Chapter 1: Numbers
At the age of thirteen, Orr considered himself something of a child prodigy. He was slim, wore glasses, couldn't be bothered to clean up his room, but spent hours every day building mammoth structures out of legos, pieces from the various erector sets he owned, and whatever else he happened to find. Four years earlier, his father had brought home a large sheet of tin, and the two had all but sheared and blowtorched it into oblivion, but the resulting inch by inch by inch cube was still a pinnacle on Orr's highest constructs. It currently sat proudly atop a structure two feet larger than Orr himself, which had taken three erector sets and a full third of Orr's lego collection to piece together. Every morning, the sun would shine on the tiny silver block-shaped thing atop the faded-yellow-and-red-and-green-and-silver behemoth that was the pride of Orr's thirteen-year-old life. Orr even prided in its mismatched colors, in the way that the method of the structure's creation was apparent to the eye. It hid nothing about its maker's engineering genius like the tools of our everyday life do.
Of course, as any builder knows, especially one with such limited resources as Orr, almost as relishing as the construction of elegant structures is their ultimate collapse. This, too, took some creativity if the doomed piece of machinery had been designed with as much care as Orr applied to his. A card castle may disappear if one card is tittered too much, but Orr's behemoth, or a clock radio, or a broken camera, these things require a great deal of care to bring down to their component bits, at least if one expects those bits to be functionable after the demise of their whole. Orr had a hammer ready whenever some new piece of equipment arrived (usually from one of his parents) that needed decimating, but he almost never used it. The household screwdrivers were almost always enough. With his own structures, he applied supreme care, analyzing them for as much as an hour beforehand to decide what could be removed and would could not so that the structure would never collapse under its own weight and ruin his beloved pieces. The entire process would take hours, often days, and several more hours if one counted cleaning up the mess that destruction inevitably brings with it (which, of course, Orr always neglected to count).
These activities are perhaps not enough to give one the conclusion of one's own genius, but the proper parental motivation, along with good grades and boredom in school, can have an altogether too-powerful effect on one's self-esteem. And, Orr's self-esteem had surged and swelled, until it encompassed and even overshadowed the structures on which it was partly based. Orr would later come to recognize and despise this ego. He would blame his parents, or more often, himself for not being able to overcome it, but the recognition itself would serve as a further foundation of his intelligence. After all, not all egotists are cunning and self-reflective enough to see their egotism. For now, however, Orr's ego and the question of the morality of possessing it were far from Orr's mind. His chief concern was to be a child prodigy, that is, to produce wondrous things before intelligence could be dimmed by public school.
Orr spent the first two weeks of his third month as a thirteen-year-old pondering what sort of genius he could bring to the world. His obvious first hope was to revolutionize engineering. His experiments with construction and destruction must lead him to a new method, a new idea, that no one had before considered. He started attempting different techniques. He laid a circular base, then a triangular one. He tried two bases that connected higher up, then tried three as a triangle, four as a square, tried connecting them at various levels and made the connectors different shapes. He even, very briefly, experimented with color coordination, but gave it up as unimportant after a few minutes. The giant structure that was currently honored with the silver cube as its pinnacle was roughly hexagonal in that it had six bases, each quite close to its neighbors, such that only four or five legos were enough to connect them. They connected throughout, at various levels, sometimes to their neighbors and sometimes to pillars that lived farther away. At the top, all six came spiraling together (or as close to spiraling as they could given Orr's very non-curved pieces), and came to a small plateau, only two legos wide, on which the cube rested contentedly. The structure represented a nest in its complexity, but it lacked a middle other than the pinnacle and the bridges from one pillar to the next, and before the pinnacle was added, it had looked from above like the three-dimensional equivalent to a very rough sketch of a Star of David surrounded by a circle.
It was grand; it was sprawling; it was complex almost to the point of overwhelming the senses; Orr had never been more proud of anything, nothing he'd built, no grade he'd gotten, not even his first successful venture atop a bicycle, had brought him as much pleasure as the nest in the corner of his room. A few seconds after adorning it, finally, with the cube, and inspecting it thoroughly, with the engineer's glitter in his eye, for any problems that had hitherto evaded his vision, he bounded downstairs, accosted his father in the living room, and demanded his presence at a momentous unveiling. His voice was shrill, even for a boy just entering puberty. His hands were clammy. This, he thought, is the beginning. He assumed that, on seeing it, his father would call the media, or bring around some real experts, and soon, Orr would be standing in front of a podium, smiling to camera flashes and the beginning of a fame-filled life. This image appeared so readily in his head and was so detailed (his tie would be striped red and white), that despite his better judgment, he thought it a prophetic vision about to be made manifest.
Father and son climbed the stairs, Orr well ahead. When Orr reached the middle landing between the two floors, he stopped and turned back to his father. His father was craning his head to the side as he climbed. The words from the television (which his father was attempting to watch) were clearly audible, but Orr could not make them out. At every word, his mind raced more quickly. He imagined his father looking at him on the television, but before it could be made out properly, he started imagining further structures, more complicated. He could not make them out, for they flooded by too quickly, but they were in his head, and they would soon appear less ephemerally, he was sure. His father was bent at a 135 degree angle, which was as close to ninety as his body would take him. He sighed, straightened up, and continued up the stairs more quickly. Orr's fingers twitched, longing to pull his father more quickly, but his brain was too preoccupied to pay them heed.
His father reached the landing, and the two continued up.
Orr felt numb. His brain had started to work so hard that, instead of coming to him quickly, thoughts inched by and took several seconds to form at all. The air felt thick. The inside of his head itched. As he climbed, he managed to convince his hand to scratch the outside, hoping that it would help. It did not. He tried to smile to himself, but that only made his throat itch. He managed to convince his saliva to go down his throat, to soothe it. He reached the top of the stairs. He forgot to wait for his father this time and just continued to his room, holding the banister for support.
This, he thought, is the beginning.
The carpet was crunchy under his bare foot. It felt foreign, and his room looked suddenly foreign, as if through a shimmering portal and not a doorframe with chipping paint. He tried to smile as his fingers brushed against the frame, and he turned to the structure in the corner. It glimmered in the artificial light, the cube as beautiful as ever.
From behind him, he heard his father whistle. "It's big!"
Orr took this statement to be a compliment, though not as awed as he would have liked. It was pretty big, though. He said, "Yeah."
His father said, "It must have taken you all day!"
It had taken the him the afternoons and evenings of Thursday and Friday combined, which Orr figured added up to just about an entire day, so he said, "Just about."
His father turned to him, looked down, smiled, and said, "I'm proud of you, Orr!" He then continued to turn and started moving back down the hall.
The inside of Orr's stomach started to tickle. He blurted, "Wait!"
His father stopped and looked back. "Yeah?"
"Is...?" Nothing was coming to Orr's head, but his stomach had begun to feel cramped, and it was becoming more so by the second. "Is...?"
The words "Is it good?" popped into his head. But, he know that they were wrong. He wanted to ask something else. He wanted to ask if it was something, something indefinable, something brilliant, something new and revolutionary. He really wanted to ask a question about himself, not about the structure, but the only words that made the journey into his inner ear were, "Is it good?" His voice was even feeble in his head, so it was no surprise to him when it sounded feeble coming out of his mouth.
His father smiled. "Oh yeah. Best thing you've built so far!" He walked back down the stairs and, presumably, into the living room.
Orr could not help but feel cheated by his father's untrained eye. But, he was also young, and though he spent the night brooding, the next morning brought a new idea into his head. It came to him as he lay in bed, trying to count how long he'd spent on the structure residing in the corner of his room. He figured that it must have taken about fifteen hours, but it had felt like a lot longer than that, like a journey, an adventure that was just coming to its close. So, he tried to measure it in minutes instead, which came to about nine hundred. That sounded better to him.
He stared at his structure: nine hundred minutes shone in front of him, glistening in the morning sun.
Then, he started to wonder how many seconds it had taken him. That came to about fifty-six thousand. He picked up his watch from the floor and watched a second go by. He thought, fifty-six thousand of those. He watched another second go by, then ten seconds. It felt like a long time, but more than five-thousand of those pieces of time had gone by during the construction of the thing in the corner. That was when the feeling of "adventure completed" started to sink in. He watched another ten seconds go by, and a shiver traveled down his spine. That had really been a long time.
Then, he started to wonder why, in hours, it hadn't seemed long enough, but in ten-seconds it had. What was special about ten-seconds?
He thought about this question for several seconds, watching them slowly move by. The movement was so fluid. Ten seconds couldn't mean anything in particular. But, neither could one second. It was all part of the same fluid movement. A minute, a second, a ten-second: they were all just ways of dividing the same fluid movement. He started counting two-and-a-half seconds to prove it to himself, and time continued to flow, just the same as before, in the same fluid movement. He measured a full minute in two-and-a-half seconds. There were twenty-four of them. Another shiver flowed down Orr's back. Against, he wondered briefly if he were prophetic, but now in a mathematical sense. Had he felt that two-and-a-half seconds would fit so perfectly into sixty? Two-and-a-half had come so easily to him, as if from elsewhere than his own brain. And, yet, it went into a minute as many times as an hour goes into a day.
But, as soon as he started to think about hours and days, thoughts of prophecy were expelled from his brain. What if I did the same thing with hours? Can I measure two-and-a-half hours instead of one?
He had soon developed entirely new systems of time. Two-and-a-half seconds going into fifteen-minutes going into three-quarters-of-an-hour into three-days into six-days into sixty-days (here, he stopped even trying to match the words of our customary demarcation) into six-hundred-days into twelve-hundred-days, and so forth.
He got up and grabbed a mechanical pencil and a spiral notebook, half of the pages of which were devoted to coordinate systems dressed with unenlightening, often squiggly, shapes, along with (starting in the margins, but creeping slowly onto entire pages) several doodles of demon-like pencil-gray eyes above tantalizing smiles and the occasional button or upside-down-seven nose. Upon returning to the bed, he opened the spiral to its inside back cover, pushed with extracted more lead from the pencil with a relished push to the eraser, and began a new system of time.
It started in the same way as before, but now named. A two-and-a-half-second was called a "secant."
There were eight secants to an "octant," and ten octants to a "suptant."
A whole dozen of the rather large suptants added up into an entirely new wording scheme; they filled a single "thirdian," three of which made an "wholian."
Here occurred another jump by factor of twelve, as twelve wholians were exactly enough to count as a single "dame."
... And, so forth...
The timing scheme entertained Orr throughout the morning. He, in all, had 9 units of measurement, the last of which (a "warp") made up a single non-leap year. Though he had a brief vision of establishing these as the new measures of time, it was short-lived. After all, the only advantage of this system (other than the fact that you didn't have to bother with variable-sized months) was the fact that it was different than the current one. If his became popular, a new genius could one day come forth and have as much right to topple it for a new one, or maybe in favor of the current system. Secant time, sadly, was destined to give pleasure and ease the life only of its inventor. But, to this fate it attached most readily, for it kept Orr preoccupied and happy until well after lunch. And, when his stomach finally forced him to abandon his bed, he had both completely filled the inside and outside covers of the front and back of his notebook (partly, of course, with block-letter logos) and completely forgotten about the cube-adorned beast in the corner of his room.
Secant time also proved to have another, far less obvious, repercussion, the result of which would eventually cause secant time itself to all but fade from Orr's memory.
On Sunday afternoon, after a day of relishing the freedom of weekends by watching television, playing Duke Nukem on his computer and doodling a few more secant-time logos (one of which was even accompanied by two drop-like eyes, thinking critically about the uselessness of the existing demarcation system for time, while sitting atop a flat-lining mouth), he opened his secant-time notebook one more time, but this time into the middle, heaved a mammoth textbook out of his backpack, and began working on his geometry homework.
The work was just beginning to be top-heavy with proofs, each of which would take Orr as many as three minutes to think through, and another ten to scrawl correctly, in as little space as possible, into the notebook. The result was so close to illegible that three of the five days of the previous week, Orr's teacher had asked him to remain after class and explain to her his solutions for the problems that she couldn't read, and the last time, he had found that he couldn't read them either.
Luckily, this weekend's homework was only half proofs, and half geometrical word problems. So, he found himself all but done after an hour. Only the last problem remained, a word problem, measuring a lamppost's height given its shadow and a person of a given hight a given distance away whose shadow ends with the lamppost's. Orr had seen several quite similar problems already, and would have solved it quite easily, but for a little snag: the units between the person's size and the shadow's length were different (the former in feet, the latter in meters). Aggravated, Orr flipped to the front of the textbook for its cheat-sheet on unit conversions. His eyes flew up and down the page, and registered "foot," and "meter," as they were supposed to, but also"secant," which appeared several times lower down the page. The connection sent his brain in two directions concurrently, both to: Oh, so that's where I'd seen that word before! and to: Wait, but feet and meters are just as arbitrary as seconds and minutes! For all its usefulness in self-understanding, the first realization quickly died out to the second. The world, with all of its units, came crashing down on Orr. Feet, meters, yards, liters, pounds, quarts, all collapsed from their heavenly stature of givens. Sure, Orr had known that there were two accepted scales for measuring anything, the American and the metric, but only now did he realize that the fact that there were two proved just how arbitrary the systems actually were. Length, weight, volume, temperature, one by one Orr realized that they were continuous, and that splitting them up in any particular way was arbitrary. He could invent secant length, secant weight, secant volume, and secant temperature.if he wanted to, each system just as able to measure as the ones currently accepted.
But then, he wondered, why do we use the systems we use? How did we come to decide on sixty-second minutes, and on twelve-inch feet? Why do some people use the metric system, but we don't?
These are difficult questions, especially for a thirteen-year-old, genius or not. And, they set Orr pacing around his room, barely able to avoid the neglected textbook that sat on the floor, still open to the front inside cover. When pacing proved ineffectual, he tried sitting on his bed, nibbling absent-mindedly on his lower lip. Then, he tried lying down and staring at the ceiling. The tiny bubbles of dried paint looked like a vast Universe of stars, making arbitrary constellations: a giant man, a goat, the letters "CM." But, no amount of staring at the Universe can help to explain why people make the decisions they do. Indeed, the vast network of stars in the Universe, much less the bits of dried paint on Orr's ceiling, are not nearly as complex as the social systems that brought the units of measurement to their current state. In fact, they can barely be compared to the the neural system that once resided in the cranium of Anders Celsius. So, it should be of little surprise that the paint, for all its complexity, did not help Orr a single bit, and he soon got up, disgruntled, still full of questions, and disappointed that his own massive heap of neurons hadn't come through for him, and finished his homework problem.
Orr's lentil soup dinner helped to spur his brain slightly, and on his way upstairs, he came upon a realization. It was by no means an answer, but it was a step in a more fruitful direction than his previous steps around his room. His thought was about the metric system, something about which most high school nerds occasionally joke, that is: mathematically, it is extremely simple because it follows a set pattern. "Kilo" means "one thousand," no matter what single unit it precedes. "Centi" means "one hundredth," no matter which single unit it precedes. And, so forth. Furthermore, each prefix is a factor of ten away from its neighboring prefixes. In other words, much about the metric system, even if it is arbitrary, makes sense. It helps to quantify numbers of bigger and bigger size easily.
Orr reached his room, and looked at the monster in the corner. Last morning, though, time had failed him. Not seconds nor minutes nor hours had been the proper unit for measuring the time it had taken him to build that beast. The problem was in the system's inability to appropriately signify the largeness of something that seemed really big to him. And, the other systems of units would have to fall into similar problems. The problem, in other words, was that the units didn't take context into account. The time that he had spent on his monster could be measured in small hours or big seconds, but the seconds didn't seem big enough individually, and the number of hours was too small. The time that he had actually spent on it was two "mini-adventure" days or, perhaps less romantically, fifteen "small project" hours. Those days, those hours, they weren't normal hours, they were hours in a context.
In this way, Orr came upon his final child prodigy invention, and the one that he would be most proud of in later years, beyond the monster in the corner and beyond secant time. He called the idea (and the products of it) "units that feel right" or UTFRs.
UTFRs are two-part units. The first is a context that gives an idea of how the person on whose experience the value is based experienced it, whether it felt big or small, scalding, lukewarm, freezing, and so forth. It is usually given in the form of a category of experience most people can appreciate, so the hottest day Orr had ever experienced (on a vacation to Israel when he was 10) could have as its first part, 'burn-your-hand-on-a-stove.' The second part is the quantitative unit, just as regular units have, like "centimeter," or "kilowatt." Put together, we have, for example, "10 trying-to-fall-asleep-before-a-test minutes."
Orr, after pacing around his room, thinking through what a UTFR would be, grabbed the same old geometry notebook that was still lying splayed on the floor, flipped to the last page, scrawled "UTFR" on the top, in big, hurried letters, and proceeded to his bed, where he sat and began coming up with useful contexts.
At first, many of his contexts were, unsurprisingly, based on his personal experiences. The longest ten minutes of which he could dream were "10 being-pinched-in-the-cheek-by-your-aunt minutes." The longest one-hundred feet were "100 carry-your-bike-home-after-a-bad-crash feet." And, the biggest two-thousand liters were "2000 curdled-milk-mixed-with-salt-and-sugar liters." This last, as he scribbled it onto the page, brought a giggle, wriggling through his throat. Once, at the age of five, his parents had both been working late, so when they arrived home, were too exhausted to keep an eye on Orr. At some point in the night, he became thirsty, and wanted a glass of milk, and, not wanting to bother his collapsed parents, he trampled down the stairs himself, pulled the fridge open, balanced the milk (unaware of the newly-purchased milk higher up in the fridge), found a cup in one of the cupboards, and poured himself some milk. It smelled awful (much, one might say, like curdled milk), and looked lumpy. Not remembering how his mother had made it look so perfect, he tried mixing it thoroughly with his finger and then thought, It doesn't smell sweet enough. Obviously, the solution was to add sugar, so he grabbed the salt shaker and proceeded to pour about half of its contents into the milk. He then mixed it some more with his finger, and smelled it again. Unsatisfied, and confused, his attention landed on the shaker, and he decided that such a large supply of sugar should not be wasted all into the milk. So, he poured himself a small handful and took a big lick. It was, needless to say, a rather disappointing to shock, and caused the rest of the salt in his hand, to fall to the floor. Disappointed, he looked back into the cupboards, found a bag of sugar, and hoped for the best as he opened it, and poured as much as would fit into the cup. A few finger-mixing seconds later, and the young Orr decided that he would have to ask his mother how she managed to get his milk so tasty, so that he could do it by himself next time, for this mixture still did not look or smell right. But, again, not wanting to be a bother, he decided to live with what he had made, drink it, and be done with milk for the night. This decision, the failed attempt to chug the putrid shake he had invented, and the final expurgation of it and dinner, left such a lasting impression on Orr that he still associated the curdled-milk-with-salt-and-sugar mixture as such a concentration of horror that not even scientists could condense anything further.
At this point, however, Orr hit a snag: there had not been two-thousand liters of curdled milk with salt and sugar. There had only been a single cup. So, he couldn't really know what two-thousand liters of curdled milk with salt and sugar really looked like, and without that piece of knowledge, 2000 curdled-milk-mixed-with-salt-and-sugar liters lacked some of the expressive power of a different contextual unit. With this realization in mind, Orr began trying to find contextual units that matched the measure that they tried to describe. So, "100 walking-home-from-school hours" would not be allowed, as nobody would ever walk home from school for 100 hours. On the other hand, "100 being-in-school-for-two-weeks hours" would be allowed, since two weeks of school added up to about one-hundred hours.
But, Orr had not really experienced so many kinds of movements that could fill up the entire spectrum, nor so many kinds of volume or temperature. So, he started to abstract. He imagined how big a football stadium was, and thought "x football-stadium-filled-with-water quarts" would be a good way to describe how big "x quarts" was... assuming he could figure out how big "x quarts" was to begin with. Similarly, the circumference of the Earth could describe length; the temperature of the sun could describe a really high temperature, and so forth.
After developing these two methods for creating contexts (based on personal experience without regard to the value, and based on common experience with regard to the value), Orr's brain turned itself off and refused to continue. After having invented a new way to measure time, and then a new way to measure everything, in the space of two days, it was exhausted, and demanded fit compensation in sleep. This request was accompanied by a helpful knock on the door by his father reminding him that it was now 9:30pm and time for bed.And, after a few minutes preparation, it took Orr little time to fall asleep. And, his dreams were filled with speeches of congratulations given by strange units of measurement (represented somehow by big squares with mustaches and sashes), in front of cheering crowds of the same.
Orr awoke an hour before his alarm clock. His brain, and the various award ceremonies and numeric adventures of its invention, had sent him tossing and turning throughout the night, such that his exhaustion, despite his suddenly gittery brain, was almost enough to let him back asleep. But, after a few minutes' vain attempt, he rolled, stood, and dressed in a stupor. Then, made his way downstairs to the computer room to twiddle online for some clever contextual units. As the computer turned on, he tripped over to the kitchen, found some cereal and a bowl, and brought them as aids to wakefulness. He returned, closing the door cautiously behind him, and started the audible explosion of dialing up. Google was just becoming popular; AOL was still considered Internet royalty by middle class America; and the song of connecting to the Internet was a largely unconscious annoyance. So, Orr poured himself some cereal and began munching as he waited. And, when, minutes later, the inelegant, fragile behemoth that was AOL 2.0 finally succeeded in its handshaking with the rest of the online world, Orr started searching for miscellaneous facts.
The sun's temperature (on the surface) is 11000 degrees Fahrenheit. A good estimate for a bull's weight is 450 pounds. The average wait time at an emergency room is 222 minutes. The distance between Alaska and Florida is about 5000 (nautical) miles. The volume of the Earth is about 1.0832 x 1021 meters cubed. And, so forth.
Orr compiled a list of about five contextual units each for temperature, weight, length, and volume ranging from the extreme (volume of the Earth) to the benign (weight of a hot dog), when he faintly heard his alarm go off upstairs, and hurried to turn off the computer, and spread around the crumbs left by his casual breakfast eating before his parents awoke, in the hopes of avoiding his mother's wrath at dirtying the carpet. Then he bounded up the stairs, two-and-three at a time, to turn off the alarm and get ready for school.