruits of whose all too brief summer fill only one octavo volume. It may also be truly said of Riemann that he touched nothing that he did not in some measure revolutionize. One of the most original mathematicians of modern times, Riemann unfortunately inherited a poor constitution, and he died before he had reaped a tithe of the golden harvests in his fertile mind. Had he been born a century later than he was, medical science could probably have leased him twenty or thirty more years of life, and mathematics would not now be waiting for his successor.

Georg Friedrich Bernhard Riemann, the son of a Lutheran pastor, and the second of six children (two boys, four girls), was born in the little village of Breselenz, in Hanover, Germany, on September 17, 1826. His father had fought in the Napoleonic wars, and on settling down to a less barbarous mode of living had married Charlotte Ebell, daughter of a court councillor. Hanover in 1826 was not exactly prosperous, and the circumstances of an obscure country parson with a wife and six children to feed and clothe were far from affluent. It is claimed by some biographers, apparently with justice, that the frail health and early deaths of most of the Riemann children were the result of undernourishment in their youth and were not due to poor stamina. The mother also died before her children were grown.

In spite of poverty the home life was happy, and Riemann always retained the warmest affection—and homesickness, when he was absent—for all his lovable family. From his earliest years he was a
timid, diffident soul with a horror of speaking in public or attracting attention to himself. In later life this chronic shyness proved a very serious handicap and occasioned him much agonized misery till he overcame it by diligent preparation for every public utterance he was likely to make. The engaging bashfulness of Riemann's boyhood and early manhood, which endeared him to all who met him, was in strange contrast to the ruthless boldness of his matured scientific thought. Supreme in the world of his own creation, he realized his transcendent powers and shrank from nobody, real or imaginary.

While Riemann was still an infant his father was transferred to the pastorate of Quickborn. There young Riemann received his first instruction, from his father, who appears to have been an excellent teacher. From the very first lessons Bernhard showed an unquenchable thirst for learning. His earliest interests were historical, particularly in the romantic and tragic history of Poland. As a boy of five Bernhard gave his father no peace about unhappy Poland, but demanded to be told over and over again the legend of that heroic country's gallant (and at times slightly fatuous) struggles for liberty and, in the late Woodrow Wilson's rich, fruity phrase, “self-determination.”

Arithmetic, begun at about six, offered something less harrowing for the sensitive young boy to dwell on. His inborn mathematical genius now asserted itself. Bernhard not only solved all the problems shoved at him, but invented more difficult teasers to exasperate his brother and sisters. Already the creative impulse in mathematics dominated the boy's mind. At the age of ten he received instruction in more advanced arithmetic and geometry from a professional teacher, one Schulz, a fairly good pedagogue. Schulz soon found himself following his pupil, who often had better solutions than he.

At fourteen Riemann went to stay with his grandmother at Hanover, where he entered his first Gymnasium, in the upper third class. Here he endured his first overwhelming loneliness. His shyness made him the butt of his schoolfellows and drove him in upon his own resources. After a temporary setback his schoolwork was uniformly excellent, but it gave him no comfort, and his only solace was the joy of buying such inconsiderable presents as his pocket money would permit, to send home to his parents and brother and sisters on their birthdays. One present for his parents he invented and made himself, an original perpetual calendar, much to the astonishment of his incredulous
schoolfellows. On the death of his grandmother two years later, Riemann was transferred to the Gymnasium at Lüneburg, where he studied till he was prepared, at the age of nineteen, to enter the University of Göttingen. At Lüneburg Riemann was within walking distance of home. He took full advantage of his opportunities to escape to the warmth of his own fireside. These years of his secondary education, while his health was still fair, were the happiest of his life. The tramps back and forth between the Gymnasium and Quickborn taxed his strength, but in spite of his mother's anxiety that he might wear himself out, Riemann continued to over-exert himself in order that he might be with his family as often as possible.