Why Beauty is Truth (61 page)

What you can do with real numbers, you can also do with complex numbers, so now you get the complex projective plane. And if those work, why not try the quaternions or the octonions?

There are problems—the obvious methods don't work, because of the lack of commutativity. But in 1949, the mathematical physicist Pascual Jordan found a meaningful way to construct an octonionic projective plane with 16 real dimensions. In 1950, the group theorist Armand Borel proved that the second exceptional Lie group F
₄
is the symmetry group of the octonionic projective plane—much like the complex plane, but formed from two 8-dimensional “rulers” labeled with octonions, not real numbers.

So now there was an octonionic explanation of two of the five exceptional Lie groups. What about the other three—E
₆
, E
₇
, and E
₈
?

The view of the exceptional Lie groups as brutal acts of a malicious deity was fairly widespread until 1959, when Hans Freudenthal and Jacques Tits independently invented the “magic square,” and explained E
₆
, E
₇
, and E
₈
.

The rows and columns of the magic square correspond to the four normed division algebras. Given any two normed division algebras, you look in the corresponding row and column, and what the magic square gives you—following a technical mathematical recipe—is a Lie group. Some of these groups are straightforward; for example, the Lie group corresponding to the real row and the real column is the group SO(3) of rotations in three-dimensional space. If both row and column correspond to the quaternions, you get the group SO(12) of rotations in twelve-dimensional space, which to mathematicians is just as familiar. But if you look in the octonion row or column, the entries are the exceptional Lie groups F
₄
, E
₆
, E
₇
, and E
₈
. The missing exceptional group G
₂
is also intimately associated with the octonions—as we've already seen, it is their symmetry group.

So now the general opinion is that the exceptional Lie groups exist because of the wisdom of the deity in permitting the octonions to exist. We should have known. As Einstein remarked, the Lord is subtle but not malicious. All five exceptional Lie groups are the symmetries of various octonionic geometries.

Around 1956, the Russian geometer Boris Rosenfeld, perhaps thinking about the magic square, conjectured that the three remaining exceptional groups E
₆
, E
₇
, and E
₈
are also the symmetry groups of projective planes. In place of the octonions, however, you have to use the following structures:

 

•  For E
₆
: the
bioctonions
, built from complex numbers and octonions.

•  For E
₇
: the
quateroctonions
, built from quaternions and octonions.

•  For E
₈
: the
octooctonions
, built from octonions and octonions.

 

The only slight snag was that no one knew how to define sensible projective planes over such combinations of number systems. But there was some evidence that the idea made sense. As matters currently stand, we can now prove Rosenfeld's conjecture, but only by making use of the groups to construct the projective planes. This is not very satisfactory, because the idea was to go the other way, from the projective planes to the groups. Still, it's a start. In fact, for E
₆
and E
₇
there now exist independent ways to construct the projective planes. Only E
₈
is still holding out.

Were it not for the octonions, the Lie group story would be more straightforward, as Killing originally hoped, but nowhere near as interesting. Not that we mortals get to choose: the octonions, and all the associated paraphernalia, are
there.
And in some obscure way, the existence of the universe may depend on them.

The connection between the octonions and life, the universe, and everything emerges from string theory. The key feature is the need for extra dimensions to hold the strings. Those extra dimensions can in principle have lots of shapes, and the big question is to find the
right
shape. In old-fashioned quantum theory, a key principle is symmetry, and that's the case in string theory too. So of course Lie groups get in on the act. Everything hinges on those Lie groups of symmetries, and again the exceptional
groups stick out—not as sore thumbs, but as opportunities for unusual coincidences that could help make the physics work.

Which gets us back to the octonions.

Here's an example of their influence. In the 1980s, physicists noticed that a rather nice relationship occurs in space-times of 3, 4, 6, and 10 dimensions. Vectors (directed lengths) and spinors (algebraic gadgets originally created by Paul Dirac in his theory of electron spin) are very neatly related in these dimensions, and only these. Why? It turns out that the vector–spinor relationship holds precisely when the dimension of space-time is 2 greater than that of a normed division algebra. Subtract 2 from 3, 4, 6, and 10, and what you get is 1, 2, 4, and 8.

The mathematical point is that in 3-, 4-, 6-, and 10-dimensional string theory, every spinor can be represented using two numbers in the associated normed division algebra. This doesn't happen for any other number of dimensions, and it has a number of nice consequences for physics. So we have four candidate string theories here: real, complex, quaternionic, and octonionic. And it so happens that among these possible string theories, the one that is currently thought to have the best chance of corresponding to reality is the 10-dimensional one,
specified by the octonions.
If this 10-dimensional theory really does correspond to reality, then our universe is built from octonions.

And that's not the only place where these strange “numbers,” barely clinging to that name because they just satisfy enough of the rules of algebra, are influential. That fashionable new candidate string theory, M-theory, involves 11-dimensional space-time. In order to reduce the perceptible part of space-time from 11 dimensions to the familiar 4, we have to throw away 7 by rolling them up so tightly that they can't be detected. And how do you do that for 11-dimensional supergravity? You make use of the exceptional Lie group G
₂
, the symmetry group of the octonions.

There they are again: no longer quaint Victoriana but a hefty clue to a possible Theory of Everything. It's an octonionic world.

W
as Keats right? Is beauty truth, and truth beauty?

The two are intimately connected, possibly because our minds react similarly to both. But what works in mathematics need not work in physics, and vice versa. The relationship between mathematics and physics is deep, subtle, and puzzling. It is a philosophical conundrum of the highest order—how science has uncovered apparent “laws” in nature, and why nature seems to speak in the language of mathematics.

Is the universe genuinely mathematical? Are its apparent mathematical features mere human inventions? Or does it seem mathematical to us because mathematics is the deepest aspect of its infinitely complex nature that we are able to understand?

Mathematics is not some disembodied version of ultimate truth, as many used to think. If anything emerges from our tale, it is that mathematics is created by people. We can readily identify with their triumphs and their tribulations. Who could fail to be moved by the appalling deaths of Abel and Galois, both at the age of 21? One was deeply loved but never earned enough money to marry; the other, brilliant and unstable, fell in love but was rejected, and perhaps died because of that love. Today's medical advances would have saved Abel, and might even have helped Hamilton stay sober.

Because mathematicians are human and live ordinary human lives, the creation of new mathematics is partly a social process. But neither mathematics nor science is
wholly
the result of social processes, as social relativists often claim. Both must respect external constraints: logic, in the case of mathematics, and experiment, in the case of science. However
desperately mathematicians might want to trisect an angle by Euclidean methods, the plain fact is that it is impossible. However strongly physicists might want Newton's law of gravity to be the ultimate description of the universe, the motion of the perihelion of Mercury proves that it's not.

This is why mathematicians are so stubbornly logical, and obsessed by concerns that most people could not care less about. Does it really
matter
whether you can solve a quintic by radicals?

History's verdict on this question is unequivocal. It does matter. It may not matter directly for everyday life, but it surely matters to humanity as a whole—not because anything important rests on being able to solve quintic equations, but because understanding why we can't opens a secret doorway to a new mathematical world. If Galois and his predecessors had not been obsessed with understanding the conditions under which an equation can be solved by radicals, humanity's discovery of group theory would have been greatly delayed, and perhaps might not have happened.

You may not encounter groups in your kitchen or on your drive to work, but without them today's science would be severely curtailed, and our lives would be far different. Not so much in gadgetry like jumbo jets or GPS navigation or cell phones—though those are part of the story too—but in insight into nature. No one could have predicted that a pedantic question about equations could reveal the deep structure of the physical world, but that is what happened.

The clear message of history is a simple one. Research on deep mathematical issues should not be rejected or denigrated merely because those issues seem to have no direct practical use. Good mathematics is more valuable than gold, and where it comes from is mostly irrelevant. What counts is where it leads.

The astonishing thing is that the best mathematics usually leads somewhere unexpected, and a lot of it turns out to be vital for science and technology, even though it was originally invented for some totally different purpose. The ellipse, studied by the Greeks as a section of a cone, was the clue that led, via Kepler, from Tycho Brahe's observations of the motion of Mars to Newton's theory of gravity. Matrix theory, whose inventor Cayley apologized for its uselessness, became an essential tool in statistics, economics, and virtually every branch of science. The octonions may be the inspiration for a Theory of Everything. Of course, the theory of
superstrings may turn out to be just a pretty piece of mathematics with no relevance to physics. If so, the existing uses of symmetry in quantum theory still demonstrate that group theory provides deep insights into nature, even though it was developed to answer a question in pure mathematics.