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More About This Textbook
Overview
From the Temple of Apollo to the Arabian desert, and from the winding canals of Venice to the medieval halls of Oxford, Dewdney searches through highlights in the history of mathematics for an answer to the timeless question
Editorial Reviews
From the Publisher
“…a lighthearted read, which may entertain the expert and instruct the layman…”(Mathematika)Publishers Weekly  Publisher's Weekly
Does the realm of numbers exist independently of numbers' manifestations in physical structures, awaiting our discovery? Or has humanity created mathematics and then found that it applies wondrously well to creation? Dewdney (200% of Nothing: The Armchair Universe) spins an absorbing narrative in which he searches the globe for answers to these questions. In order to consult colleagues on the nature of math, the author travels in his imagination to the temple of Miletus, where Pythagoras once worked; to the Jordanian desert under a night sky; to the damp palazzos of Venice; and finally to the goldentinted spires of Oxford. Dewdney refers to important mathematical discoveries, many of which were made by scientists in different cultures independently of one another, in an attempt to puzzle out whether we discover or create mathematics. Unfortunately, too much attention is given to the history of math, which many other writers have addressed, while not enough is devoted to the application of the author's fictional discussions in order to answer his questions. Intriguingly, at the end, Dewdney turns to the ideas of "essential content," "the holos" and even a cosmos "permeated with consciousness," for possible ultimate answers. Throughout, his plotting and dialogue work well, though he miscasts himself, albeit effectively, as a mathematical na f. There's not much new here for the mathematically sophisticated. Those less informed, but interested in the history of mathematical discovery and the deep issues Dewdney raises, will, however, find the book to be an amenable introduction to a difficult subject. (Mar.)Library Journal
Dewdney, a computer science professor and author of several popular works on science and mathematics (Yes, We Have No Neutrons, Wiley, 1997), addresses two closely related, longpondered questions. Why is mathematics so uncannily effective in describing the physical universe? Is "new" mathematics invented, or is it a preexisting something that is discovered? Dewdney's approach is to offer a fictional account of his visits with four fictional contemporary scholars in Europe and Egypt. He explores these fundamental questions via discussions of the mathematical work of Pythagoras, the medieval Arab mathematicians, modern theoretical physicists, and modern mathematicians. The mathematical portions of his chapters should be understandable even to lay readers, yet the material is quite useful in exploring the deep issues that lie at the heart of the book. Dewdney is apparently tempted to believe that mathematics is revealed through discovery rather than invention. He leaves his readers with much to ponder. This is an excellent popular introduction to some fundamental questions in the philosophy of science; strongly recommended for both public and academic libraries.Jack W. Weigel, Univ. of Michigan Lib., Ann ArborBooknews
Starting his virtual pasttofuture tour in Greek mathematician Pythagorus' neighborhood, Dewdney (computer science, U. of Western Ontario), the author of , makes stops at apt venues in pursuing the conundrum of which came first: the laws of math or the universe. No bibliography. Annotation c. by Book News, Inc., Portland, Or.Product Details
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Chapter One
Death of a Dream
Izmir, Turkey, June 22, 1995
* * *
A strange day! I spent it in Miletus, or what used to be Miletus in 500 B.C. It was there, if anywhere, that mathematics became a science. Miletus was the unrivaled center of commerce, of philosophy, and of the arts. Here lived Thales, the first scientist; Anaximander, the philosopher; and Timotheus, the poet. Here, too, the great Pythagoras visited from his native Samos, to learn and to teach.
First, let me backtrack, however. I arrived yesterday in Athens and changed planes for a onehour flight across the Aegean, which took me to Izmir, Turkey, an hour's drive north of Miletus. This morning, in Izmir, I rented a Fiat Uno and drove south through a succession of beautiful valleys in supernaturally hot weather. I came at last to the Aegean coast, following it through relentless humidity until I arrived at Miletus, a collection of ruins with numerous tourist signs and a weedfilled parking lot. The site of the ancient city has been largely lost to silting and erosion by the Meander, the prototype of every river that twists and turns. I parked outside a fencedoff area and made my way through partial restorations of the ancient city to the temple of Apollo. There, some tourists were gathering their bags and cameras, departing singly and in groups for a waiting bus.
There are times, when visiting old places, that hints of an ancient presence overwhelm you, like ghosts, in broad daylight. You cannot be with atour group to have this feeling, you must be alone. It came over me as I stood before the temple of Apollo Delphineus, its steps and pillars haunted by memories not my own. I looked around for Petros Pygonopolis, the man I was supposed to meet, but there was no one in sight.
As I ascended the steps to the temple floor, I saw a man kneeling as if in prayer, hunched over the perfectly fitted, square paving stones. As I quietly approached, I could see that he was measuring the stones with a bronze ruler. He was a large man, with a shock of black hair, gray at the temples, and an olive complexion. He looked strangely out of place, for he wore an elegant white suit. When I cleared my throat, he looked up, startled. His stared at me confusedly from beneath bushy brows, then retrieved a pair of glasses, which he put on. His face was suddenly wreathed in a generous smile. Who could it be but Pygonopolis? He rose to dust the knees of his trousers, then bowed.
"You must be Dewdney! Excuse me for not seeing you. Thank the gods those tourists have finally gone. This is not the thing to wear for fieldwork, no? Of course not!" he answered his own question. "I am Petros Pygonopolis, historian of science and specialist in Greek mathematicsthat is, ancient Greek mathematics."
We shook hands and stepped back to examine each another.
"Welcome to Miletus, and welcome to your intellectual roots," Pygonopolis continued. "The questions you have asked in your letter are the right ones, in my humble view. Is mathematics discovered or is it invented? Does it have an independent existence? How refreshing that people can still ask such questions! The answers, such as they are, begin with what I was just doing, measuring these stones with this ruler." He held up the bronze strip. "It is the pechya, or cubit, that we Greeks once used."
While most personable and charming, Pygonopolis had a nervous and agitated air, as though something crucial hinged on our meeting. Was it because few people took much interest in his work?
"That's a strange ruler," I said. "It has no marks on it."
"It has no marks," Pygonopolis explained, "because I am not measuring these stones in the usual way. I am not interested in the dimensions of the stones or of the building itself. I am simply curious to know in what unit the builders worked. If measurements come out even with this pechya ruler, then the builders must have used the pechya. If not, I will try some other unit." Pygonopolis glanced at an assortment of bronze rulers leaning against a restored pillar. The ancient Greeks, he explained, had no less than 20 different units of measure.
"Yet even while I satisfy my curiosity, I am following in the footsteps of the great Pythagoras himself." Leaving this mysterious remark hanging in the air, he brandished the ruler he was holding. "Let us see if the temple was based on the pechya. If it doesn't work, I will try next the pygon. I will start all over, so you can see what I am up to."
He strode to the rear of the temple, then knelt once again on the floor, ruler and pencil in hand. He laid the ruler on one of the great, square paving stones, its end flush with one side. The ruler stretched a little more than halfway across the stone. Pygonopolis made a small pencil mark on the stone at the front end of the ruler, then slid it expertly along its own length until the back end met the pencil line exactly.
The front end of the ruler now fell well beyond the crack between stones. I ventured aloud that perhaps the pechya was not the proper measure for this temple. Pygonopolis only grunted something about waiting and seeing. He made a new mark, shifted the ruler again, and seemed unperturbed when its front end stopped nowhere near the next crack. He proceeded to march the ruler along the row of stones toward the front of the temple, talking as he went.
"Sooner or later, if this is the right unit, the front of the ruler will again come even with one of the cracks. Of course, the pechya may be incorrect.... Well, well. What do you know!"
Near the portico, the front end of his ruler had met one of the cracks exactly. Excitedly, Pygonopolis took a small notebook from his breast pocket and made an entry.
"This was lucky," he commented. "There was no guarantee that the first ruler I should try would work. Let us see: In the process of measuring, of waiting until the ruler and the stones came even once again, I crossed 5 stones. At the same time, I measured 8 pechyas. From these facts we can deduce the exact size of the stones in pechyas, can we not?"
He stared at me expectantly. I would subsequently learn that whenever he ended a question this way, he expected me to answer it. Hastily, I set myself thinking. If 5 stones were, all together, 8 pechyas wide, then 1 stone must be 1/5 of this distance, or 8/5 of a pechya. I blurted the answer. "Eight fifths of a pechya or, if you prefer, 1 3/5 pechyas."
"Yes and no. I had better say something about ancient arithmetic. The classical Greeks had no sophisticated number system like ours. Their way of writing numbers symbolically was roughly equivalent to Roman numerals and not at all suited for calculation of any kind. Moreover, they had no way to express fractions such as 8/5. Instead, they would speak of ratios of integers, such as 8to5.
"The important thing to notice here is that the measurement process came out even. With the ruler, I was measuring off an everincreasing distance in pechyas. Meanwhile, I was traversing an everincreasing distance in stones. Then, suddenly, the two distances coincided. Whenever this happens, you have a common measure, a certain length in which both measurements are integers, what some people call whole numbers. The whole numbers in this case are 8 and 5. The common measure is 1/5 of a pechya. The pechya consists of 5 of these units, and each stone consists of 8 of them."
"So, did the builders of this temple work in fifths of a pechya?" I asked.
"It is entirely possible," stated Pygonopolis, "but what really interests me is not what unit the builders worked in, but something much deeper. In the end, it is not the pechya that counts but another measure, a special unit in which all measurements would come out as integers."
"I don't quite follow!" I interjected. I was becoming somewhat confused.
Pygonopolis suddenly leaned forward with a conspiratorial air. "It is entirely possible," he continued in a hushed voice, while he looked around him, "that the young Pythagoras himself stood in this very temple and measured these stones. He once did what I have just done. He was not determining the measure of the temple, either, but something far more profound." At this, the irrepressible Pygonopolis hurried me to the front of the temple where we could gaze at the Aegean Sea.
"Look over there!" He pointed to a long, mountainous island just across a strait to the west. "That is Samos, where Pythagoras was born, about 582 B.C." Pygonopolis swept his arms expansively along the strait. "At that time, this whole area, from north to south, was known as Ionia, a loose confederacy of Greek cities. Here, in the temple of Apollo Delphineus, we stand in the middle of Miletus, the most powerful city of Ionia, a center of trade and home to many philosophers in the real sense, men who interested themselves in everything. Here lived the great Thales, mathematician and teacher of the young Pythagoras. Thales was a merchant and a great traveler. From Egypt, from Arabia, and from the far Indus, he brought the mathematical riches that would become the foundations of Greek mathematics. And none was more influential in laying these foundations than Pythagoras himself. But there's much more to this story than mathematics, make no mistake!
"Somehow, perhaps through the influence of Thales, Pythagoras became convinced of an amazing doctrine, one that bears directly on your question concerning the independent existence of mathematics. Not only did mathematics have an independent existence, as far as Pythagoras was concerned, but it also had a powerful influence on existence itself, answering your second question. Pythagoras believed that what we call the real world was not merely measured by number, not merely described by number, but it was actually made of numberand, I might add, not just any numbers, but whole numbers, or integers. You could call it the integral universe. You could even call it a kind of digital universe.
"Can you imagine what this means? The whole idea is far more audacious than the timid doctrine of Democritus who, 100 years later, proposed a world made of atomshard, indivisible units. These were material units, after all, whereas the units Pythagoras proposed were immaterial, the integers. Can you imagine anything more immaterial than numbers? What a concept! Believe me, my friend, we are still catching up to Pythagoras."
These ideas swept around me, flooding me in a turbulent current. It was more than I had bargained for. There was also something of the impresario about Pygonopolis, something I could not wholly trust. We sat down on the temple steps, gazing out at Samos, while Pygonopolis caught his breath. Slowly, Miletus of old seemed to come alive around us, haunted by ideas that would never die.
"I have reason to believe that Pythagoras came here and to other places where he could experiment with, uh, commensurability Ah, English! What an ugly word is commensurability. You know English, so you know what means this word, do you not?"
"Umm, let's see." I struggled to recall the definition. "Two lengths are commensurable if they have a common measure?"
"Just so. The pechya and one of those stones have commensurable lengths because they have a common measure, the 1/5 pechya."
I interrupted, "If you will permit a remark, most people see no need for such a difficult concept as commensurability because they think that any two lengths have a common measure, do they not?" (He had me doing it.)
"Just so. And for this they can surely be forgiven, for Pythagoras himself certainly thought so at one time. But I am getting ahead of myself.
"Commensurability is more easily grasped if you turn things around for a moment. Start with the unit. Suppose I have some unit, it doesn't matter which unit, perhaps it is very small. If I make two integer lengths out of this unit, any two integral lengths, the lengths will be commensurable. Suppose the lengths are 5 units and 8 units. If your ruler is 5 units long and the stones are 8 units wide, your measuring process is absolutely guaranteed to come out even, as it did when I measured the temple floor. As I moved the ruler into successive positions, I measured off an accumulating total length in fifths of a pechya:
"Now the widths of the stones were also adding up as I crossed them:
"You see, I arrived at a common number, 40. Sooner or later, the 5unit ruler matched the 8unit stones. The measuring process eventually came out even because the two lengths have a common unit. It had nothing to do with the integers themselves, as long as they are integers.
"This ruler is commensurable with those stones behind us because my measurements finally came out even. The connection is not obvious, of course. I will explain it later. The point is that the end of the ruler finally met a crack. Yet in a mathematical sense, there was no builtin guarantee that the ruler would ever do that, even if the temple floor extended infinitely! If the ruler ever meets a crack on an infinitely tiled floor, the two lengths, that of the ruler and that of the tile, are commensurable. They would turn out to have a common measure, like our 1/5 pechya.
"But we have been very sloppy. We must put some meat on the bones of your definition by providing a precise test for the commensurability of two lengths. We will dispense with the stones altogether and speak instead of just two rulers. They are not real rulers, of course, just two strips of metal, each having a specific length. We will say that one of the rulers has length X and the other length Y. You may substitute any two specific measurements for X and Y that you like. What I am about to say will apply just as well to those two lengths.
[ILLUSTRATION OMITTED]
"This commensurability test is like a kind of game. We play the game with the two rulers. We begin by placing the back ends of the two rulers even with each other. We then slide the shorter of the two rulers ahead until its rear end comes exactly even to where its front end was. In fact, I have just given you the only rule of the game: Always take the ruler whose front end is behind and slide it forward by precisely its own length. That's it. The question is, will the two rulers ever come out even, their front ends matching? If they do, you win. The two lengths X and Y are, uh, commensurable. If the two rulers never come even, you lose. In such a case, the rulers are not commensurable."
"Hmmm," I grumbled. "You could end up playing forever, could you not?"
"Theoretically, of course, but we know this only as the beneficiaries of modern mathematics. We know that there are pairs of lengths for which the ruler game will never end, but Pythagoras did not know that. He knew, of course, that it was a theoretical possibility, but he believed that it would never happen. He believed that the world was structured in such a way that no matter what rulers you started with, you would always win the ruler game.
"As I mentioned earlier, the Pythagorean universe was based on integers. In practical terms, this meant that all lengths, whether of stones, rulers, or anything else, were ultimately integers. There was a fundamental unit in which all things would prove to have an integral measure. One possible test of this theory would be the ruler game. In such a world, it must always end in victory.
"This concept of a fundamental unit unified arithmetic and geometry in a particularly simple way. Arithmetic is about numbers, and geometry is about lengths. For every length, there was a privileged number, an integer, which expressed it. And every integer would, sooner or later, turn out to be the length of something or other.
"For Pythagoras, as well as for Thales and other early Greeks, arithmetic and geometry were already regarded as aspects of the same fundamental reality. A basket of figs always contained some definite number of figs, and a stone always had a definite size. Now the first kind of number was an integer. But what sort of number could one assign to the stone? Every ruler gave a different length, depending on the units it employed, and rarely did the dimension of a stone turn out to be an exact integer. It was far from obvious that there existed a privileged ruler, one marked off in these fundamental units I have been speaking of, by which the length of that stone, of all stones and everything else, would come out as integers."
Pygonopolis paused. "What I'm going to tell you now, you must listen carefully. Never mind the tape recorder. You will see all of Greek mathematics spin from this story, like the Golden Fleece of the sun.
"First I am going to show how Pythagoras would have proved the intimate connection between the ruler game and his integral universe. But that is hardly more than a sideshow compared to what follows. His integral universe collapsed when he discovered a pair of incommensurable lengths. For Pythagoras, it was a firstclass crisis. A certain little diagram from Egypt had two lengths that, it could be proved, were not commensurable."
Clouds were gathering over Samos. Pygonopolis cast a worried glance at them.
"To begin with, what is the connection between the ruler game and the integer universe? Briefly, it is this. In the integral universe, you always won the ruler game. Conversely, if you always won the ruler game, you must be in an integral universe. It would not have taken Pythagoras long to prove that."
He had paused for breath again, so I interjected, "I'm curious to know how Pythagoras could have arrived at such a proof if the early Greeks had no algebra and could not even multiply or divide numbers, let alone symbols."
"We moderns might use X and Y to represent the unknown lengths, then apply algebra to prove the result. You are quite right to point out that the early Greeks did not have algebra, or an efficient number system. But they had something nearly as good when it came to proving results. As far as numbers are concerned, Pythagoras used a kind of symbolic geometry in which numbers were represented by configurations of dots. The configurations might be lines, triangles, or rectangles, all made of dots. For example, you could represent the number 10 by 10 dots in a row, by a rectangle 2 dots wide and 5 dots high, or even by the famed tetractys figure, a triangle with 4 dots on the base row, 3 dots in the next row, 2 in the next, and 1 on top, forming the apex of the triangle. Which representation you used for a number would depend on what you wanted to do with it.
[ILLUSTRATION OMITTED]
"To represent algebraic ideas, such as ratios and products among unknown quantities, Pythagoras would use a geometrical figure, perhaps one that showed the successful outcome of the ruler game for two particular rulers. The diagram would show the positions taken up by the two rulers on their way to the final, successful outcome.
"By the way, I have no doubt that much of Greek mathematics, given its dependence on diagrams and geometry, was developed with Earth itself as the chalkboard. Archimedes is said to have been killed by a Roman soldier while pondering a problematic figure in the dirt. I sincerely hope there are no Roman soldiers about just now!" Pygonopolis sketched the following figure:
[ILLUSTRATION OMITTED]
"To be sure, no one knows just how Pythagoras proved things. Only one thing is certain. The use of diagrams as part of formal proofs marks the singular success of Greek mathematics. It is a great strain to hold a detailed image of a problem in mind while pondering its components. To relieve the brain of this burden, the ancient Greeks learned to render the diagram with suitable precision in the dirt. This was a technological breakthrough of sorts. Their genius lay in applying geometrical thinking of one kind or another to these diagrams, replacing the algebra they did not possess by geometrical logic, which they did possess.
"Here is a case in point. How much easier it is to reason about the ruler game with such a diagram before one! Pythagoras would stare at it for at least a few minutes, mumbling to himself about the two lengths. Sooner or later, he would say Aha! He had found a proof that the ratio of the lengths of the long ruler to the short one was a ratio of two integers. From there it was but a short step to deducing the existence of a common measure, as we shall see."
"In his first and crucial step, Pythagoras would have matched each short ruler in the upper row with a corresponding long ruler in the lower one, noting that by the time he had counted his way to the end of the lower row, he would still be short of the end in the upper onelike so."
Pygonopolis marked off the corresponding rulers with X's in the figure.
[ILLUSTRATION OMITTED]
"Now we have the lower row all crossed out and only a portion of the upper row so treated. However, because both rows contain the same number of crossedout rulers, the ratio of the lengths of these rows must be the same as the ratio of the lengths of the rulers that compose them. Is that not so?"
It was, I said, perfectly clear. Dividing both integers of the ratio by this number would have no effect on the ratio. Although this was modern thinking, or seemed to be, I let the point pass. Presumably, the early Greeks had a geometric proof of this idea.
"Now, see how pretty this is! Pythagoras next imagined the long rulers of the upper row all shrinking until they had the length of the short rulers." Hastily, he drew a new figure in the dirt.
[ILLUSTRATION OMITTED]
"Now you see what is going on, do you not?"
When I noticed that the shortened lower row had the same length as the crossedout upper row in the previous diagram, I felt like saying "aha" myself. Mutely, I pointed to the two short rows, one in each diagram.
"Exactly. They are equal! In both cases, the ratio of the longer row to the shorter one is the same. In the previous figure we saw that this was simply the ratio of the long ruler's length to that of the short one. In the second figure, it is the ratio of the number of short rulers to the number of long ones. But these two numbers are integers. Therefore the ratio of the lengths of the two rulers is an integer ratio."
Pygonopolis had delivered the main proof, but there was something left. I pressed him to explain why the integer ratio meant the two lengths had a common measure.
"That part is the easiest. Simply divide the longer ruler into as many equal units as the larger of the two integers entering the ratio. Similarly, divide the shorter ruler into as many equal units as the smaller of the two integers. Because the ratio of the lengths equals the ratio of the number of units composing each ruler, the two kinds of units must be the same."
The proof had not been a difficult one, but my head spun a little, as if I had received a brain transplant. Early Greek mathematics had a very different feel from modern, algebraic reasoning. I ventured a question: "You have just shown me how Pythagoras might have proved the ruler gamethat is, winning the ruler game was tantamount to the existence of a common measure. We in the modern age might proceed differently. We would work with the symbolic ratio X/Y and use algebra to prove the result. I feel a bit foolish asking this, but I must. Why should two entirely different systems of thought arrive at the same conclusion?"
Rather than becoming impatient with my question, as I feared he might, Pygonopolis looked pleased.
"This illustrates how two completely different trains of mathematical thought arrive at the same station, so to speak. It is a firstclass phenomenon, when you think about it. Two completely different approaches to a problem, our modern algebraic approach and the old geometric one, lead to precisely the same result. Is it a coincidence? If you view mathematics as a purely cultural activity, you will miss a crucial point: It is not, in my view, a coincidence." Then he laughed.
"When some people talk about the cultural element in Greek mathematics, I fear they imagine Pythagoras dancing on the beach like Zorba, with a bouzouki playing in the background."
A brief rumble of distant thunder rolled across the strait from Samos, where clouds were gathering. Pygonopolis shivered slightly, staring down in silence at the diagrams. This was my opening.
"If it is not a coincidence, what is it?" I asked.
"It is essentially the phenomenon of independent discovery, the same idea finding a completely different expression by two people or groups of people separated by space, by time, or by culture. The phenomenon has been repeated thousands of times throughout the history of mathematics, and it points to something very special going on in mathematics. I suppose my own beliefs on this point are not very different from those of Pythagoras. For even after his integral universe was shattered, Pythagoras continued to believe that mathematics had an independent existence, although not in a material sense. But what, I ask, did he call it?
"Pythagoras was a mystic in the traditional sensesomeone who practiced inner discipline to arrive at new levels of understanding. Perhaps I will say more about that tomorrow. In the meantime, I can tell you only my opinion: He surely had a name for the place where mathematics exists. I have tried to imagine what this name might be. My best guess is the Holos."
"The Holos?" I repeated, as this was an unfamiliar word.
"The Holos is the place of mathematics. It stands in a special relation to the cosmos. Holos the source, cosmos the manifestation."
Pygonopolis paused, breathless again. The new word echoed in my mind. The holos, the holos, a beautiful word, pronounced with the Greek letter chi, a rasping H, followed by an ululation.
"Earlier you described the Pythagorean universe of integers," I remarked, "but all along you've been hinting at a tragedy. What happened?"
"As I said before," he responded in a patient tone, "the major underpinning of the integer universe, as Pythagoras imagined it, was what we could call the hypothesis of cosmic commensurability: Every two lengths were commensurable, not just in practice, but also in principle. There can be little doubt that during the time that Pythagoras believed the hypothesis, he bent every effort to proving it. He worked geometrically, trying one approach after another, but all his efforts came to nothing. No matter how much he wanted the hypothesis to be true, he could not prove it. Nevertheless he continued to imagine that the integers, specifically the number one, was the atomos from which the gods made everything. Ah, what a blow it was!"
"What happened?" I asked.
"His supreme vision was shattered when Pythagoras found the first pair of incommensurable magnitudes. Perhaps it was his old teacher, Thales, who suggested that Pythagoras check the commensurability of the side of a square with its diagonal. See, here it is.
[ILLUSTRATION OMITTED]
"If the universe was based on integers, all pairs of lengths would be commensurable, including the two lengths in this innocentlooking little diagram. One of the lengths is the side of the square, all four sides having the same length. The only other length in the figure is that of the diagonal. It doesn't matter what size you draw the figure, as we are concerned only with the ratio of the two lengths. Was it an integer ratio or not?
"Pythagoras may have puzzled over this longer than he should have. Sometimes mathematicians are slow to discover the truth about a pet idea because they fondly imagine it to be true and are continually trying to find a proof of it. They never seriously seek to disprove it. But now Pythagoras had a test case to consider. How long did it take him to realize that it was what we call a counterexample?
"One day it came. The discovery staggered him, for it brought the integerbased cosmos to a huge nothing. Once he had gotten over his shock, he felt immense gratitude that at last the question of commensurability had been settledin the negative, as it turned out. Up to this time, Greek mathematics recognized only two kinds of numbers, the integers and their ratios. Now there appeared to be a mysterious third kind of number, one that called for a revision in thinking. A new world had opened.
"Here's where the cultural element comes into play: His gratitude was so great, he went to a templeperhaps this very oneand sacrificed an ox. We moderns do not understand sacrifice, by the way. Imagine feeling so grateful for some wonderful event that, to relieve your heart of its burden of joy, you buy a Mercedes and set it on fire!
"The argument that Pythagoras used to show the incommensurability of the side of a square with its diagonal is quite simple when you write it in modern symbolism, but we will prove it more or less in the way that Pythagoras did. We will not use algebra, then, but we will allow letter names for parts of the diagram. In particular, call the short side X and the long side Y. These, you will agree, are not algebraic variables. We begin with the very figure that Thales showed Pythagoras."
Pygonopolis stabbed Thales's figure with his stick.
"Thales had been to Egypt and had learned many wonderful things from the Egyptian priests, including this interesting little fact about the side of a square and its diagonal." Pygonopolis drew a second square tilted in relation to the first. One of the sides of the new square was the diagonal of the first one.
[ILLUSTRATION OMITTED]
"The Egyptians, who labored under the same restrictions as the Greeks, had been clever enough to discover a curious relationship between the two squares. The larger one has twice the area of the smaller one. "The Egyptian proof was simple. You simply add 2 new lines, like so, and realize that the large square has been divided into 4 small triangles, while the small square is already divided into 2 of the same triangles: 4 is twice 2. Quod erat demonstrandum, as it says in the old texts."
[ILLUSTRATION OMITTED]
Pygonopolis carefully smoothed away the two construction lines, restoring the earlier figure. Absentmindedly, he brushed his hand on his suit, then swore. "Agh," he said, "what a stupid thing to do. See what I've done!" He spent a few moments brushing the smudge on his suit, frowning.
"After a certain amount of the usual fumbling around that mathematicians go through, Pythagoras found the first step of his proof. If he assumed that X and Y were commensurable, then both X and Y had integer lengths in the unit of their commensurability. He also insisted that these numbers be the smallest ones having this property; this meant that the integers could not have a common factor.
"He could visualize not only the lines X and Y as rows of dots (the units) but also both squares made up of those dots. In particular, the number of dots in the large square was an even number, being twice the number of dots in the small square. Pythagoras then asked himself, `Can an odd number be squared to produce an even one?'"
"My dear Professor Pygonopolis," I interjected. "I thought you said that Pythagoras had no algebra, and I assumed that meant no squaring."
"No, no, no, Dewdney. As I explained earlier, the ancient Greeks could multiply by geometry, and this meant the operation of squaring, as well. In this case, he drew a number as a row of dots. To square the number geometrically, he made a square of it, literally. He added more rows of dots above the first, as many as it took to produce a square shape. In fact, this is where the English word squaring comes from. In any case, the total number of dots in the square is the product of the number of dots along the bottom and along one vertical side.
"Pythagoras undoubtedly already knew, and had proved, that when you squared an odd number in this manner, the total number
[ILLUSTRATION OMITTED]
of dots in the square was always odd. And, when you squared an even number, the result was always even. Now, according to the Egyptians, the larger square had twice the area of the smaller one. This meant that its area, the number of dots in it, was even. But, as we have already seen, this could only have been the case if the length of the side being squarednamely Ywas also an even number.
"Now the pace picks up. If Y had an even number of dots, then its square would have not merely a twofold (even) number of dots but a fourfold one. In modern language, this means that the square of Y was a multiple of 4.
"Now recall the Egyptian theorem: The square of Y was twice the square of X. Yet the square of Y is also a fourfold number, a multiple of 4. This meant that the square of X must have been a twofold number, or multiple of 2. You see where this is going, do you not?"
"Are you going to apply the same reasoning all over again to X?" I guessed.
"Exactly. Pythagoras could now apply the same reasoning to X as he had to Y, concluding finally that both lengths consisted of an even number of the fundamental units that made them up. This meant that if you cut each of the two integers in half, you would get new, smaller integers with the same property: Their ratio would be X to Y again. Because the integers in question were already the smallest possible, however, this was a contradiction. The logic refused to cooperate further. The machine had ground to a halt. In such cases, the Greek mathematicians, no less than we moderns, knew that one of the assumptions going into the proof must be wrong. There was only one assumption madethat the lengths of X and of Y were commensurable. The contradiction meant that they couldn't be."
Pygonopolis sighed, seemingly to catch his breath. "Can you imagine? Can you just imagine this moment for Pythagoras? There was no doubting the new result. Instead of proving the longsought theorem, `Every pair of lengths is commensurable,' he had proved exactly the reverse: `There exists a pair of incommensurable lengths.' Although it doomed his doctrine, at least in its existing form, I dare say that Pythagoras was secretly delighted. He sensed higher ground ahead, as though scaling Olympus itself. The numerical atomos was deeper and more complicated than he had thought. There was another kind of number lurking in the holos and, therefore, in the cosmos. It was not an integer, nor was it a ratio of integers. We moderns call such numbers irrationals, meaning only that they are not rational numbers."
Storm clouds were gathering across the strait with Samos. Thunder rolled more and more frequently across the sea.
It was now late afternoon. Pygonopolis picked up his hat and strolled into the temple. I stood in a trance, watching him collect his briefcase, then stride out to the steps again, smiling broadly.
"I feel as though I have overwhelmed you in some way. It is exciting, this story of Pythagoras and the incommensurables. It shows so many things about early mathematics, yet we only have to stretch our imagination a little to understand how the early mathematicians must have worked."
I nodded as we parted for our separate cars. The rain fell heavily as we each drove north to Izmir. I had been hoping to ponder the things I had learned today as I drove but had no calm moments for reflection. Instead, I was hard put to keep up with Pygonopolis's rented Mercedes. He drove like a maniac, in spite of the bad road conditions. I was vastly relieved when we reached the outskirts of Izmir.
We had supper this night, Pygonopolis and I. We dined on Mediterranean seafood in the hotel. By asking Pygonopolis about the holos, eating as he talked, I succeeded in finishing long before he did.
"But just what is the holos, do you think?" I asked.
"The holos is the place where all of mathematics, both known and as yet unknown, exists," he responded cheerfully. "Here are the definitions, the axioms, the rules of deduction, the theorems, and the proofs. Here, too, are all the numbers, number systems, sets, families of sets, and on and on and on."
"But what I really want to know," I pushed my point, "is how such things can exist. Do they exist independently, like this chair?"
"The answer is subtle," he said, "for the existence of these things seems to depend on a human mind, yet it doesn't. Think of the number 3, for example. Wherever there are 3 of anything, the number 3 is also present, and not just as a concept.
"Here, for example, are 3 prawns left on my plate. Just 3. The threeness of these prawns will control how many more prawns I can eat without ordering more. In short, I cannot eat more prawns than are on my plate. Not only is the threeness of the prawns manifest to our senses and minds, but it also has an operational significance that goes beyond my conception of threeness. My inability to eat more than 3 prawns from my plate has nothing to do with my conception of threeness, or even the fact that it is me sitting here. Anyone else would face the same options. Likewise, if I order 10 more prawns, I may calculate that the number of prawns on my plate will then be 13. In this and in many other ways both simple and complex does mathematics control the world. In this way do holos and cosmos interlock."
"If I understand you correctly," I followed up, "the holos is a real place, although not in our ordinary sense of being locatable in our universe, or cosmos. Yet it also controls, to some degree at least, what goes on in our world, in the cosmos. What I'm not clear about is how much of this theory is due to Petros Pygonopolis and how much to Pythagoras."
"The theory of the holos, as I describe it, is entirely my own. An extended fantasy, if you like. Yet I cannot help but believe that Pythagoras thought of mathematics in pretty much those terms. He had seen how numbers evanesce when you consider what it is that unites all collections of 3 things. He had also seen how lines vanish as they are drawn ever more accurately, finely incised on smooth, flat stones. He had witnessed then, as anyone can today, how these concepts retreat when they are pursued, as though fleeing back to the holos. Yet they advance to take control at other times."
"This afternoon you made a joke about Pythagoras dancing to bouzouki music on the beach." I had not understood this earlier reference. "I assume he didn't actually do this, so what role did early Greek culture actually play in Greek mathematics?"
"Let me make an analogy. Mathematics is like the wheel. Almost every culture has its wheel, and the wheels made by different cultures all look different. An Egyptian chariot wheel is quite different from that on a medieval European oxcart, and both wheels are again different from those on a modern automobile. Yet all wheels operate on exactly the same principles.
"Nevertheless, cultural clues are crucial to understanding Greek mathematicsnot so much its validity or universality as its direction. On one hand, you will find nothing discovered by my forebears that could not have been discovered by a Solomon Islander. But you might also find that the Solomon Islander would not be very interested in Pythagoras's problems, so he would be very unlikely to pursue them. It is early Greek culture that molded Pythagoras's mind. At the center of this culture resided the gods. He accepted them as perfectly real, and his deepest ponderings on the ultimate nature of the cosmos necessarily included the gods. He believed in the idea of a controlling presence such as Zeus, assisted by other presences in the pantheon. But this was the horse that pulled his cart and not the cart itself, so to speak. It motivated his search, even inspired it, but it did not play any role in what Pythagoras actually discoveredunless of course ..."
Pygonopolis had begun to perspire heavily after finishing his meal. He paused to pass a handkerchief over his face.
"Let us just say that as far as Pythagoras was concerned, the controlling presence had left clues about itself, and Pythagoras, eager to scale Olympus, cast himself in the role of cultural hero. He saw in mathematics the path to knowledge such as only gods enjoy. Logical consequences resided in the very fabric of existence. It was surely how the gods worked."
Pygonopolis stared intently at me with his dark brown eyes. The hair on my neck stood up, and I felt, just for a moment, that he knew far more than he was telling. It occurred to me that he, Pygonopolis, actually believed in the gods of oldbut then, abruptly, the mood passed.
"Tomorrow morning, we will take the plane for Athens. It is late."
Pygonopolis looked at his watch and, in so doing, revealed the palm of his left hand. There was a tattoo therea small blue star. I glanced away quickly as Pygonopolis looked back at me from the watch.
As we parted, be observed, "You cannot know what it means to me to have a real listener."
Table of Contents
First Chapter
Izmir, Turkey, June 22, 1995
A strange day! I spent it in Miletus, or what used to be Miletus in 500 B. C. It was there, if anywhere, that mathematics became a science. Miletus was the unrivaled center of commerce, of philosophy, and of the arts. Here lived Thales, the first scientist; Anaximander, the philosopher; and Timotheus, the poet. Here, too, the great Pythagoras visited from his native Samos, to learn and to teach.
First, let me backtrack, however. I arrived yesterday in Athens and changed planes for a onehour flight across the Aegean, which took me to Izmir, Turkey, an hour's drive north of Miletus. This morning, in Izmir, I rented a Fiat Uno and drove south through succession of beautiful valleys in supernaturally hot weather. I came at last to the Aegean coast, following it through relentless humidity until I arrived at Miletus, a collection of ruins with numerous tourist signs and a weedfilled parking lot. The site of the ancient city has been largely lost to silting and erosion by the Meander, the prototype of every river that twists and turns. I parked outside fencedoff area and made my way through partial restorations of the ancient city to the temple of Apollo. There, some tourists were gathering their bags and cameras, departing singly and in groups for a waiting bus.
There are times, when visiting old places, that hints of an ancient presence overwhelm you, like ghosts, in broad daylight. You cannot be with a tour group to have this feeling, you must be alone. It came over me as I stood before the temple of Apollo Delphineus, its steps and pillars haunted by memories not my own. I looked around for Petros Pygonopolis, the man I was supposed to meet, but there was no one in sight.
As I ascended the steps to the temple floor, I saw a man kneeling as if in prayer, hunched over the perfectly fitted, square paving stones. As I quietly approached, I could see that he was measuring the stones with a bronze ruler. He was a large man, with a shock of black hair, gray at the temples, and an olive complexion. He looked strangely out of place, for he wore an elegant white suit. When cleared my throat, he looked up, startled. His stared at me confusedly from beneath bushy brows, then retrieved a pair of glasses, which he put on. His face was suddenly wreathed in a generous smile. Who could it be but Pygonopolis? He rose to dust the knees of his trousers, then bowed.
"You must be Dewdney! Excuse me for not seeing you. Thank the gods those tourists have finally gone. This is not the thing to wear for fieldwork, no? Of course not!" he answered his own question. "I am Petros Pygonopolis, historian of science and specialist in Greek mathematicsthat is, ancient Greek mathematics. "
We shook hands and stepped back to examine each another.
"Welcome to Miletus, and welcome to your intellectual roots," Pygonopolis continued. "The questions you have asked in your letter are the right ones, in my humble view. Is mathematics discovered or is it invented? Does it have an independent existence? How refreshing that people can still ask such questions! The answers, such as they are, begin with what I was just doing, measuring these stones with this ruler." He held up the bronze strip. "It is the pechya, or cubit, that we Greeks once used."
While most personable and charming, Pygonopolis had a nervous and agitated air, as though something crucial hinged on our meeting. Was it because few people took much interest in his work?
"That's a strange ruler," I said. "It has no marks on it."
"It has no marks," Pygonopolis explained, "because I am not measuring these stones in the usual way. I am not interested in the dimensions of the stones or of the building itself. I am simply curious to know in what unit the builders worked. If measurements come out even with this pechya ruler, then the builders must have used the pechya. If not, I will try some other unit." Pygonopolis glanced at an assortment of bronze rulers leaning against a restored pillar. The ancient Greeks, he explained, had no less than 20 different units of measure.
"Yet even while I satisfy my curiosity, I am following in the footsteps of the great Pythagoras himself." Leaving this mysterious remark hanging in the air, he brandished the ruler he was holding. "Let us see if the temple was based on the pechya. If it doesn't work, I will try next the pygon. I will start all over, so you can see what I am up to."
He strode to the rear of the temple, then knelt once again on the floor, ruler and pencil in hand. He laid the ruler on one of the great, square paving stones, its end flush with one side. The ruler stretched a little more than halfway across the stone. Pygonopolis made a small pencil mark on the stone at the front end of the ruler, then slid it expertly along its own length until the back end met the pencil line exactly.
The front end of the ruler now fell well beyond the crack between stones. I ventured aloud that perhaps the pechya was not the proper measure for this temple. Pygonopolis only grunted something about waiting and seeing. He made a new mark, shifted the ruler again, and seemed unperturbed when its front end stopped nowhere near the next crack. He proceeded to march the ruler along the row of stones toward the front of the temple, talking as he went.
"Sooner or later, if this is the right unit, the front of the ruler will again come even with one of the cracks. Of course, the pechya may be incorrect. . . . Well, well. What do you know!"
Near the portico, the front end of his ruler had met one of the cracks exactly. Excitedly, Pygonopolis took a small notebook from his breast pocket and made an entry.
"This was lucky," he commented. "There was no guarantee that the first ruler I should try would work. Let us see: In the process of measuring, of waiting until the ruler and the stones came even once again, I crossed 5 stones. At the same time, I measured 8 pechyas. From these facts we can deduce the exact size of the stones in pechyas, can we not?"
He stared at me expectantly. I would subsequently learn that whenever he ended a question this way, he expected me to answer it. Hastily, I set myself thinking. If 5 stones were, all together, 8 pechyas wide, then 1 stone must be 1 /5 of this distance, or 8 /5 of a pechya. I blurted the answer. "Eight fifths of a pechya or, if you prefer, 1 3 /5 pechyas."
"Yes and no. I had better say something about ancient arithmetic. The classical Greeks had no sophisticated number system like ours. Their way of writing numbers symbolically was roughly equivalent to Roman numerals and not at all suited for calculation of any kind. Moreover, they had no way to express fractions such as 8 /5 . Instead, they would speak of ratios of integers, such as 8to5.
"The important thing to notice here is that the measurement process came out even. With the ruler, I was measuring off an ever increasing distance in pechyas. Meanwhile, I was traversing an everincreasing distance in stones. Then, suddenly, the two distances coincided. Whenever this happens, you have a common measure, a certain length in which both measurements are integers, what some people call whole numbers. The whole numbers in this case are 8 and 5. The common measure is 1 /5 of a pechya. The pechya consists of 5 of these units, and each stone consists of 8 of them."
"So, did the builders of this temple work in fifths of a pechya?" I asked.
"It is entirely possible," stated Pygonopolis, "but what really interests me is not what unit the builders worked in, but something much deeper. In the end, it is not the pechya that counts but another measure, a special unit in which all measurements would come out as integers."
"I don't quite follow!" I interjected. I was becoming somewhat confused.
Pygonopolis suddenly leaned forward with a conspiratorial air. "It is entirely possible," he continued in a hushed voice, while he looked around him, "that the young Pythagoras himself stood in this very temple and measured these stones. He once did what have just done. He was not determining the measure of the temple, either, but something far more profound." At this, the irrepressible Pygonopolis hurried me to the front of the temple where we could gaze at the Aegean Sea.
"Look over there!" He pointed to a long, mountainous island just across a strait to the west. "That is Samos, where Pythagoras was born, about 582 B. C." Pygonopolis swept his arms expansively along the strait. "At that time, this whole area, from north to south, was known as Ionia, a loose confederacy of Greek cities. Here, in the temple of Apollo Delphineus, we stand in the middle of Miletus, the most powerful city of Ionia, a center of trade and home to many philosophers in the real sense, men who interested themselves in everything. Here lived the great Thales, mathematician and teacher of the young Pythagoras. Thales was a merchant and a great traveler. From Egypt, from Arabia, and from the far Indus, he brought the mathematical riches that would become the foundations of Greek mathematics. And none was more influential in laying these foundations than Pythagoras himself. But there's much more to this story than mathematics, make no mistake!
"Somehow, perhaps through the influence of Thales, Pythagoras became convinced of an amazing doctrine, one that bears directly on your question concerning the independent existence of mathematics. Not only did mathematics have an independent existence, as far as Pythagoras was concerned, but it also had a powerful influence on existence itself, answering your second question. Pythagoras believed that what we call the real world was not merely measured by number, not merely described by number, but it was actually made of numberand, I might add, not just any numbers, but whole numbers, or integers. You could call it the integral universe.
You could even call it a kind of digital universe. "Can you imagine what this means? The whole idea is far more audacious than the timid doctrine of Democritus who, 100 years later, proposed a world made of atomshard, indivisible units. These were material units, after all, whereas the units Pythagoras proposed were immaterial, the integers. Can you imagine anything more immaterial than numbers? What a concept! Believe me, my friend, we are still catching up to Pythagoras."
These ideas swept around me, flooding me in a turbulent current. It was more than I had bargained for. There was also something of the impresario about Pygonopolis, something I could not wholly trust. We sat down on the temple steps, gazing out at Samos, while Pygonopolis caught his breath. Slowly, Miletus of old seemed to come alive around us, haunted by ideas that would never die. "I have reason to believe that Pythagoras came here and to other places where he could experiment with, uh, commensurability. Ah, English! What an ugly word is commensurability. You know English, so you know what means this word, do you not?"
"Umm, let's see." I struggled to recall the definition. "Two lengths are commensurable if they have a common measure?"
"Just so. The pechya and one of those stones have commensurable lengths because they have a common measure, the 1 /5 pechya."
I interrupted, "If you will permit a remark, most people see no need for such a difficult concept as commensurability because they think that any two lengths have a common measure, do they not?" (He had me doing it.)
"Just so. And for this they can surely be forgiven, for Pythagoras himself certainly thought so at one time. But I am getting ahead of myself.
"Commensurability is more easily grasped if you turn things around for a moment. Start with the unit. Suppose I have some unit, it doesn't matter which unit, perhaps it is very small. If I make two integer lengths out of this unit, any two integral lengths, the lengths will be commensurable. Suppose the lengths are 5 units and 8 units. If your ruler is 5 units long and the stones are 8 units wide, your measuring process is absolutely guaranteed to come out even, as it did when I measured the temple floor. As I moved the ruler into successive positions, I measured off an accumulating total length in fifths of a pechya:
5 10 15 20 25 30 35 40
"Now the widths of the stones were also adding up as I crossed them:
8 16 24 32 40
"You see, I arrived at a common number, 40. Sooner or later, the 5unit ruler matched the 8unit stones. The measuring process eventually came out even because the two lengths have a common unit. It had nothing to do with the integers themselves, as long as they are integers.
"This ruler is commensurable with those stones behind us because my measurements finally came out even. The connection is not obvious, of course. I will explain it later. The point is that the end of the ruler finally met a crack. Yet in a mathematical sense, there was no builtin guarantee that the ruler would ever do that, even if the temple floor extended infinitely! If the ruler ever meets a crack on an infinitely tiled floor, the two lengths, that of the ruler and that of the tile, are commensurable. They would turn out to have a common measure, like our 1 /5 pechya.
"But we have been very sloppy. We must put some meat on the bones of your definition by providing a precise test for the commensurability of two lengths. We will dispense with the stones altogether and speak instead of just two rulers. They are not real rulers, of course, just two strips of metal, each having a specific length. We will say that one of the rulers has length X and the other length Y. You may substitute any two specific measurements for X and Y that you like. What I am about to say will apply just as well to those two lengths.
The Ruler Game
"This commensurability test is like a kind of game. We play the game with the two rulers. We begin by placing the back ends of the two rulers even with each other. We then slide the shorter of the two rulers ahead until its rear end comes exactly even to where its front end was. In fact, I have just given you the only rule of the game: Always take the ruler whose front end is behind and slide it forward by precisely its own length. That's it. The question is, will the two rulers ever come out even, their front ends matching? If they do, you win. The two lengths X and Y are, uh, commensurable.
If the two rulers never come even, you lose. In such a case, the rulers are not commensurable." "Hmmm," I grumbled. "You could end up playing forever, could you not?"
"Theoretically, of course, but we know this only as the beneficiaries of modern mathematics. We know that there are pairs of lengths for which the ruler game will never end, but Pythagoras did not know that. He knew, of course, that it was a theoretical possibility, but he believed that it would never happen. He believed that the world was structured in such a way that no matter what rulers you started with, you would always win the ruler game.
"As I mentioned earlier, the Pythagorean universe was based on integers. In practical terms, this meant that all lengths, whether of stones, rulers, or anything else, were ultimately integers. There was a fundamental unit in which all things would prove to have an integral measure. One possible test of this theory would be the ruler game. In such a world, it must always end in victory.
"This concept of a fundamental unit unified arithmetic and geometry in a particularly simple way. Arithmetic is about numbers, and geometry is about lengths. For every length, there was a privileged number, an integer, which expressed it. And every integer would, sooner or later, turn out to be the length of something or other.
"For Pythagoras, as well as for Thales and other early Greeks, arithmetic and geometry were already regarded as aspects of the same fundamental reality. A basket of figs always contained some definite number of figs, and a stone always had a definite size. Now the first kind of number was an integer. But what sort of number could one assign to the stone? Every ruler gave a different length, depending on the units it employed, and rarely did the dimension of a stone turn out to be an exact integer. It was far from obvious that there existed a privileged ruler, one marked off in these fundamental units I have been speaking of, by which the length of that stone, of all stones and everything else, would come out as integers."
Pygonopolis paused. "What I'm going to tell you now, you must listen carefully. Never mind the tape recorder. You will see all of Greek mathematics spin from this story, like the Golden Fleece of the sun.
"First I am going to show how Pythagoras would have proved the intimate connection between the ruler game and his integral universe. But that is hardly more than a sideshow compared to what follows. His integral universe collapsed when he discovered a pair of incommensurable lengths. For Pythagoras, it was a firstclass crisis. A certain little diagram from Egypt had two lengths that, it could be proved, were not commensurable."
Clouds were gathering over Samos. Pygonopolis cast a worried glance at them. "To begin with, what is the connection between the ruler game and the integer universe? Briefly, it is this. In the integral universe, you always won the ruler game. Conversely, if you always won the ruler game, you must be in an integral universe. It would not have taken Pythagoras long to prove that."
He had paused for breath again, so I interjected, "I'm curious to know how Pythagoras could have arrived at such a proof if the early Greeks had no algebra and could not even multiply or divide numbers, let alone symbols."
"We moderns might use X and Y to represent the unknown lengths, then apply algebra to prove the result. You are quite right to point out that the early Greeks did not have algebra, or an efficient number system. But they had something nearly as good when it came to proving results. As far as numbers are concerned, Pythagoras used a kind of symbolic geometry in which numbers were represented by configurations of dots. The configurations might be lines, triangles, or rectangles, all made of dots. For example, you could represent the number 10 by 10 dots in a row, by a rectangle 2 dots wide and 5 dots high, or even by the famed tetractys figure, a triangle with 4 dots on the base row, 3 dots in the next row, 2 in the next, and 1 on top, forming the apex of the triangle. Which representation you used for a number would depend on what you wanted to do with it.
Dot Diagrams for the Number 10
"To represent algebraic ideas, such as ratios and products among unknown quantities, Pythagoras would use a geometrical figure, perhaps one that showed the successful outcome of the ruler game for two particular rulers. The diagram would show the positions taken up by the two rulers on their way to the final, successful outcome.
"By the way, I have no doubt that much of Greek mathematics, given its dependence on diagrams and geometry, was developed with Earth itself as the chalkboard. Archimedes is said to have been killed by a Roman soldier while pondering a problematic figure in the dirt. I sincerely hope there are no Roman soldiers about just now!" Pygonopolis sketched the following figure:
The Two Rulers Come Even
"To be sure, no one knows just how Pythagoras proved things. Only one thing is certain. The use of diagrams as part of formal proofs marks the singular success of Greek mathematics. It is a great strain to hold a detailed image of a problem in mind while pondering its components. To relieve the brain of this burden, the ancient Greeks learned to render the diagram with suitable precision in the dirt. This was a technological breakthrough of sorts. Their genius lay in applying geometrical thinking of one kind or another to these diagrams, replacing the algebra they did not possess by geometrical logic, which they did possess.
"Here is a case in point. How much easier it is to reason about the ruler game with such a diagram before one! Pythagoras would stare at it for at least a few minutes, mumbling to himself about the two lengths. Sooner or later, he would say Aha! He had found a proof that the ratio of the lengths of the long ruler to the short one was a ratio of two integers. From there it was but a short step to deducing the existence of a common measure, as we shall see."
"In his first and crucial step, Pythagoras would have matched each short ruler in the upper row with a corresponding long ruler in the lower one, noting that by the time he had counted his way to the end of the lower row, he would still be short of the end in the upper onelike so."
Pygonopolis marked off the corresponding rulers with X's in the figure.
"Now we have the lower row all crossed out and only a portion of the upper row so treated. However, because both rows contain the same number of crossedout rulers, the ratio of the lengths of these rows must be the same as the ratio of the lengths of the rulers that compose them. Is that not so?"
It was, I said, perfectly clear. Dividing both integers of the ratio by this number would have no effect on the ratio. Although this was modern thinking, or seemed to be, I let the point pass. Presumably, the early Greeks had a geometric proof of this idea.
"Now, see how pretty this is! Pythagoras next imagined the long rulers of the upper row all shrinking until they had the length of the short rulers." Hastily, he drew a new figure in the dirt.
Shrinking the Longer Rulers "Now you see what is going on, do you not?"
When I noticed that the shortened lower row had the same length as the crossedout upper row in the previous diagram, I felt like saying "aha" myself. Mutely, I pointed to the two short rows, one in each diagram.
"Exactly. They are equal! In both cases, the ratio of the longer row to the shorter one is the same. In the previous figure we saw that this was simply the ratio of the long ruler's length to that of the short one. In the second figure, it is the ratio of the number of short rulers to the number of long ones. But these two numbers are integers. Therefore the ratio of the lengths of the two rulers is an integer ratio."
Pygonopolis had delivered the main proof, but there was something left. I pressed him to explain why the integer ratio meant the two lengths had a common measure.
"That part is the easiest. Simply divide the longer ruler into as many equal units as the larger of the two integers entering the ratio. Similarly, divide the shorter ruler into as many equal units as the smaller of the two integers. Because the ratio of the lengths equals the ratio of the number of units composing each ruler, the two kinds of units must be the same."
The proof had not been a difficult one, but my head spun a little, as if I had received a brain transplant. Early Greek mathematics had a very different feel from modern, algebraic reasoning. I ventured a question: "You have just shown me how Pythagoras might have proved the ruler gamethat is, winning the ruler game was tantamount to the existence of a common measure. We in the modern age might proceed differently. We would work with the symbolic ratio X/ Y and use algebra to prove the result. I feel a bit foolish asking this, but I must. Why should two entirely different systems of thought arrive at the same conclusion?"
Rather than becoming impatient with my question, as I feared he might, Pygonopolis looked pleased.
"This illustrates how two completely different trains of mathematical thought arrive at the same station, so to speak. It is a firstclass phenomenon, when you think about it. Two completely different approaches to a problem, our modern algebraic approach and the old geometric one, lead to precisely the same result. Is it a coincidence? If you view mathematics as a purely cultural activity, you will miss a crucial point: It is not, in my view, a coincidence." Then he laughed.
"When some people talk about the cultural element in Greek mathematics, I fear they imagine Pythagoras dancing on the beach like Zorba, with a bouzouki playing in the background."
A brief rumble of distant thunder rolled across the strait from Samos, where clouds were gathering. Pygonopolis shivered slightly, staring down in silence at the diagrams. This was my opening. "If it is not a coincidence, what is it?" I asked.
"It is essentially the phenomenon of independent discovery, the same idea finding a completely different expression by two people or groups of people separated by space, by time, or by culture. The phenomenon has been repeated thousands of times throughout the history of mathematics, and it points to something very special going on in mathematics. I suppose my own beliefs on this point are not very different from those of Pythagoras. For even after his integral universe was shattered, Pythagoras continued to believe that mathematics had an independent existence, although not in material sense. But what, I ask, did he call it?
"Pythagoras was a mystic in the traditional sensesomeone who practiced inner discipline to arrive at new levels of understanding. Perhaps I will say more about that tomorrow. In the meantime, I can tell you only my opinion: He surely had a name for the place where mathematics exists. I have tried to imagine what this name might be. My best guess is the Holos."
"The Holos?" I repeated, as this was an unfamiliar word.
"The Holos is the place of mathematics. It stands in a special relation to the cosmos. Holos the source, cosmos the manifestation." Pygonopolis paused, breathless again. The new word echoed in my mind. The holos, the holos, a beautiful word, pronounced with the Greek letter chi, a rasping H, followed by an ululation. "Earlier you described the Pythagorean universe of integers," I remarked, "but all along you've been hinting at a tragedy. What happened?"
"As I said before," he responded in a patient tone, "the major underpinning of the integer universe, as Pythagoras imagined it, was what we could call the hypothesis of cosmic commensurability: Every two lengths were commensurable, not just in practice, but also in principle. There can be little doubt that during the time that Pythagoras believed the hypothesis, he bent every effort to proving it. He worked geometrically, trying one approach after another, but all his efforts came to nothing. No matter how much he wanted the hypothesis to be true, he could not prove it. Nevertheless he continued to imagine that the integers, specifically the number one, was the atomos from which the gods made everything. Ah, what a blow it was!"
"What happened?" I asked.
"His supreme vision was shattered when Pythagoras found the first pair of incommensurable magnitudes. Perhaps it was his old teacher, Thales, who suggested that Pythagoras check the commensurability of the side of a square with its diagonal. See, here it is.
Square with Diagonal "If the universe was based on integers, all pairs of lengths would be commensurable, including the two lengths in this innocentlooking little diagram. One of the lengths is the side of the square, all four sides having the same length. The only other length in the figure is that of the diagonal. It doesn't matter what size you draw the figure, as we are concerned only with the ratio of the two lengths. Was it an integer ratio or not?
"Pythagoras may have puzzled over this longer than he should have. Sometimes mathematicians are slow to discover the truth about a pet idea because they fondly imagine it to be true and are continually trying to find a proof of it. They never seriously seek to disprove it. But now Pythagoras had a test case to consider. How long did it take him to realize that it was what we call a counterexample?
"One day it came. The discovery staggered him, for it brought the integerbased cosmos to a huge nothing. Once he had gotten over his shock, he felt immense gratitude that at last the question of commensurability had been settledin the negative, as it turned out. Up to this time, Greek mathematics recognized only two kinds of numbers, the integers and their ratios. Now there appeared to be a mysterious third kind of number, one that called for a revision in thinking. A new world had opened.
"Here's where the cultural element comes into play: His gratitude was so great, he went to a templeperhaps this very one and sacrificed an ox. We moderns do not understand sacrifice, by the way. Imagine feeling so grateful for some wonderful event that, to relieve your heart of its burden of joy, you buy a Mercedes and set it on fire!
"The argument that Pythagoras used to show the incommensurability of the side of a square with its diagonal is quite simple when you write it in modern symbolism, but we will prove it more or less in the way that Pythagoras did. We will not use algebra, then, but we will allow letter names for parts of the diagram. In particular, call the short side X and the long side Y. These, you will agree, are not algebraic variables. We begin with the very figure that Thales showed Pythagoras."
Pygonopolis stabbed Thales's figure with his stick.
"Thales had been to Egypt and had learned many wonderful things from the Egyptian priests, including this interesting little fact about the side of a square and its diagonal." Pygonopolis drew a second square tilted in relation to the first. One of the sides of the new square was the diagonal of the first one.
"The Egyptians, who labored under the same restrictions as the Greeks, had been clever enough to discover a curious relationship between the two squares. The larger one has twice the area of the smaller one. "The Egyptian proof was simple. You simply add new lines, like so, and realize that the large square has been divided into 4 small triangles, while the small square is already divided into 2 of the same triangles: 4 is twice 2. Quod erat demonstrandum, as it says in the old texts."
Egyptian Proof of Relation Pygonopolis carefully smoothed away the two construction lines, restoring the earlier figure. Absentmindedly, he brushed his hand on his suit, then swore. "Agh," he said, "what a stupid thing to do. See what I've done!" He spent a few moments brushing the smudge on his suit, frowning.
"After a certain amount of the usual fumbling around that mathematicians go through, Pythagoras found the first step of his proof. If he assumed that X and Y were commensurable, then both X and Y had integer lengths in the unit of their commensurability. He also insisted that these numbers be the smallest ones having this property; this meant that the integers could not have a common factor. "He could visualize not only the lines X and Y as rows of dots (the units) but also both squares made up of those dots. In particular, the number of dots in the large square was an even number, being twice the number of dots in the small square. Pythagoras then asked himself, 'Can an odd number be squared to produce an even one? '"
"My dear Professor Pygonopolis," I interjected. "I thought you said that Pythagoras had no algebra, and I assumed that meant no squaring."
"No, no, no, Dewdney. As I explained earlier, the ancient Greeks could multiply by geometry, and this meant the operation of squaring, as well. In this case, he drew a number as a row of dots. To square the number geometrically, he made a square of it, literally. He added more rows of dots above the first, as many as it took to produce a square shape. In fact, this is where the English word squaring comes from. In any case, the total number of dots in the square is the product of the number of dots along the bottom and along one vertical side.
"Pythagoras undoubtedly already knew, and had proved, that when you squared an odd number in this manner, the total number of dots in the square was always odd. And, when you squared an even number, the result was always even. Now, according to the Egyptians, the larger square had twice the area of the smaller one. This meant that its area, the number of dots in it, was even. But, as we have already seen, this could only have been the case if the length of the side being squarednamely Ywas also an even number.
"Now the pace picks up. If Y had an even number of dots, then its square would have not merely a twofold (even) number of dots but a fourfold one. In modern language, this means that the square of Y was a multiple of 4.
"Now recall the Egyptian theorem: The square of Y was twice the square of X. Yet the square of Y is also a fourfold number, multiple of 4. This meant that the square of X must have been a twofold number, or multiple of 2. You see where this is going, do you not?"
"Are you going to apply the same reasoning all over again to X?" I guessed.
"Exactly. Pythagoras could now apply the same reasoning to X as he had to Y, concluding finally that both lengths consisted of an even number of the fundamental units that made them up. This meant that if you cut each of the two integers in half, you would get new, smaller integers with the same property: Their ratio would be X to Y again. Because the integers in question were already the smallest possible, however, this was a contradiction. The logic refused to cooperate further. The machine had ground to a halt. In such cases, the Greek mathematicians, no less than we moderns, knew that one of the assumptions going into the proof must be wrong. There was only one assumption madethat the lengths of X and of Y were commensurable. The contradiction meant that they couldn't be."
Pygonopolis sighed, seemingly to catch his breath. "Can you imagine? Can you just imagine this moment for Pythagoras? There was no doubting the new result. Instead of proving the longsought theorem, 'Every pair of lengths is commensurable, ' he had proved exactly the reverse: 'There exists a pair of incommensurable lengths. ' Although it doomed his doctrine, at least in its existing form, I dare say that Pythagoras was secretly delighted. He sensed higher ground ahead, as though scaling Olympus itself. The numerical atomos was deeper and more complicated than he had thought. There was another kind of number lurking in the holos and, therefore, in the cosmos. It was not an integer, nor was it a ratio of integers. We moderns call such numbers irrationals, meaning only that they are not rational numbers."
Storm clouds were gathering across the strait with Samos. Thunder rolled more and more frequently across the sea.
It was now late afternoon. Pygonopolis picked up his hat and strolled into the temple. I stood in a trance, watching him collect his briefcase, then stride out to the steps again, smiling broadly.
"I feel as though I have overwhelmed you in some way. It is exciting, this story of Pythagoras and the incommensurables. It shows so many things about early mathematics, yet we only have to stretch our imagination a little to understand how the early mathematicians must have worked."
I nodded as we parted for our separate cars. The rain fell heavily as we each drove north to Izmir. I had been hoping to ponder the things I had learned today as I drove but had no calm moments for reflection. Instead, I was hard put to keep up with Pygonopolis's rented Mercedes. He drove like a maniac, in spite of the bad road conditions. I was vastly relieved when we reached the outskirts of Izmir.
We had supper this night, Pygonopolis and I. We dined on Mediterranean seafood in the hotel. By asking Pygonopolis about the holos, eating as he talked, I succeeded in finishing long before he did. "But just what is the holos, do you think?" I asked.
"The holos is the place where all of mathematics, both known and as yet unknown, exists," he responded cheerfully. "Here are the definitions, the axioms, the rules of deduction, the theorems, and the proofs. Here, too, are all the numbers, number systems, sets, families of sets, and on and on and on."
"But what I really want to know," I pushed my point, "is how such things can exist. Do they exist independently, like this chair?"
"The answer is subtle," he said, "for the existence of these things seems to depend on a human mind, yet it doesn't. Think of the number 3, for example. Wherever there are 3 of anything, the number 3 is also present, and not just as a concept. "Here, for example, are 3 prawns left on my plate. Just 3. The threeness of these prawns will control how many more prawns can eat without ordering more. In short, I cannot eat more prawns than are on my plate. Not only is the threeness of the prawns manifest to our senses and minds, but it also has an operational significance that goes beyond my conception of threeness. My inability to eat more than 3 prawns from my plate has nothing to do with my conception of threeness, or even the fact that it is me sitting here. Anyone else would face the same options. Likewise, if I order 10 more prawns, I may calculate that the number of prawns on my plate will then be 13. In this and in many other ways both simple and complex does mathematics control the world. In this way do holos and cosmos interlock."
"If I understand you correctly," I followed up, "the holos is a real place, although not in our ordinary sense of being locatable in our universe, or cosmos. Yet it also controls, to some degree at least, what goes on in our world, in the cosmos. What I'm not clear about is how much of this theory is due to Petros Pygonopolis and how much to Pythagoras."
"The theory of the holos, as I describe it, is entirely my own. An extended fantasy, if you like. Yet I cannot help but believe that Pythagoras thought of mathematics in pretty much those terms. He had seen how numbers evanesce when you consider what it is that unites all collections of 3 things. He had also seen how lines vanish as they are drawn ever more accurately, finely incised on smooth, flat stones. He had witnessed then, as anyone can today, how these concepts retreat when they are pursued, as though fleeing back to the holos. Yet they advance to take control at other times."
"This afternoon you made a joke about Pythagoras dancing to bouzouki music on the beach." I had not understood this earlier reference. "I assume he didn't actually do this, so what role did early Greek culture actually play in Greek mathematics?" "Let me make an analogy. Mathematics is like the wheel. Almost every culture has its wheel, and the wheels made by different cultures all look different. An Egyptian chariot wheel is quite different from that on a medieval European oxcart, and both wheels are again different from those on a modern automobile. Yet all wheels operate on exactly the same principles.
"Nevertheless, cultural clues are crucial to understanding Greek mathematicsnot so much its validity or universality as its direction. On one hand, you will find nothing discovered by my forebears that could not have been discovered by a Solomon Islander. But you might also find that the Solomon Islander would not be very interested in Pythagoras's problems, so he would be very unlikely to pursue them. It is early Greek culture that molded Pythagoras's mind. At the center of this culture resided the gods. He accepted them as perfectly real, and his deepest ponderings on the ultimate nature of the cosmos necessarily included the gods. He believed in the idea of a controlling presence such as Zeus, assisted by other presences in the pantheon. But this was the horse that pulled his cart and not the cart itself, so to speak. It motivated his search, even inspired it, but it did not play any role in what Pythagoras actually discoveredunless of course . . . "
Pygonopolis had begun to perspire heavily after finishing his meal. He paused to pass a handkerchief over his face. "Let us just say that as far as Pythagoras was concerned, the controlling presence had left clues about itself, and Pythagoras, eager to scale Olympus, cast himself in the role of cultural hero. He saw in mathematics the path to knowledge such as only gods enjoy. Logical consequences resided in the very fabric of existence. It was surely how the gods worked."
Pygonopolis stared intently at me with his dark brown eyes. The hair on my neck stood up, and I felt, just for a moment, that he knew far more than he was telling. It occurred to me that he, Pygonopolis, actually believed in the gods of oldbut then, abruptly, the mood passed.
"Tomorrow morning, we will take the plane for Athens. It is late."
Pygonopolis looked at his watch and, in so doing, revealed the palm of his left hand. There was a tattoo therea small blue star. I glanced away quickly as Pygonopolis looked back at me from the watch.
As we parted, be observed, "You cannot know what it means to me to have a real listener."