“Okay. That makes sense. But why triangles? Angles and sides and all that. Why is a right triangle so important?”
Pop stubbed out one cigarette and fired up another one. “Because the Greeks discovered that they could say beautiful, simple, elegant things about right triangles. And because they could build houses using right triangles.”
“Houses again! How come it’s always about houses? Why not start with a-a cloud? Why not invent a formula for finding the volume of a cloud?”
“Too messy,” his father said. “ Euclid started with something easy and useful. In all fairness, he was fond of squares and circles too.”
“Why did he get to make up his own rules? They’re wrong!”
“The one about parallel lines may not always work. The others have stood up pretty well,” Pop said mildly.
“But what about two points making a line? I could make a system where they don’t, couldn’t I?”
“Attack the system at your own risk. I’m going to tell you a story.” Commercials had taken the place of football on the TV in the wall unit across from his father’s desk. He muted the sound and said, “A long time ago there was a genius named Pythagoras. He was a genius because he made some discoveries about the integers that no one had ever made before. These discoveries were so elegant, so incredible, that numbers became a religion. The Pythagoreans believed, for instance, that the cosmos formed from a one. It split into the integers, which formed themselves into geometrical shapes, and finally became air, earth, fire, and water. All Nature, all Reality, grew from Number.”
“Is it true?”
“I’m a linguist,” Pop said, “so I wouldn’t turn to Number. I suppose I could found a religion that said that in the beginning was the Word. Wait a minute, I’m already Episcopalian.” Elliott’s mother laughed.
“So the Pythagoreans were an important cult. The most important belief they held was that all Nature came from whole numbers, by which I mean integers and ratios of integers, what we call fractions today.
“Then one day something terrible happened. One of the Pythagoreans, maybe the Master himself, made a new discovery.” The football game came back on, but Pop was rolling now and his eyes went to the screen but his voice stayed with the story.
“They had just discovered the formula for finding the hypotenuse of a right-angle triangle,” he said. “A squared plus B squared equals C squared. Can you imagine how they must have felt, sitting in the shade on a summer’s day, looking at each other when they found this wonderful formula?” Elliott thought of the bearded men in white robes, sitting on steps by white columns, clapping each other on the back. It must have felt like winning the Super Bowl.
“Then somebody said, ‘Let’s try that triangle out with a side that measures a single unit, a one,’ ” Pop said. “They tried it out. And a devil sprang out! Because one squared plus one squared equals two. Therefore the hypotenuse was the square root of two.” He leaned toward Elliott and said in a chilling theatrical whisper, “And that number couldn’t exist.”
“Wow!” Elliott said.
“That thing, that square root of two, couldn’t be described as an integer or as a ratio. It completely contradicted the beautiful universe the Pythagoreans had constructed. Now they had a choice-to accept this ugly thing into their system and work with it, or to try to suppress the fact that it existed. To lie about it, because the Pythagorean religion could not encompass something as ill-formed, as unlocatable as this.”
“So what did they do?” Elliott’s mother said.
“They swore the whole brotherhood to strict secrecy. This secret made a mockery of their beliefs. Now their religion was based on a lie.”
“What happened?” Elliott asked. He lay on the rug, his head propped in his hands, near the fireplace, the book forgotten. It was almost nine, but he wasn’t feeling sleepy, he was all fired up.
“A young man named Hippasus leaked the secret,” Pop said. “And you know what happened then?”
“What?”
“They killed him. Set fire to a ship he was on near Calabria. Sunk it.”
“The Pythagoreans did that?” Elliott gasped.
“Never underestimate the passion of a mathematician,” Pop said. “Of course, the secret was already out. Nowadays we call those ugly numbers the irrational numbers.”
“We let those numbers in?”
“And even uglier things. The imaginary numbers. The transcendents. The transfinites.”
“Poor Hippasus,” his mother said. She dog-eared her page and went into the kitchen.
“Those numbers aren’t real,” Elliott said. “Not like One and Two.”
“Prove they don’t exist and I’ll give you a canoe,” his father said.
In this way Elliott learned that what his intuition told him was only acceptable to other people if he could show them a proof. Elliott became obsessed with mathematical proofs. He had found his own language, a language his father couldn’t learn any more than Elliott could remember the conjugation of a Sanskrit verb.
The proofs of the main theorems of mathematics contained absolute certainty, a certainty that existed nowhere else in his universe of home and school.
A fever overtook him. The proofs burned into his eyes late at night.
Proofs were the rewards of playing this particular game of arithmetic, but he never forgot that other, more difficult games waited in the murk of the future for him to discover.
“El,” Pop called. Elliott put his memories aside, set his bowl into the stainless-steel sink, and went into the living room. Pop never went upstairs anymore; the muscles in his legs had become too weak. Pop was barefoot. His back had hunched in some indefinable way. How strange. He was growing old as well as sick.
As he looked at his father carefully taking out the ad supplements, then putting the newspaper back in order so he could read it in sequence, he felt again the burning pressure to work, to find, as quickly as possible. Pop was only fifty-five, but he had been ill for five years now and his sharp mind had changed in some way hard to describe. It was as though only small things mattered to him anymore, the Zeros, the Ones.
His mother’s clock ticked on the mantel, next to a picture of his father shaking hands with Noam Chomsky at a podium somewhere.
“I don’t know why, but I feel so cold,” Pop said.
“How about a bath?”
“I need a little push.” Elliott pushed his chair into the adjoining bedroom, pulled down the curtains, and got the water running in the tub. Pop had a special tub where you opened the waterproof door and stepped in and sat down on the bench. So far, he could manage.
“Think I’ll go upstairs and do some work,” Elliott said.
“You look tired. You haven’t told me about the conference at Lake Tahoe.”
“Well, lots of presentations that didn’t interest me much. Nothing new, really,” Elliott lied.
“Did you see any old friends?”
“I did see a couple of guys from MIT, but I didn’t know them well.”
“What are they doing these days?”
“One works at Lawrence Livermore Lab in California. The other one went to Los Alamos.” He was used to lying about his activities, so the words came out very naturally.
“Oh. Physicists. That’s nice.”
“By the way, I got paid on the consulting work I did last spring. It’ll keep us going until Christmas.”
His father said, “You have found such an interesting career, all this flitting about, doing your consulting. It’s wonderful that you can spend so much time with me.”
“It’s my home. I doubt I could work anywhere else. I wouldn’t want to leave you, Pop. We get along.”
“My good fortune, that you love it here so much.”
“I do have to go to town for a couple of hours in the morning. Gloria will be here, though. Do you want anything special at the store if I leave before you’re up?”
“How about some of those Paul Newman chocolate wafers. Those are so good.”