The Rationality of Numbers

“Reason is immortal, all else mortal”
-Pythagoras

The student had been studying the problem for months, staring at the straight lines and the numbers until some epiphany would come and all would be well. He had well deteriorated in the past three months, seldom wasting his time with such luxuries as food, water, or sleep. No, none of that was important compared to the solution. There had to be a solution.

He had drawn and redrawn squares of different sizes, lines of different lengths and found that for some, the outcome was simple and straight forward, but for most he found that the outcome did not actually exist. It made no sense to him, nor anyone else in the Brotherhood, all confounded by this anomaly. Even the Master had spent weeks racking his brains. He had torn his robes, he had eaten sand, he had slammed his head against whatever hard, flat object was near, as they all did, but still the answer eluded them all.
Entering his fourth month on the problem, the Student found that the other Brothers had moved on to other problems to remove themselves from the tortuous unanswerable. They begged him to follow suit, but he refused. Triangles of different sizes and proportions littered his home and his mind, until his wife was compelled to return to her mother's home.
He continued recklessly and madly, seeking the solution that would bring him sanity. At times, the townspeople would hear him in the middle of the night mumbling incoherently in a ridiculous tone, at first just numbers, but soon devolving into a stream of babbling interspersed with insane laughter. The families nearby, worried for their children's sake, also moved away leaving him alone to consider the problem in his solitary madness.
He roamed about his abandoned corner of town, eating the leaves and bark from the trees and carving triangles into each oak in the neighborhood, until he finally became agoraphobic. Sealing himself in his home, he became obsessed with being clean and scrubbed himself raw with a sponge he had made before with some cork and some twigs from the tree in his yard. At first the twigs would break, but after a while, the jagged, little things became too small to break anymore.
One night, a thousand years after the problem began, he stared at the triangles for too long. They took a form before him, they took on a meaning which they had previously lost.
A dog barked outside, and whatever sanity remained within him snapped and was lost forever.
His mouth fell open, and tears welled in his eyes. His torn skin seemed to draw itself tighter about him, particularly about his decrepit face, but he felt cool everywhere. He stopped mumbling, scratching; he nearly stopped breathing. He believed that he had found the answer at last.

Three days later, he emerged from his abandoned corner of town and sought out his friends in the Brotherhood. They stood in awe of him, knowing that days before, he was a blithering maniac, but now he seemed so composed and in control of his senses. He was even smiling. His body was still worn with signs of the madness creeping through his intestines, still lacerated and strained, but his demeanor was that of a normal, healthy gentleman of his rightful age.
The Brothers were so excited to finally have him back from the edge of lunacy where he could converse and share his genuine insights with him once more that they immediately told the Master of the Student's apparent resurrection from his malady. The Master declared that they would sail on the Mediterranean that afternoon to celebrate his return. The Student was extremely pleased and eager to espouse what his madness had wrought: his epiphany and his terrible proof.

The sea was a deep blue that did not bother to reflect the sun as it set on the horizon. The land was flat and far away, along with the world. It became less of a certainty and more of an absurdity, which is why the Brotherhood often sailed for celebrations: to remove themselves from the absurdity of the less intelligent world. On the sea, they were free from the constraints placed on them by their dimwitted fellow man: while on land, they would be heralded as witches, but on this boat, they only heralded themselves, and they could identify however they saw fit.
The wine was passed around and their tongues loosened as ideas flowed freely like the water carrying the ship on the crests of its gentle waves. Finally the Student decided to unveil his epiphany, and his madness was laid bare for all.
With a knife, he drew a square on the deck of the boat, one unit by one unit. The Brothers gathered round, remembering the fateful problem that had driven him mad in the first place and whispering secret prayers that he would not fall away again, not now, not here. He drew the fated diagonal and pointed at one half of the square, the loathsome right triangle, and triumphantly shouted the incantation of the formula that each had come to know and hate, that each had felt a tinge of madness from studying:
“One square added to one square is equal to two. Now, what square alone also equals two?”
The question the Brothers hoped they would never hear again was once more uttered. They stared, some hopeful that this all-to-recent lunatic may have found the piece that connects all, that somehow while he wandered the back of his brain all those years, he had managed to unlock some treasure of unknown knowledge. Others were incredulous.
With a smile, the Student carved a strange symbol that looked like a lightening bolt piercing the number two. The most vulgar of the incredulous Brothers spoke up asking, “What the hell is that?” Still smiling, not even bothering to look up from his insane creation, he chuckled, more to himself than to anyone else, that it was, “an irrational two.”
His joke was not well received. He began scrawling more ridiculous characters into the deck, but the crowd that had gathered around was breaking and falling away, whispering about his remaining malady. Some reached out to touch him, their sympathy overwhelming them, but the Student paid no heed to them, but instead continued scratching his bizarre language and mumbling his babbling lunacy. Presently, a Brother happened to look down at the ramblings, and when he saw and understood, he cried out and fell backwards. He went into a cold sweat, breathing heavy breaths, and pointing at the scribblings, a look on his face of fright beyond fright.
The Master came forward to see what had so disturbed the young student and driven him to such a state so instantaneously. He crouched beside the Student and read over the absurd proof. The Brotherhood once again crowded around to observe.
On the deck, in a wild scrawl was the following:

√2 = m/n assume that m/n is a fraction in lowest terms
2 = m2/n2
m2 = 2n2 m must be a multiple of 2, call it 2q
4q2 = 2n2
2q2 = n2 n must be a multiple of 2 as well
The Student fiddled expectantly with his knife, waiting for someone to utter the inevitable statement. His breath hung on the air, waiting for some fool to say what everyone saw but dared not mention. He cut his hands wringing the sharp instrument, but he paid no mind. Soon, he thought, they will know that my madness was warranted, and my genius assured. Finally, after the sun hung just so for days, a Brother whispered the annotation that made all things clear:
“But... that's impossible. How can m/n be a fraction in lowest terms when both numbers must be divisible by two?”
“Exactly!” the Student shouted, leaping to his feet in wild exultation. “Only an irrational number would act that way, and so it is irrational! And this irrational number's square alone is equal to two!” He began to dance wildly about the boat, while the other Brothers simply stared at the deck in awe and horror. This madman, this lunatic, with his terrible proof had rent the world asunder. The goddamn fool, if numbers cannot be rational, then there is nothing on Earth that is.
This fear of the mystical “irrational number” gripped the Brotherhood one by one as they looked up from the proof and focused their eyes on the mental dwarf whose mind had wrought such foolishness. Locked in the mirth of his lunacy, he did not notice when their faces became stern, nor when they made the decision that he must die for entertaining the very idea that the world could be anything but rational. When they rushed him, he believed they were going to celebrate with him, but in an instant, they had thrown him off the side of the ship.
Only the Master remained behind to contemplate the proof, and as he heard the splash of the Student when he hit the water, he reached out with his cancerous hand and whispered to himself, “I never would have believed that the rationality of Numbers so closely mimics the rationality of Men had I not seen the proof with my own eyes.”