The Axiom Of Choice
By Mary Lynn Reed
Around two a.m., I closed my laptop, went to the chalkboard, and started the construction over from the beginning. I knew the first sixteen steps. If I started over enough times, transcribing each step from memory—pushing hard against the automatic adrenaline intoxication that flooded my senses every time—if I did this enough, one day the seventeenth step would spring forth like a gift from a divine oracle. I believed that with every cell of my existence.
At four, I put the chalk down and stretched out on the torn, lumpy couch in my office, pulled the coffee-stained afghan over my legs and closed my eyes, my neck muscles twitching, my jaw clenched. I lay awake, wishing for four more months of winter. It was almost April. The students would return on Monday, filled with spring fever and lust for the semester's completion. I was locked in permanent February, still craving its harshness, the desolation of work unfinished.
"It's not an obsession," I said. "It's a passion."
Sitting across from me at The Happy Greek, Lucy scraped her fork against her plate and I cringed. She was the Chair of the History department and my best friend, but sometimes I couldn't bear to look at her.
"Like your passion for Sheila," Lucy said. "Did she get that restraining order filed?"
"I'm serious, Clem. You need some help." Lucy stabbed an olive with her fork and glared across the table.
I raised one eyebrow; her face flushed. It was so easy to rattle her. She'd been without a lover for a long time.
"It's not healthy," Lucy said. "Seriously."
"Lesbians, especially old ones—like us—don't have sex with our bodies," I said. "Sure, it's nice when it happens, but it isn't a requirement anymore. Is it?"
Lucy took a long sip of water and looked away. She fiddled with her napkin, crumpling the corner then smoothing it back into place, crumpling it again. She wasn't an unattractive woman; she'd just given up. She wore the same green cardigan six days a week. I stared at the frayed threads on her sleeve then glanced down at myself: jeans I bought fifteen years ago at a flea market, sweatshirt discolored from the sun and layers of persistent chalk dust.
"Something to say?" I said.
Her chest rose and fell. "Nothing you will discuss."
The waiter stepped in and asked if we'd be interested in dessert. I wanted to say no, ask for the check, and get back to mathematics. The work gnawed at the edge of my mind, creeping inside everything.
"I'd like some coffee," Lucy said.
I ordered Earl Grey and a baklava, offered her a smile.
"Tell me about the construction. I want to hear your progress."
"You know I love you, Luce."
She looked past me and out the window, sipped her coffee.
"I'm stuck on step seventeen," I said.
I woke up at three on Monday, tossed and turned until five. I brewed tea and peered out the front blinds, thick with dust. The sky was diffused grey and the lawn was spotted white, a reluctant final snow. I stepped outside in my sweats and sneakers to retrieve the newspaper.
I sipped tea slowly, steam filling my nostrils; I hoped for the scent of an idea, something that wasn't in my laptop or on the chalkboard or on the pad of yellow paper next to my bed.
The theorem I was trying to prove was not new. I'd produced ten different proofs over a span of fifteen years. The result itself had ceased to be interesting to anyone, including me, long ago. The original work was fifty pages of my doctoral dissertation, an existence argument hinging on several inductive leaps of mathematical faith.
To know an object exists and to construct it with your bare hands—those were two very different things. That's what obsessed me: the search for a constructive proof. My final contribution to mathematics, I assumed. It would not revolutionize the discipline. I doubted it would garner even a wrinkle of notice within my tiny sub-field of algebraic number theory. But the need to create, to build, to produce—it had burrowed its way under my skin, haunting my days and nights, leaving me, occasionally, quaking with fear. The uncertain certainty. Blinding frustration.
I walked to school with my thin jacket open against the relentless wind and tried to picture my life once I finally conquered the construction. Ten years from now. Twenty. I had no idea how long it would take. Or what I would be, after.
Ralph stopped by my office at lunchtime. I was eating peanut butter with my finger straight out of the jar.
"Attractive," Ralph said, both his hands buried in corduroy pants. Ralph was an assistant professor, our youngest new hire. He looked like a baby in his father's grown up clothes.
"This is what happens when you get tenure," I said. "Be afraid."
Ralph sat down under the chalkboard. "It looks like someone's been living in here."
I looked around my office. Empty pizza boxes and aluminum cans stacked in the corner. Notebooks and yellow legal pads full of scribbled calculations on every surface.
"I worked a little over the break," I said.
Ralph nodded. "I've been reading Kronecker."
"God made the integers, all else is the work of man."
"He was a bit of a nut, don't you think?"
"And Cantor wasn't? Transfinite induction? You don't really buy that crap, do you? We've built mathematics up on towers of sand, my friend. Where is the concrete? The steel support beams?"
"All of mathematics unravels if you take the hard line of constructivism," Ralph said. "You know that. The Axiom of Choice, the Well-Ordering Principle, Zorn's lemma. These are things that must be accepted. The alternative is madness."
Ralph was an analyst and a devout follower of Weierstrass; he loved to taunt me. Denigrate my newfound constructivist religion. There was no reasoning with him.
"What good is something if you can't see it? To know the truth, but not be able to touch it?" I said. "You can't axiomatize something into existence."
Ralph stood up and tossed the pieces of chalk onto the metal tray. "You're losing it, professor. You need a new problem. Come join us on the computational biology project. Put some of that constructive energy to good use. Solve the protein folding problem. Find a cure for cancer. Nobody cares about those damned polynomials that are driving you insane."
"Science already has enough mathematicians sucking on her left tit. I'm fine where I am."
Ralph started to leave, then stuck his head back in. "One last question: without the Axiom of Choice, how do you know your theorem is true? If you believe your own rhetoric, all of your inductive proofs are suspect, right? Perhaps you can't construct the thing because it doesn't really exist—"I picked up a dusty eraser and threw it at his head.
At midnight, I picked up the phone and called Sheila. The phone rang ten times and no answer.
I waited ten minutes, called again. Ten rings, still no answer. I poured myself a glass of water and paced in the short hall between my bedroom and cluttered living room. I called again. Fifteen rings, no answer.
At one I went to bed, but didn't sleep. I pictured Sheila lying on her down comforter, hands folded behind her head, ignoring the phone.
At 1:30, I called again. Twenty rings, no answer. I pictured Sheila at a lesbian bar in Chicago, flirting with the bartender, ordering gin and tonic with that naughty little girl smile.
At two, I started to replay our affair, from the first day we met, at the bar in Chicago. I bought her a drink and she slid her stool closer to mine. I kept buying her drinks. When the bar closed, we walked to her car, she wrote her number on a napkin with a felt tip pen. I pressed the napkin in my pocket and clumsily kissed her cheek. It wasn't a great moment of passion, and now, I had to admit, it felt like nothing. Like something that could have easily been overlooked. But I called her the next week and she said she was happy to hear my voice. The calls got longer and more frequent. For three months, we were something. Infrequent lovers. Occasional friends.
At four, I looked at the clock and put my hand on the phone, but didn't call. I fell asleep, my hand resting on the receiver, thinking that tomorrow I would go back to step fifteen. Step sixteen of the construction had been an error. I could see that now. Take a step back and start again.
***The next day I stayed home, worked continuously on the new idea. Step back in order to step forward. It was working. I was up to twenty constructive steps before noon. I had a foothold. It was unraveling. The foundation crumbled as I gripped the pen tightly and wrote out each step again, with conviction.
When I heard the knock on my front door, I didn't jump up to answer it. I was lying on the floor in the living room resting my eyes and my mind from the non-stop, tilt-o-whirl they'd been spinning on for days. I heard the key turning the knob, the door squeaking open, and I knew it was Lucy. No one else had a key.
"Clem?" Lucy looked straight ahead, not noticing me on the floor.
I saw her upside down, the angle contorting her form, making her asymmetric and ominous. "I'm here," I said.
"Jesus," she startled, looked down and stepped over me. "What the hell are you doing?"
"Ralph called me," she said.
I sat up and turned around to face her. "He worries too much for such a young guy."
Lucy sat down in a chair, inches from me. She reached out and brushed hair out of my face. "You look like hell," she said.
She took off her coat and leaned back. I rested my chin on her knees. "I think I have it," I said.
Her eyes brightened. "Your proof?"
"Constructive proof," I said. "It's almost there."
She sighed. "Almost?"
"I know, I know. I've said that before."
Lucy sat up and brushed me away. "I'm going to clean up a little." She stood, rolled up her sleeves.
"You don't have to do that."
"Somebody does," she said, gathering my dirty socks and T-shirts off the sofa.
I woke up in bed naked with Lucy, her head on my shoulder. I peered under the covers; she was naked too. Her breathing was heavy and calm and she didn't stir. I couldn't remember what happened. I didn't remember sex. So many years of chaste friendship and I didn't remember us crossing the line. My arm was around her shoulder and I could smell the faint fragrance of her cucumber-melon shampoo. I matched my breathing with hers, closed my eyes, remembered the construction. I'd finished it. Lucy was cooking beef stew and vacuuming my bedroom. That's all I could remember. I had no idea what happened after that.
Mary Lynn Reed lives and writes near Washington, D.C. Her short
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