“Pi is a damned good fake of a random number,” Gregory said. “I just wish it were not as good a fake. It would make our lives a lot easier.”
Around the three hundred millionth decimal place of pi, the digits go 88888888—eight eights come up in a row. Does this mean anything? It seems to be random noise. Later, ten sixes erupt: 6666666666. Only more noise. Somewhere past the half-billion mark appears the string 123456789. It’s an accident, as it were. “We do not have a good, clear, crystallized idea of randomness,” Gregory said. “It cannot be that pi is truly random. Actually, a truly random sequence of numbers has not yet been discovered.”
He explained that the “random” combinations of a slot machine in a casino are not random at all. They’re generated by simple computer programs, and, according to Gregory, the pattern is easy to figure out. “You might need only five consecutive tries on a slot machine to figure out the pattern,” he said.
“Why don’t you go to Las Vegas and make some money this way?” I asked.
“Eh.” Gregory shrugged, leaning on his cane.
“But look, this is not interesting,” David said. Besides, he pointed out, Gregory’s health would be threatened by a trip to Las Vegas.
No one knew what happened to the digits of pi in the deeper regions, as the number resolved toward infinity. Did the digits turn into nothing but eights and fives, say? Did they show a predominance of sevens? In fact, no one knew if a digit simply stopped appearing in pi. For example, there might be no more fives in pi after a certain point. Almost certainly, pi doesn’t do this, Gregory Chudnovsky thinks, because it would be stupid, and nature isn’t stupid. Nevertheless, no one has ever been able to prove or disprove it. “We know very little about transcendental numbers,” Gregory said.
If you take a string of digits from the square root of two and you compare it to a string of digits from pi, they look the same. There’s no way to tell them apart just by looking at the digits. Even so, the two numbers have completely distinct properties. Pi and the square root of two are as different from each other as a Rembrandt is from a Picasso, but human beings don’t have the ability to tell the two numbers apart by looking at their digits. (A sufficiently intelligent race of beings could probably do it easily.) Distressingly, the number pi makes the smartest humans into blockheads.
Even if the brothers couldn’t prove anything about the digits of pi, they felt that by looking at them through the window of their machine they might have a chance of at least seeing something that could lead to an important conjecture about pi or about transcendental numbers as a class. You can learn a lot about all cats by looking closely at one of them. So if you wanted to look closely at pi, how much of it could you see with a very large supercomputer? What if you turned the entire universe into a computer? What if you took every particle of matter in the universe and used all of it to build a computer? What then? How much pi could you see? Naturally, the brothers had considered this project. They had imagined a supercomputer built from the whole universe.
Here’s how they estimated the machine’s size. It has been calculated that there may be around 1079 electrons and protons in the observable universe. This is the so-called Eddington number of the universe. (Sir Arthur Stanley Eddington, an astrophysicist, first came up with it.) The Eddington number is the digit 1 followed by seventy-nine zeros: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. Ten vigintsextillion. The Eddington number. It gives you an idea of the power of the device that the Chudnovskys referred to as the Eddington machine.
The Eddington machine was the entire universe turned into a computer. It was made of all the atoms in the universe. If the Chudnovsky brothers could figure out how to program it with FORTRAN, they might make it churn toward pi.
“In order to study the sequence of pi, you have to store it in the Eddington machine’s memory,” Gregory said. To be realistic, the brothers felt that a practical Eddington machine wouldn’t be able to store more than about 1077 digits of pi. That’s only a hundredth of the Eddington number. Now, what if the digits of pi were to begin to show regularity only beyond 1077 digits? Suppose, for example, that pi were only to begin manifesting a regularity starting at 10100 decimal places? That number is known as a googol. If the design in pi appeared only after a googol of digits, then not even the largest possible computer would ever be able to penetrate pi far enough to reveal any order in it. Pi would look totally disordered to the universe, even if it contained a slow, vast, delicate structure. A mere googol of pi might be only the first warp and weft, the first knot in a colored thread, in a limitless tapestry woven into gardens of delight and cities and towers and unicorns and unimaginable beasts and impenetrable mazes and unworldly cosmogonies, all invisible forever to us. It may never be possible, in principle, to see the design in the digits of pi. Not even nature itself may know the nature of pi.
“If pi doesn’t show systematic behavior until more than ten to the seventy-seven decimal places, it would really be a disaster,” Gregory said. “It would actually be horrifying.”
“I wouldn’t give up,” David said. “There might be some way of leaping over the barrier—”
“And of attacking the son of a bitch,” Gregory said.
* * *
THE BROTHERS first came in contact with the membrane that divides the dreamlike earth from the perfect and beautiful world of mathematical reality when they were boys, growing up in Kiev. Their father, Volf, gave David a book entitled What is Mathematics?, written by Richard Courant and Herbert Robbins, two American mathematicians. The book is a classic. Millions of copies of it had been printed in unauthorized Russian and Chinese editions alone. (Robbins wrote most of the book, while Courant got ownership of the copyright and collected most of the royalties but paid almost none of the money to Robbins.) After reading it, David decided to become a mathematician. Gregory soon followed his brother’s footsteps into the nature beyond nature.
Gregory’s first publication, in a Soviet math journal, came when he was sixteen years old: “Some Results in the Theory of Infinitely Long Expressions.” Already you can see where he was headed. David, sensing his younger brother’s power, encouraged him to grapple with central problems in mathematics. In 1900, at the dawn of the twentieth century, the German mathematician David Hilbert had proposed a series of twenty-three great problems in mathematics that remained to be solved, and he’d challenged his colleagues, and future generations, to solve them. They became known as the Hilbert problems. At the age of seventeen, Gregory Chudnovsky made his first major discovery when he solved Hilbert’s Tenth Problem. To solve a Hilbert problem would be an achievement for a lifetime; Gregory was a high school student who had read a few books on mathematics. Strangely, a young Russian mathematician named Yuri Matyasevich had also just solved Hilbert’s Tenth Problem, but Gregory hadn’t heard the news. Eventually, Matyasevich said that the Chudnovsky method was the preferred way to solve Hilbert’s Tenth Problem.
The brothers enrolled at Kiev State University, and took their PhDs at the Ukrainian Academy of Sciences. At first, they published their papers separately, but as Gregory’s health declined, they began collaborating. They lived with their parents in Kiev until the family decided to try to take Gregory abroad for medical treatment. In 1976, Volf and Malka applied to the government of the USSR for permission to emigrate. Volf was immediately fired from his job.
It was a totalitarian state. The KGB began tailing the brothers. “I had twelve KGB agents on my tail,” David told me. “No, look, I’m not kidding! They shadowed me around the clock in two cars, six agents in each car—three in the front seat and three in the backseat. That was how the KGB operated.” One day in 1976, David was walking down the street when KGB officers attacked him, fracturing his skull. He nearly died. He didn’t dare go to the hospital; he went home instead. “If I had gone to the hospital, I would have died for sure,” he said. “The hospital was run by the state. I would forget to breathe.
”
One July day, plainclothesmen from the KGB accosted Volf and Malka on a street corner and beat them up. They broke Malka’s arm and fractured her skull. David took his mother to the hospital, where he found that the doctors feared the KGB. “The doctor in the emergency room said there was no fracture,” David recalled.
By this time, the Chudnovskys were quite well known to mathematicians in the United States. Edwin Hewitt, a mathematician at the University of Washington, in Seattle, had collaborated with Gregory on a paper. He brought the Chudnovskys’ case to the attention of Senator Henry M. “Scoop” Jackson—a powerful politician from Washington State—and Jackson began putting pressure on the Soviets to let the Chudnovsky family leave the country. Not long before that, two members of a French parliamentary delegation made an unofficial visit to Kiev to see what was going on with the Chudnovskys. One of the visitors was Nicole Lannegrace, who would later become David’s wife. The Soviet government unexpectedly let the Chudnovskys go. “That summer when I was getting killed by the KGB, I could never have imagined that the next year I would be in Paris in love, or that I would wind up in New York, married to a beautiful Frenchwoman,” David said.
* * *
IF PI IS TRULY RANDOM, then at times pi will appear to be orderly. Therefore, if pi is random it contains accidental order. For example, somewhere in pi a sequence may run 070707070707070707 for as many digits as there are atoms in the sun. It’s just an accident. Somewhere else the exact same sequence may appear, only this time interrupted, just once, by the digit 3. Another accident. Every possible arrangement of digits probably erupts in pi, though this has never been proved. “Even if pi is not truly random, you can still assume that you get every string of digits in pi,” Gregory told me. In this respect, pi is like the Library of Babel in the story by Jorge Luis Borges. In that story, Borges imagined a library of vast size that contained all possible books.
You could find all possible books in pi. If you were to assign letters of the alphabet to combinations of digits—for example, the letter a might be 12, the letter b might be 34—you could turn the digits of pi into letters. (It doesn’t matter what digits are assigned to what letters—the combination could be anything.) You could do this with all alphabets and ideograms in all languages. Then pi could be turned into strings of written words. Then, if you could look far enough into pi, you would probably find the expression “See the U.S.A. in a Chevrolet!” Elsewhere, you would find Christ’s Sermon on the Mount in his native Aramaic tongue, and you would find versions of the Sermon on the Mount that are blasphemy. Also, you would find a guide to the pawnshops of Lubbock, Texas. It might or might not be accurate. Even so, somewhere else you would find the accurate guide to Lubbock’s pawnshops…if you could look far enough into pi. You would find, somewhere in pi, the unwritten book about the sea that James Joyce supposedly intended to tackle after he finished Finnegans Wake. You would find the collected transcripts of Saturday Night Live rendered into Etruscan. You would find a Google-searchable version of the entire Internet with every page on it exactly as it existed at midnight on July 1, 2007, and another version of the Internet from thirty seconds later. Each occurrence of an apparently ordered string in pi, such as the words “Ruin hath taught me thus to ruminate, / That Time will come and take my love away,” is followed by unimaginable deserts of babble. No book and none but the shortest poems will ever actually be seen in pi, for it is infinitesimally unlikely that even as brief a text as an English sonnet will appear in the first 1077 digits of pi, which is the longest piece of pi that can be calculated in this universe.
Anything that can be produced by a simple method is orderly. Pi can be produced by very simple methods; it is orderly, for sure. Yet the distinction between chance and fixity dissolves in pi. The deep connection between order and disorder, between cacophony and harmony, seems to be tantalizingly almost visible in pi, but not quite. “We are looking for some rules that will distinguish the digits of pi from other numbers,” Gregory said. “Think of games for children. If I give you the sequence one, two, three, four, five, can you tell me what the next digit is? A child can do it: the next digit is six. What if I gave you a sequence of a million digits from pi? Could you tell me the next digit just by looking at it? Why does pi look totally unpredictable, with the highest complexity? For all we know, we may never find out the rule in pi.”
* * *
HERBERT ROBBINS, the coauthor of What Is Mathematics?, the book that had turned the Chudnovsky brothers on to math, was an emeritus professor of mathematical statistics at Columbia University and had become friends with the Chudnovskys. He lived in a rectilinear house with a lot of glass in it, in the woods near Princeton, New Jersey. Robbins was a tall, restless man in his seventies, with a loud voice, furrowed cheeks, and penetrating eyes. One day, he stretched himself out on a daybed in a garden room in his house and played with a rubber band, making a harp across his fingertips.
“It is a very difficult philosophical question, the question of what ‘random’ is,” Robbins said. He plucked the rubber band with his thumb, boink, boink. “Everyone knows the famous remark of Albert Einstein, that God does not throw dice. Einstein just would not believe that there is an element of randomness in the construction of the world. The question of whether the universe is a random process or is determined in some way is a basic philosophical question that has nothing to do with mathematics. The question is important. People consider it when they decide what to do with their lives. It concerns religion. It is the question of whether our fate will be revealed or whether we live by blind chance. My God, how many people have been murdered over an answer to that question! Mathematics is a lesser activity than religion in the sense that we’ve agreed not to kill each other but to discuss things.”
Robbins got up from the daybed and sat in an armchair. Then he stood up and paced the room, and sat at a table, and moved himself to a couch, and went back to the table, and finally returned to the daybed. The man was in constant motion.
“Mathematics is broken into tiny specialties today, but Gregory Chudnovsky is a generalist who knows the whole of mathematics as well as anyone,” he said as he moved around. “He’s like Mozart. I happen to think that his and David’s pi project is a will-o’-the-wisp, but what do I know? Gregory seems to be asking questions that can’t be answered. To ask for the system in pi is like asking, ‘Is there life after death?’ When you die, you’ll find out. Most mathematicians are not interested in the digits of pi. In order for a mathematician to become interested in a problem, there has to be a possibility of solving it. Gregory likes to do things that are impossible.”
The Chudnovsky brothers were operating on their own, and they were looking more and more unemployable. Columbia University was never going to make them full-fledged members of the faculty, never give them tenure. This had become obvious. The John D. and Catherine T. MacArthur Foundation awarded Gregory Chudnovsky a “genius” fellowship. The brothers had won other fashionable and distinguished prizes, but there was a problem in their résumé, which was that Gregory had to lie in bed most of the time. The ugly truth was that Gregory Chudnovsky couldn’t get an academic job because he was physically disabled. But there were other, more perplexing reasons that had led the Chudnovskys to pursue their work in solitude. They had been living on modest grants from the National Science Foundation and various other research agencies and, of course, on their wives’ salaries. Christine’s father, Gonzalo Pardo, who was a professor of dentistry, had also chipped in. He had built the steel frame for m zero in his basement, using a wrench and a hacksaw.
The brothers’ solitary mode of existence had become known to mathematicians around the world as the Chudnovsky Problem. Herbert Robbins eventually decided to try to solve it. He was a member of the National Academy of Sciences, and he sent a letter to all of the mathematicians in the academy:
I fear that unless a decent and honorable position in the American educational research system is found for the brothers soon, a
personal and scientific tragedy will take place for which all American mathematicians will share responsibility.
There wasn’t much of a response. Robbins got three replies to his letter. One, from a professor of mathematics at an Ivy League university, complained about David Chudnovsky’s personality. He remarked that “when David learns to be less overbearing,” the brothers might have better luck.
Then Edwin Hewitt, the mathematician who had helped get the Chudnovsky family out of the Soviet Union, got mad, and erupted in a letter to colleagues:
The Chudnovsky situation is a national disgrace. Everyone says, “Oh, what a crying shame” & then suggests that they be placed at somebody else’s institution. No one seems to want the admittedly burdensome task of caring for the Chudnovsky family.
The brothers, because they insisted that they were one mathematician divided between two bodies, would have to be hired as a pair. Gregory would refuse to take any job unless David got a job, too, and vice versa. To hire them, a math department would have to create two openings. And Gregory couldn’t teach classes in the normal way, because he was more or less confined to bed. And he might die, leaving the Chudnovsky Mathematician bereft of half its brain.
“The Chudnovskys are people the world is not able to cope with, and they are not making it any easier for the world,” Herbert Robbins said. “Even so, this vast educational system of ours has poured the Chudnovskys out on the sand, to waste. When I go up to that apartment and sit by Gregory’s bed, I think, My God, when I was a mathematics student at Harvard I was in contact with people far less interesting than this. I’m grieving about it.”
* * *
“TWO BILLION DIGITS OF PI? Where do they keep them?” Samuel Eilenberg said scornfully. Eilenberg was a distinguished topologist and emeritus professor of mathematics at Columbia University.
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