Men of Mathematics (90 page)

What might have happened to Germany had Kronecker not abandoned finance for mathematics also offers a wide field for speculation. His business abilities were of a high order; he was an ardent patriot with an uncanny insight into European diplomacy and a shrewd cynicism—his admirers called it realism—regarding the unexpressed sentiments cherished by the great Powers for one another.

At first a liberal like so many intellectual young Jews, Kronecker quickly became a rock-ribbed conservative when he saw which side his own abundant bread was buttered on—after his financial exploits,
and proclaimed himself a loyal supporter of that callous old truth-doctor Bismarck. The famous episode of the Ems telegram which, according to some, was the electric spark that touched off the Franco-Prussian war in 1870, had Kronecker's warm approval, and his grasp of the situation was so firm that
before
the battle of Weissenburg, when even the military geniuses of Germany were doubtful as to the outcome of their bold challenging of France, Kronecker confidently predicted the success of the entire campaign and was proved right in detail. At the time, and indeed all his life, he was on cordial terms with the leading French mathematicians, and he was clear-headed enough not to let his political opinions cloud his just perception of his scientific rivals' merits. It is perhaps as well that so realistic a man as Kronecker cast his lot with mathematics.

Leopold Kronecker's life was easy from the day of his birth. The son of prosperous Jewish parents, he was born on December
7, 1823,
at Liegnitz, Prussia. By an unaccountable oversight Kronecker's official biographers (Heinrich Weber and Adolf Kneser) omit all mention of Leopold's mother, although he probably had one, and concentrate on the father, who owned a flourishing mercantile business. The father was a well educated man with an unquenchable thirst for philosophy which he passed on to Leopold. There was another son, Hugo, seventeen years younger than Leopold, who became a distinguished physiologist and professor at Berne. Leopold's early education under a private tutor was supervised by the father; Hugo's upbringing later became the loving duty of Leopold.

In the second stage of his education at the preparatory school for the Gymnasium Leopold was strongly influenced by the co-rector Werner, a man with philosophical and theological leanings, who later taught Kronecker when he entered the Gymnasium. Among other things Kronecker imbibed from Werner was a liberal draught of Christian theology, for which he acquired a lifelong enthusiasm. With what looks like his usual caution, Kronecker did not embrace the Christian faith till practically on his deathbed when, having seen that it did his six children no noticeable mischief, he permitted himself to be converted from Judaism to evangelical Christianity in his sixty eighth year.

Another of Kronecker's teachers at the Gymnasium also influenced him profoundly and became his lifelong friend, Ernst Eduard Kummer (1810-1893), subsequently professor at the University of Berlin and
one of the most original mathematicians Germany has produced, of whom more will be said in connection with Dedekind. These three, Kronecker senior, Werner, and Kummer, capitalized Leopold's immense native abilities, formed his mind, and charted the future course of his life so cunningly that he could not have departed from it if he had wished.

Already in this early stage of his education we note an outstanding feature of Kronecker's genial character, his ability to get along with people and his instinct for forming lasting friendships with men who had risen in the world or were to rise, and who would be useful to him either in business or mathematics. This genius for friendships of the right sort, which is one of the successful business man's distinguishing traits, was one of Kronecker's more valuable assets and he never mislaid it. He was not consciously mercenary, nor was he a snob; he was merely one of those lucky mortals who is more at ease with the successful than with the unsuccessful.

Kronecker's performance at school was uniformly brilliant and many-sided. In addition to the Greek and Latin classics which he mastered with ease and for which he retained a lifelong liking, he shone in Hebrew, philosophy, and mathematics. His mathematical talent appeared early under the expert guidance of Kummer, from whom he received special instruction. Young Kronecker however did not concentrate to any great extent on mathematics, although it was obvious that his greatest talent lay in that field, but set himself to acquiring a broad liberal education commensurate with his manifold abilities. In addition to his formal studies he took music lessons and became an accomplished pianist and vocalist. Music, he declared when he was an old man, is the finest of all the fine arts, with the possible exception of mathematics, which he likened to poetry. These many interests he retained throughout his life. In none of them was he a mere dabbler: his love of the classics of antiquity bore tangible fruit in his affiliation with Graeca, a society dedicated to the translation and popularization of the Greek classics; his keen appreciation of art made him an acute critic of painting and sculpture, and his beautiful house in Berlin became a rendezvous for musicians, among them Felix Mendelssohn.

Entering the University of Berlin in the spring of
1841,
Kronecker continued his broad education but began to concentrate on mathematics. Berlin at that time boasted Dirichlet (
1805-1859),
Jacobi (
18041851)
and Steiner
(1796-1863)
on its mathematical faculty;
Eisenstein
(1823-1852),
the same age as Kronecker, also was about, and the two became friends.

The influence of Dirichlet on Kronecker's mathematical tastes (particularly in the application of analysis to the theory of numbers) is clear all through his mature writings. Steiner seems to have made no impression on him; Kronecker had no feeling for geometry. Jacobi gave him a taste for elliptic functions which he was to cultivate with striking originality and brilliant success, chiefly in novel applications of magical beauty to the theory of numbers.

Kronecker's university career was a repetition on a larger scale of his years at school: he attended lectures on the classics and the sciences and indulged his bent for philosophy by profounder studies than any he had as yet undertaken, particularly in the system of Hegel. The last is emphasized because some curious and competent reader may be moved to seek the origin of Kronecker's mathematical heresies in the abstrusities of Hegel's dialectic—a quest wholly beyond the powers of the present writer. Nevertheless there is a strange similarity between some of the weird unorthodoxies of recent doubts concerning the self-consistency of mathematics—doubts for which Kronecker's “revolution” was partly responsible—and the subtleties of Hegel's system. The ideal candidate for such an undertaking would be a Marxian communist with a sound training in Polish many-valued logic, though in what incense tree this rare bird is to be sought God only knows.

Following the usual custom of German students, Kronecker did not spend all his time at Berlin but moved about. Part of his course was pursued at the University of Bonn, where his old teacher and friend Kummer had taken the chair of mathematics. During Kronecker's residence at Bonn the University authorities were in the midst of a futile war to suppress the student societies whose chief object was the fostering of drinking, duelling, and brawling in general. With his customary astuteness, Kronecker allied himself secretly with the students and thereby made many friends who were later to prove useful.

*  *  *

Kronecker's dissertation, accepted by Berlin for his Ph.D. in
1845,
was inspired by Kummer's work in the theory of numbers and dealt with the units in certain algebraic number fields. Although the problem is one of extreme difficulty when it comes to actually exhibiting the units, its nature can be understood from the following rough de
scription of the
general
problem of units (for
any
algebraic number field, not merely for the
special
fields which interested Kummer and Kronecker). This sketch may also serve to make more intelligible some of the allusions in the present and subsequent chapters to the work of Kummer, Kronecker, and Dedekind in the higher arithmetic. The matter is quite simple but requires several preliminary definitions.

The common whole numbers 1, 2, 3, . . . are called the (positive) rational integers. If
m
is any rational integer, it is the root of an algebraic equation of the
first
degree, whose coefficients are
rational integers,
namely
x—m
= 0. This, among other properties of the rational integers, suggested the
generalization
of the concept of integers to the “numbers” defined as roots of algebraic equations. Thus if
r
is a root of the equation

x
ⁿ
+
a
₁
x
^n-1
+ . . . +
a
_n-1
x
+
a
_n
= 0,

where the
a's
are rational integers (positive or negative), and if further r satisfies no equation of degree less than
n,
all of whose coefficients are rational integers and whose leading coefficient is 1 (as it is in the above equation, namely the coefficient of the highest power,
x
ⁿ
,
of
x
in the equation is 1), then r is called an
algebraic integer
of
degree n.
For example,
is an algebraic integer of degree 2, because it is a root of
x
²
−2x +
6
= 0, and is not a root of any equation of degree less than 2 with coefficients of the prescribed kind; in fact
is the root of
and the last coefficient,
is not a rational integer.

If in the above definition of an algebraic
integer
of degree
n
we suppress the requirement that the leading coefficient be 1, and say that it can be any rational integer (other than zero, which is considered an integer), a root of the equation is then called an
algebraic number
of
degree n.
Thus
is
^an
algebraic
number
of degree 2, but is not an algebraic
integer;
it is a root of 2x
²
—
2x
+ 3 = 0.