Why Beauty is Truth (36 page)

Hamilton kept writing to her, through her relatives. By 1853 she had renewed contact, sending him a small gift. Hamilton responded by going to
see her, bearing a copy of his book on quaternions. Two weeks later, she was dead, and Hamilton was grief-stricken. His life became more and more disorderly; uneaten food was found mixed with his mathematical papers after his death, which occurred in 1865—attributed to gout, a common disease of heavy drinkers.

Hamilton believed quaternions to be the Holy Grail of algebra and physics—the true generalization of complex numbers to higher dimensions, and the key to geometry and physics in space. Of course, space has three dimensions, while quaternions have four, but Hamilton spotted a natural subsystem with three dimensions. These were the “imaginary” quaternions
bi
+
cj
+
dk
. Geometrically, the symbols
i
,
j
,
k
can be interpreted as rotations about three mutually perpendicular axes in space, although there are some subtleties: basically, you have to work in a geometry where a full circle contains 720°, not 360°. This quirk aside, you can see why Hamilton found them useful for geometry and physics.

The missing “real” quaternions behaved just like real numbers. You couldn't eliminate them altogether, because they were likely to turn up whenever you carried out algebraic calculations, even if you started with imaginary quaternions. If it had been possible to stay solely within the domain of imaginary quaternions, there would have been a sensible three-dimensional algebra, and Hamilton's quest would have succeeded. The four-dimensional system of quaternions was the next best thing, and the natural three-dimensional system embedded rather tidily inside it was just as useful as a purely three-dimensional algebra would have been.

Hamilton devoted the rest of his life to quaternions, developing their mathematics and promoting their applications to physics. A few devoted followers sung their praises. They founded a school of quaternionists, and when Hamilton died the reins were taken up by Peter Tait in Edinburgh and Benjamin Peirce at Harvard.

Others, however, disliked quaternions—partly for their artificiality, but mostly because they believed they had found something better. The most prominent of the dissenters were the Prussian Hermann Grassmann and the American Josiah Willard Gibbs, now recognized as the creators of “vector algebra.” Both of them invented useful types of algebra with any number of dimensions. In their work there was no limit to four dimensions or to the three-dimensional subset of imaginary quaternions. The algebraic
properties of these vector systems were not as elegant as Hamilton's quaternions. You couldn't divide one vector by another, for instance. But Grassmann and Gibbs preferred general concepts that worked, even if they lacked a few of the usual features of numbers. It may have been impossible to divide one vector by another, but who
cared?

Hamilton went to his grave believing that quaternions were his greatest contribution to science and mathematics. For the next hundred years, hardly anyone save Tait and Peirce would have agreed, and quaternions remained an obsolete backwater of Victorian algebra. If you wanted an example of the sterility of mathematics for its own sake, quaternions were just the ticket. Even in university courses on pure mathematics, quaternions never appeared or were shown as a curiosity. According to Bell,

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 19th Century as the discovery of fluxions was for the close of the seventeenth.” Never was a great mathematician so hopelessly wrong.

Really?

Quaternions may not have developed quite along the lines that Hamilton laid down, but their importance grows every year. They have become absolutely fundamental to mathematics, and we will see that the quaternions and their generalizations are fundamental to physics, too. Hamilton's obsession opened the door to vast tracts of modern algebra and mathematical physics.

Never was a quasi-historian so hopelessly wrong.

Hamilton may have exaggerated the applications of quaternions, and tortured them into performing tricks to which they were not really suited, but his faith in their importance is beginning to appear justified. Quaternions have developed a strange habit of turning up in the most unlikely places. One reason is that they are unique. They can be characterized by a few reasonable, relatively simple properties—a selection of the “laws of
arithmetic,” omitting only one important law—and they constitute the only mathematical system with that list of properties.

This statement requires unpacking.

The only number system that is familiar to most people on the planet is the real numbers. You can add, subtract, multiply, and divide real numbers, and your result is always a real number. Of course, division by zero is not tolerated, but aside from that necessary limitation, you can apply lengthy series of arithmetic operations without ever leaving the system of real numbers.

Mathematicians call such a system a
field.
There are many other fields, such as the rationals and the complex numbers, but the real field is special. It is the only field with two further properties: it is ordered, and it is complete.

“Ordered” means that the numbers occur in a linear order. The reals are strung out along a line, with negative numbers to the left and positive numbers to the right. There are other ordered fields, such as the rational numbers, but unlike the other ordered fields the reals are also complete. This extra property (whose full statement is somewhat technical) is the one that allows numbers like
and π to exist. Basically, the completeness property says that infinite decimals make sense.

It can be proved that the real numbers constitute the only complete ordered field. That is why they play such a central role in mathematics. They are the only context in which arithmetic, “greater than,” and basic operations of calculus can be carried out.

The complex numbers extend the real numbers by throwing in a new kind of number, the square root of minus one. But the price we pay for being able to take square roots of negative numbers is the loss of order. The complex numbers are a complete system but are spread out across a plane rather than aligned in a single orderly sequence.

The plane is two-dimensional, and two is a finite integer. The complex numbers are the only field that contains the real numbers and has finite dimension—other than the real numbers themselves, with dimension one. This implies that the complex numbers, too, are unique. For many important purposes, the complex numbers are the only gadget that can do the job. Their uniqueness makes them indispensable.

The quaternions arise when we try to extend the complex numbers, increasing the dimension (while keeping it finite) and retaining as many of the laws of algebra as possible. The laws we want to keep are all the usual properties of addition and subtraction, most of the properties of multiplication,
and the possibility of dividing by anything other than zero. The sacrifice this time is more serious; it is what caused Hamilton so much heartache. You have to abandon the commutative law of multiplication. You just have to accept that as a brutal fact, and move on. When you get used to it, you wonder why you ever expected the commutative law to hold in any case, and start to think it a minor miracle that it holds for the complex numbers.

Any system with this mix of properties, commutative or not, is called a
division algebra.

The real numbers and the complex numbers are division algebras, because we don't rule out commutativity of multiplication, we just don't demand it. Every field is a division algebra. But some division algebras are not fields, and the first to be discovered was the quaternions. In 1898, Adolf Hurwitz proved that the system of quaternions is also unique. The quaternions are the
only
finite-dimensional division algebra that contains the real numbers and is not equal either to the real numbers or the complex numbers.

There is a curious pattern here. The dimensions of the reals, complexes, and quaternions are 1, 2, and 4. This looks suspiciously like the start of a sequence, the powers of 2. A natural continuation would be 8, 16, 32, and so on.

Are there interesting algebraic systems with those dimensions?

Yes and no. But you'll have to wait to see why, because the story of symmetry now enters a new phase: connections with differential equations, the most widely used way to model the physical world, and the language in which most of the physicists' laws of nature are couched.

Again, the deepest aspects of the theory boil down to symmetry, but with a new twist. Now the symmetry groups are not finite, but “continuous.” Mathematics was about to be enriched by one of the most influential programs of research ever conducted.

M
arius Sophus Lie studied science only because his poor eyesight disqualified him from any military profession. When Sophus, as he came to be called, graduated from the University of Christiania in 1865, he had taken a few mathematics courses, including one on Galois theory given by the Norwegian Ludwig Sylow, but he showed no special talent in the subject. For a while he dithered—he knew that he wanted an academic career but was unsure whether it should be in botany, zoology, or perhaps astronomy.

The library records at the university show him taking out more and more books on mathematical topics. In 1867, in the middle of the night, he was struck by a vision of his life's work. His friend Ernst Motzfeldt was astonished to be woken from sleep by an excited Lie, who was shouting “I have found it, it is quite simple!”

What he had found was a new way to think about geometry.

Lie began to study the works of the great geometers, such as the German Julius Plücker and the Frenchman Jean-Victor Poncelet. From Plücker he got the idea of geometries whose underlying elements are not Euclid's familiar points but other objects—lines, planes, circles. He published a paper outlining his big idea in 1869, at his own expense. Like Galois and Abel before him, he discovered that his ideas were too revolutionary for the old guard, and the regular journals did not wish to publish his researches. But Ernst refused to let his friend become discouraged, and kept him working on his geometry. Eventually, one of Lie's
papers was published in a prestigious journal and was favorably received. It gained Lie a scholarship. Now he had the money to travel, visit leading mathematicians, and discuss his ideas with them. He went to the hotbeds of Prussian and German mathematics, Göttingen and Berlin, and talked to the algebraists Leopold Kronecker and Ernst Kummer and the analyst Karl Weierstrass. He was impressed by Kummer's way of doing mathematics, less so by Weierstrass's.