Men of Mathematics (102 page)

Anyone who will ponder a little on the foregoing bare outline of Dedekind's creation will see that what he did demanded penetrating
insight and a mind gifted far above the ordinary good mathematical mind in the power of abstraction. Dedekind was a mathematician after Gauss' own heart:
“At nostro quidem judicio hujusmodi veritates ex notionibus potius quam ex notationibus hauriri debeant”
(But in our opinion such truths [arithmetical] should be derived from notions rather than from notations). Dedekind always relied on his head rather than on an ingenious symbolism and expert manipulations of formulas to get him forward. If ever a man put notions into mathematics, Dedekind did, and the wisdom of his preference for creative ideas over sterile symbols is now apparent although it may not have been during his lifetime. The longer mathematics lives the more abstract—and therefore, possibly, also the more practical—it becomes.

I
. If
x
^p
+ y
^p
= z
^p
, then
x
^p
= z
^p
−y
^p
and resolving
z
^p
–
y
^p
, into its
p
factors of the first degree, we get

x
^p
= (z-y) (z-ry) (z-r
²
y) . . . (z-r
^p-1
y),

in which r is a “
p
th root of unity” (other than l), namely
r
^p
– 1 = 0, with
r
not equal to 1. The algebraic integers in the field of degree
p
generated by r are those which Kummer introduced into the study of Fermat's equation, and which led him to the invention of his “ideal numbers” to restore unique factorization in the field—an integer in such a field is not uniquely the product of primes in the field for
all
primes
p
.

II
. †The “infinite” in Kummer's title is still (1936) unjustified; “many” should be put for “infinite."

III
. No adequate biography of Dedekind has yet appeared. A life was to have been included in the third volume of his collected works (1932), but was not, owing to the death of the editor in chief (Robert Fricke). The account here is based on Landau's commemorative address. Note that, following the good old Teutonic custom of some German biographers, Landau omits all mention of Dedekind's mother. This no doubt is in accordance with the theory of the “three K's” propounded by the late Kaiser of Germany and heartily endorsed by Adolf Hitler: “A woman's whole duty is comprised in the three big K's—Kissing, Kooking [cooking is spelt with a K in German], and Kids.” Still, one would like to know at least the maiden name of a great man's mother.

A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.
—H
ENRI
P
OINCARÉ

I
N THE
History of his Life and Times
the astrologer William Lilly (1602-1681) records an amusing—if incredible—account of the meeting between John Napier (1550-1617), of Merchiston, the inventor of logarithms, and Henry Briggs (1561-1631) of Gresham College, London, who computed the first table of common logarithms. One John Marr, “an excellent mathematician and geometrician,” had gone “into Scotland before Mr. Briggs, purposely to be there when these two so learned persons should meet. Mr. Briggs appoints a certain day when to meet in Edinburgh; but failing thereof, the lord Napier was doubtful he would not come. It happened one day as John Marr and the lord Napier were speaking of Mr. Briggs: ‘Ah John (said Merchiston), Mr. Briggs will not now come.' At the very moment one knocks at the gate; John Marr hastens down, and it proved Mr. Briggs to his great contentment. He brings Mr. Briggs up into my lord's chamber, where almost
one quarter of an hour was spent,
each beholding other with admiration,
before one word was spoke.”

Recalling this legend Sylvester tells how he himself went after Briggs' world record for flabbergasted admiration when, in 1885, he called on the author of numerous astonishingly mature and marvellously original papers on a new branch of analysis which had been swamping the editors of mathematical journals since the early 1880's.

“I quite entered into Briggs' feelings at his interview with Napier,” Sylvester confesses, “when I recently paid a visit to Poincaré [18541912] in his airy perch in the Rue Gay-Lussac. . . . In the presence of that mighty reservoir of pent-up intellectual force my tongue at first refused its office, and it was not until I had taken some time (it
may be two or three minutes) to peruse and absorb as it were the idea of his external youthful lineaments that I found myself in a condition to speak.”

Elsewhere Sylvester records his bewilderment when, after having toiled up the three flights of narrow stairs leading to Poincaré's “airy perch,” he paused, mopping his magnificent bald head, in astonishment at beholding a mere boy, “so blond, so young,” as the author of the deluge of papers which had heralded the advent of a successor to Cauchy.

A second anecdote may give some idea of the respect in which Poincaré's work is held by those in a position to appreciate its scope. Asked by some patriotic British brass hat in the rabidly nationalistic days of the World War—when it was obligatory on all academic patriots to exalt their esthetic allies and debase their boorish enemies—who was the greatest man France had produced in modern times, Bertrand Russell answered instantly, “Poincaré.” “What!
That
man?” his uninformed interlocutor exclaimed, believing Russell meant Raymond Poincaré, President of the French Republic. “Oh,” Russell explained when he understood the other's dismay, “I was thinking of Raymond's cousin,
Henri
Poincaré.”

Poincaré was the last man to take practically all mathematics, both pure and applied, as his province. It is generally believed that it would be impossible for any human being starting today to understand comprehensively, much less do creative work of high quality in more than two of the four main divisions of mathematics—arithmetic, algebra, geometry, analysis, to say nothing of astronomy and mathematical physics. However, even in the 1880's, when Poincaré's great career opened, it was commonly thought that Gauss was the last of the mathematical universalists, so it may not prove impossible for some future Poincaré once more to cover the entire field.

As mathematics evolves it both expands and contracts, somewhat like one of Lemaître's models of the universe. At present the phase is one of explosive expansion, and it is quite impossible for any man to familiarize himself with the entire inchoate mass of mathematics that has been dumped on the world since the year 1900. But already in certain important sectors a most welcome tendency toward contraction is plainly apparent. This is so, for example, in algebra, where the wholesale introduction of postulational methods is making the
subject at once more abstract, more general, and less disconnected. Unexpected similarities—in some instances amounting to disguised identity—are being disclosed by the modern attack, and it is conceivable that the next generation of algebraists will not need to know much that is now considered valuable, as many of these particular, difficult things will have been subsumed under simpler general principles of wider scope. Something of this sort happened in classical mathematical physics when relativity put the complicated mathematics of the ether on the shelf.

Another example of this contraction in the midst of expansion is the rapidly growing use of the tensor calculus in preference to that of numerous special brands of vector analysis. Such generalizations and condensations are often hard for older men to grasp at first and frequently have a severe struggle to survive, but in the end it is usually realized that general methods are essentially simpler and easier to handle than miscellaneous collections of ingenious tricks devised for special problems. When mathematicians assert that such a thing as the tensor calculus is easy—at least in comparison with some of the algorithms that preceded it—they are not trying to appear superior or mysterious but are stating a valuable truth which any student can verify for himself. This quality of inclusive generality was a distinguishing trait of Poincaré's vast output.

If abstractness and generality have obvious advantages of the kind indicated, it is also true that they sometimes have serious drawbacks for those who must be interested in details. Of what immediate use is it to a working physicist to know that a particular differential equation occurring in his work is solvable, because some pure mathematician has proved that it is, when neither he nor the mathematician can perform the Herculean labor demanded by a numerical solution capable of application to specific problems?

To take an example from a field in which Poincaré did some of his most original work, consider a homogeneous, incompressible fluid mass held together by the gravitation of its particles and rotating about an axis. Under what conditions will the motion be stable and what will be the possible shapes of such a stably rotating fluid? Mac-Laurin, Jacobi, and others proved that certain ellipsoids will be stable; Poincaré, using more intuitive, “less arithmetical” methods than his predecessors, once thought he had determined the criteria for the
stability of a pear-shaped body. But he had made a slip. His methods were not adapted to numerical computation and later workers, including G. H. Darwin, son of the famous Charles, undeterred by the horrific jungles of algebra and arithmetic that must be cleared out of the way before a definite conclusion can be reached, undertook a decisive solution.
^I

The man interested in the evolution of binary stars is more comfortable if the findings of the mathematicians are presented to him in a form to which he can apply a calculating machine. And since Kronecker's fiat of “no construction, no existence,” some pure mathematicians themselves have been less enthusiastic than they were in Poincaré's day for existence theorems which are not constructive. Poincaré's scorn for the kind of detail that users of mathematics demand and must have before they can get on with their work was one of the most important contributory causes to his universality. Another was his extraordinarily comprehensive grasp of all the machinery of the theory of functions of a complex variable. In this he had no equal. And it may be noted that Poincaré turned his universality to magnificent use in disclosing hitherto unsuspected connections between distant branches of mathematics, for example between (continuous) groups and linear algebra.

One more characteristic of Poincaré's outlook must be recalled for completeness before we go on to his life: few mathematicians have had the breadth of philosophical vision that Poincaré had, and none is his superior in the gift of clear exposition. Probably he had always been deeply interested in the philosophical implications of science and mathematics, but it was only in 1902, when his greatness as a technical mathematician was established beyond all cavil, that he turned as a side-interest to what may be called the popular appeal of mathematics and let himself go in a sincere enthusiasm to share with nonprofessionals the meaning and human importance of his subject. Here his liking for the general in preference to the particular aided him in
telling intelligent outsiders what is of more than technical importance in mathematics without talking down to his audience. Twenty or thirty years ago workmen and shopgirls could be seen in the parks and cafés of Paris avidly reading one or other of Poincaré's popular masterpieces in its cheap print and shabby paper cover. The same works in a richer format could also be found—well thumbed and evidently read—on the tables of the professedly cultured. These books were translated into English, German, Spanish, Hungarian, Swedish, and Japanese. Poincaré spoke the universal languages of mathematics and science to all in accents which they recognized. His style, peculiarly his own, loses much by translation.

For the literary excellence of his popular writings Poincaré was accorded the highest honor a French writer can get, membership in the literary section of the Institut. It has been somewhat spitefully said by envious novelists that Poincaré achieved this distinction, unique for a man of science, because one of the functions of the (literary) Academy is the constant compilation of a definitive dictionary of the French language, and the universal Poincaré was obviously the man to help out the poets and grammarians in their struggle to tell the world what automorphic functions are. Impartial opinion, based on a study of Poincaré's writings, agrees that the mathematician deserved no less than he got.

Closely allied to his interest in the philosophy of mathematics was Poincaré's preoccupation with the psychology of mathematical creation. How do mathematicians make their discoveries? Poincaré will tell us later his own observations on this mystery in one of the most interesting narratives of personal discovery that was ever written. The upshot seems to be that mathematical discoveries more or less make themselves after a long spell of hard labor on the part of the mathematician. As in literature—according to Dante Gabriel Rossetti—“a certain amount of fundamental brainwork” is necessary before a poem can mature, so in mathematics there is no discovery without preliminary drudgery, but this is by no means the whole story. All “explanations” of creativeness that fail to provide a recipe whereby a gifted human being can create are open to suspicion. Poincaré's excursion into practical psychology, like some others in the same direction, failed to bring back the Golden Fleece, but it did at least suggest that such a thing is not wholly mythical and may some
day be found when human beings grow intelligent enough to understand their own bodies.