A Doubter's Almanac (61 page)

“Of course.”

“Well,” I said. “I’m sure you got a deal on it.”

“I didn’t. I bought it new. But yes, at least it was a little bit on sale.” She smiled. “People change, Hans.”

“I know they do.”

“And you can, too.”

“Thanks, Mom.”

“You’re welcome.” She slid a piece of buttered toast across the table.

“I was just wondering,” I said, “about the sweater.”

“I know. I know you were. It was
your
money, if that’s something else you were wondering. Mostly, anyway.”

“Of course, Mom. It’s fine. That’s why I earned it.”

For a time, then, we ate in silence. When I finished the toast, she buttered another slice for me.

“By the way,” I said. “What did you mean by I can change, too?”

“Just because your father couldn’t—that doesn’t mean that you can’t. You’ve got
me
in you, too, you know.” She smiled again. “That’s another reason I did it. To show you that it can be done.”

“Really?”

“Sort of, really.”

“Well,” I said. “Thank you.”

“You’re welcome, again.”

“But Mom, what about the things you used to tell us? About thrift and discipline? About being frugal? I
believed
you.”

“And now you’re all grown, aren’t you?”

—

O
NE LAST THING:
it happened a few weeks ago, at the Wides. A warm mid-September afternoon when we were out there for another picnic. There was a steady wind from the south, and the moths and butterflies were bumping around on it. We’d been aiming the slingshot at a stump of a beech, probably 125 yards to the north, and letting the balloons ride the wind for the extra distance. The high sun was heating the day, which added to our reach.

It was Emmy’s turn to shoot. On the round before, Niels had come within four or five yards of the target, and he was eager to go again. But Emmy took her time. In these types of affairs, she knows how to agitate her competition. She tossed a blade of sedge into the air, checking the wind. She looked across the water into the trees, which were rustling at their crowns. She looked up into the sky, doing whatever she does. She’s always been an efficient calculator, but there’s something in her character that also relies on intuition, especially at moments like these. I don’t know exactly what she sees when she looks out into the world, or even what she’s looking for; but she always appears to gather some inscrutable shade of information to which the rest of us aren’t privy. The same way my father used to.

When she finally did let it fly, the water balloon vaulted out of its sheath like a missile from its launchpad, instantly transforming its trigonometric fate into a glinting leftward-canted ellipse that elongated obliquely in the breeze. I knew, the moment it peaked, that it would hit the target.

But that’s not why I mention it.

Both kids can get off impressive shots—I hardly even remark on them anymore. But on this particular afternoon, as Emmy’s balloon climbed swiftly to the river’s midpoint and transcended its own apex, I saw it explode into a thousand glittering pieces that shot off in every direction into the sky.

The strange thing is that a moment later I heard it hit the tree on the far side. When I turned, the leaves of the beech were rustling. A few of them floated down. I blinked. I looked back into the air above the river. In my eye, the fireworks continued. Shimmering translucencies of white ember arcing within an oscilloptic polygon, still poised at the apogee. Then they began organizing themselves into a wavering, sinusoidal curve—first dully, then brightly—like an old TV coming on. Finally they peaked. For those who wish it described mathematically—and I still remember it mathematically—I then observed, for perhaps a second and a half, a mobile Lissajous figure continually transforming itself, first homotopically, then homeomorphically, then entirely, into a strange, scattering set of flaming runes, the entire white-hot conflagration sparkling and growing brighter as it fell.

It persisted until I turned my head.

—

W
E CALL IT
intuition because we don’t have a better word for it.

A few days after I quit Physico, when I was still wandering Manhattan during the daylight hours like a man let out of prison, Audra suggested I go back to see my father. At that point, I’d been home from my second visit for less than a week. I couldn’t decide whether I felt the way I did because of Dad or because I didn’t have a job anymore.

“Well,” she said, “why don’t you just go out there again and find out?”

Two days later, when I pulled into the cove, I could see Cle and Paulie and Mom inside the cabin, preparing the feast they’d promised me if I came back. All three of them were in the kitchen, sidestepping one another as they gathered the dishes. Dad was outside, standing in a haze of smoke at the end of the dock, waving a pair of barbecue tongs at the car. When I walked down to him, he raised his glass. “The prodigal son returns.”

“Yes he does.”

He took a polite-sized swallow of bourbon. “Here’s to you, then.”

“And to you, Dad.”

A longer swallow. “And here’s to our singularities!”

“Which singularities would those be?”

“Well,” he said. “You quit your job, didn’t you? And your old man still feels like a million bucks.”

In mathematics, singularities are reversals—the points at which the graph makes a sharp turn.

“To our health,” he said. One more gulp. “And to our new freedom.”

Not a half hour later, as I was loading silverware into the picnic basket in the kitchen, I watched him hurry along the dock to help Cle and my mother, who were walking down with their platters. At the stairs, he reached forward and guided them up. (Mom first, then Cle—the order all of them always acknowledged.) He didn’t know I was watching—or maybe he did—and as they moved out toward the grill, he stayed at the top of the steps, gazing after them. I have to say, he did look well again, eyeing the women who’d just waltzed him back into his life.

Paulie left the house then, carrying a pitcher of lemonade, and as she came up the beach, Dad waited for her, too, at the top of the stairs. She didn’t look happy, exactly, but I remember that she looked optimistic. I hadn’t seen her like that in a while. She’d been out here now without me for the week, and she and Dad seemed to be friendly in a way that they hadn’t ever been before. She climbed a couple of steps, and he reached forward from the edge of the dock and held out his hand. She looked up for a moment, then grinned shyly and took it. Something about the expression on her face—the hopefulness in it—made me look back down into the picnic basket. That’s why I didn’t see it happen.

I wish it hadn’t been Paulie.

The sound was like a branch breaking off somewhere in the woods. Then the smash of the pitcher. When I looked up, Paulie had fallen onto the sand. On the dock, Dad was staggering backward. He looked down at his hand, which stuck out strangely from his sleeve. For a moment he seemed confused. Then he let out a howl. My mother came running. Dad grabbed his elbow, bellowing now, and careened forward to the top of the stairs, where he stumbled, missed the railing, and fell sidelong onto the beach.

Proof by Exhaustion

T
HE SCAN CAME
up covered with blots: metastases too numerous to count. In the emergency room, Dr. Gandapur steadied my mother with his hand.

The arm had snapped when he’d tried to help my sister up the stairs, then shattered when he’d landed on the ground.

They didn’t operate. Instead, he was put in a cast that went all the way over his shoulder. Only his fingertips showed at the end. They were purple. The doctor hadn’t even been able to straighten the bones, so the cast bent a second time, midway up the forearm, as though he’d been given another elbow.

But the next day, when we brought him back to the cabin, he sat down on the couch with a cigarette and a glass and used his good hand to light the cigarette. He dragged on it, set it down, and lifted the glass.

—

I
N THE DRIVEWAY,
Dr. Gandapur reached out the window of the old Mercedes and handed me a bag. “He’ll need this now. As you know, I cannot return every day.”

I looked at the label. “He won’t like it.”

“I know he won’t. But I will keep the dose to a minimum.”

“Thank you, Doctor.”

Glancing back at the cabin, he said, “Pardon me for asking, but are Mr. and Mrs. Biettermann still visiting?”

“Mrs. is.”

“And Mr.—is he around at all?”

“He’s back in New York.”

The doctor remained still for a moment, looking carefully up at my face. Then he said, “Just make sure you keep it upstairs.”

—

A
UDRA,
I
KNOW,
thinks I have a hard time speaking about the things that matter to me. And Matthew, at Stillwater, thought the same. I know they both believe that I don’t want to reveal myself, or perhaps that I don’t know what my feelings truly are.

Well, they’re right: I don’t.

But not because I don’t think about them. I do. A mathematician goes to great lengths to define things. A plane in mathematics is not merely a flat surface but a flat surface of infinite thinness and size. Trivial? Not to us. When I say
plane,
I’m not thinking of a tabletop or sheet of glass or a piece of paper. You might point to any one of these objects; but all of them are precisely that:
objects
. They exist in the world. And because they do, they are defined by their breadth and reach. To a mathematician, a tabletop is no more a plane than a slice of rum cake is. In the world we know, in fact, the only thing that can actually be called a plane—or a portion of one, anyhow—is a shadow.

You see?

Words fail us. Even the world fails us.

Are there not a thousand forms of sorrow? Is the sorrow of death the same as the sorrow of knowing the pain in a child’s future? What about the melancholy of music? Is it the same as the melancholy of a summer dusk? Is the loss I was feeling for my father the same I would have felt for a man better-fit to the world, a man who might have thrown a baseball with me or taken me out in the mornings to fish? Both we call grief. I don’t think we have words for our feelings any more than we have words for our thoughts. I don’t even believe that we actually do the things we call
thinking
and
feeling
. We do something, but it is only out of crudeness that we call it thinking; and when we do the other thing, we call it feeling. But I can tell you, if you asked Archimedes in the third century
B.C.
, or Brahmagupta in the seventh
A.D.
, or Hilbert in the twentieth, when they’d first known that they’d solved their great problem, I suspect they’d all say they had a
feeling
.

Maybe that’s why mathematicians like blackboards. Words steer, while equations mostly follow. When the terms of mathematics fail, we invent new ones. Euclid. Diophantus. Viète. Descartes.

What I’ve often believed about people like my father—and like me, and my sister, and my daughter, and quite possibly my son—is that we’ll always be in chase. In chase of the next question, which we’re usually familiar with because it was the answer to the previous one. Everything builds. Increment upon increment. There’s no proof in mathematics that can’t be broken down into steps basic enough for a child of reasoning age to follow. The trick is accumulating the steps, each one so trivial that it can be comprehended by the crippled thing we call the mind. Concentration, if you will. This is all we have. Desire creates the concentration.

I’ve often thought that the remarkable thing about problems like Fermat’s or Poincaré’s—or even Malosz’s—is not that they were eventually solved but that for so long they were not. Really, each one is nothing but grains of sand. The length of time it took to solve them was in good part dependent on probability: there are so many more wrong ways to stack the grains than there are right ways.

Under our old mulberry tree, when my father was first showing me differential calculus, he once said that the discovery that shapes can be described with incrementally smaller shapes, that anything at all could be approximated in such a simple manner, had guided him in much of his thinking.

The thought comes back to me now.

Does one grow wise in increments? By fractioning a life and then summing it? By stacking sand? An infant, in his first sleepiness, must let go of the world; a man must learn to die. What comes between are the grains of sand. Ambition. Loss. Envy. Desire. Hatred. Love. Tenderness. Joy. Shame. Loneliness. Ecstasy. Ache. Surrender.

Live long enough and you will solve them all.

But how to solve the grief I felt for my father in those last days? We think that our sorrow, like the planes we know in this world, has borders. But does it? When he came home from the hospital that evening, he sat back on the leather couch, his expression dull, his one undamaged hand set loosely around a glass. He was the same man I’d always known. The same shadowed eyes. The same slightly bent features. Yet I also knew that in some frame of time that already existed and had almost been reached he was already gone. So what? If you go out along another dimension, you can come back at any other point in time. For a long while by then, when I looked at him on the couch, I’d been seeing him no longer there, seeing not my father but the empty space in which he’d once lived as though I were looking into the future. Well, was I? Does the soul, like the plane, have only a truncated representation in our world? Do we give it edges and dimension only so that we may say we understand it?

As soon as we conform anything to language, we’ve changed it. Use a word and you’ve altered the world. The poets know this. It’s what they try so hard to avoid.

I don’t have much patience with religion, or even with what at Stillwater they liked to call the spiritual life; but nonetheless, it is part of a mathematician’s job to not rule out a possibility until it’s disproven. Could the thing I felt when I thought about living on this earth without my father merely have been the first scald that one feels when at long last one lays one’s hand upon the infinite? Not the bounded thing whose edges we see but the other thing, the thing whose truth can only be approached if we ignore what we think we know?