The Proof

The Proof

I am a mathematician.

As a mathematician, I am often asked what it is that I do. A rather profound question indeed. Would that there existed an equally profound answer. Or even a simple enough answer. You see, I don’t really do anything. Most mathematical research is conducted in tandem with professorial duties. Hence, the majority of my time is taken up by spewing the standard sophomore spiel to semi-interested second-years. A little Newton and Leibnitz, some Cauchy and Euler to firm things up, and, if I’m feeling particularly adventurous, a hint of Riemann. Fascinating, isn’t it?

When free from the drudgery of recall and relate, I try to think. Oh, for a truly original idea, something that would shake up the foundations of the mathematical edifice! That is the realm of the truly great. And the truly young—mathematics, more than anything else, is called the young man’s game. The most revered prize in the field of mathematics, the Fields Medal, is famously awarded only to a mathematician who has made a remarkable discovery and is less than forty years of age.

Me, I am thirty-eight now. Nearing the point that someone decided was all-important. What if I’d been born on the 29th of February? With a birthday only once every four years, I would probably have been eligible even after my death! Or if no one knew when was it that I was born? They’d have to give me the benefit of the doubt.

Of course, I’d have to have a remarkable discovery too.

This is the story of such a discovery.

I believe it all started when the Upstart published a startlingly original proof of the Four Colour Theorem. Although the theorem had been proved before, rather recently, in fact, this proof outlined a bold new approach that could be applied to a host of similar fields of enquiry. There were even murmurs of the Fields committee having come to a unanimous decision about the award.

I didn’t leave my room for days.

When I finally met the world anew, my colleagues, my rivals, lesser men all, murmured jealousy and despair, just within earshot. I could not deny them, even to myself. It was jealousy, yes. But not for the paper, no. It was for the Temptress—a young, dark-eyed beauty with raven tresses. I had courted her, with decorum, of course, sobriety and propriety ever the logician’s watchword. But when has logic ever helped to decrypt the mysteries of love?

And so it was that she fell for the Upstart, a brash man on the better side of thirty, who took her dancing, of all things. I had long sought to woo her with tales of my forays into the realms of the mathematical arts—I showed her my papers. And then my magnum opus, my greatest work. The vixen, she begged me to borrow it. I should have known better—but the words of such a gorgeous creature were as wine to me, the loneliness in my heart was as disorienting as an attack of vertigo—it seemed logical to give her the script.

And she gave it to him.

When I finally got around to reading his proof, it was as I feared. The braggart had taken the fruits of my labour, called them his own, and used them to prove something as relatively trivial as the Four Colour Theorem. My work was a treatise on mathematical structure—it was a treatise on logic itself. The first example of work in the field I was dealing with was Gödel’s set of theorems on incompleteness in 1931. Put loosely, they spoke of what it was and what was not possible to do in a given mathematical structure—a framework of axioms. When I delved into his work, I realised there was a gold mine to be exploited—the consequences of his work were used to resolve the issue of whether it was possible to axiomatise the entire field of mathematics. The hope was that henceforth the mathematician’s job would be as a technician, an application of axioms and simple theorems to resolve the hardest of problems. What Gödel did was to prove that this could never be. The implication was that there would always be statements in any given mathematical structure that could neither be proved nor disproved. And that a mathematician remained in every sense of the word, an artist, one who, more than anything else, had to let free his imagination.

It would take a creature as base as him, I suppose, to attempt to profiteer from true beauty.

But I still had some hope—the Old Man.

There was one person in the entire department—in the entire university, even, who seemed to understand me. He saw beauty in what I did, and he believed in me when no one else had. His age was great—his hair was white and his skin wrinkled. But his eyes were as sharp as a man’s ever were. To underestimate him was to imperil yourself—he had a biting wit and an irascible tongue. His favourite anecdote was that of a dinner party a long, long time ago, when Professor Wolfgang Pauli, the great physicist, remarked sadly at a very poorly written manuscript submitted by a junior—‘Das ist nicht nur nicht richtig, es ist nicht einmal falsch!’(Not only is that not right, but it is not even wrong!) He used to speak admiringly of him. His idolatry of the physicist extended even to his work—you could never catch the Old Man on a mathematical error.

But a strict adherence to rigour cages the imagination, and so it was that he had never produced an original paper on anything. He’d solved a few problems, and in correcting other people’s papers, there was no one better. In this too, thus he was as Pauli—the mind of a true genius, but a mind which had managed to manacle itself in its own constructs.

I had given him a copy of my manuscript—he could testify that the Upstart had stolen my work. I felt as though there was hope yet.

But it was not to be.

I decided to go and visit him, so as to seek his counsel. As I reached his rooms, I became aware of an unhealthy odour in the air—the sickly-sweet smell of something slightly off. Seeing the door ajar, I decided to walk in.

His living quarters appeared to be deserted. His dishes lay unwashed—this accounted at least in part for the odour in the air. As I entered his study, I became aware of a musty gloom; the shades were drawn. In the dark, I made out a chair with its back to the door, where I stood, a stool drawn untidily close to it—the carpet had been upset on the floor. There was a huge desk at the far left—it was empty. I crossed the room and pulled back the shades. The light from the sun blinded me temporarily. I turned around, and as my vision cleared, I leapt back in shock.

The Old Man sat on the sofa. His skin seemed even more lined, his eyes sunken. He had never seemed older. However, it was his eyes that caused me the greatest unease. For the first time since I’d known him, his eyes were bleary and unfocused. As I neared him, I saw that he had soiled himself. In all likelihood, he had been sitting here for several days. Professor? Sir? Are you alright?”

It was as though I’d never spoken—there was no reaction. I could see him breathing—but his utter expressionlessness and lack of response seemed inhuman. I could see tear tracks in his face—which, again, was as blank as a wet slate.

On the stool before him, there was my own manuscript, a register, and a piece of paper. I thought I recognised his handwriting, and the piece of paper seemed to be addressed to me; hence, despite everything, I picked it up.

…the work … it implies a flaw in any logical structure … there is no point in trying to understand ANYTHING. Man’s progress is bound to die—logic no longer has meaning … NOTHING DOES…

His untidy scrawl was extremely hard to decipher—apart from the handwriting itself, it was as though his hand kept veering off the page. There were stray marks everywhere.

The manuscript on the stool made sense. But what was in the register? I picked it up.

Within it—I recognised a listing of some of the theorems I’d proved. Then the Old Man had tried to prove something—I couldn’t make out in a glance. He seemed to be interpreting the theorems in an even more general manner than I’d meant them to. His handwriting started to degenerate—it became more and more random, as though he could no longer see the lines on the page. It seemed as though now he was trying to disprove his earlier work—it made no sense.

I decided to postpone a more thorough appraisal of his work. I called an ambulance—people came running up. I explained how I’d come across him—I made no mention of the note or the proof.

Later, in my quarters, I tried to make sense of what I’d seen. According to the Old Man’s note, and what I saw, an implication of the work that I did was to prove the pointlessness of the application of logic. Also, the generality of the proof meant that it was no longer possible for a rational person to complete a logical thought.

What was it that could shake a person’s belief so strongly in existence? It was as though the hubris of man had come a complete circle. We had made great discoveries, yes. But a proof of a flaw in logic?

How could logic disprove logic? It implied an inconsistency in our system of thought, in our understanding of reason, cause and effect. It was as though there existed a huge blind spot, and as though understanding the proof meant losing belief in sentient existence.

It was all too much for me. I had to read the proof.

With a feeling of dread and more than a little excitement, I started. I had to admire the perfection of the professor’s application of my work—he truly understood the philosophy behind it. Three hours later, at about the three-fourth mark, I cast my mind ahead and suddenly seemed to understand what the professor suffered from. The register fell from my hands—my vision seemed to flicker, then everything settled.

As long as I did not continue, I could hopefully delude myself. The professor could have made an error. I could think of possible ways in which the expected proof could fail. After an eternity of thought, I put down the register. I didn’t dare to continue.

Logic, you see, is an assumption. That is well-known to be true. What the professor had found was a different construct that subsumed logic as only one method of thought. It is hard to conceive—it requires an incredible amount of imagination—it was as though the Old Man had been saving up a life’s worth of creative energy to do what he did. I know this seems to be incredulous—but think of those times in which something obvious lay undiscovered in front of your eyes, and it was as though it were revealed by a swish of a magic wand. Mathematicians will be reminded of the term an ‘aha!’ proof.

I thought of the implications of all I’d seen that day. After the Old Man’s breakdown, I knew I could no longer pursue Mathematics. I could no longer work in Natural Philosophy—Physics, Chemistry, Biology—it was all now fatal to me—my belief in logic was shaken. Another reason why I couldn’t do so was that if I were to start thinking about the proof again and inadvertently complete it, I would share the professor’s fate.

However, in my state of heartbreak, there was one thing I did want—revenge on the Upstart. The possession of the deadly proof was turning my mind; I wanted to be rid of it. I could achieve both ends at one go.

I asked the Temptress to meet me once more. She was surprised I wanted to meet her—the only reason she came was out of a sense of guilt, I suppose. I told her that the Old Man had left behind an incredible manuscript, that it was going to win me the prize. I opened a bottle of wine and proceeded to become drunk. I had seen the glint of avarice in her eyes; I merely facilitated what in her mind she had already achieved.

I read Upstart’s obituary a month later. ‘Brilliant mind…elegant and masterful mathematician…’

He had apparently hung himself. Suicide was a sure marker of mental illness, was it not? People drew parallels to other great mathematicians who had committed suicide or had suffered devastating psychiatric disorders. Me? I hadn’t laughed so hard in a very long time.

Unfortunately, the Old Man too passed away. The doctors said that it was as though he had utterly and completely lost ‘the will to live’.

I am no longer a mathematician. I now work as a small-time artist and musician. It’s much safer, you see.