Killing Pythagoras (Mediterranean Prize Winner 2015) (43 page)

Irrational numbers

…

They are numbers that cannot be expressed as a quotient—or fraction—of two whole numbers. One of their characteristics is that their decimal places are infinite and non-repeating.

Their discovery caused the biggest crisis in the history of mathematics.

The Babylonians and the Egyptians worked out approximations to some irrational numbers without realizing they could never achieve an exact result. The Greeks, too, worked along similar lines, but it was they who eventually discovered the existence of irrational numbers.

To the Pythagoreans, fractions expressed the proportion, or ratio, between two whole numbers. This reflected the reality of the world as they perceived it: all elements in nature maintained an exact ratio among themselves. That conclusion derived from the fact that they were only aware of rational numbers—which could be expressed as fractions of whole numbers—and from their empirical research. They had made some discoveries where the ratio was exact, and they trusted they could unravel all the mysteries of the universe in the same way.

The discovery of irrational numbers remains a mystery. Aristotle claims they were discovered when Pythagoras’ theorem was applied to a triangle whose hypotenuse was the square root of two, an irrational number. What we do know is that it was a dramatic discovery for Greek mathematics, and especially for the Pythagoreans, whose scientific philosophy was based on the belief that only rational numbers existed.

Greek mathematicians had been following a path that was suddenly blocked. Their bafflement was absolute, and threw them into a state of creative paralysis that took several decades to overcome.

…

Encyclopedia Mathematica.
Socram Ofisis. 1926.

CHAPTER 93

July 17
th
, 510 B.C.

Akenon left the communal building and headed for Pythagoras’ house. That morning, the philosopher had asked him to attend a meeting with the most prominent members of the community.

He didn’t explain the reason for the meeting
, thought Akenon, intrigued.

That had been one of the few times they had spoken since he had returned from Sybaris and told Pythagoras what had happened with Crisipo.

But I’m not the only person Pythagoras has stayed away from in the past few days
, he thought as he walked.

After Aristomachus’ suicide, Pythagoras had frequently left the community to walk in the woods, immersed in his thoughts. Akenon imagined that the philosopher had been thinking out certain decisions during those solitary walks, and that he had summoned them to a meeting now to inform them of those decisions.

When he entered the room there was only one free chair left. He took it and waited in silence next to Evander and Milo.

After a while, Pythagoras looked up.

“I’ve gathered you here to talk about my succession.”

The attendees remained silent, waiting anxiously for the venerable philosopher to continue. To everyone’s ears, the word
succession
had held a sorrowful tone of defeat and farewell.

Pythagoras looked weary, but he took a breath and continued in his deep voice.

“I’ve sent messengers to all our communities. In ten days, we’ll hold an assembly at Milo’s villa. I expect all the grand masters of the School to attend, as well as many masters from the highest levels.”

Milo nodded in silence. He had a country house near Croton which he made available to the brotherhood whenever Pythagoras needed it. It was where they held their large conventions. This meeting would be the most important in the history of the brotherhood.

“At that assembly,” continued Pythagoras, “I will name the people who will succeed me as heads of the School. My initial idea was that one person would assume my role of the past thirty years. However, the murders of several candidates and the grave threats looming over us all, have led me to decide on a different system of governance for the brotherhood.”

His audience was taken aback. Pythagoras looked at them one by one before proceeding.

“I’m going to appoint a committee where each member will have a different role, though the weight of your votes will be the same on all matters affecting the overall functioning of the School. I will also ratify those masters appointed to lead each community. Moreover, I’ll establish a second governing body, subordinate to the main committee, made up of grand masters from all the communities.” His expression became grave. “I won’t deceive you. The function of the second body will be to guarantee the survival and unity of the brotherhood should there be a new attack resulting in new deaths.”

Akenon clenched his jaw when he heard that. He was furious with himself for not having managed to get Crisipo to reveal the masked man’s whereabouts.

“Evander,” said Pythagoras, turning to the grand master, “you’ll be chiefly responsible for political matters in the committee. I hope to be able to help you with this task for a few more years.”

“Yes, master.” Evander bowed his head humbly, conscious that it was still too soon for him to assume that responsibility.

“Hippocreon will support and advise you from the beginning, and especially when I am no longer with you.”

The taciturn Hippocreon nodded in agreement. Though he detested politics, he was aware there were difficult times ahead, and would do whatever was in his power for the good of the brotherhood.

Pythagoras paused momentarily to order his thoughts. However, what came to his mind was the memory of the grand masters he had lost: Cleomenides, Daaruk, Orestes, and Aristomachus.

Four of my six candidates have died
.

The last, Aristomachus, had committed suicide when Pythagoras still hadn’t recovered from the loss of Orestes. Aristomachus’ death had especially affected him. He had always been like an insecure child, a genius at mathematics with an overly sensitive soul. Besides, he had been the best mathematician left in the School, and should have been the person in charge of the committee’s academic branch.

Pythagoras continued immersed in his thoughts, not realizing that the rest of the attendees were waiting for him to continue. Aristomachus’ suicide had come with terrible revelations. Whoever had pushed him to suicide, whoever had sent the parchment, had terrifying power over the minds of men. He had already demonstrated it when he incited the members of the brotherhood to kill Orestes,
but the power he had exercised over Aristomachus was more befitting a god than a human being.

Another issue was that the enemy had made a discovery that placed him far above Pythagoras’ own capabilities. What had become clear to him was that, at least in mathematics, he was no more than a novice compared to the killer.

The discovery itself was something Pythagoras doubted he would ever recover from. In his letter to Aristomachus, the enemy had revealed, again with brilliant simplicity, something that demolished Pythagoras’ entire concept of the world. He had believed that in the universe, in the
cosmos
, all things were in proportion to each other, a relationship which could be known and manipulated with the mathematical tools they were developing. The enemy had destroyed his hopes of predicting and controlling the mysteries of nature. With the discovery of irrational numbers, he had opened a door to the vastness of infinity.

I thought we had made significant advances in the conquest of knowledge, but in fact we’re teetering on the edge of a limitless abyss
.

Pythagoras remained silent, his gaze abstracted, a perplexed look on his face. His listeners looked at each other, unsure of what to do. Finally, Akenon coughed deliberately, and the philosopher seemed to wake. A look of alarm flashed across his face.

No one must know what I’m thinking
.

He had decided that, for now, he wouldn’t reveal the discovery of irrational numbers. Aristomachus had committed suicide to eliminate all trace of their existence, including in his own mind. It had been a desperate attempt to protect the School, prompted by the enemy’s perverse words. Pythagoras wasn’t about to kill himself, but, for now, he would try to keep it secret from the brotherhood. If it were made public at this juncture, all the members of the School would be thrown into confusion as he had been.
It could mean the disintegration of the brotherhood
.

Of course, the murderer could make his discovery public whenever he chose to, but there was still a chance they might catch him before that happened. On the other hand, Pythagoras realized the existence of irrational numbers was simply a reality.

They’re a fact. Someone will inevitably discover them again sooner or later. The path to knowledge necessarily leads to irrational numbers, an unmanageable infinity
. He shook his head slowly, unaware of his own movements.
What can we do?

He had no answer to that question, which he had been asking himself constantly for the past week.

“Milo,” he continued at last in a hoarse voice, “you will also be on the committee. You’re not at the level of master, but you’re one of our most faithful and valiant brothers. No one is as highly regarded as you in Croton, you’re one of the most influential members of the Council of Three Hundred, and the army is loyal to you.”

Milo was moved.

“I’ll do anything I can, master.”

Pythagoras turned to his wife.

“Theano, you’ll be responsible for most of the academic affairs of the School, and you’ll take on the role of political advisor. Your prudence and wisdom have always been a source of pride to the School.”

“Dear husband,” replied Theano in a calm, melodious tone, “I will always be at your service and that of our brotherhood. I’ll gladly be part of the committee, as I hope you will for many years to come.”

Theano’s words slightly relaxed the tension on Pythagoras’ face.

“As for Akenon and Ariadne,” he continued, “though you won’t be part of the committee, you’ll attend the meetings related to the criminal investigations.”

Akenon nodded gravely. He was thinking of the parchment Aristomachus had received just before killing himself. An inspection of it had only served to confirm it was impregnated with some fire-retardant substance. Pythagoras had been evasive about why Aristomachus had tried to burn it. Moreover, he hadn’t let Akenon see its contents, allowing him only to inspect the reverse side, and only in his presence. Akenon knew he hadn’t even shown it to Ariadne.

It must contain one of their big secrets.

Akenon looked up at Ariadne, sitting in front of him. Scarcely a word had passed between them since his return from taking Crisipo to Glaucus’ palace. That had been almost a week ago. Their eyes met and he smiled at her. Ariadne hesitated a moment, then averted her gaze, making Akenon feel as if he had just been slapped.

She was aware that her behavior over the past few days had been much more reserved, but she preferred that to running the risk that someone might discover the secret she was so carefully guarding.

Even though she knew her mother’s parchment on pregnancy by heart, now and then she opened it in the privacy of her room and reread its contents. She was fascinated to find in her own body the changes and symptoms it described. She also read with apprehensive excitement everything that would happen in the future.

Unconsciously, she touched her abdomen. She knew there were certain herbs that would end her pregnancy, but she had decided to keep her child.

The meeting continued with details of the forthcoming assembly at Milo’s house. Ariadne stopped paying attention to her surroundings, as she had often done lately, and remained focused on her pregnancy. She had to find the right moment to tell her father, but had been unable to spend any time alone with him for quite a while. Besides, in recent days he had seemed so dejected she didn’t want to give him another cause for concern.

She sneaked a look at Akenon, realizing the pregnancy had caused an irrevocable rift between them. Feeling a life growing inside her had intensified her need to protect herself from the world. If she thought about it logically, she could see that the walls separating her from Akenon were in some way unreal, since they were constructed of traumas, insecurities, and fears. Nevertheless, being aware of what was happening to her did nothing to help her change it. The pregnancy had made those walls thicker than ever.

I hope Akenon goes back to Carthage before the pregnancy becomes obvious
.