A mathematician once dismissed the very idea that people outside his circle could ever understand the true essence of the field. Mathematics is an art form, like music or painting. Translating math into the English language, he said, is harder than translating Chinese poetry. The beauty is lost, the elegance, and a proof that is a thing of ineffable iridescence becomes reduced to a baffling or mundane-sounding bottom line.
Others echoed that sentiment. Fritz John, a mathematician at New York University, said he wanted neither fame nor fortune but merely “the grudging admiration of a few close friends.”
But even if the rest of us cannot appreciate mathematics as an art form, are we really shut out? Articles in The New York Times may not give the details of proofs, but they reveal a rich world that can be exciting, surprising, and can even tug at the heartstrings. They even address the age-old question, What is mathematics? Is it discovered or is it invented? Art or science? If it is art, then why, as George Johnson wrote in one of the articles that opens this book, does the universe appear to follow mathematical laws?
Yet if we put philosophy aside, the variety of mathematical questions, the scope of its inquiries as reported in the Times, can be stunning. There are articles that give mathematical solutions to everyday questions—why do heavy objects rise to the top of a container? The strawberries in your jam are all at the top of the jar so when you get toward the bottom, all you have is thick syrup. Brazil nuts are at the top of the mixed nuts can. A mathematical discovery tells why.
Or what about the woman who won the New Jersey lottery twice in four months? The odds of that happening were widely reported as 1 in 17 trillion. But actually, statisticians calculated, they were more like 1 in 30. A Times article tells how to reason through such questions.
And does arthritis pain really respond to changes in the weather? Statisticians answer that one, too, with an analysis that makes sense but confounds perceptions. Another article tells of a surprising result in controlling traffic jams—mathematicians can prove that sometimes closing streets actually improves traffic flow.
But while those articles can make us look at the world differently, they are not about results that rocked mathematics. If you asked mathematicians which proofs were most important to them, many would cite the surprising, drama-laden tales of the search for solutions to some of their most famous problems. And the story of Fermat’s Last Theorem would be at the top of many lists. Nearly 400 years ago, a French mathematician scrawled the problem in the margins of a book, saying he had a simple proof but no space to write it. Ever since, mathematicians tried to solve it, to no avail. Some famous mathematicians said they would not even take it up—it was a fool’s errand, they would just waste precious years of their lives only to end up empty-handed.
Then, in 1993, a young mathematician, Andrew Wiles, announced that he had solved Fermat. What followed was elation, followed by intense questioning. It was a complicated proof, relying on recently discovered mathematics that few truly understood. And as mathematicians scrutinized Wiles’s work, they found a hole in the proof, which Wiles then desperately tried to fix. He retreated to a barren office in his attic, where he had secretly done his work, attempting to make the proof whole again. It was a year of drama that ended well, but the rollercoaster tale, told in the pages of the Times as it happened, was an unforgettable story of pride and ambition, talent and determination.
Not all mathematics is logical, and researchers have wondered how to describe the unpredictable, like the famous analogy of the Butterfly Effect, described by mathematician Edward Lorenz in a lecture in 1972 titled “Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” As mathematicians tried to understand such phenomena, they developed the new fields, or newly named fields, chaos theory and catastrophe theory. And with those fields came debate. Were the results overstated? How predictable were some of these events? And what can the research say about important questions, like global warming, which involve some of the same uncertainties?
Other articles involve discoveries that changed the way we live and work. In the 1980s, a few mathematicians had a brilliant idea for making unbreakable secret codes. There are some problems that are simple to check once you have the answer, but pretty much impossible to solve unless you have centuries or millennia to run your computer. One such problem is factoring—figuring out which prime numbers can be multiplied together to make a particular number. You can easily multiply, say 3 times 5 times 11 times 13 to get 2,145, but it is a bit more work to start with 2,145 and find those factors. The codes, though, use enormous numbers, not simple ones like 2,145. And with huge numbers, there is no easy way to get the factors. So, suppose you made a code that required you to factor a large number if you wanted to break it. You could send a message by doing the equivalent of multiplying a large group of prime numbers together. And no one could illicitly read that message without factoring the resulting huge number.
The idea was so powerful that the federal government got alarmed at mathematicians’ proclivities to publish all of their work, leading to a difficult national debate about how much can or should be revealed. Some said it was important for the codes to be public in order to use them to keep sensitive information, like credit card transactions, private. Others said it was important to keep coding methods secret so enemies could not use them and make codes that the government could not break. In the end, the methods became an integral part of today’s online world, allowing, for example, the secure Web sites we use when we shop online. The Times articles tell the story of the discoveries and the wrenching debate.
Although many fields of science today involve huge teams of researchers, mathematicians often work alone. One brilliant person can change a field. And many of these mathematicians have stories and insights that can be unforgettable. The Times articles include the haunting story of Srinivasa Ramanujan, born in the 19th century in a small town in India, who died at age 32. He left “a strange, raw legacy, about 4,000 formulas written on the pages of three notebooks and some scrap paper.” His extraordinary story—how he was discovered, how inventive he was—is a tale like no other. “He seemed to have functioned in a way unlike anybody else we know of,” one mathematician said. “He had such a feel for things that they just flowed out of his brain. Perhaps he didn’t see them in way that’s translatable. It’s like watching somebody at a feast that you haven’t been invited to.”
Contemporary mathematicians can tell us what it might feel like to be part of their feast. Leonard Adleman, one of the inventors of the new type of secret code, discovered in graduate school that mathematics “is less related to accounting than it is to philosophy.” While many “think of mathematics as some kind of practical art,” he said, “the point when you become a mathematician is when you somehow see through this and see the beauty and power of mathematics.”
Some, not surprisingly, are just odd people, geniuses but eccentric almost beyond belief. That, at least, describes Paul Erdos, a Hungarian mathematician who had no home and no job. Other mathematicians invited him into their homes, feeding and housing him—and collaborating with him—until he moved on to another mathematician’s home. He also took on the question that opens this book. Are mathematical truths discovered or invented? Erdos said they were discovered. As I wrote in an obituary about this unforgettable man, Erdos “spoke of a Great Book in the sky, maintained by God, that contained the most elegant proofs of every mathematical problem. He used to joke about what he might find if he could just have a glimpse of that book.”
Gina Kolata