the universe
in zero words
DANA MACKENZIE
the universe in zero words
THE STORY OF MATHEMATICS AS TOLD THROUGH EQUATIONS
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright © 2012 by Elwin Street Productions
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Published in 2012 in the United States, Canada, the United Kingdom, and Europe by Princeton University Press
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Library of Congress Control Number 2011936364
ISBN 978-0-691-15282-0
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Printed in China
10 9 8 7 6 5 4 3 2 1
contents
preface
6
introduction: the abacist versus the algorist
10
part one: equations of antiquity
16
1. Why we believe in arithmetic: the world’s simplest equation
20
2. Resisting a new concept: the discovery of zero
26
3. The square of the hypotenuse: the Pythagorean theorem
30
4. The circle game: the discovery of π
40
5. From Zeno’s paradoxes to the idea of infinity
46
6. A matter of leverage: laws of levers
52
part two: equations in the age of exploration
56
7. The stammerer’s secret: Cardano’s formula
60
8. Order in the heavens: Kepler’s laws of planetary motion
68
9. Writing for eternity: Fermat’s Last Theorem
74
10. An unexplored continent: the fundamental theorem of calculus
80
11. Of apples, legends … and comets: Newton’s laws
90
12. The great explorer: Euler’s theorems
96
part three: equations in a promethean age
104
13. The new algebra: Hamilton and quaternions
108
14. Two shooting stars: group theory
114
15. The geometry of whales and ants: non-Euclidean geometry
122
16. In primes we trust: the prime number theorem
128
17. The idea of spectra: Fourier series
134
18. A god’s-eye view of light: Maxwell’s equations
142
part four: equations in our own time
150
19. The photoelectric effect: quanta and relativity
154
20. From a bad cigar to Westminster Abbey: Dirac’s formula
164
21. The empire-builder: the Chern-Gauss-Bonnet equation
174
22. A little bit infinite: the Continuum Hypothesis
182
23. Theories of chaos: Lorenz equations
194
24. Taming the tiger: the Black-Scholes equation
204
conclusion: what of the future?
214
acknowledgments
218
bibliography
219
index
222
preface
In this book I hope to lift the veil of mystery and secrecy that surrounds mathematics and equations, so that those who are interested can see what lies underneath.
Firstly, let me briefly address some questions of terminology. The words “equation,” “formula,” and “identity” are all used in mathematics, and have slightly different shades of meaning. “Formulas” tend to be a little bit more utilitarian; you use a formula to solve an equation. “Identities” are somewhat less deep and have the connotation of something that can be proved purely by symbolic manipulation. For the purposes of this book, though, I am not going to insist on any such distinctions.
You will also frequently encounter the words “axiom,” “theorem,” “hypothesis,” and “conjecture,” in this book. An axiom is a statement that mathematicians assume as an unproven fact. They may do so because they genuinely believe it is a universal truth, or they may do it just as a convenient starting point.
A theorem is the gold standard of mathematical truth; it is a statement that has been formally deduced from a specific axiom system. It is not subject to experimental error or intellectual fashion … except for the fact that the axiom system itself may go out of fashion. Revolutions do occur in mathematics. Usually they occur not because theorems are incorrect but because the assumptions they are based on are judged to be too restrictive, too loose, too imprecise, or not close enough to reality.
A hypothesis or conjecture (the words are synonymous) is a mathematical statement that has not been proved yet, but has substantial evidence in its favor. The evidence may come from similar but weaker theorems, empirical observations or computer experiments. Nevertheless, in mathematics a fact can never be proven by empirical evidence, plausibility, or a statistical test. This is a rule that distinguishes mathematics from the empirical sciences including physics, biology, and chemistry.
THE CHOICE OF EQUATIONS was necessarily a matter of individual taste and preference. Some equations are almost obligatory, such as Einstein’s equation E = mc2, probably the most famous equation of all. Other equations will be unfamiliar to all but the most savvy readers, such as the Continuum Hypothesis. Here are some of the criteria I have used to decide what makes an equation great.
1. It is surprising. A great equation tells us something that we did not know before. It may look like a work of alchemy, transforming one quantity into another one that at first seems completely different, yet every step can be explained and justified. The only magic is in the human mind that can discover such connections.
2. It is concise. A great equation has the spare aesthetic of Japanese calligraphy; it contains nothing but the essentials. It says something simple and powerful.
3. It is consequential. I discarded several equations that I consider to be beautiful, inspiring mathematics—but which in the end have significance only for a few cognoscenti. The equations that make the deepest impression are the ones that revolutionize mathematics, change our view of the world, or change the material possibilities of our lives.
4. It is universal. One of the great attractions of math is that an equation proven today will remain true forever. It is not subject to the whims of fashion, it is the same across the globe, and it cannot be censored or legislated.
Some of the equations presented here are not mathematical theorems, but physical “laws” or theories, for example, Maxwell’s equations. Physical theories are generally confirmed by induction from data, or the “scientific method,” rather than by deduction from a set of axioms. Unlike mathematical theorems, they are subject to empirical evidence and statistical testing, and occasionally, when
more sensitive experiments come along, they are proved wanting.
The fact is that mathematics has two faces. First, it is a body of knowledge in its own right; and second, it is a language for expressing knowledge about the universe. If you look at equations merely as a means of conveying scientific information, then you are missing the way that mathematics can unbind our mental straitjackets. If you look at equations only as abstract nuggets of wisdom, then you are missing the subtle guidance nature gives us to ask the “right” questions.
LEOPOLD KRONECKER, a nineteenth-century German mathematician, once said “God created the integers; all else is the work of man”. Although it is not entirely clear how literally one should take his witticism, historically he is far from alone in suggesting a divine origin for mathematics. In ancient Mesopotamia, it was a gift from Nisaba, the patron goddess of scribes. “Nisaba, the woman radiant with joy, the true woman, the scribe, the lady who knows everything, guides your fingers on the clay,” wrote a scribe in the twentieth century BC. “Nisaba generously bestowed upon you the measuring rod, the surveyor’s gleaming line, the yardstick, and the tablets which confer wisdom.” On Babylonian mathematical tablets, the solution to a problem was never complete until the solver wrote, “Praise Nisaba!” at the end.
According to the ancient Chinese, the originator of mathematics was Fu Xi, the legendary first emperor of China. He is often depicted holding a carpenter’s square. “Fu Xi created the eight trigrams in remote antiquity to communicate the virtues of the gods,” wrote the third-century mathematician Liu Hui. In addition, he says, Fu Xi “invented the nine-nines algorithm to coordinate the variations in the hexagrams.” The “trigrams” and “hexagrams” are the basic units of Chinese calligraphy; thus, in a loose sense, Fu Xi is being credited with the invention of writing, while, the “nine-nines algorithm” means the multiplication table. Thus, mathematics was not only divinely inspired, but was invented at the same time as written language.
We can already discern in these accounts three distinct branches of mathematics, which have continued to flow abundantly over the centuries since then. The first branch is arithmetic or algebra, the science of quantity; the second is geometry, the science of shape; and the third is applied mathematics, the science of translating mathematics into solutions to concrete problems of engineering, physics, and economics.
A fourth wellspring is not apparent in the above quotes, and that is the science of the infinite—the analysis of both infinitely large and infinitely small quantities, which are essential to understand any process of continuous motion or change. Mathematicians simply call this branch of mathematics “analysis,” even though the rest of the world interprets this word to mean something quite different.
Thus, I consider the four main tributaries of mathematics to be Algebra, Geometry, Applied Mathematics, and Analysis. All four of them mingle together and cooperate in a most wonderful way, and witnessing this interaction is one of the great joys of being a mathematician. Nearly every mathematician finds himself or herself drawn more to one of these tributaries than the others, but the beauty and power of the subject undoubtedly derives from all four. For that reason, the four chapters in this book each have a theme, or “storyline” running throughout, relating to the evolution of the four branches over the ages.
introduction
the abacist versus the algorist
One afternoon in Rio de Janeiro, the Nobel Prize-winning physicist Richard Feynman was eating dinner in his favorite restaurant. It wasn’t actually dinnertime yet, so the dining room was quiet … until the abacus salesman walked in. The waiters, who were presumably not interested in buying an abacus, challenged the salesman to prove that he could do arithmetic faster than their customer. Feynman agreed to the challenge.
At first, the contest wasn’t even close. On the addition problems, Feynman wrote, the abacus salesman “beat me hollow.” He would have the answer before Feynman even finished writing down the numbers. But then the salesman started getting cocky. He challenged Feynman to multiplication problems. Feynman still lost to the abacus, but not by as much. The salesman, not satisfied with his narrow margin of victory, challenged Feynman to harder and harder problems, and got more and more flustered. Finally he played his trump card. “Raios cubicos!” the salesman said. “Cube roots!”
Obviously, by this point the competition was more about pride than about selling an abacus. It’s difficult to imagine why a restaurant manager would ever need to compute a cube root. But Feynman agreed, provided that the waiters, who were watching the competition and enjoying it immensely, would choose the number. The number they picked was 1729.03.
The abacist set to work with a passion, hunching over the abacus, his fingers flying too fast for the eye to follow. Meanwhile, Feynman writes, he was just sitting there. The waiters asked him what he was doing, and he tapped his head: “Thinking!” Within a few seconds, Feynman had written down five digits of the answer (12.002). After a while, the abacus salesman triumphantly announced “12!” and then a few minutes later, “12.0!” By this time Feynman had added several more digits to his answer. The waiters laughed at the salesman, who left in humiliation, beaten by the power of pure thought.
Like all good tales, Feynman’s duel with the abacist has many layers of meaning. On the most superficial level, it is a story about genius; the Nobel Prize winner beating the machine. However, Feynman’s intention when he told this story about himself was quite different. He was not a boastful man. In the context of his book, the point of the story was that ordinary people—not Nobel Prize winners, not geniuses—could do just the same thing as he did, with a little bit of number sense and mathematical knowledge. There were two secrets behind his seemingly magical feat. First, he needed to know that 1728 was a perfect cube: 123 = 1728 (not common knowledge, perhaps, but it’s something most physicists would be aware of, because a cubic foot is 123 or 1728 cubic inches.) And he needed to know a famous equation from calculus, called Taylor’s formula—a very general approximation method that allows you to go from the exact equation:
Equations are the lifeblood of mathematics and science. They are the brush strokes that mathematicians use to create their art, or the secret code that they use to express their ideas about the universe. That is not to say that equations are the only tool that mathematicians use; words and diagrams are important, too. Nevertheless, when push comes to shove—for instance, when they have to compute the cube root of 1729.03—equations convey information with an economy and precision that words or abaci can never match.
The rest of the world, outside of science, does not speak the language of equations, and thus a vast cultural gap has emerged between those who understand them and those who do not. This book is an attempt to build a bridge across that chasm. It is intended for the reader who would like to understand mathematics on its own terms, and who would like to appreciate mathematics as an art. Surely we would not attempt to discuss the works of Rembrandt or Van Gogh without actually looking at their paintings. Why, then, should we talk about Isaac Newton or Albert Einstein without exhibiting their “paintings”? The following chapters will try to explain in words—even if words are feeble and inaccurate—what these equations mean and why they are justly treasured by those who know them.
Let’s go back now to Richard Feynman and that abacus salesman, because there is more to say about them. In all likelihood, neither of them knew that they were playing out a scene that had already been enacted centuries before, when Arabic numerals first arrived in Europe.
When the new number system appeared around the beginning of the thirteenth century, many people were deeply suspicious of it. They had to learn nine new and unfamiliar symbols: 1, 2, 3, 4, 5, 6, 7, 8, and 9—or, to be more precise, they had to learn the somewhat distorted thirteenth-century versions thereof. The new symbols looked to some people like occult runes, instead of the nice solid Roman letters (I, V, X, etc.) they were accustomed to. To make things worse, they were Arabic—not even Chris
tian—which made them appear even more suspicious to a deeply religious society. And finally, they included an innovation that was especially hard to grasp: the number zero, a something that meant nothing.
Nevertheless, Arabic numbers had an undeniable power. Unlike Roman numerals, which were useful for writing numbers but impractical for calculating with them, the decimal place-value system made it possible to do both. In a sense, Arabic numbers democratized mathematics. In many ancient societies, only a specially trained class of scribes could do arithmetic. With decimal notation, you did not need special training or special tools, only your brain and a pen.
The struggle between the old and new number systems went on for a very long time—well over two centuries. And, in fact, open competitions were held between abacists (people who used mechanical tools to do arithmetic) and algorists (people who used the new algorithmic methods). So Feynman and the abacus salesman were re-fighting a very old duel!
WE KNOW HOW the battle ended. Nowadays, everyone in Western society uses decimal numbers. Grade school students learn the algorithms for adding, subtracting, multiplying, and dividing. So clearly, the algorists won. But Feynman’s story shows that the reasons may not be as simple as you think. On some problems, the abacists were undoubtedly faster. Remember that the abacus salesman “beat him hollow” at addition. But the decimal system provides a deeper insight into numbers than a mechanical device does. So the harder the problem, the better the algorist will perform. As science progressed during the Renaissance, mathematicians would need to perform even more sophisticated calculations than cube roots. Thus, the algorists won for two reasons: at the high end, the decimal system was more compatible with advanced mathematics; while at the low end, the decimal system empowered everyone to do arithmetic.
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