Is God a Mathematician?

By Mario Livio
Simon & Schuster. 320 pp. $26
Feb. 8, 2008


Chapter One

A MYSTERY

A few years ago, I was giving a talk at Cornell University. One of my PowerPoint slides read: "Is God a mathematician?" As soon as that slide appeared, I heard a student in the front row gasp: "Oh God, I hope not!"

My rhetorical question was neither a philosophical attempt to define God for my audience nor a shrewd scheme to intimidate the math phobics. Rather, I was simply presenting a mystery with which some of the most original minds have struggled for centuries - the apparent omnipresence and omnipotent powers of mathematics. These are the type of characteristics one normally associates only with a deity. As the British physicist James Jeans (1877-1946) once put it: "The universe appears to have been designed by a pure mathematician." Mathematics appears to be almost too effective in describing and explaining not only the cosmos at large, but even some of the most chaotic of human enterprises.

Whether physicists are attempting to formulate theories of the universe, stock market analysts are scratching their heads to predict the next market crash, neurobiologists are constructing models of brain function, or military intelligence statisticians are trying to optimize resource allocation, they are all using mathematics. Furthermore, even though they may be applying formalisms developed in different branches of mathematics, they are still referring to the same global, coherent mathematics. What is it that gives mathematics such incredible powers? Or, as Einstein once wondered: "How is it possible that mathematics, a product of human thought that is independent of experience [the emphasis is mine], fits so excellently the objects of physical reality?"

This sense of utter bewilderment is not new. Some of the philosophers in ancient Greece, Pythagoras and Plato in particular, were already in awe of the apparent ability of mathematics to shape and guide the universe, while existing, as it seemed, above the powers of humans to alter, direct, or influence it. The English political philosopher Thomas Hobbes (1588-1679) could not hide his admiration either. In Leviathan, Hobbes's impressive exposition of what he regarded as the foundation of society and government, he singled out geometry as the paradigm of rational argument:

Seeing then that truth consisteth in the right ordering of names in our affirmations, a man that seeketh precise truth had need to remember what every name he uses stands for, and to place it accordingly; or else he will find himself entangled in words, as a bird in lime twigs; the more he struggles, the more belimed. And therefore in geometry (which is the only science that it hath pleased God hitherto to bestow on mankind), men begin at settling the significations of their words; which settling of significations, they call definitions, and place them in the beginning of their reckoning.

Millennia of impressive mathematical research and erudite philosophical speculation have done relatively little to shed light on the enigma of the power of mathematics. If anything, the mystery has in some sense even deepened. Renowned Oxford mathematical physicist Roger Penrose, for instance, now perceives not just a single, but a triple mystery. Penrose identifies three different "worlds": the world of our conscious perceptions, the physical world, and the Platonic world of mathematical forms. The first world is the home of all of our mental images - how we perceive the faces of our children, how we enjoy a breathtaking sunset, or how we react to the horrifying images of war. This is also the world that contains love, jealousy, and prejudices, as well as our perception of music, of the smells of food, and of fear. The second world is the one we normally refer to as physical reality. Real flowers, aspirin tablets, white clouds, and jet airplanes reside in this world, as do galaxies, planets, atoms, baboon hearts, and human brains. The Platonic world of mathematical forms, which to Penrose has an actual reality comparable to that of the physical and the mental worlds, is the motherland of mathematics. This is where you will find the natural numbers 1, 2, 3, 4, ..., all the shapes and theorems of Euclidean geometry, Newton's laws of motion, string theory, catastrophe theory, and mathematical models of stock market behavior. And now, Penrose observes, come the three mysteries. First, the world of physical reality seems to obey laws that actually reside in the world of mathematical forms. This was the puzzle that left Einstein perplexed. Physics Nobel laureate Eugene Wigner (1902-95) was equally dumbfounded:

The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

Second, the perceiving minds themselves - the dwelling of our conscious perceptions - somehow managed to emerge from the physical world. How was mind literally born out of matter? Would we ever be able to formulate a theory of the workings of consciousness that would be as coherent and as convincing as, say, our current theory of electromagnetism? Finally, the circle is mysteriously closed. Those perceiving minds were miraculously able to gain access to the mathematical world by discovering or creating and articulating a treasury of abstract mathematical forms and concepts.

Penrose does not offer an explanation for any of the three mysteries. Rather, he laconically concludes: "No doubt there are not really three worlds but one, the true nature of which we do not even glimpse at present." This is a much more humble admission than the response of the schoolmaster in the play Forty Years On (written by the English author Alan Bennett) to a somewhat similar question:

Foster: I'm still a bit hazy about the Trinity, sir.

Schoolmaster: Three in one, one in three, perfectly straightforward. Any doubts about that see your maths master.

The puzzle is even more entangled than I have just indicated. There are actually two sides to the success of mathematics in explaining the world around us (a success that Wigner dubbed "the unreasonable effectiveness of mathematics"), one more astonishing than the other. First, there is an aspect one might call "active." When physicists wander through nature's labyrinth, they light their way by mathematics - the tools they use and develop, the models they construct, and the explanations they conjure are all mathematical in nature. This, on the face of it, is a miracle in itself. Newton observed a falling apple, the Moon, and tides on the beaches (I'm not even sure if he ever saw those!), not mathematical equations. Yet he was somehow able to extract from all of these natural phenomena, clear, concise, and unbelievably accurate mathematical laws of nature. Similarly, when the Scottish physicist James Clerk Maxwell (1831-79) extended the framework of classical physics to include all the electric and magnetic phenomena that were known in the 1860s, he did so by means of just four mathematical equations. Think about this for a moment. The explanation of a collection of experimental results in electromagnetism and light, which had previously taken volumes to describe, was reduced to four succinct equations. Einstein's general relativity is even more astounding - it is a perfect example of an extraordinarily precise, self-consistent mathematical theory of something as fundamental as the structure of space and time.

But there is also a "passive" side to the mysterious effectiveness of mathematics, and it is so surprising that the "active" aspect pales by comparison. Concepts and relations explored by mathematicians only for pure reasons - with absolutely no application in mind - turn out decades (or sometimes centuries) later to be the unexpected solutions to problems grounded in physical reality! How is that possible? Take for instance the somewhat amusing case of the eccentric British mathematician Godfrey Harold Hardy (1877-1947). Hardy was so proud of the fact that his work consisted of nothing but pure mathematics that he emphatically declared: "No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world." Guess what - he was wrong. One of his works was reincarnated as the Hardy-Weinberg law (named after Hardy and the German physician Wilhelm Weinberg [1862-1937]), a fundamental principle used by geneticists to study the evolution of populations. Put simply, the Hardy-Weinberg law states that if a large population is mating totally at random (and migration, mutation, and selection do not occur), then the genetic constitution remains constant from one generation to the next. Even Hardy's seemingly abstract work on number theory - the study of the properties of the natural numbers - found unexpected applications. In 1973, the British mathematician Clifford Cocks used the theory of numbers to create a breakthrough in cryptography - the development of codes. Cocks's discovery made another statement by Hardy obsolete. In his famous book A Mathematician's Apology, published in 1940, Hardy pronounced: "No one has yet discovered any war-like purpose to be served by the theory of numbers." Clearly, Hardy was yet again in error. Codes have been absolutely essential for military communications. So even Hardy, one of the most vocal critics of applied mathematics, was "dragged" (probably kicking and screaming, if he had been alive) into producing useful mathematical theories.

But this is only the tip of the iceberg. Kepler and Newton discovered that the planets in our solar system follow orbits in the shape of ellipses - the very curves studied by the Greek mathematician Menaechmus (fl. ca. 350 BC) two millennia earlier. The new types of geometries outlined by Georg Friedrich Bernhard Riemann (1826-66) in a classic lecture in 1854 turned out to be precisely the tools that Einstein needed to explain the cosmic fabric. A mathematical "language" called group theory, developed by the young prodigy Évariste Galois (1811-32) simply to determine the solvability of algebraic equations, has today become the language used by physicists, engineers, linguists, and even anthropologists to describe all the symmetries of the world. Moreover, the concept of mathematical symmetry patterns has, in some sense, turned the entire scientific process on its head. For centuries the route to understanding the workings of the cosmos started with a collection of experimental or observational facts, from which, by trial and error, scientists attempted to formulate general laws of nature. The scheme was to begin with local observations and build the jigsaw puzzle piece by piece. With the recognition in the twentieth century that well-defined mathematical designs underlie the structure of the subatomic world, modern-day physicists started to do precisely the opposite. They put the mathematical symmetry principles first, insisting that the laws of nature and indeed the basic building blocks of matter should follow certain patterns, and they deduced the general laws from these requirements. How does nature know to obey these abstract mathematical symmetries?

In 1975, Mitch Feigenbaum, then a young mathematical physicist at Los Alamos National Laboratory, was playing with his HP-65 pocket calculator. He was examining the behavior of a simple equation. He noticed that a sequence of numbers that appeared in the calculations was getting closer and closer to a particular number: 4.669 ... To his amazement, when he examined other equations, the same curious number appeared again. Feigenbaum soon concluded that his discovery represented something universal, which somehow marked the transition from order to chaos, even though he had no explanation for it. Not surprisingly, physicists were very skeptical at first. After all, why should the same number characterize the behavior of what appeared to be rather different systems? After six months of professional refereeing, Feigenbaum's first paper on the topic was rejected. Not much later, however, experiments showed that when liquid helium is heated from below it behaves precisely as predicted by Feigenbaum's universal solution. And this was not the only system found to act this way. Feigenbaum's astonishing number showed up in the transition from the orderly flow of a fluid to turbulence, and even in the behavior of water dripping from a tap.

The list of such "anticipations" by mathematicians of the needs of various disciplines of later generations just goes on and on. One of the most fascinating examples of the mysterious and unexpected interplay between mathematics and the real (physical) world is provided by the story of knot theory - the mathematical study of knots. A mathematical knot resembles an ordinary knot in a string, with the string's ends spliced together. That is, a mathematical knot is a closed curve with no loose ends. Oddly, the main impetus for the development of mathematical knot theory came from an incorrect model for the atom that was developed in the nineteenth century. Once that model was abandoned - only two decades after its conception - knot theory continued to evolve as a relatively obscure branch of pure mathematics. Amazingly, this abstract endeavor suddenly found extensive modern applications in topics ranging from the molecular structure of DNA to string theory - the attempt to unify the subatomic world with gravity. I shall return to this remarkable tale in chapter 8, because its circular history is perhaps the best demonstration of how branches of mathematics can emerge from attempts to explain physical reality, then how they wander into the abstract realm of mathematics, only to eventually return unexpectedly to their ancestral origins.

Discovered or Invented?

Even the brief description I have presented so far already provides overwhelming evidence of a universe that is either governed by mathematics or, at the very least, susceptible to analysis through mathematics. As this book will show, much, and perhaps all, of the human enterprise also seems to emerge from an underlying mathematical facility, even where least expected. Examine, for instance, an example from the world of finance - the Black-Scholes option pricing formula (1973). The Black-Scholes model won its originators (Myron Scholes and Robert Carhart Merton; Fischer Black passed away before the prize was awarded) the Nobel Memorial Prize in economics. The key equation in the model enables the understanding of stock option pricing (options are financial instruments that allow bidders to buy or sell stocks at a future point in time, at agreed-upon prices). Here, however, comes a surprising fact. At the heart of this model lies a phenomenon that had been studied by physicists for decades - Brownian motion, the state of agitated motion exhibited by tiny particles such as pollen suspended in water or smoke particles in the air. Then, as if that were not enough, the same equation also applies to the motion of hundreds of thousands of stars in star clusters. Isn't this, in the language of Alice in Wonderland, "curiouser and curiouser"? After all, whatever the cosmos may be doing, business and finance are definitely worlds created by the human mind.

Or, take a common problem encountered by electronic board manufacturers and designers of computers. They use laser drills to make tens of thousands of holes in their boards. In order to minimize the cost, the computer designers do not want their drills to behave as "accidental tourists." Rather, the problem is to find the shortest "tour" among the holes, that visits each hole position exactly once. As it turns out, mathematicians have investigated this exact problem, known as the traveling salesman problem, since the 1920s. Basically, if a salesperson or a politician on the campaign trail needs to travel in the most economical way to a given number of cities, and the cost of travel between each pair of cities is known, then the traveler must somehow figure out the cheapest way of visiting all the cities and returning to his or her starting point. The traveling salesman problem was solved for 49 cities in the United States in 1954. By 2004, it was solved for 24,978 towns in Sweden. In other words, the electronics industry, companies routing trucks for parcel pickups, and even Japanese manufacturers of pinball-like pachinko machines (which have to hammer thousands of nails) have to rely on mathematics for something as simple as drilling, scheduling, or the physical design of computers.

(Continues...)



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