Is God a Mathematician?

Preface

1.
A Mystery

2.
Mystics: The Numerologist and the Philosopher

3.
Magicians: The Master and the Heretic

4.
Magicians: The Skeptic and the Giant

5.
Statisticians and Probabilists: The Science of Uncertainty

6.
Geometers: Future Shock

7.
Logicians: Thinking About Reasoning

8.
Unreasonable Effectiveness?

9.
On the Human Mind, Mathematics, and the Universe

Notes

Bibliography

Credits

When you work in cosmology—the study of the cosmos at large—one of the facts of life becomes the weekly letter, e-mail, or fax from someone who wants to describe to you
his
own theory of the universe (yes, they are invariably men). The biggest mistake you can make is to politely answer that you would like to learn more. This immediately results in an endless barrage of messages. So how can you prevent the assault? A particular tactic that I found to be quite effective (short of the impolite act of not answering at all) is to point out the true fact that as long as the theory is not precisely formulated in the language of mathematics, it is impossible to assess its relevance. This response stops most amateur cosmologists in their tracks. The reality is that without mathematics, modern-day cosmologists could not have progressed even one step in attempting to understand the laws of nature. Mathematics provides the solid scaffolding that holds together any theory of the universe. This may not sound so surprising until you realize that the nature of mathematics itself is not entirely clear. As the British philosopher Sir Michael Dummett once put it: “The two most abstract of the intellectual disciplines, philosophy and mathematics, give rise to the same perplexity: what are they
about
? The perplexity does not arise solely out of ignorance: even the practitioners of these subjects may find it difficult to answer the question.”

In this book I humbly try to clarify both some aspects of the essence of mathematics and, in particular, the nature of the relation between mathematics and the world we observe. The book is definitely not meant to represent a comprehensive history of mathematics. Rather, I chronologically follow the evolution of some concepts that have direct implications for understanding the role of mathematics in our grasp of the cosmos.

Many people have contributed, directly and indirectly, over a long
period of time, to the ideas presented in this book. I would like to thank Sir Michael Atiyah, Gia Dvali, Freeman Dyson, Hillel Gauchman, David Gross, Sir Roger Penrose, Lord Martin Rees, Raman Sundrum, Max Tegmark, Steven Weinberg, and Stephen Wolfram for very helpful exchanges. I am indebted to Dorothy Morgenstern Thomas for allowing me to use the complete text of Oscar Morgenstern’s account of Kurt Gödel’s experience with the U.S. Immigration and Naturalization Service. William Christens-Barry, Keith Knox, Roger Easton, and in particular Will Noel were kind enough to give me detailed explanations of their efforts to decipher the Archimedes Palimpsest. Special thanks are due to Laura Garbolino for providing me with crucial materials and rare files regarding the history of mathematics. I also thank the special collections departments of the Johns Hopkins University, the University of Chicago, and the Bibliothèque nationale de France, Paris, for finding some rare manuscripts for me.

I am grateful to Stefano Casertano for his help with difficult translations from Latin, and to Elizabeth Fraser and Jill Lagerstrom for their invaluable bibliographic and linguistic support (always with a smile).

Special thanks are due to Sharon Toolan for her professional help in the preparation of the manuscript for print, and to Ann Feild, Krista Wildt, and Stacey Benn for drawing some of the figures.

Every author should consider herself or himself fortunate to receive from their spouse the type of continuous support and patience that I have received from my wife, Sofie, during the long period of the writing of this book.

Finally, I would like to thank my agent, Susan Rabiner, without whose encouragement this book would have never happened. I am also deeply indebted to my editor, Bob Bender, for his careful reading of the manuscript and his insightful comments, to Johanna Li for her invaluable support with the production of the book, to Loretta Denner and Amy Ryan for copyediting, to Victoria Meyer and Katie Grinch for promoting the book, and to the entire production and marketing team at Simon & Schuster for all their hard work.

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:

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:

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:

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 math
ematical 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.