Is God a Mathematician? (2 page)

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:

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 3
²
4
²
5
²
, 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:

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.

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:

Interestingly, 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.

There is little doubt that Pythagoras was born in the early sixth century BC on the island of Samos, just off the coast of modern-day Turkey. He may have traveled extensively early in life, especially to Egypt and perhaps Babylon, where he would have received at least part of his mathematical education. Eventually he emigrated to a small Greek colony in Croton, near the southern tip of Italy, where an enthusiastic group of students and followers quickly gathered around him.

The Greek historian Herodotus (ca. 485–425 BC) referred to Pythagoras as “the most able philosopher among the Greeks,” and the pre-Socratic philosopher and poet Empedocles (ca. 492–432 BC) added in admiration: “But there was among them a man of prodigious knowledge, who acquired the profoundest wealth of understanding and was the greatest master of skilled arts of every kind; for whenever he willed with all his heart, he could with ease discern each and every truth in his ten—nay, twenty men’s lives.” Still, not all were equally impressed. In comments that appear to stem from some personal rivalry, the philosopher Heraclitus of Ephesus (ca. 535–475 BC) acknowledges Pythagoras’s broad knowledge, but he is also quick to add disparagingly: “Much learning does not teach wisdom; otherwise it would have taught Hesiod [a Greek poet who lived around 700 BC] and Pythagoras.”

Pythagoras and the early Pythagoreans were neither mathematicians nor scientists in the strict sense of these terms. Rather, a metaphysical philosophy of the meaning of numbers lay at the heart of their doctrines. To the Pythagoreans, numbers were both living entities and universal principles, permeating everything from the heavens to human ethics. In other words, numbers had two distinct, complementary aspects. On one hand, they had a tangible physical existence; on the other, they were abstract prescriptions on which everything
was founded. For instance, the
monad
(the number 1) was understood both as the generator of all other numbers, an entity as real as water, air, and fire that participated in the structure of the physical world, and as an idea—the metaphysical unity at the source of all creation. The English historian of philosophy Thomas Stanley (1625–78) described beautifully (if in seventeenth century English) the two meanings that the Pythagoreans associated with numbers:

So numbers were not simply tools to denote quantities or amounts. Rather, numbers had to be discovered, and they were the formative agents that are active in nature. Everything in the universe, from material objects such as the Earth to abstract concepts such as justice, was number through and through.

The fact that someone would find numbers fascinating is perhaps not surprising in itself. After all, even the ordinary numbers encountered in everyday life have interesting properties. Take the number of days in a year—365. You can easily check that 365 is equal to the sums of three consecutive squares: 365=10
²
+ 11
²
+ 12
²
. But this is not all; it is also equal to the sum of the next two squares (365 13
²
+ 14
²
)! Or, examine the number of days in the lunar month—28. This number is the sum of all of its divisors (the numbers that divide it with no remainder): 28= 1 + 2 + 4 + 7 + 14. Numbers with this special property are called
perfect numbers
(the first four perfect numbers are 6, 28, 496, 8218). Note also that 28 is the sum of the cubes of the first two odd numbers: 28 = 1
³
+ 3
³
. Even a number as widely used in our decimal system as 100 has its own peculiarities: 100 = 1
³
+ 2
³
+ 3
³
+ 4
³
.

OK, so numbers can be intriguing. Still, one may wonder what was the origin of the Pythagorean doctrine of numbers? How did the idea arise that not only do all things possess number, but that all things
are
numbers? Since Pythagoras either wrote nothing down or his writings have been destroyed, it is not easy to answer this question. The surviving impression of Pythagoras’s reasoning is based on a small number of pre-Platonic fragments and on much later, less reliable discussions, mostly by Platonic and Aristotelian philosophers. The picture that emerges from assembling the different clues suggests that the explanation of the obsession with numbers may be found in the preoccupation of the Pythagoreans with two apparently unrelated activities: experiments in music and observations of the heavens.

To understand how those mysterious connections among numbers, the heavens, and music materialized, we have to start from the interesting observation that the Pythagoreans had a way of
figuring
numbers by means of pebbles or dots. For instance, they arranged the natural numbers 1, 2, 3, 4,…as collections of pebbles to form triangles (as in figure 1). In particular, the triangle constructed out of the first four integers (arranged in a triangle of ten pebbles) was called the
Tetraktys
(meaning quaternary, or “fourness”), and was taken by the Pythagoreans to symbolize perfection and the elements that comprise it. This fact was documented in a story about Pythagoras by the Greek satirical author Lucian (ca. AD 120–80). Pythagoras asks someone to count. As the man counts “1, 2, 3, 4,” Pythagoras interrupts him, “Do you see? What you take for 4 is 10, a perfect triangle and our oath.” The Neoplatonic philosopher Iamblichus (ca. AD 250–325) tells us that the oath of the Pythagoreans was indeed:

Figure 1

Why was the Tetraktys so revered? Because to the eyes of the sixth century BC Pythagoreans, it seemed to outline the entire nature of the universe. In geometry—the springboard to the Greeks’ epochal revolution in thought—the number 1 represented a point •, 2 represented a line
, 3 represented a surface
, and 4 represented a three-dimensional tetrahedral solid
. The Tetraktys therefore appeared to encompass all the perceived dimensions of space.

But that was only the beginning. The Tetraktys made an unexpected appearance even in the scientific approach to music. Pythagoras and the Pythagoreans are generally credited with the discovery that dividing a string by simple consecutive integers produces harmonious and consonant intervals—a fact figuring in any performance by a string quartet. When two similar strings are plucked simultaneously, the resulting sound is pleasing when the lengths of the strings are in simple proportions. For instance, strings of equal length (1:1 ratio) produce a unison; a ratio of 1:2 produces the octave; 2:3 gives the perfect fifth; and 3:4 the perfect fourth. In addition to its all-embracing spatial attributes, therefore, the Tetraktys could also be seen as representing the mathematical ratios that underlie the harmony of the musical scale. This apparently magical union of space and music generated for the Pythagoreans a powerful symbol and gave them a feeling of
harmonia
(“fitting together”) of the
kosmos
(“the beautiful order of things”).