Is God a Mathematician? Read online

  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.

  Mathematics has even penetrated into areas not traditionally associated with the exact sciences. For instance, there is a Journal of Mathematical Sociology (which in 2006 was in its thirtieth volume) that is oriented toward a mathematical understanding of complex social structures, organizations, and informal groups. The journal articles address topics ranging from a mathematical model for predicting public opinion to one predicting interaction in social groups.

  Going in the other direction—from mathematics into the humanities—the field of computational linguistics, which originally involved only computer scientists, has now become an interdisciplinary research effort that brings together linguists, cognitive psychologists, logicians, and artificial intelligence experts, to study the intricacies of languages that have evolved naturally.

  Is this some mischievous trick played on us, such that all the human struggles to grasp and comprehend ultimately lead to uncovering the more and more subtle fields of mathematics upon which the universe and we, its complex creatures, were all created? Is mathematics, as educators like to say, the hidden textbook—the one the professor teaches from—while giving his or her students a much lesser version so that he or she will seem all the wiser? Or, to use the biblical metaphor, is mathematics in some sense the ultimate fruit of the tree of knowledge?

  As I noted briefly at the beginning of this chapter, the unreasonable effectiveness of mathematics creates many intriguing puzzles: Does mathematics have an existence that is entirely independent of the human mind? In other words, are we merely discovering mathematical verities, just as astronomers discover previously unknown galaxies? Or, is mathematics nothing but a human invention? If mathematics indeed exists in some abstract fairyland, what is the relation between this mystical world and physical reality? How does the human brain, with its known limitations, gain access to such an immutable world, outside of space and time? On the other hand, if mathematics is merely a human invention and it has no existence outside our minds, how can we explain the fact that the invention of so many mathematical truths miraculously anticipated questions about the cosmos and human life not even posed until many centuries later? These are not easy questions. As I will show abundantly in this book, even modern-day mathematicians, cognitive scientists, and philosophers don’t agree on the answers. In 1989, the French mathematician Alain Connes, winner of two of the most prestigious prizes in mathematics, the Fields Medal (1982) and the Crafoord Prize (2001), expressed his views very clearly:

  Take prime numbers [those divisible only by one and themselves], for example, which as far as I’m concerned, constitute a more stable reality than the material reality that surrounds us. The working mathematician can be likened to an explorer who sets out to discover the world. One discovers basic facts from experience. In doing simple calculations, for example, one realizes that the series of prime numbers seems to go on without end. The mathematician’s job, then, is to demonstrate that there exists an infinity of prime numbers. This is, of course, an old result due to Euclid. One of the most interesting consequences of this proof is that if someone claims one day to have found the greatest prime number, it will be easy to show that he’s wrong. The same is true for any proof. We run up therefore against a reality every bit as incontestable as physical reality.

  Martin Gardner, the famous author of numerous texts in recreational mathematics, also takes the side of mathematics as a discovery. To him, there is no question that numbers and mathematics have their own existence, whether humans know about them or not. He once wittily remarked: “If two dinosaurs joined two other dinosaurs in a clearing, there would be four there, even though no humans were around to observe it, and the beasts were too stupid to know it.” As Connes emphasized, supporters of the “mathematics-as-a-discovery” perspective (which, as we shall see, conforms with the Platonic view) point out that once any particular mathematical concept has been grasped, say the natural numbers 1, 2, 3, 4,…, then we are up against undeniable facts, such as 32 42 52, irrespective of what we think about these relations. This gives at least the impression that we are in contact with an existing reality.

  Others disagree. While reviewing a book in which Connes presented his ideas, the British mathematician Sir
Michael Atiyah (who won the Fields Medal in 1966 and the Abel Prize in 2004) remarked:

  Any mathematician must sympathize with Connes. We all feel that the integers, or circles, really exist in some abstract sense and the Platonic view [which will be described in detail in chapter 2] is extremely seductive. But can we really defend it? Had the universe been one dimensional or even discrete it is difficult to see how geometry could have evolved. It might seem that with the integers we are on firmer ground, and that counting is really a primordial notion. But let us imagine that intelligence had resided, not in mankind, but in some vast solitary and isolated jelly-fish, buried deep in the depths of the Pacific Ocean. It would have no experience of individual objects, only with the surrounding water. Motion, temperature and pressure would provide its basic sensory data. In such a pure continuum the discrete would not arise and there would be nothing to count.

  Atiyah therefore believes that “man has created [the emphasis is mine] mathematics by idealizing and abstracting elements of the physical world.” Linguist George Lakoff and psychologist Rafael Núñez agree. In their book Where Mathematics Comes From, they conclude: “Mathematics is a natural part of being human. It arises from our bodies, our brains, and our everyday experiences in the world.”

  The viewpoint of Atiyah, Lakoff, and Núñez raises another interesting question. If mathematics is entirely a human invention, is it truly universal? In other words, if extraterrestrial intelligent civilizations exist, would they invent the same mathematics? Carl Sagan (1934–96) used to think that the answer to the last question was in the affirmative. In his book Cosmos, when he discussed what type of signal an intelligent civilization would transmit into space, he said: “It is extremely unlikely that any natural physical process could transmit radio messages containing prime numbers only. If we received such a message we would deduce a civilization out there that was at least fond of prime numbers.” But how certain is that? In his recent book A New Kind of Science, mathematical physicist Stephen Wolfram argued that what we call “our mathematics” may represent just one possibility out of a rich variety of “flavors” of mathematics. For instance, instead of using rules based on mathematical equations to describe nature, we could use different types of rules, embodied in simple computer programs. Furthermore, some cosmologists have recently discussed even the possibility that our universe is but one member of a multiverse—a huge ensemble of universes. If such a multiverse indeed exists, would we really expect the other universes to have the same mathematics?

  Molecular biologists and cognitive scientists bring to the table yet another perspective, based on studies of the faculties of the brain. To some of these researchers, mathematics is not very different from language. In other words, in this “cognitive” scenario, after eons during which humans stared at two hands, two eyes, and two breasts, an abstract definition of the number 2 has emerged, much in the same way that the word “bird” has come to represent many two-winged animals that can fly. In the words of the French neuroscientist Jean-Pierre Changeux: “For me the axiomatic method [used, for instance, in Euclidean geometry] is the expression of cerebral faculties connected with the use of the human brain. For what characterizes language is precisely its generative character.” But, if mathematics is just another language, how can we explain the fact that while children study languages easily, many of them find it so hard to study mathematics? The Scottish child prodigy Marjory Fleming (1803–11) charmingly described the type of difficulties students encounter with mathematics. Fleming, who never lived to see her ninth birthday, left journals that comprise more than nine thousand words of prose and five hundred lines of verse. In one place she complains: “I am now going to tell you the horrible and wretched plague that my multiplication table gives me; you can’t conceive it. The most devilish thing is 8 times 8 and 7 times 7; it is what nature itself can’t endure.”

  A few of the elements in the intricate questions I have presented can be recast into a different form: Is there any difference in basic kind between mathematics and other expressions of the human mind, such as the visual arts or music? If there isn’t, why does mathematics exhibit an imposing coherence and self-consistency that does not appear to exist in any other human creation? Euclid’s geometry, for instance, remains as correct today (where it applies) as it was in 300 BC; it represents “truths” that are forced upon us. By contrast, we are neither compelled today to listen to the same music the ancient Greeks listened to nor to adhere to Aristotle’s naïve model of the cosmos.

  Very few scientific subjects today still make use of ideas that can be three thousand years old. On the other hand, the latest research in mathematics may refer to theorems that were published last year, or last week, but it may also use the formula for the surface area of a sphere proved by Archimedes around 250 BC! The nineteenth century knot model of the atom survived for barely two decades because new discoveries proved elements of the theory to be in error. This is how science progresses. Newton gave credit (or not! see chapter 4) for his great vision to those giants upon whose shoulders he stood. He might also have apologized to those giants whose work he had made obsolete.

  This is not the pattern in mathematics. Even though the formalism needed to prove certain results might have changed, the mathematical results themselves do not change. In fact, as mathematician and author Ian Stewart once put it, “There is a word in mathematics for previous results that are later changed—they are simply called mistakes.” And such mistakes are judged to be mistakes not because of new findings, as in the other sciences, but because of a more careful and rigorous reference to the same old mathematical truths. Does this indeed make mathematics God’s native tongue?

  If you think that understanding whether mathematics was invented or discovered is not that important, consider how loaded the difference between “invented” and “discovered” becomes in the question: Was God invented or discovered? Or even more provocatively: Did God create humans in his own image, or did humans invent God in their own image?

  I will attempt to tackle many of these intriguing questions (and quite a few additional ones) and their tantalizing answers in this book. In the process, I shall review insights gained from the works of some of the greatest mathematicians, physicists, philosophers, cognitive scientists, and linguists of past and present centuries. I shall also seek the opinions, caveats, and reservations of many modern thinkers. We start this exciting journey with the groundbreaking perspective of some of the very early philosophers.

  CHAPTER 2

  MYSTICS: THE NUMEROLOGIST AND THE PHILOSOPHER

  Humans have always been driven by a desire to understand the cosmos. Their efforts to get to the bottom of “What does it all mean?” far exceeded those needed for mere survival, improvement in the economic situation, or the quality of life. This does not mean that everybody has always actively engaged in the search for some natural or metaphysical order. Individuals struggling to make ends meet can rarely afford the luxury of contemplating the meaning of life. In the gallery of those who hunted for patterns underlying the universe’s perceived complexity, a few stood head and shoulders above the rest.

  To many, the name of the French mathematician, scientist, and philosopher René Descartes (1596–1650) is synonymous with the birth of the modern age in the philosophy of science. Descartes was one of the principal architects of the shift from a description of the natural world in terms of properties directly perceived by our senses to explanations expressed through mathematically well-defined quantities. Instead of vaguely characterized feelings, smells, colors, and sensations, Descartes wanted scientific explanations to probe to the very fundamental microlevel, and to use the language of mathematics:

  I recognize no matter in corporeal things apart from that which the geometers call quantity, and take as the object of their demonstrations…And since all natural phenomena can be explained in this way, I do not think that any other principles are either admissible or desirable in physics.

  Interes
tingly, Descartes excluded from his grand scientific vision the realms of “thought and mind,” which he regarded as independent of the mathematically explicable world of matter. While there is no doubt that Descartes was one of the most influential thinkers of the past four centuries (and I shall return to him in chapter 4), he was not the first to have exalted mathematics to a central position. Believe it or not, sweeping ideas of a cosmos permeated and governed by mathematics—ideas that in some sense went even further than those of Descartes—had first been expressed, albeit with a strong mystical flavor, more than two millennia earlier. The person to whom legend ascribes the perception that the human soul is “at music” when engaged in pure mathematics was the enigmatic Pythagoras.

  Pythagoras

  Pythagoras (ca. 572–497 BC) may have been the first person who was both an influential natural philosopher and a charismatic spiritual philosopher—a scientist and a religious thinker. In fact, he is credited with introducing the words “philosophy,” meaning love of wisdom, and “mathematics”—the learned disciplines. Even though none of Pythagoras’s own writings have survived (if these writings ever existed, since much was communicated orally), we do have three detailed, if only partially reliable, biographies of Pythagoras from the third century. A fourth, anonymous one was preserved in the writings of the Byzantine patriarch and philosopher Photius (ca. AD 820–91). The main problem with attempting to assess Pythagoras’s personal contributions lies in the fact that his followers and disciples—the Pythagoreans—invariably attribute all their ideas to him. Consequently, even Aristotle (384–322 BC) finds it difficult to identify which portions of the Pythagorean philosophy can safely be ascribed to Pythagoras himself, and he generally refers to “the Pythagoreans” or “the so-called Pythagoreans.” Nevertheless, given Pythagoras’s fame in later tradition, it is generally assumed that he was the originator of at least some of the Pythagorean theories to which Plato and even Copernicus felt indebted.