At the fundamental level nature, for whatever reason, prefers beauty.
— physicist David Gross, director of the Institute for Theoretical Physics at University of California, Santa Barbara
Mathematics seems to have astonishing power to tell us how things work, why things are the way they are, and what the universe would tell us if we could only learn to listen. This comes as a surprise from a branch of human activity that is supposed to be abstract, objective, and devoid of sentiment.
Yet, the way we view ourselves is closely connected to what we know (or think we know) about objective aspects of nature. Math tells us truths not only about how gravity works (the better to build bridges), but also universal truths that influence how we think and feel (the better to build societies). Physicist Frank Oppenheimer liked to call these the “sentimental” fruits of science.
True, math does all the things we learned in school: building bridges and balancing checkbooks and calculating the odds of winning the lottery. But it also sheds light on those muddles of the mind that keep not only scientists up at night, but also artists and actors and poets and schoolteachers and psychologists and lovers and parents: How can we make sense of nature, including human nature? What is the nature of truth?
People search for the answers in God and in equations (sometimes both at the same time), by writing plays and studying ants. Curiously, the same methods of thinking that helped reveal light as an undulating electromagnetic field can also help sort out the causes of various social problems. The same approaches to proof that physicists used to establish the reality of a particle called the top quark are brought to the courtroom in the trial of O.J. Simpson.
It’s a heady notion: Mathematics—that seemingly dry stuff—has so much relevance to the deep philosophical ideas that are the foundations of society. By learning how it works, we can get a better grip on everything from obscure aspects of physics to methods of fashioning fairer divorce settlements.
That’s one reason I’ve tried in this book to directly relate ideas from mathematics to problem solving in unlikely places—from life on Mars to the riddle of the Unabomber. It’s an attempt to demonstrate how mathematics informs the kinds of questions people really think and worry about. If I could accomplish one thing in this book, it would be to show that an interest in the quality of life is in no way diminished by quantitative arguments. Quantity and quality are inseparable. Scientists and mathematicians, as well as saints and philosophers, search for the fundamental how’s and why’s of existence. And although they have different standards of evidence and proof, quantitative insights do help us understand qualitative problems.
Of course, mathematical tools do not substitute for the insights of artists and actors and economists and psychologists and historians and writers and spiritual leaders. But they can supply badly needed fresh perspectives.
This book is structured in five (not equal) parts. In the introductory chapter (What’s Math Got to Do with It?), I present the idea that mathematics is not about numbers so much as it is a way of thinking, a way of framing questions that allows us to turn things inside out and upside down to get a better sense of their true nature. Mathematicians know this, of course, but most people outside the profession do not. The chapter tours some of the unexpected territory covered by mathematics—from daily headlines to the Golden Rule.
The first section (Where Mind Meets Math) demonstrates some of the reasons we need math to help sift through the confusion. In the first place, numbers don’t speak for themselves, because our all too human brains get in the way. Certain kinds of relationships that ought to be plain to everyone simply can’t penetrate the veil that physiology and experience puts between knowledge and truth. Indeed, these mental filters make it difficult (perhaps impossible) for human brains to perceive things the way they really are (whatever that is). They are necessary parts of human psychology and physiology, so there is no point trying to “cure” them. However, it helps a great deal to be aware of them—in the same way as it helps if your car’s steering pulls to the left, to compensate by pulling to the right.
The second section (Interpreting the Physical World) explores some of the obstacles to clear seeing that are thrown up by aspects of physical reality itself (not that the muddiness in our minds can ever be completely separated from the messiness of reality). Signals scrambled by persistent interference and changing context, qualities that melt into quantities before our eyes (and vice versa), complex webs of influences that can be impossible to untangle, the elusiveness of observation, and the hazards of prediction—all make the art of getting sense from information a challenge even for the most mathematically adept.
The third section (Interpreting the Social World) gives a taste of how math has illuminated human questions such as fairness. For example, a branch of mathematics called game theory suggests that following the Golden Rule is not only a moral way to behave, but also an effective strategy for getting results.
The fourth and longest section (The Mathematics of Truth) is the heart of the book’s premise. It’s about some of the ways that mathematics can (and does) frequently reveal surprising fundamental relationships—between causes and effects, for example, evidence and proof, truth and beauty. The juiciest part of all—the payoff (at least for this writer)—is the story of how a young mathematician named Emmy Noether figured out how to make Albert Einstein’s general relativity consistent by showing the link between symmetry and the fundamental, unchanging laws of nature. In other words, the same properties that make a snowflake appealing underlie the laws that control the universe. Truth and beauty are two sides of a coin.