Men of Mathematics (100 page)

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.

*  *  *

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 Gauss
^III
. 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.

In later life Dedekind regretted that the mathematical instruction available during his student years at Göttingen, while adequate for the rather low requirements for a state teacher's certificate, was inconsiderable as a preparation for a mathematical career. Subjects of living interest were not touched upon, and Dedekind had to spend two years of hard labor after taking his degree to get up by himself elliptic functions, modern geometry, higher algebra, and mathematical physics—all of which at the time were being brilliantly expounded at Berlin by Jacobi, Steiner, and Dirichlet. In
1852
Dedekind got his doctor's degree (at the age of twenty one) from Gauss for a short dissertation on Eulerian integrals. There is no need to explain what this was: the dissertation was a useful, independent piece of work, but it betrayed no such genius as is evident on every page of many of Dedekind's later works. Gauss' verdict on the dissertation will be of interest: “The memoir prepared by Herr Dedekind is concerned with a research in the integral calculus, which is by no means commonplace. The author evinces not only a very good knowledge of the relevant field, but also such an independence as augurs favorably for his future achievement. As a test essay for admission to the examination I find the memoir completely satisfying.” Gauss evidently saw more in the dissertation than some later critics have detected; possibly his close contact with the young author enabled him to read between the lines. However, the report, even as it stands, is more or less the usual perfunctory politeness customary in accepting a passable dissertation,
and we do not know whether Gauss really foresaw Dedekind's penetrating originality.

In 1854 Dedekind was appointed lecturer
(Privatdozent)
at Göttingen, a position which he held for four years. On the death of Gauss in 1855 Dirichlet moved from Berlin to Göttingen. For the remaining three years of his stay at Göttingen, Dedekind attended Dirichlet's most important lectures. Later he was to edit Dirichlet's famous treatise on the theory of numbers and add to it the epoch-making “Eleventh Supplement” containing an outline of his own theory of algebraic numbers. He also became a friend of the great Riemann, then beginning his career. Dedekind's university lectures were for the most part elementary, but in 1857-8 he gave a course (to two students, Selling and Auwers) on the Galois theory of equations. This was probably the first time that the Galois theory had appeared formally in a university course. Dedekind was one of the first to appreciate the fundamental importance of the concept of a group in algebra and arithmetic. In this early work Dedekind already exhibited two of the leading characteristics of his later thought, abstractness and generality. Instead of regarding a finite group from the standpoint offered by its representation in terms of substitutions (see chapters on Galois and Cauchy), Dedekind defined groups by means of their postulates (substantially as described in Chapter 15) and sought to derive their properties from this distillation of their essence. This is in the modern manner: abstractness and therefore generality. The second characteristic, generality, is, as just implied, a consequence of the first.

At the age of twenty six Dedekind was appointed (in 1857) ordinary professor at the Zurich polytechnic, where he stayed five years, returning in 1862 to Brunswick as professor at the technical high school. There he stuck for half a century. The most important task for Dedekind's official biographer—provided one is unearthed—will be to explain (not explain away) the singular fact that Dedekind occupied a relatively obscure position for fifty years while men who were not fit to lace his shoes filled important and influential university chairs. To say that Dedekind preferred obscurity is one explanation. Those who believe it should leave the stock market severely alone, for as surely as God made little lambs they will be fleeced.

Till his death (1916) in his eighty fifth year Dedekind remained fresh of mind and robust of body. He never married, but lived with his sister Julie, remembered as a novelist, till her death in 1914. His
other sister, Mathilde, died in 1860; his brother became a distinguished jurist.

Such are the bare facts of any importance in Dedekind's material career. He lived so long that although some of his work (his theory of irrational numbers, described presently) had been familiar to all students of analysis for a generation before his death, he himself had become almost a legend and many classed him with the shadowy dead. Twelve years before his death, Teubner's
Calendar for Mathematicians
listed Dedekind as having died on September 4, 1899, much to Dedekind's amusement. The day, September 4, might possibly prove to be correct, he wrote to the editor, but the year certainly was wrong. “According to my own memorandum I passed this day in perfect health and enjoyed a very stimulating conversation on 'system and theory' with my luncheon guest and honored friend Georg Cantor of Halle.”

*  *  *

Dedekind's mathematical activity impinged almost wholly on the domain of number in its widest sense. We have space for only two of his greatest achievements and we shall describe first his fundamental contribution, that of the “Dedekind cut,” to the theory of irrational numbers and hence to the foundations of analysis. This being of the very first importance we may recall briefly the nature of the matter. If
a, b
are common whole numbers, the fraction
a/b
is called a rational number; if no whole numbers m,
n
exist such that a certain “number”
N
is expressible as
m/n,
_y
then
N
is called an irrational number. Thus
are irrational numbers. If an irrational number be expressed in the decimal notation the digits following the decimal point exhibit no regularities—there is no “period” which repeats, as in the decimal representations of a rational number, say 13/11, = 1.181818 . . . , where the “18” repeats indefinitely. How then, if the representation is entirely lawless, are decimals equivalent to irrationals to be defined, let alone manipulated? Have we even any clear conception of what an irrational number is? Eudoxus thought he had, and Dedekind's definition of equality between numbers, rational or irrational, is identical with that of Eudoxus (see Chapter 2).

If two rational numbers are equal, it is no doubt obvious that their square roots are equal. Thus 2X3 and 6 are equal; so also then are
and
But it is
not
obvious that
and hence that
The un-obviousness of this simple
assumed equality,
taken for granted in school arithmetic, is evident if we visualize what the equality implies: the “lawless” square roots of
2, 3,
6
are to be extracted, the first two of these are then to be multiplied together, and the result is to come out equal to the third. As not one of these three roots can be extracted exactly, no matter to how many decimal places the computation is carried, it is clear that the verification by multiplication as just described will never be complete. The whole human race toiling incessantly through all its existence could never
prove
in this way that
Closer and closer approximations to equality would be attained as time went on, but finality would continue to recede. To make these concepts of “approximation” and “equality” precise, or to replace our first crude conceptions of irrationals by sharper descriptions which will obviate the difficulties indicated, was the task Dedekind set himself in the early
1870
's—his work on
Continuity and Irrational Numbers
was published in
1872.