Why Beauty is Truth Page 18

  Might there be an undiscovered extension of the number system to three dimensions? Hamilton’s formalization of complex numbers as pairs of real numbers suggested a way to approach this proposal: try to set up a number system based on triples (x, y, z). The problem was that no one had worked out an algebra of triples. Hamilton decided to try.

  Adding triples was easy: you could take a hint from complex numbers and just add corresponding coordinates. This kind of arithmetic, known today as “vector addition,” obeys very pleasant rules, and there is only one sensible way to do it.

  The bugbear was multiplication. Even for the complex numbers, multiplication does not work like addition. You do not multiply two pairs of real numbers by separately multiplying their first and second components. If you do, a lot of pleasant things happen—but two fatally unpleasant ones happen as well.

  The first is that there is no longer a square root of minus one.

  The second is that you can multiply two nonzero numbers together and get zero. Such “divisors of zero” play merry hell with all of the usual algebraic methods, such as ways to solve equations.

  For the complex numbers we can overcome this obstacle by choosing a less obvious rule for multiplication, which is what Hamilton did. But when he tried similar tricks on triples of numbers, he got a horrible shock. Try as he might, he could not avoid some fatal defect. He could get a square root for minus one, but only by introducing divisors of zero. Getting rid of divisors of zero seemed to be completely impossible, whatever else he did.

  If you’re thinking that this sounds a bit like attempts to solve the quintic, you’re onto something. If many capable mathematicians try something and fail, it is conceivable that it may be impossible. If there is one big thing mathematics has taught us, it is that many problems do not have solutions. You can’t find a fraction whose square is 2. You can’t trisect an angle with straightedge and compass. You can’t solve the quintic by radicals. Mathematics has limits. Maybe you can’t construct a three-dimensional algebra with all the nice properties you would like it to have.

  If you’re serious about finding out whether that is indeed the case, a program of research opens up. First you need to specify the properties you want your three-dimensional algebra to have. Then you must analyze the consequences of those properties. Given enough information from that program, you can search for features that such an algebra must have if it does exist, and reasons why it might not exist.

  At least, that’s what you would do today. Hamilton’s approach was not so systematic. He tacitly assumed that his algebra must have “all” reasonable properties, and suddenly realized that one of them might have to be dispensed with. More significantly, he realized that an algebra of three dimensions was not in the cards. The closest he could get was four. Quadruples, not triples.

  Back to those elusive rules of algebra. When mathematicians do algebraic calculations, they rearrange symbols in systematic ways. Recall that the original Arabic name “al-jabr” means “restoration”—what nowadays we call “move the term to the other side of the equation and change its sign.” Only within the last 150 years have mathematicians bothered to make explicit lists of the rules behind such manipulations, deriving other well-known rules as logical consequences. This axiomatic approach does for algebra what Euclid did for geometry, and it took mathematicians only two thousand years to get the idea.

  To set the scene, we can focus on three of these rules, all related to multiplication. (Addition is similar but more straightforward; multiplication is where everything starts to go pear-shaped.) Children learning their multiplication tables eventually notice some duplication of effort. Not only does three times four make twelve: so does four times three. If you multiply two numbers together you get the same result whichever one comes first. This fact is called the commutative law, and in symbols it tells us that ab = ba for any numbers a and b. This rule also holds in the extended system of complex numbers. You can prove this by examining Hamilton’s formulas for how to multiply pairs.

  A subtler law is the associative law, which says that when you multiply three numbers together in the same order, it makes no difference where you start. For example, suppose I want to work out 2 × 3 × 5. I can start with 2 × 3, getting 6, and then multiply 6 by 5. Alternatively, I can start with 3 × 5, which is 15, and then multiply 2 by 15. Either method yields the same result, namely 30. The associative law states that this is always the case; in symbols it says that (ab) c = a (bc), where the parentheses show the two ways to do the multiplication. Again, this rule holds for both real and complex numbers, and this can be proved using Hamilton’s formulas.

  A final, very useful rule—let me call it the division law, although you will find it in the textbooks as “existence of a multiplicative inverse”—states that you can always divide any number whatsoever by any nonzero number. There are good reasons to forbid division by zero: the main one is that it seldom makes sense.

  We saw earlier that you can manufacture an algebra of triples using an “obvious” form of multiplication. This system satisfies the commutative law and the associative law. But it fails to obey the division law.

  Hamilton’s great inspiration, reached after hours of fruitless searching and calculation, was this: it is possible to form a new number system in which both the associative law and the division law are valid, but you have to sacrifice the commutative law. Even then, you can’t do it with triples of real numbers. You have to use quadruples. There is no “sensible” three-dimensional algebra, but there is a fairly nice four-dimensional one. It is the only one of its kind, and it falls short of the ideal in just one respect: the commutative law fails.

  Does that matter? Hamilton’s biggest mental block was in thinking that the commutative law was essential. All that changed in an instant when, inspired by who knows what, he suddenly understood how to multiply quadruples. The date was 16 October 1843. Hamilton and his wife were walking along the towpath of the Royal Canal, heading for a meeting of the prestigious Royal Irish Academy in Dublin. His subconscious mind must have been churning away at the problem of three-dimensional algebra, because inspiration suddenly struck. “I then and there felt the galvanic circuit of thought close,” he wrote in a subsequent letter, “and the sparks which fell from it were the fundamental equations between i, j, k; exactly such as I have used them ever since.”

  So overcome was Hamilton that he immediately carved the formulas into the stonework of Broome Bridge (he called it “Brougham”). The bridge survives, but not the carving—though there is a commemorative plaque. The formulas also survive:

  i2 = j2 = k2 = ijk = –1

  These are very pretty formulas, with a lot of symmetry. But what you are probably wondering is, where are the quadruples?

  Complex numbers can be written as pairs (x, y), but they are usually written as x + iy where i = . In the same manner, the numbers Hamilton had in mind could be written either as quadruples (x, y, z, w) or as a combination x + iy + jz + kw. Hamilton’s formulas use the second notation; if you are of a formal turn of mind, you may prefer to use quadruples instead.

  Hamilton called his new numbers quaternions. He proved that they obey the associative law and—remarkably, as it later transpired—the division law. But not the commutative law. The rules for multiplying quaternions imply that ij = k, but ji = –k.

  The system of quaternions contains a copy of the complex numbers, the quaternions of the form x + iy. Hamilton’s formulas show that –1 does not have just two square roots, i and –i. It also has j, –j, k, and –k. In fact there are infinitely many different square roots of minus one in the quaternion system.

  So along with the commutative law, we have also lost the rule that a quadratic equation has two solutions. Fortunately, by the time quaternions were invented, the focus of algebra had shifted away from the solution of equations. The advantages of quaternions greatly outweighed their defects. You just had to get used to them.

  In 1845, Thomas Disney visited Hamilt
on and brought his daughter, William’s childhood love Catherine, with him. By then she had lost her first husband and married again. The encounter reopened old wounds, and Hamilton’s reliance on alcohol became more severe. He made such a complete fool of himself at a scientific dinner in Dublin that he went on the wagon and drank only water for the next two years. But when the astronomer George Airy began taunting him for his abstinence, Hamilton responded by downing alcohol in large quantities. From then on he was a chronic alcoholic.

  Two uncles died, and a friend and colleague committed suicide; then Catherine started writing to him, which made his depression worse. She quickly realized that what she was doing was not proper for a respectable married woman, and made a half-hearted attempt to kill herself. She separated from her husband and went to live with her mother.

  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 la
w 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.

  10

  THE WOULD-BE SOLDIER AND THE WEAKLY BOOKWORM

  Marius 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.