Men of Mathematics (69 page)

But on attempting to symbolize the behavior of vectors in three dimensional space so as to preserve those properties of vectors which are of use in physics, particularly in the combination of rotations, Hamilton was held up for years by an unforeseen difficulty whose very nature he for long did not even suspect. We may glance in passing at one of the clues he followed. That this led him anywhere—as he insisted it did—is all the more remarkable as it is now almost universally regarded as an absurdity, or at best a metaphysical speculation without foundation in history or in mathematical experience.

Objecting to the purely abstract, postulational formulation of algebra advocated by his British contemporaries, Hamilton sought to found algebra on something “more real,” and for this strictly meaningless enterprise he drew on his knowledge of Kant's mistaken notions—exploded by the creation of non-Euclidean geometry—of space
as “a pure form of sensuous intuition.” Indeed Hamilton, who seems to have been unacquainted with non-Euclidean geometry, followed Kant in believing that “Time and space are two sources of knowledge from which various
a priori
synthetical cognitions can be derived. Of this, pure mathematics gives a splendid example in the case of our cognition of space and its various relations. As they are both pure forms of sensuous intuition, they render synthetic propositions
a priori
possible.” Of course any not utterly illiterate mathematician today knows that Kant was mistaken in this conception of mathematics, but in the 1840's, when Hamilton was on his way to quaternions, the Kantian philosophy of mathematics still made sense to those—and they were nearly all—who had never heard of Lobatchewsky. By what looks like a bad mathematical pun, Hamilton applied the Kantian doctrine to algebra and drew the remarkable conclusion that, since geometry is the science of space, and since time and space are “pure sensuous forms of intuition,” therefore the rest of mathematics must belong to time, and he wasted much of his own time in elaborating the bizarre doctrine that
algebra is the science of pure time.

This queer crotchet has attracted many philosophers, and quite recently it has been exhumed and solemnly dissected by owlish metaphysicians seeking the philosopher's stone in the gall bladder of mathematics. Just because “algebra as the science of pure time” is of no earthly mathematical significance, it will continue to be discussed with animation till time itself ends. The opinion of a great mathematician on the “pure time” aspect of algebra may be of interest. “I cannot myself recognize the connection of algebra with the notion of time,” Cayley confessed; “granting that the notion of continuous progression presents itself and is of importance, I do not see that it is in anywise the fundamental notion of the science.”

Hamilton's difficulties in trying to construct an algebra of vectors and rotations for three-dimensional space were rooted in his subconscious conviction that the most important laws of common algebra must persist in the algebra he was seeking. How were vectors in three-dimensional space to be multiplied together?

To sense the difficulty of the problem it is essential to bear in mind (see Chapter on Gauss) that
ordinary complex numbers a + bi (i
=
) had been given a simple interpretation in terms of
rotations in a plane,
and further that
complex numbers obey all the rules of common algebra,
in particular the
commutative law of multiplication:
if
A, B
are
any complex numbers, then
A
×
B = B
×
A,
whether
A, B
are interpreted
algebraically, or in terms of rotations in a plane.
It was but human then to anticipate that
the same commutative law
would hold for the
generalizations of complex numbers
which represent
rotations in space of three dimensions.

Hamilton's great discovery—or invention—was an algebra, one of the “natural” algebras of rotations in space of three dimensions, in which the commutative law of multiplication does not hold. In this Hamiltonian algebra of
quaternions
(as he called his invention), a multiplication appears in which
A
×
B
is
not
equal to
B
×
A
but to
minus B × A,
that is,
A
×
B = -B
×
A.

That a consistent, practically useful system of algebra could be constructed in defiance of the commutative law of multiplication was a discovery of the first order, comparable, perhaps, to the conception of non-Euclidean geometry. Hamilton himself was so impressed by the magnitude of what suddenly dawned on his mind (after fifteen years of fruitless thought) one day (October
16, 1843)
when he was out walking with his wife that he carved the fundamental formulas of the new algebra in the stone of the bridge on which he found himself at the moment. His great invention showed algebraists the way to other algebras until today, following Hamilton's lead, mathematicians manufacture algebras practically at will by negating one or more of the postulates for a field and developing the consequences. Some of these “algebras” are extremely useful; the general theories embracing swarms of them include Hamilton's great invention as a mere detail, although a highly important one.

In line with Hamilton's quaternions the numerous brands of
vector analysis
favored by physicists of the past two generations sprang into being. Today all of these, including quaternions,
so far as physical applications are concerned,
are being swept aside by the incomparably simpler and more general
tensor analysis
which came into vogue with general relativity in
1915.
Something will be said about this later.

In the meantime it is sufficient to remark that Hamilton's deepest tragedy was neither alcohol nor marriage but his obstinate belief that quaternions held the key to the mathematics of the physical universe. History has shown that Hamilton tragically deceived himself when he insisted “. . . I still must assert that this discovery appears to me to be as important for the middle of the nineteenth century as the discovery
of fluxions [the calculus] was for the close of the seventeenth.” Never was a great mathematician so hopelessly wrong.

*  *  *

The last twenty two years of Hamilton's life were devoted almost exclusively to the elaboration of quaternions, including their application to dynamics, astronomy, and the wave theory of light, and his voluminous correspondence. The style of the overdeveloped
Elements of Quaternions,
published the year after Hamilton's death, shows plainly the effects of the author's mode of life. After his death from gout on September 2, 1865 in the sixty first year of his age, it was found that Hamilton had left behind a mass of papers in indescribable confusion and about sixty huge manuscript books full of mathematics. An adequate edition of his works is now in progress. The state of his papers testified to the domestic difficulties under which the last third of his life had been lived: innumerable dinner plates with the remains of desiccated, unviolated chops were found buried in the mountainous piles of papers, and dishes enough to supply a large household were dug out from the confusion. During his last period Hamilton lived as a recluse, ignoring the meals shoved at him as he worked, obsessed by the dream that the last tremendous effort of his magnificent genius would immortalize both himself and his beloved Ireland, and stand forever unshaken as the greatest mathematical contribution to science since the
Principia
of Newton.

His early work, on which his imperishable glory rests, he came to regard as a thing of but little moment in the shadow of what he believed was his masterpiece. To the end he was humble and devout, and wholly without anxiety for his scientific reputation. “I have very long admired Ptolemy's description of his great astronomical master, Hipparchus, as
a labor-loving and truth-loving man. Be such my epitaph.”

I
. The date on his tombstone is August 4, 1805. Actually he was born at midnight; hence the confusion in dates. Hamilton, who had a passion for accuracy in such trifles, chose August 3rd until in later life he shifted to August 4th for sentimental reasons.

Against stupidity the gods themselves fight unvictorious.
—S
CHILLER

A
BEL WAS DONE TO DEATH
by poverty, Galois by stupidity. In all the history of science there is no completer example of the triumph of crass stupidity over untamable genius than is afforded by the all too brief life of Évariste Galois. The record of his misfortunes might well stand as a sinister monument to all self-assured pedagogues, unscrupulous politicians, and conceited academicians. Galois was no “ineffectual angel,” but even his magnificent powers were shattered before the massed stupidity aligned against him, and he beat his life out fighting one unconquerable fool after another.

The first eleven years of Galois' life were happy. His parents lived in the little village of Bourg-la-Reine, just outside Paris, where Évariste was born on October 25, 1811. Nicolas-Gabriel Galois, the father of Évariste, was a relic of the eighteenth century, cultivated, intellectual, saturated with philosophy, a passionate hater of royalty and an ardent lover of liberty. During the Hundred Days after Napoleon's escape from Elba, Galois was elected mayor of the village. After Waterloo he retained his office and served faithfully under the King, backing the villagers against the priest and delighting social gatherings with the old-fashioned rhymes which he composed himself. These harmless activities were later to prove the amiable man's undoing. From his father, Évariste acquired the trick of rhyming and a hatred of tyranny and baseness.

Until the age of twelve Galois had no teacher but his mother, Adélaïde-Marie Demante. Several of the traits of Galois' character were inherited from his mother, who came from a long line of distinguished jurists. Her father appears to have been somewhat of a Tartar. He gave his daughter a thorough classical and religious education, which she in turn passed on to her eldest son, not as she had
received it, but fused into a virile stoicism in her own independent mind. She had not rejected Christianity, nor had she accepted it without question; she had merely contrasted its teachings with those of Seneca and Cicero, reducing all to their basic morality. Her friends remembered her as a woman of strong character with a mind of her own, generous, with a marked vein of originality, quizzical, and, at times, inclined to be paradoxical. She died in 1872 at the age of eighty four. To the last she retained the full vigor of her mind. She, like her husband, hated tyranny.

There is no record of mathematical talent on either side of Galois' family. His own mathematical genius came on him like an explosion, probably at early adolescence. As a child he was affectionate and rather serious, although he entered readily enough into the gaiety of the recurrent celebrations in his father's honor, even composing rhymes and dialogues to entertain the guests. All this changed under the first stings of petty persecution and stupid misunderstanding, not by his parents, but by his teachers.