The morning was beautiful, as always, and the sun's gentle caresses awoke the two people sleeping on the grassy sward. Their slumber had been restful, and they arose easily, backs unbent by care, worry, or sleeping on the ground. No alarms, no schedules, no deadlines - that was all to come later. Each was tall and slender, formed without blemish by the the Master Of All Craftsmen.
They were in their usual sleeping place, if something could be usual after only a week. Still, it was their first and only week of existence, and it seemed already an eternity since the Dreamer Whose Dreams Come True had created them. The slept in a small, flat area in the midst of a collection of rolling hills whose gentle swells were far too perfect to have been formed by crude geologic action.
Their names were One and Zero, which might have seemed odd had they known any other named beings besides themselves and their Creator, who was both He of A Thousand Names and The Nameless One - something which they were having a bit of trouble understanding. Fortunately, it had become clear that any phrase given a grand enough intonation referred to He Who Upholds The Heavens Without Pillars Above Us. The first bolt of lightning ever to scorch the verdant landscape had, in fact, happened the previous day when One had gotten a little too into the spirit of things and referred to the Supreme Being as "The Voice Who Doesn't Sleep, Eat, Or Ever Seem To Have Much Fun." He had been talking to Zero, but He Who Listens Even When We Are Just Kidding Around had not taken it well.
Still, life was pretty good for One and Zero. The sun was always shining, the wind always gentle, and there was a pervasive feeling of well-being. It was true that things had been a little boring - there just wasn't that much to do, besides enjoying their world - but The Almighty One had announced that they would see something new today, and they were excited about it.
"I wonder what He Who I Am Running Out Of Names For is going to show us today?" said One.
"Me too", replied Zero. She had a vague image in her mind of something soft and warm and furry, but didn't have any idea what new wonder might have those attributes. One and Zero knew a lot of adjectives, particularly superlatives, but they were a little short on nouns.
"What time do you think we'll get to see it?", said One, as he leisurely scratched the bottom of his foot. "Is The Creator going to wait until our lunch meeting? That would be..." his words trailed off as he searched madly for an appropriately negative phrase, but there were none to be found. "...uh...not...very...wonderful or perfect."
"What should we do until then?" said Zero, who was carefully inspecting her toenails. Despite all their free time, the two still hadn't examined every part of themselves - new things were so rare and thus exciting that they rationed the pleasure, viewing a fresh area each day. When they finished with themselves, they planned to switch and examine each other, which looked like it was going to be just as much fun - there were clearly some major differences.
She glanced up from her inspection and met One's glance. She knew that glint well, and the words left their lips in perfect synch:
"Number Theory!"
One and Zero were both very good at number theory. They had a natural feel for numbers which probably stemmed from the fact that they were numbers themselves. It was their major recreation, the one game which The Perfect One had given them to pass the time. Fortunately, they enjoyed it a great deal. They had derived a great deal, starting from a few axioms and definitions which they had worked out with the help of their Eternal Teacher.
That first day had been wonderful. They had not been formed suddenly, or painfully, it had been a slow coalescing. Neither could pinpoint the moment of consciousness, but both had known when they were past it. They had talked about many things with their Lord. He had explained that their purpose, as the primeval One and Zero, was to eventually generate the rest of the numbers. Once they had finished this task, they could rest forever, idling away their time however they wished.
Of course, he had said a lot more than that. At first, it had been rather difficult to understand him. He had babbled a lot (if The All-Encompassing Wonder could be said to babble) about elliptic curves and imaginary numbers, Cambridge and the IAS. As these were all nouns without referents, they had been pretty much ignored, although a game they occasionally played was to guess what those strange words meant, based on their sound and texture. They had been promised that once they understood numbers pretty well, they would be shown a new kind called "rational", so it was a word that they particularly liked to discuss.
God hadn't babbled at them for that long. After a while, he had suddenly become much more concise and clear, as if he'd gone off and practiced. He taught them languages, both for human communication and for mathematics. He created them a world, and showed them all the colors which it possessed. He seemed All Powerful and All Knowing, although he sometimes came off as kind of a stuffy old coot.
He had taught them that everything was number, that every blade of grass, every cloud in the sky, everything that could be named could also be counted. Zero was a little angry at first over being the "lesser quantity". She pointed out that One could be added to numbers to increase them, while no matter how many times you added Zero, the number would not change. Their Lord had then explained multiplication to them, and pointed out that One caused no change when multiplying, while Zero could reduce any number to herself. Thus they were each equal in their own domain. Zero had then asked the obvious question, what number was the Holy One? He had sighed, and replied that he could not not answer that in a way they would understand. Perhaps some day, when they had completed their assigned task, he could explain it.
So the morning drifted by as they expected many to in the future. They were relaxed and unworried as they conjectured, experimented, and proved. Modular arithmetic, abstract algebra, combinatorics, all were fertile fields for exploration. Still, a modern mathematical observer would have sensed that something was missing, and that something was the concept of infinity. To One and Zero, all sets had a maximum and a minimum - it was a trivial observation. They knew naught of countability, limits or continuity. God had created the integers, and they were good, but they had never stopped to consider how many there were. LIke finding the multiplicative inverse of zero, it was simply something that was not done. Still, one could prove an awful lot without infinity, and they saw no gap.
Finally the perfect sun reached its maximum height, and as usual, a table flashed into being, laden with food and creaking under its weight. One and Zero had no idea from whence came the fruits and meats which they ate, they saw them only in final form, as cooked foods conjured from the ether. As they ate, a nebulous form shimmered at the far end of the table, and the voice of He Who Cannot Speak Quietly boomed out:
"Today I will show you the great binary tree of knowledge, which enumerates the many numbers which you will help to create. It is a large and wondrous thing, but be warned: you must not eat its fruit! It has drastic effects, and this paradise would be forever forbidden to you if you took so much as a bite."
This seemed a bit strange to One and Zero - nothing had ever been forbidden to them before. They had sort of gotten the impression that the whole world was created for them, and so anything they did was alright.
"But why did you create this tree with fruit if we aren't supposed to eat it?" questioned Zero. "That doesn't make any sense."
Not for the first time, He Who Was Usually Confident let out what sounded like a sigh. "Its a very complicated issue. I didn't really create this tree, it is eternal and everlasting, and the fruit is a part of its nature."
"But aren't you all-powerful? Didn't you create everything?"
"No. That is not quite true. I am not going to claim that I am All-Powerful. I created your world, and everything in it that you have seen so far, but there are some things that are as immutable as I, and this tree is one of them. Remember when you asked me what number I was? Like that, it is a complicated issue."
One and Zero continued to eat as they digested this new concept. Something that The Formerly Almighty didn't have complete control over? A temptation they would have to resist? This was all new and strange, and they felt a momentary flicker in their glowing health. Still, it was rather a beautiful day, and the grilled swordfish (or so He Who Knows The Names Of All The Junk He Makes had told them) was delicious, and their good mood quickly returned.
When they had finished eating and the table had disappeared, they followed their Lord in a direction they had not yet gone. Long before they reached the Tree, they could see it towering in the distance like a huge inverted pyramid. It grew larger and larger until finally they crested a hill and saw it looming in front of them. The base was squat and huge - twenty people could not have linked arms around it. It rose a few feet off the ground, and then bifurcated into two smaller branches. These branches then split into two, and the tree continued to divide, with twice as many branches at each level, until they became so fine and tangled and multitudinous that they seemed like a solid mass.
Just as unusual was the pattern of stripes. The base of the tree was white, and each of the next two branches had two stripes - one was white and then black, the other white and then white. In fact, as One and Zero examined the tree more closely, they realized that at each branching, the new pair of branches bore the same stripe pattern as the single branch below, with the addition of a single stripe on the rightmost side which was white on one branch and black on the other. This fascinating pattern was repeated on every branch they could see. The analytical minds of One and Zero went immediately to work on this pattern. It seemed that at every level, each branch had the same number of stripes, and that number depended on how many levels up the tree one was looking. At each level, every possible pattern that began with a white stripe and had the correct number of stripes was visible, and thus every possible pattern of stripes that began with white could be found on some branch.
Their study was interrupted by the voice of their Lord And Master, who proclaimed: "This is the Binary Tree of Knowledge, one of the fundamental things in your world. You may examine it, climb on it, and study it, but you may not eat the apples and oranges which it bears, or you will be forever barred from this place I have created for you. That is all." With that ominous proclamation, his shimmery form vanished, and One and Zero were left to explore their new wonder.
They had not noticed the fruit which this strange tree bore, but now that it had been mentioned and forbidden, they saw that hanging from every branch was an apple on one side and an orange on the other. The fruits grew smaller and smaller as the branches divided, until they disappeared into the canopy formed by the unimaginably small branches. One went so far as to smell the fruit, but he could not divine any reason why it was so important.
The rest of the day was spent in happily exploring the Tree, climbing it as high as the thinning of the divided branches would allow, puzzling over its construction. They began to explore and discover ideas about binary numbers and bases using this notion of representing numbers in strange and useful ways.
Their mood was ebullient throughout the day, but as dusk fell, they lay in the grass as usual, and talked over their new experiences. At first the discussion stayed with mathematics, but eventually it was drawn to the disturbing new ideas which had entered their world. The Tree was was the first capitalized entity besides themselves and the Glorious One that had come into their little world, and suddenly the landscape of power had shifted. Whatever this Tree was, it was somehow above the domain of The No Longer Omnipotent. One and Zero drifted off into their first uneasy sleep, wondering what the morrow might bring.
Despite their worries, the next few days were lighthearted and merry. Shortly after they awoke, they would walk to the Tree, which seemed to move closer with each passing day. They continued to explore themselves and each other, and to extend their mathematics into increasingly abstruse realms. They spent less time talking out their theorems and more meditating upon them, pondering their proofs for hours beneath the shade of the Tree. The subject of the forbidden fruit was never mentioned, although occasionally when One or Zero's back was turned, the other would sniff it surriptitously, wondering what could make it so dangerous. Each evening they would return to their sleeping area to discuss the days events. Their conversation strayed occasionally back to the changes in their world, but for the most part they were content. This was not to last.
The change was signalled by The Omnipresent One announcing at lunch that he would be busy for the next few days, and would be leaving them to study the tree in peace. One and Zero exchanged nervous glances at this - a few days was a significant fraction of their lives, and they had never gone a day without seeing Him. It seemed a bit odd to leave them alone with this fruit that was so dangerous, too. This was all a bit unsettling.
The next morning, they crested the small hill just before the Tree, and saw before them a gigantic serpent, whose head dangled from the lowest branches of the tree and rested on the ground. Its scales shimmered with iridescence, flickering from ebony to garnet to malachite as the light played across them. The creatures monstrous tail was intertwined with the branches - weaving through each level, climbing ever higher until it disappeared into the haze. Just before the head were two huge hoods, each of which had in its center a circle of skin which was a dull matte black, making the whole thing appear to have a figure-eight pattern. After the second hood, the head narrowed triangularly, ending in a scaly mouth from which a long forked tongue kept emerging to taste the air.
As they moved closer, the immensity of the creature struck them. At its thickest point, a little below the head, its girth was almost as great as the first branches of the tree, and each hood was the size of Zero. It seemed strangely lethargic, as if it were dead or sleeping, but had they looked closely they would have seen the glimmer of life in its obsidian eyes. For the first few minutes they kept their distance from this strange thing, but as they grew bold, they examined it more closely. One finally became brave enough to lay a hand upon it, at which touch a tremor ran through its body. He snatched his hand away, and the head of the great beast rose from the ground, and it looked at the pair.
"Ssssooooo......" it hissed, in a strangely sensual way, with its sinuous tongue stabbing out at them. One and Zero instinctively reached out for each other, clasping hands for reassurance.
"He has created you, yes? He has shown you this tree, see? Still, there is so much he has not said."
"What do you mean? Who are you?" replied One. His short existence had taught him little about communicating with strange beasts, and certainly nothing about how to deal with this. "Why are you on the tree? Were you sent by our Protector?"
The serpents breath rasped between its teeth, like dry leaves scraping along the concrete, as it laughed "Heee, heee, heee. You say Protector, yes? He is called such? So little you even suspect. He hasn't even told you not to speak with strangers - always wise."
"I don't understand. What are you? What don't we suspect? Tell us!", said One, angry at not being answered. The Lord always answered him promptly, even if his meaning was often unclear.
"Softly, my sweet, softly. First we must take little steps towards understanding. Your subject for today is: where is the rattle at the end of my tail?"
As the final sounds of the question hissed from his mouth, the snake's eyelids closed, and he lay quiet. One and Zero tried asking, begging, and yelling, but all was for naught. They tried climbing the tree, an act which necessitated stepping on the body of the snake, which was warm and springy and very unlike the wood of the branches. As far up as they could see, its tail extended. It passed once through each level, alternating which side of each branch it passed, creating a woven effect that they had previously discovered how to create with reeds and grass. At the end of each level, the tail bent in a gentle curve, and moved up to the next. Like the tree, it got smaller and smaller as it went up, without any sign of ever stopping.
Throughout the day they tried to return to their normal activities of puzzling and solving, but the sight of the snake was a continual distraction. After One and Zero had returned to their sleeping place and lain down, they discussed the question of how far up the tree the snake went. Both agreed that it seemed as though the tail of the snake went up the tree forever, passing all of the numbered branches. This seemed disturbing, but neither could figure out why or how. They had their most uneasy sleep yet. For the first time since the beginning of the world (which admittedly was just a couple of weeks back) One awoke in the middle of the night, troubled by a dream.
He had dreamt that God was forcing him to climb the snake all the way to the top, and the higher he got, the harder it was to hold on to the tiny tail, and he felt like no matter how much he climbed, he was no closer to the rattle. He glanced towards the tree, which lately had been just visible on the horizon, but it was too dark to see much, and so he closed his eyes and managed to drift into a restless slumber. In the morning, he awoke grumpy from his interrupted sleep. It was a new experience, and not a pleasant one. He snapped at Zero several times during breakfast when she tried to make conversation, until she gave up and concentrated on her own uneasy thoughts.
As they walked towards the Tree, both were watching it carefully, hoping to see the snake coiled there again. They both saw it at the same moment, and their paces quickened. It was awake this time, and seemed to expect them, as if it had been waiting for them.
"Ssssoooo....What did you see yesterday? A solution to my question? Did you see the end of my tail?" it hissed, moving its head back and forth to look at their faces.
One replied, a little angrily "No, we didn't. I don't think it has an end! As if it just goes up forever! But that is impossible, it has to end somewhere. Where does it end?"
"Questions, so many questions, small one. First, riddle me this: What number does my rattle represent? When you understand that, perhaps you will see where it is."
One and Zero puzzled over this strange question. While it was true that their Lord had taught them that number was everything and everything was number, they had taken it partially as a metaphor. Still, the question remained unanswered. Where was his rattle?
Again, the snake fell silent, and they were left even more confused than before. This time, they did not even try to reason out mathematics as usual, instead they concentrated on this new puzzle. If every branch on the tree represented a number, as it seemed too, then the snakes rattle lived above and beyond all numbers, somehow transcending them. The tree itself did not have this property. It was true that its branches contained numbers greater than any number you might care to name, but also for any number on its branch, there was a greater one. This tail, on the other hand, which dared to exist above all numbers - it was something new, something different.
The prodigal pair's keenly developed mathematical intuition sensed a conceptual breakthrough. There was something about that concept, a tail that was somewhere but nowhere, that existed above all numbers, that promised to open a floodgate of mathematical knowledge. And mathematical knowledge was, after all, both their main recreation and the seeming purpose of their existence. Yet it dodged elusively no matter how hard they tried to pursue it, refusing to be grasped. They struggled all day, and they struggled all night, but the dim light of pre-dawn found them only more tired and not a whit wiser. For the first night ever, they had stayed by the tree, for the first night ever they had not slept in their usual place, for the first night ever they had hardly slept at all. They were fascinated by the snake and his ideas, by the sensation of vast and untapped realms awaiting discovery.
As dawn slowly approached, they watched the horizon and the snake with bleary but eager eyes, wondering when he would awake and reveal to them these secrets. Even when the sun crested the horizon, he continued to lie quiet. Since it was earlier than they had ever before talked to him they waited. The sun moved behind a large cloud, and it grew chilly for a moment. Finally, a beam of light pierced a rent in the cloud, drawing a golden circle of light around the snake's head. A ripple ran through its body, like the first time One had touched it, and its eyes opened lazily.
"Ssssoooo...how curious you must be. It is so early to be discussing strange and wondrous subjects."
Zero spoke this time. "We have been trying to figure out where your tail is, and what it represents. We cannot quite understand it. We can sense that there is much to be learned, but our minds cannot seem to make the leap. Can you tell us?"
"Sorry. It is not for me to teach."
"What? What do you mean. You said you would help us to understand!"
"Yes, I can help you seek knowledge, just not teach you directly. You must consume some fruit."
One and Zero recoiled in shock. They had wondered, of course, why they must not eat the fruit, but they had not expected to find out! Pure mystery had not been quite tempting enough, but a half-known mystery, a mystery whose important they could grasp, that was another thing entirely. To risk the wrath of their Creator - what an awful thing to face? But yet, if that was the only way to learn of these marvels, how could they not take it? They had been created to learn and discover mathematics. Right?
The snake clearly found delight in their discord. His huge head moved back and forth, tongue constantly darting out as he watched their discussion.
"Look, I know we want to learn this, but how do we know we won't figure it out on our own?" said Zero, reasonably. "We have lots of time to try. Who knows what terrible things could happen if we break the Almighty's command? We could even wait until we have finished our task."
One agreed with her, but he did not want to give up so easily, so he argued on anyway. "But are you sure we will have another chance? How long will the tree be here? Will the fruit always be ripe? What if He finds out about us talking to old Snake over there. They don't seem to exactly be friends."
They argued for a while, but since One didn't really want to eat the fruit, they eventually agreed that they should leave it be. Strangely enough, the snake did not seem unhappy. He whispered: "Yes, yes, you have so long to think about it. For when your task is complete, you will be where my rattle is, yes?"
Zero understand what he meant immediately, and she turned to One with even more shock than when the snake had suggested eating the fruit. "Oh no! Our task. Complete. Our task is to generate all of the numbers! Don't you understand? When we finish that task, we will be with his rattle, because we will never finish! It would be like climbing his body to reach the end. The Great Deceiver didn't tell us that!"
One's dream returned to him vividly - the horror of climbing and climbing, the snake getting smaller and smaller, but never reaching the top, because there was no top to reach. He looked at Zero, and it was clear that a decision had been made. They walked purposefully towards the tree, as the snake watched with delight, hissing madly. One took the apple from the left side of the lowest branch, and Zero the orange from the right side. Zero tore her orange in half, and bit into it just as One bit into the apple.
The ground beneath them shook, and clouds rushed in from nowhere, turning the sky to black, but neither noticed. They were deluged with new ideas and sensations, definitions and concepts. Aleph-null, cardinality, infinity's of all shapes and kinds and sizes, suddenly they understood how frail and limited their understanding of mathematics had been. The epiphany had struck, and it was a glorious burning thing that started at the feet and boiled blood all the way up to their fevered brains. Suddenly there was a mighty crack, and the tree shattered in two, splitting exactly down the middle, each half falling to one side, but both missing the dumbstruck couple.
Then God appeared, and He was not happy. The snake was nowhere to be seen, so the tableau which lay before him consisted of his Tree, broken in two, and his creations, in the midst of defying his commands.
"WHAT HAVE YOU DONE? WHY HAVE YOU BROKEN MY TREE, AND EATEN MY FRUIT!" He shouted, in a voice that boomed and thundered, shaking all in creation that had just been shaken and much that had not. It was a voice which could not be ignored, and One and Zero reluctantly answered. The cold horror of their apostasy and the glorious heat of their new-found knowledge fought to a lukewarm standstill. Zero answered, reasonably but boldly, for one should not reason with God: "Well, your Tree isn't broken - now you have two trees, each as large as the original! Anyway, why did you lie to us about ever being able to finish creating all the numbers?"
"SILENCE! YOU DO NOT QUESTION ME! YOU HAVE EATEN OF THE FORBIDDEN FRUIT Of INFINITY, AND YOU SHALL LIVE IN THE PARADISE OF FINITE NUMBERS NO LONGER. I CAST YOU DOWN INTO THE WORLD, TO SURVIVE AS BEST YOU CAN AS MORTALS!"
With these final words, the ground split beneath their feet, and they fell, twisting and tumbling, until they landed in the real world. Armed with their hard-earned knowledge, they made a place for themselves. Life was difficult and unpleasant, but the joys of math and this strange terrible concept of infinity helped them manage. They lived happily for years, producing many numbers, and they never spoke with He Who We Don't Really Miss At All again, although they continued to make up new names for Him.
Last Modified: February 27th, 2000
Patri Friedman / patri@izzy.com