Incompleteness

I

Every afternoon at twenty past five I go downstairs and make the bed, even though I have had all day to do it. He walks in at half past five. With the bedroom door open, the bed is right in your line of sight as you enter the flat. He finds me leaving the bedroom, flushed, perhaps with pleasure to see him, or perhaps with the exertion of making the bed quickly. Some days I don’t shower until after lunch. I fill in number puzzles in the newspaper with all the wrong numbers and leave them on the coffee table. He doesn’t notice, which is mildly surprising given that he is a mathematician. But he gave up numbers years ago.

When we get upstairs, he says, “I think today I might have found it.” During early summer he is very busy marking exam papers. This is his favourite time of year. Four years ago he became convinced that someone would contact him via a manuscript, and this has become the focus of his life.

“There’s a funfair over on the heath,” I say.

“I find them rather sinister,” he says, and takes a sheaf of papers from his briefcase. He removes one sheet from the top of the pile and then places the rest on a shelf already sagging under the weight of paper. He takes copies of the exam papers he marks. I know he is not allowed to do this.

“Look.”

I peer over his shoulder. This manuscript is an original written in blue ink. It looks untidy. There are arrows pointing to boxes containing logical derivation and some sections are crossed out. At the bottom of the page the writing gets smaller and smaller as though the reader is getting further and further away. We go through this routine every few days. He likes to show me. There is kindness in the act. He points to question eight and says, “You see?”

Here is my understanding of things.

There is an important theorem of mathematics stating that every system of logic contains a true proposition that is unprovable within that system. The proposition is able to say something about itself that you would not expect it to be able to say. It speaks with a voice coming from outside itself, from a loudspeaker piped from a bigger world. My husband describes the proof of this theorem as "elegant". It makes me think of the cover of a storybook I had as a child. On the front a boy sat reading a copy of the very same book I held in my hand. And so on the cover of his book sat the boy reading the book, and so on. I remember when I looked at it I had the sensation of floating in cold, dark water.

“I bought lamb chops for dinner,” I say, straightening up.

Four years ago, whilst working on an extension of this theorem, he began to posit all manner of worlds within worlds. My lamb chops are now segmented in higher dimensions, like Russian dolls. The world became full of unending echoes. It is clear that under such circumstances it is difficult for a man to retain a normal, logical mind.

Now, he is waiting for a message. For confirmation. From someone who has this special property of being able to step outside themselves, this ability to see from without rather than within.

“This is showing me something. This is not a usual response to the problem I set.” He looks at me with wide eyes, and I notice that there is sleep in the corner of one.

“Candidate 8128. I need to find him.”

Or her, I think.

II

The central offices of the university are characterised by a sense of age, of wrinkling parchment and skin. A sense of propriety guards the information the offices contain as effectively as the locks on the door. Tonight there is only the hum of a solitary computer left on. The man tests the door gently, then stands with his forehead resting on the pane for a long time. Finally the sound of breaking glass breaks the silence like a sneeze in a library. He whispers, “Sorry.”

III

The man at the crossing waits until the very last moment, looking uncertainly at the stationary traffic as though considering outcomes. He starts to cross just as the cars move, at the moment of maximum danger. Drivers swear inaudibly through glass, calling him a fucking moron. But this being university territory, it is safer to assume that the untidy man now hurrying away from the town centre is a man of learning and not a drunk.

The door is opened by a boy wearing some kind of makeshift pyjamas. He opens the door wide and regards the man standing on the step with some surprise.

“I thought you’d be pizza,” he says, holding out a hand full of change by way of explanation.

“I’m looking for Stephen Haynes.”

“Oh.” He turns back into the house, pausing to shout “Steve!” at the foot of the stairs. The door remains open and the man waits. A clothed boy, or young man, comes down the stairs. He waves slightly.

“Hi. Are you Stephen?”

“Yes.”

“I’m Professor Cartwright. I am a logic tutor. I don’t think I have taught you, actually. You are a second year mathematics student?”

“No. And yes.” Stephen agrees.

The man on the step shifts from one leg to another as though keeping warm, fidgets, opens and closes his briefcase. Then he stops abruptly, as though something important has just occurred to him.

“Will you talk to me about your discoveries?”

“My discoveries?”

“Yes. I think you have found something of great interest to me.” He smiles and puts his hand to his mouth, as though struggling to contain an exciting secret.

The boy thinks about this and frowns. “Have you spoken to my tutor?”

“I have read some of your work.” He steps towards the boy, who retreats further into the house. “I think I understand what you are trying to say. What I need to know is, how far does it go?”

IV

It was the realisation that 8128 is a perfect number that seemed to rouse him into a determination of action. A perfect number is the sum of its proper divisors. I did not stop him. He spends a lot of time in his office now, rereading the manuscript like an old love letter. The authorities have informed him that he is suspended from his duties pending an inquiry. There is remorse, not for the scandal but for the proof that eluded him. He feels that if he had taken a gentler approach he might not have frightened the boy away. I feel that if he hadn’t broken into the university offices and harassed one of the students then perhaps things might have turned out better.

Today he is meeting with the Dean of the university. I go into his office and pick up the exam paper. The writing is smudged in several places, no doubt by my husband’s trembling fingers. I take it into the kitchen, put the plug in the sink and fill it with water. When he comes home he finds the manuscript bobbing in the water, bleeding blue ink.

He doesn’t look at me. He says, “It doesn’t matter. I memorised it.”

I start cooking dinner.