Is God a Mathematician? Read Free Page A

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