by E. T. Bell
The British school changed all this, although they were unable to take the final step and prove that their postulates for common algebra will never lead to a contradiction. That step was taken only in our own generation by the German workers in the foundations of mathematics. In this connection it must be kept in mind that algebra deals only with finite processes; when infinite processes enter, as for example in summing an infinite series, we are thrust out of algebra into another domain. This is emphasized because the usual elementary text labelled “Algebra” contains a great deal—infinite geometric progressions, for instance—that is not algebra in the modern meaning of the word.
The nature of what Hamilton did in his creation of quaternions will show up more clearly against the background of a set of postulates (taken from L. E. Dickson’s Algebras and Their Arithmetics, Chicago, 1923) for common algebra or, as it is technically called, a field (English writers sometimes use corpus as the equivalent of the German Körper or French corps).
“A field F is a system consisting of a set S of elements a, b, c, . . . and two operations, called addition and multiplication which may be performed upon any two (equal or distinct) elements a and b of S, taken in that order, to produce uniquely determined elements a ⊕ b and a ⊙ b of S, such that postulates I-V are satisfied. For simplicity we shall write a + b for a ⊕ b, and ab for a ⊙ b, and call them the sum and product, respectively, of a and b. Moreover, elements of S will be called elements of F.
“I. If a and b are any two elements of F, a + b and ab are uniquely determined elements of F, and
b + a = a + b, ba = ab.
“II. If a, b, c are any three elements of F,
(a + b) + c = a + (b + c), (ab)c = a(bc), a(b + c) = ab + ac.
“III. There exist in F two distinct elements, denoted by 0, 1, such that if a is any element of F, a + 0 = a, a1 = a (whence 0 + a = a, 1a = a, by I).
“IV. Whatever be the element a of F, there exists in Fan element x such that a + x = 0 (whence x + a = 0 by I).
“V. Whatever be the element a (distinct from 0) of F, there exists in F an element y such that ay = 1 (whence = ya, by I).”
From these simple postulates the whole of common algebra follows. A word or two about some of the statements may be helpful to those who have not seen algebra for years. In II, the statement (a + b) + c = a + (b + c), called the associative law of addition, says that if a and b are added, and to this sum is added c, the result is the same as if a and the sum of b and c are added. Similarly, with respect to multiplication, for the second statement in II. The third statement in II is called the distributive law. In III a “zero” and “unity” are postulated; in IV, the postulated x gives the negative of a; and the first parenthetical remark in V forbids “division by zero.” The demands in Postulate I are called the commutative laws of addition and multiplication respectively.
Such a set of postulates may be regarded as a distillation of experience. Centuries of working with numbers and getting useful results according to the rules of arithmetic—empirically arrived at—suggested most of the rules embodied in these precise postulates, but once the suggestions of experience are understood, the interpretation (here common arithmetic) furnished by experience is deliberately suppressed or forgotten, and the system defined by the postulates is developed abstractly, on its own merits, by common logic plus mathematical tact.
Notice in particular IV, which postulates the existence of negatives. We do not attempt to deduce the existence of negatives from the behavior of positives. When negative numbers first appeared in experience, as in debits instead of credits, they, as numbers, were held in the same abhorrence as “unnatural” monstrosities as were later the “imaginary” numbers etc., arising from the formal solution of equations such as x2 + 1 = 0, x2 + 2 = 0, etc. If the reader will glance back at what Gauss did for complex numbers he will appreciate more fully the complete simplicity of the following partial statement of Hamilton’s original way of stripping “imaginaries” of their silly, purely imaginary mystery. This simple thing was one of the steps which led Hamilton to his quaternions, although strictly it has nothing to do with them. It is the method and the point of view behind this ingenious recasting of the algebra of complex numbers which are of importance for the sequel.
If as usual i denotes , a “complex number” is a number of the type a + bi, where a, b are “real numbers” or, if preferred, and more generally, elements of the field F defined by the above postulates. Instead of regarding a + bi as one “number,” Hamilton conceived it as an ordered couple of “numbers,” and he designated this couple by writing it (a, b). He then proceeded to impose definitions of sum and product on these couples, as suggested by the formal rules of combination sublimated from the experience of algebraists in manipulating complex numbers as if the laws of common algebra did in fact hold for them. One advantage of this new way of approaching complex numbers was this: the definitions for sum and product of couples were seen to be instances of the general, abstract definitions of sum and product as in a field. Hence, if the consistency of the system defined by the postulates for a field is proved, the like follows, without further proof, for complex numbers and the usual rules by which they are combined. It will be sufficient to state the definitions of sum and product in Hamilton’s theory of complex numbers considered as couples (a, b) (c, d), etc.
The sum of (a, b) and (c, d) is (a + b, c + d); their product is (ac – bd, ad + be). In the last, the minus sign is as in a field; namely, the element x postulated in IV is denoted by – a. To the 0, 1 of a field correspond here the couples (0, 0), (1, 0). With these definitions it is easily verified that Hamilton’s couples satisfy all the stated postulates for a field. But they also accord with the formal rules for manipulating complex numbers. Thus, to (a, b), (c, d) -correspond respectively a + bi, c + di, and the formal “sum” of these two is (a + c) + i(b + d), to which corresponds the couple (a + c, b + d). Again, formal multiplication of a + bi, c + id gives (ac – bd) + i(ad + be), to which corresponds the couple (ac – bd, ad + be). If this sort of thing is new to any reader, it will repay a second inspection, as it is an example of the way in which modern mathematics eliminates mystery. So long as there is a shred of mystery attached to any concept that concept is not mathematical.
Having disposed of complex numbers by couples, Hamilton sought to extend his device to ordered triples and quadruples. Without some idea of what is sought to be accomplished such an undertaking is of course so vague as to be meaningless. Hamilton’s object was to invent an algebra which would do for rotations in space of three dimensions what complex numbers, or his couples, do for rotations in space of two dimensions, both spaces being Euclidean as in elementary geometry. Now, a complex number a + bi can be thought of as representing a vector, that is, a line segment having both length and direction, as is evident from the diagram, in which the directed segment (indicated by the arrow) represents the vector OP.
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.
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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.
CHAPTER TWENTY
Genius and Stupidity
GALOIS
Against stupidity the gods themselves fight unvictorious.—SCHILLER
ABEL 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.
