After we finished lunch, and were fortified with Gatorade, the brothers and I went into the chamber of m zero, into a pool of thick heat. The room enveloped us like noon on the Amazon, and it teemed with hidden activity. The machine clicked, the red lights flashed, the air conditioner hummed, and you could hear dozens of whispering fans. Gregory leaned on his cane and stared into the machine. “Frankly, I don’t know what it’s doing right now. It’s doing some algebra, and I think it’s also backing up some pieces of pi.”
“Sit down, Gregory, or you will fall,” David said.
“What is it doing now, Dave?”
“It’s blinking.”
“It will die soon.”
“Gregory, I heard a funny noise.”
“You really heard it? Oh, God, it’s going to be like the last time.”
“That’s it!”
“We are dead! It crashed!”
“Sit down, Gregory, for God’s sake!”
Gregory sat on a stool and tugged at his beard. “What was I doing before the system crashed? With God’s help, I will remember.” He jotted a few notes in a notebook. David slashed open a cardboard box with his hunting knife and lifted something out of the box and plugged it into m zero. Gregory crawled under a table. “Oh, crap,” he said from beneath the table.
“Gregory! You killed the system again!”
“Dave, Dave, can you get me a flashlight?”
David handed his Mini Maglite under the table. Gregory joined some cables together and stood up. “Whoo! Very uncomfortable. David, boot it up.”
“Sit down for a moment.”
Gregory slumped in a chair.
“This monster is going on the blink,” David said, tapping a keyboard.
“It will be all right.”
On a screen, m zero declared, “The system is ready.”
“Ah,” David said.
The machine began to click, while its processors silently multiplied and joined huge numbers, heading ever deeper into pi. Gregory went off to bed, David holding him by the arm.
In his junkyard, his nest, his chamber of memory and imagination, Gregory kicked off his gentleman’s slippers, lay down on his bed, and brought into his mind’s eye the shapes of computing machines yet un-built.
IN THE NINETEENTH CENTURY, mathematicians attacked pi with the help of human computers. The most powerful of these was a man named Johann Martin Zacharias Dase. A prodigy from Hamburg, Dase could multiply large numbers in his head. He made a living exhibiting himself to crowds and hiring himself out as a computer for use by mathematicians. A mathematician once asked Dase to multiply 79,532,853 by 93,758,479, and Dase gave the right answer in fifty-four seconds. Dase extracted the square root of a hundred-digit number in fifty-two minutes, and he was able to multiply a couple of hundred-digit numbers in his head in slightly less than nine hours. Dase could do this kind of thing for weeks on end, even as he went about his daily business. He would break off a calculation at bedtime, store everything in his memory for the night, and resume the calculation in the morning. Occasionally, Dase had a system crash. In 1845, he crashed while he was trying to demonstrate his powers to an astronomer named Heinrich Christian Schumacher—he got wrong every multiplication he tried. He explained to Schumacher that he had a headache. Schumacher also noticed that Dase did not in the least
To many mathematicians, mathematical objects such as a circle seem to exist in an external, objective reality. Numbers, as well, seem to have a reality apart from time or the world. Numbers seem to transcend the universe. Numbers might even exist if the universe did not. I suspect that in their hearts most working mathematicians are Platonists, in that they accept the notion that mathematical reality stands apart from the world, and is at least as real as the world, and possibly gives shape to the world, as Plato suggested. Most mathematicians would probably agree that the ratio of the circle to its diameter exists luminously and eternally in the nature beyond nature, and would exist even if the human mind was not aware of it. Pi might exist even if God had not bothered to create it. One could imagine that pi existed before the universe came into being and will exist after the universe is gone. Pi may even exist apart from God. This is in the opinion of some mathematicians, anyway, who would argue that while there is at least some reason to doubt the existence of God, there is no good reason to doubt the existence of the circle.
“To an extent, pi is more real than the machine that is computing it,” Gregory remarked to me one day. “Plato was right. I am a Platonist. Since pi is there, it exists. What we are doing is really close to experimental physics—we are ‘observing pi.’ Observing pi is easier than studying physical phenomena, because you can prove things in mathematics, whereas you can’t prove anything in physics. And, unfortunately, the laws of physics change once every generation.”
“Is mathematics a form of art?” I asked.
“Mathematics is partially an art, even though it is a natural science,” he said. “Everything in mathematics does exist now. It’s a matter of
Pi is elusive and can be approached only through approximations. There is no equation built from whole numbers that will give an exact value for pi. If equations are trains threading the landscape of numbers, no train stops at pi. A formula that heads toward pi will never get there, though it can get ever closer to pi. It will consist of a chain of operations that never ends. It is an infinite series. In 1674, Gottfried Wilhelm Leibniz (the coinventor of calculus, along with Isaac Newton) discovered an extraordinary pattern of numbers buried in the circle. This string of numbers—the Leibniz series for pi—has been called one of the most beautiful mathematical discoveries of the seventeenth century:
In English: pi divided by four equals one minus a third plus a fifth minus a seventh plus a ninth—and so on. It seems almost musical in its harmony. You follow this chain of odd numbers out to infinity, and when you arrive there and sum the terms, you get pi. But since you never arrive at infinity, you never get pi. Mathematicians find it deeply mysterious that a chain of discrete rational numbers can connect so easily to the smooth and continuous circle.
As an experiment in “observing pi,” as Gregory put it, I got a pocket calculator and started computing the Leibniz series, to see what would happen. It was easy to do. I got answers that seemed to wander slowly toward pi. As I pushed the buttons on the calculator, the answers touched on 2.66, then 3.46, then 2.89, and 3.34, in that order. The answers landed higher than pi and lower than pi, skipping back and forth across pi, and were gradually closing in on pi. A mathematician would say that the series “converges on pi.” It converges on pi forever, playing hopscotch over pi, narrowing it down, but never landing on pi. No matter how far you take it, it never exactly touches pi. Transcendental numbers continue forever, as an endless nonrepeating string, in whatever rational form you choose to display them, whether as digits or an equation. The Leibniz series is a beautiful way to represent pi, and it is finally mysterious, because it doesn’t tell us much about pi. Looking at the Leibniz series, you feel the independence of mathematics from human culture. Surely on any world that knows pi the Leibniz series will also be known.
It is worth thinking about what a decimal place means. Each decimal place of pi is a range that shows the
