Read an Excerpt
A PLUME BOOK
EUCLID IN THE RAINFOREST
JOSEPH MAZUR is professor of mathematics at Marlboro College, where he has taught a wide range of classes in all areas of mathematics, its history, and its philosophy. He lives with his wife in Vermont.
“Joseph Mazur brilliantly explores the symbiotic relationship between the physical and the mathematical worlds. He asks the questions: How do we know that the world is what we experience it to be? Can logic guide us through the rainforest of science and math and provide us with a chance to discover the underlying foundations for their truths? In his highly original search, Mazur is a brilliant forester whose graceful pursuit leads him to understand the logical bases of human reason. Mazur has given us a stylish and seductive book that convinces the mind even as it delights the soul.”
—PEN American Center (finalist, 2005 Martha Albrand Award for First Nonfiction)
“This charming book radiates love of mathematics ... and of life. Mazur weaves elementary explanations of a wide range of essential mathematical ideas into narratives of his far-ranging travels.... This book is a treasure of human experience and intellectual excitement.”
—Choice (2005 Outstanding Academic Title)
“A rare example of a universally appealing math book.”
—Mathematics Teacher
“A delightfully entertaining adventure that weaves mathematical ideas with an amazing collection of real-life stories.”
—Science and Theology News
“An unusual book ... Like a fascinating conversation that stays within certain bounds but nevertheless moves in unexpected directions.”
—Nature
“Provocative and philosophical ... Mazur does readers the great service of setting the arcane ideas and procedures of mathematics back in the world where they belong.”
—Mathematical Intelligencer
“Mazur successfully explores how mathematical logic and proof are essential building blocks to understanding knowledge and universal truths. ... His text is devoid of complex proofs and dense mathematical language; instead, the author has drawn upon his experiences as a formative teacher to create a book rich in content that connects with real-world experiences.”
—Library Journal
Euclid in the
Rainforest
Discovering Universal Truth in Logic and Math
JOSEPH MAZUR
Introduction
I came to understand mathematics by way of a Russian novel. On the morning of my seventeenth birthday, my older brother presented me with two books that I, as he put it, “might enjoy reading.” One was a 532-page paperback of Dostoyevsky’s Crime and Punishment; the other was a 472-page algebra textbook. My brother, having the gift of quick mathematical insight, hadn’t noticed that one does not read a textbook of modern algebra in the same way that one reads a Russian novel. The next morning, I began reading the novel without ever getting out of bed and finished sometime late that night, having skipped lunch and supper. Raskolnikov, after swinging the axe with both arms and bringing the blunt side down on the old woman’s head, made me feverish and deliriously spellbound. On reaching the end of the novel, trust in my brother’s choice of mesmeric literature was so strong that I turned to my other book, expecting it to be as riveting as the first. The next morning, I pondered a sentence on the second page for hours: “Clearly, any result which can be proved or deduced from the postulates defining an integral domain will be true in any particular integral domain....”
Stuck at clearly, I dressed and once again spent the day attempting to understand my new book, but this time I did not get past page five. That summer, I struggled to get past the first few chapters. What was modern algebra about? By the fourth chapter, I was well into abstractions, having completed as many exercises as I could. But I could not understand the relevance of all these mathematical abstractions to life itself.
The joys of feeling confidence in solving problems and the emotion of witnessing beauty at the end of a proof are intense. Mathematics gradually became for me as a teenager a range of mountains to climb. Challenges of ascending through the thin air of abstraction only made vistas from the summits more magnificent. With a firm proven foothold on one peak, I could see others beckoning me higher from above misty clouds covering valley paths through rainforests of flowering ideas.
In thirty years of teaching mathematics, I have collected stories of extraordinary students and fellow mathematicians trying to reach peaks from the steepest faces. Their stories are about the climb, the view from even the smallest peak, the excitement of discovery, the investigation of unknown intellectual encounters with beauty, and the confidence of feeling certain about mathematical proof. They are human stories, ultimately not so removed from the excitement of a Russian novel. I begin to see my brother’s inadvertent point.
But this book also has another point to make about mathematics, about logic, about scientific truth. To appreciate modern mathematics, we inevitably must examine how mathematics is communicated, question what makes us feel persuaded by proofs of theorems. What is proof? It might seem odd to find that, even in mathematics, a subject envied for its precision, there is no universally accepted answer. A formal answer might be that it is an ordered list of statements ascending from an established fact (axiom, theorem and so on), each statement logically derived from the one preceding. However, mathematicians follow a more informal practice. Many theorems accepted and used in mainstream mathematics have proofs that hardly conform to any rigorous definition of proof.
Mathematics enjoys a reputation for being an intellectual pursuit that generates universal truths. But contrary to what many of us think, those truths are not communicated through airtight chains of logical arguments. The essence of proof contains something more than just pure logic, just as music is more than just musical notes. It might seem strange to think that, even though mathematics seems to be independent of culture, opinion plays a central role in the profession. How do mathematicians know when a proof is complete? Is it complete if nobody can find an error? Or does it come from an inner feeling that plays with opinion through knowledge and experience? A significant part of the feeling of being mathematically right comes from experience with logic developed by the practice of rational criticism and debate. Sometime early in the sixth century B.C., two things happened to dramatically alter the way Western civilization explained the world. The first was the use of cause and effect, as opposed to the supernatural in explaining natural phenomena; we might say that nature was first discovered then. The second was the practice of rational criticism and debate. These fresh developments occurred after a time of great political upheaval in the eastern Mediterranean, which led to profound changes in the political structures of Greek cities. Democracy in Athens meant that citizens could participate in government and law, freely debating and questioning political ideas. Before the establishment of the Greek city-state, a change in rule usually meant merely a change from one tyrant to another. Greek philosophy, according to tradition, began in 585 B.C. when Thales and other Ionian merchants traveled to Egypt and other parts of the known world. They returned rich with information about applications of mathematics related to building practices. One can imagine Thales thinking and analyzing the essence of what he had learned during those long voyages home as his ship crossed the Mediterranean and sailed up the Aegean coast back to Miletus, his hometown on the coast of modern Turkey.
For the next three hundred years, from the time of Thales and Pythagoras, the founders of Greek philosophy, through Plato and his school in Athens, to Euclid and the founding of the Museum in Alexandria, logical reasoning developed into a system of principles empowering investigations of the purely abstract immaterial world of mathematics. A third defining moment came shortly before Euclid wrote his Elements when Aristotle formalized ordinary logic. He constructed fourteen elementary models of logic, such as “All men are mortal; all heroes are men; therefore, all heroes are mortal.” By 300 B.C., the thirteen parchment rolls of Euclid’s Elements were written and logical reasoning had matured enough to be reduced to a handful of rules. Part 1 of this book, “Logic,” is about this kind of logic.
But logical reasoning could not address the weirdness of infinity. An incredulous Greek mathematician named Zeno constructed polemics against motion, which used supposedly iron chains of logic to tangle arguments into clanging self-contradictions. Plato tells us that Zeno came to Athens from Elea (on the west coast of Italy) with his lover, Parmenides, for the Panathenaea Festival. While there, presumably between events, Zeno read from his works to a very young Socrates. According to one of Zeno’s many arguments, even the swift-footed Achilles could not overtake a slow-crawling tortoise if the tortoise was given any head start. This, Zeno argued, is because the moment Achilles reaches the tortoise’s starting point, the tortoise would have moved to a spot farther ahead; at that point, the argument repeats, with the tortoise being given a new head start. Achilles would have to repeat this forever just to catch up with the tortoise. In another argument, Zeno shows that movement is impossible because, for a body to move any distance, it must first get to half the distance, then half the remaining distance, and so on, forever having to get to half of some remaining distance and, hence, never reaching the full distance.
Zeno might have raised these puzzles to provoke intellectual discourse or merely to irritate Athenian philosopher/sports fans. He was known as “the two-tongued Zeno” because he often argued both sides of his own arguments, which usually involved either the infinite or the infinitesimal and had an enduring effect on the development of geometry. It took time. Except for Zeno and Archimedes’s brief noble attempts at understanding infinity shortly after Euclid, direct confrontation with the infinite had to wait almost two thousand years until tradition-bound rules of logical reasoning were relaxed to address some of the difficulties Zeno raised. In 1629, Bonaventura Cavalieri, a student of Galileo, devised a scheme for sidestepping the issues raised by Zeno, deliberately ignoring problems with the logic of his own arguments; oddly, his arguments led to correct results. Cavalieri’s great contribution was to let intuition, rather than logic, guide mathematics. His ideas created the driving force behind the invention of calculus, the new math responsible for fantastic applications to the real world, from predicting planetary motion to the design of musical instruments.
Cavalieri’s methods relied heavily on strong intuition. For almost two hundred years, new mathematical concepts, those that outgrew the bounds of ordinary logic, were guided and accepted by intuition rather than by logic. Strong intuition carried mathematics to new and glorious heights until things began to go wrong in the eighteenth century, when inconsistencies began to sprout. By the middle of the nineteenth century, intuition and logic were at loggerheads. Theorems that were once proven by intuition were being proven false by logic. A new kind of logic was needed, one capable of working with the rich intricacies of the infinite and infinitesimal. That new logic had to wait until the late nineteenth century for the discovery of set theory, the branch of mathematics that deals with the proper way to define numbers. Set theory gives us the axioms of arithmetic and leads to deep questions concerning the foundations of mathematics itself. Set theory presented us with a general unifying language for all branches of mathematics.
Georg Cantor had the appealing title of Extraordinary Professor of Mathematics (at the University of Halle in Germany). He developed set theory in the nineteenth century to study the real numbers and, by doing so, was led to one of the most revolutionary results in mathematics: that there are different sizes of infinity. What kind of logic led to that notion? Cantor spent a great deal of time writing philosophical and theological treatises in defense of his results on the infinite because they defied intuition. At the same time, he had a passion for Elizabethan literature and spent much of his time attempting to prove that Francis Bacon wrote Shakespeare’s plays. He played on the edge of logic.
The axioms of set theory were not formulated before the beginning of the twentieth century, after many mathematicians had done a great deal of work building the correct framework for their foundations. On the other hand, Shakespeare is still credited with writing his plays.
In 1931, Kurt Gödel surprised the mathematics community by showing that the axioms of set theory were incomplete; in fact, he showed that, no matter how many new axioms are added to the system, there would always be a statement that cannot be proved or disproved within the framework of the axioms of set theory. When we say proved or disproved, we mean that nobody in the eternal future will ever be able to prove or disprove the statement. This must have been as great a shock to mathematics as Pythagoras’s discovery that one ruler cannot measure both the side and diagonal of a square. Even Zeno’s paradoxes could not rival this discovery. The persistent problem is, as it always has been, how some things never seem to end. Part 2 of this book, “Infinity,” is about the logic of infinity.
Though logicians have problems with the formalisms of axiomatic set theory, everyone readily acknowledges that we are able to count, do amazing mathematics, and verifiably build and support science on the shoulders of mathematics. Problems with formal logic do not seem to interfere with material reality. The cost of relaxing the requirements from airtight proof to plausible proof has a great benefit: It validates the scientific method. Sir Francis Bacon, the father of the scientific method (and not the author of Shakespeare’s plays), suggested that deductive reasoning is not appropriate in investigations of the material world. He argued that one could arrive at plausible general conclusions by observing special concrete cases.
Science relies on three kinds of reasoning. Surely, it relies on ordinary logic and, implicitly, on the logic of infinity, but it relies most heavily on plausible reasoning. It is based on the idea that what one finds true often enough is true. In mathematical proof, often enough means infinitely often, but scientific proof is far more relaxed. The sun rose in the sky often enough in my lifetime for me to believe it will rise again tomorrow. On the other hand, though I have never experienced a devastating earthquake, never is not often enough for me to believe that I will not experience one in the future. Whereas ordinary deductive reasoning inescapably forces the specific cases from general assumptions, plausible reasoning takes the opposite path and argues from specific observations to general (but only plausible) conclusions. No one can deny that this seems to weaken the strength of truth to the status of the plausible, but when Sir Francis Bacon introduced this notion in 1620, he changed our understanding of knowledge, and the subsequent mathematics (probability and statistics) that was built to support his idea changed science forever, although a sure mathematical footing that quantifies plausibility would have to wait another hundred and fifty years for Thomas Bayes. The story of plausible reasoning is what Part 3 of this book, “Reality,” is all about.
These are the essential forms of human reasoning and logic, what we humans consider proof that something is true: ordinary logic, which is concerned with proof and classification; the logic of infinity, which is concerned with infinity and number; and plausible reasoning, which is concerned with probability and nature—even rainforests.
* * *
Math at first sight is often intimidating—not just for novices, but even for trained scientists. Unfortunately, it is not always so clear as, say, the Hindu proof of the Pythagorean theorem, which is simply a picture and the word Behold. But in this book, I hope to show the beauty of mathematics and the pleasures of trekking through the mathematical landscape by climbing peaks to see rainforests of ideas through slow-moving clouds, just as Euclid did twenty-three hundred years ago. And after a long journey into abstraction, it’s refreshing to return to the plausible logic that rules the science of the natural world.
PART I
Logic
CHAPTER 1
The Search for Knowledge School
An Introduction to Logic and Proof
“Number theory effortlessly produces innumerable problems, which have a sweet, innocent air about them—like tempting flowers.”
“Flowers?”
“Mm-hmm.” He also says that, “it swarms with bugs waiting to bite the flower lovers who, once bitten, are inspired to excesses of effort.”
—Daphne Clair quoting Barry Mazur, Summer Seduction
The mix of fragrances from blooming yucca, violets, red jasmine, and frangipani carried on the vapors of rising mists above the humid Orinoco River are intoxicating. It is the perfume of the Venezuelan rainforests, where tobacco, banana and coffee grow wild and barbets call so loudly you can hardly hear nearby rapids. I was there at a time when one could still get a doctorate in anthropology from stories of undiscovered indigenous tribes in one of the last unknown corners of the world, a time when roads connecting Venezuelan cities and towns were unpaved or didn’t exist at all. Young with mettle in adventure and eager for cerebral exercise, it was there on the edges of the primal rainforest that I first stumbled over notions of logic and what it means to believe a proof.
Before I knew anything about what anthropologists were or what they did, I read Hamilton Rice’s accounts of his encounters with the Yanomami Indians living along the Orinoco River in southern Venezuela and decided to travel to the remote Venezuelan village of La Esmeralda on my own. It was 1960, several years before Napoleon Chagnon’s best-selling book Yanomamo: The Fierce People was published.1 I got the required inoculations, bought malaria pills, and went to the Venezuelan consulate in New York to apply for a visa.
I was disappointed to learn that it would take two months to process my visa. Thinking it would be faster to get a visa from a small consulate near Venezuela, I flew to Aruba, which is just sixteen miles north of the coast of Venezuela. I arrived there on a Sunday to find that the consulate was open only on Wednesdays. I waited three days at the edge of boredom. At 9:00 a.m. on Wednesday, I walked up the two flights of creaking stairs in a dark, alcohol-reeking, wooden office building. The sign outside the consul’s office door clearly indicated that the office would be open from 9:00 to noon. I waited. At about 11:15, a Sydney Greenstreet look-alike in a white suit and Panama hat arrived smelling of rum. He was the Venezuelan consul.
“Can I help you?” he asked scornfully.
“I would like a visa for Venezuela,” I said, wondering if he would ask me why.
“Yes, I can get you a visa, but it will take two months!” he said with a snickering smile.
Defeated by the thought that I had traveled a thousand miles to a barren island for the same deal that I could have had in New York, I thought of the crazy idea of getting a fisherman to bring me across the short stretch of water separating Aruba from the mainland. However, the consul’s snicker gave me hope, so I asked if there was any other way to speed up the process.
“Certainly!” he said. “For five dollars, you could have it today!”
I placed a five-dollar bill on his desk. “Your passport please,” he demanded. He pulled out a mechanical stamp machine from the center drawer of his desk, found an empty page in my passport, and stamped it with a visa to Venezuela. I took the next flight to Caracas.
An Englishman a bit older than me sat beside me on the plane. “Have you ever been to Caracash?” he asked with a proper English accent embroidered with an S-lisp while extending his greeting hand. “My name is Roger Hooper,” he continued with a look suggesting that I should have heard the name before. Upon learning my plans to travel to the Orinoco River, he proceeded to tell me about his dream of doing the same. He proceeded to tell me about Joseph Conrad’s Heart of Darkness, as if I had never heard of it. I was absorbed in his story almost as much as he was until a lightning storm struck without warning. Passengers around us anxiously clutched at armrests as we bounced from air pocket to air pocket. The plane pitched forward and backward, rocked from side to side, and shook with each threatening lightning bolt as it passed through torrential rain and foreboding darkness. Roger casually went on with his story, unaware that I was no longer listening. The plane came to an abrupt stop, listing to the left while torrential rain continued to pour over the windows; I could see nothing outside.
My neighbor continued talking as if nothing had happened. A minute passed before the pilot announced that, though the plane had come to a safe landing, it would not be easy to disembark. The plane was in water up to its belly. The pilot had chosen to emergency-land his plane in the safety of water rather than risk an airport landing close to rugged mountains in a severe storm. We disembarked on a makeshift ramp constructed by curious emergency personnel. As we were coming off the plane, Roger did not say a word about the event, nor a word about the inconvenience of our waterlogged luggage. He continued to lisp about the Congo, but I was processing almost nothing of what he was saying and was beginning to be annoyed. I was going to the Orinoco River, not the Congo. He said, in his first reference to our aircraft drama, that he, too, was headed for the Amazon on an adventure far more exciting than any emergency landing.
When I finally did get around to telling him my name, he blankly announced that he knew of a mathematician by that last name and asked if there was any relation. I had grown used to saying “he’s my brother” whenever people asked the same question. At that, he said that he also majored in mathematics and unleashed the grand story of Fermat and his Last Theorem, the famous theorem that says that xn + yn = zn has no nonzero whole-number solutions for x, y, and z when n is greater than 2. He could make the Caracas airport flight schedule seem like a Noel Coward play. Roger explained the theorem with the math embroidered into a fabric of supporting characters. In fact, all we really know for sure is that when Fermat died, his son Samuel discovered a copy of a translation of Diophantus’s Arithmetica in the library with a marginal note in his father’s handwriting: “I have discovered a truly remarkable proof, which this margin is too small to contain.”
Few of us are taught the rules of logic or principles of proof. Even mathematicians seem to innocently learn them indirectly through instinctive knowledge and experience of language, just as the toddler does in acquiring the elements of grammar. However, when we intellectually reflect on logic and proof, what they are and what they do, we tend to agree that, formally, a proof consists of a finite collection of well-formed statements linking back, by rules of inference, to elementary assumptions. For example, if we want to prove that the base angles of an isosceles triangle2 are equal, we must start somewhere. Proof means that belief is universal. If Euclid’s proofs are to be accepted, all parties must agree that “there is a unique line passing through any two points,” which is exactly Euclid’s first elementary assumption. This assumption, together with four others, forms a footing to support a superstructure of Euclid’s geometry.
Margin notes were typical for Pierre Fermat. He was a lawyer and judge in Toulouse, and, though considered the greatest mathematician of his day, he did mathematics only in his spare time. He didn’t publish his works, but he preferred to write them as gifted letters to friends and professional mathematicians. Pierre frequently wrote his thoughts in the margins of his books, often while listening to briefs at court. But Roger painted a fanciful picture of Samuel, his appearance as a young man with bulging eyes, an unusually long but handsome nose, a sullen face, and a tragic mouth that always seemed to be lamenting something lost. According to Roger, Samuel was asleep in his library when a howling wind blew a window open. The window frame knocked Diophantus’s Arithmetica off a sill, face down, and opened to a page biased by a pressed dry rose. And there it rested until Samuel lit a candle to see the short comment responsible for a great deal of modern number theory.
When Roger finished his story, he suggested that we should head for La Esmeralda together. I was only eighteen and Roger was ten years older. He already had a receding hairline at the sides of his head and the beginnings of a double chin that he was trying to hide behind a newly cultured beard. And, though he was definitely peculiar, insensitive, and possibly even crazy, I wanted someone to accompany me on a journey that was beginning to frighten me. Roger had the annoying habit of cracking his knuckles at odd times when he was silent. He would dovetail fingers from both hands, turn his palms outward, and push both hands forward to the sound of multiple cracks. However, he was fluent in Spanish, and I needed a translator. I had no idea what trouble I was about to get into and naively thought I could make my way into the jungle on a few provisions, a tent, and malaria pills.
At his suggestion, we hitchhiked a ride with a Venezuelan army convoy traveling five hundred miles into the jungle along dirt roads to Cabruta on the Orinoco River. It was an innocent time, when two foreign civilians could unofficially still do such things. The convoy would go no further than Cabruta, for there was no bridge to cross the river, but we accepted the ride, thinking that at least we would get to the Orinoco, even if it were a thousand miles downstream from La Esmeralda. After about two hundred and fifty miles, we came to a washed-out mountain switchback. A canvas-covered army vehicle was tottering over a precipice and about to fall into a ravine several hundred feet below. It was nighttime and lightly raining; the bugs were out to devour anyone stupid enough to stand on the road unprotected. More than twenty soldiers took positions to push the vehicle onto a more solid road surface, and my position was at the right taillight, the main attraction for mosquitoes and diving beetles in the area. My clothes were covered with tightly gripping diving beetles. Roger was thrilled. This was the adventure he had come for.
We gave up for the night when the rain started to come down heavily. The road was impassible, so we slept huddled under the canvases of the trucks. I can’t recall why the tottering vehicle was not tethered or winched to one of the other trucks, but it wasn’t. In the morning, the sun rose above several isolated mists that passed with gentle breezes. A choir of birds awakened us with what seemed to be a thousand different simultaneous songs. Several of us with stiff bodies from sleeping in awkward positions rose with the sun to hear and feel the road give way, sending the tottering vehicle to its grave at the bottom of the ravine. I looked out in time to see Roger standing on the road, peering down at the fallen truck, laughing.
A cook prepared breakfast of potatoes and eggs on a propane stove, while a small group of three young, tough soldiers and a platoon sergeant argued about how to salvage the fallen vehicle. Miguel Ramos, the sergeant, was more concerned with the contents of the truck. Eating peaches from a can, he gave a stout order to salvage the contents. So several fifty-five gallon oil drums and dozens of sealed wooden crates marked in alphanumeric code were hoisted to the road, while Jesus and two other soldiers whose names I no longer recall continued to argue about how to salvage the truck itself.
The truck weighed two tons, and the winch cable capacity was only one ton. I volunteered to say that if the truck could be straightened out, it would take only about 1.4 tons of cable tension to winch the truck back to the road.3 At the time, I thought it to simply be an academic point, since the cable’s capacity was still smaller than what was needed to hoist the entire truck. But I had to convince the group that they needed a winch with only a 1.4-ton capacity.
I used a stick to draw a diagram in the dirt, trying to explain the trigonometry of forces involved while a conspicuously colorful object, far off in the distance over the steep hillside of wildflowers, distracted my audience. I stopped my explanation to join the diversion.
It was a wild apricot tree two hundred feet away in the middle of the field, surrounded by tall, particularly beautiful flowers with very tall pistils and alternating red and black petals. In this wonderful field of yellow and blue flowers, that unusual red-and-black one was the only one swarmed with bees. No one had ever seen such a flower. For several minutes, we were captivated.
We approached the tree to find that what we thought was one apricot tree was actually two intertwined trees. As we stuffed our pockets with wild apricots, an idea leaped in front of me: If we could get the winch on the fallen truck to work, we could intertwine two winches to work together with a capacity of two tons—more force than necessary! The answer depended on whether the fallen truck would start. Luckily, a tree had broken the fall as the truck rolled backward, so the only damage was to the rear end. Jesus started the truck and got the winch working. The rest was easy. Entwined cables connected the fallen truck to a tethered truck on the road. The two winches synchronically rewound, slowly pulling the fallen truck up the dirt incline. A small piece of entirely theoretical trigonometry solved a thoroughly physical problem, a testament to the power of mathematics. The operation worked so smoothly that Jesus asked me to explain how I made my calculations. He was hooked.
And so was I. For the remainder of our trip, Roger enthusiastically taught us applications of trigonometry and beguiling mathematics.
* * *
Mathematics was not always what it is today. Its formal rigor and structure did not always rely on finite collections of well-formed statements linking back, by rules of inference, to elementary assumptions. No, in Thales’s time (around 600 B.C., a half-century before Pythagoras), when Western mathematics inherited a fortune in concrete applications from a thousand years of Babylonian and Egyptian calculations, when abstract so-called theorems in need of proof were beginning to surface on the mathematical horizons, mathematical proof was far more relaxed and casual. Arguments were persuasive but not as rigorous as those that would come in the next three hundred years, before Euclid, when the ideas for structured proof based on elementary assumptions emerged.
One of the earliest existing histories of geometry is one that comes to us from Proclus, a philosopher and historian who summarized an earlier history by Eudemus of Rhodes. According to several historians, from Proclus to Plutarch, Thales of Miletus introduced abstract geometry, a new phenomenon that was to excite further Greek creations.4 The Rhind papyrus, now in the British Museum, is a handbook of practical problems dating from about 1550 B.C. The papyrus calls itself “a guide to accurate reckoning of entering into things, knowledge of existing things all.” Everything in the papyrus is stated in terms of specific numbers and nothing is generalized. Rectangles are measured, but only when their sizes are specified; right-angled triangles5 mark out areas only when the sides are specified. During Thales’s lifetime, Egyptian mathematics was still relying on tables to solve problems from surveying to banking, just as modern bankers rely on tables or computer programs to compute mortgage payments. Thales’s revelation was that there are precise relationships between parts of figures and that such relationships enable one to find one part by knowing others.
It may seem obvious to us that the height of a right triangle, whose hypotenuse is, say, five units long, absolutely depends on the length of the base. Equally obvious is the thought that one can know what that dependency is. We know that the height is the square root of twenty-five minus the square of the length of the base. But this was not something in the Egyptian schoolboy’s textbook.
Modern Western mathematics began with Thales. We are told that he is the first to “demonstrate” the following:
1. The circle is bisected by its diameter.
2. The base angles of an isosceles triangle are equal.
3. The vertical angles of two straight lines are equal.
4. The right-angled triangle can be inscribed in a circle in such a way that its hypotenuse coincides with the diameter and its right angle sits on the circle.
These are extraordinary statements about every circle and every isosceles triangle.
He must have known that the sum of the angles of a triangle is 180°, or the sum of two right angles. The persuasive argument of his time must have been something like this: Any triangle can be split into two right triangles simply by dropping a perpendicular from the largest angle to the side opposite that largest angle.
Take any one of these two right triangles, inscribe it in a circle, and connect the right angle to the center of the circle.
You now have two isosceles triangles. But Thales knew that the base angles of an isosceles triangle are equal. Since angle ADB is a right angle, we see that the sum of all three angles must be two right angles because angle ADC + angle CDB equals one right angle, and angle DAC + angle DBC = angle ADC + angle CDB.
This could be one of the earliest examples of mathematical persuasion, but not a proof in the modern sense of the word because it simply follows a picture without inferences from axioms or established statements. To make the argument into a proof, we must infer the argument from some collection of established statements that can never be disputed. What are those established statements? We used the fact that the points of the circle are equidistant from the center to know that the triangles are isosceles. We used the fact that the base angles of an isosceles triangle are equal. And we used the fact that a right-angled triangle could be completely inscribed in a circle whose diameter is equal to the hypotenuse of the triangle. These were all facts known to Thales.
We might accept the statement that points of the circle are equidistant from its center as part of the definition of a circle. However, the statement that base angles of an isosceles triangle are equal is much harder to accept. It’s not obvious. It relies on an argument composed of several more elementary statements, one of which is that two points determine a unique line, and another that a circle of given center and radius can always be constructed.
These last two statements seem very strongly acceptable—so acceptable that we might accept them without argument. We could simply establish these as true and “prove” the statement that the sum of the angles of any triangle is two right angles by inferring it from the established statements. However, we must be careful. Suppose that the two established statements have hidden conflicts. What if, after one thousand years of building arguments on arguments that use these statements as foundations, a contradiction is implied? We must be sure that such a thing can never happen. In other words, we must be sure that the statements are independent of each other.
This means that the negation of one cannot be logically derived from the other. Discovering this was the great achievement of early Greek mathematicians in the three hundred year span between Thales and Euclid. When Thales generalized the practical mathematics of the Egyptians, other mathematicians made other general discoveries. The Pythagorean School discovered relationships between the sizes of the sides of right triangles. Eventually, the work turned to organizing general statements, systematizing and ordering statements in a network of inferential statements, one following from another.
Although mathematics appears to be built from a collection of self-evident assumptions, it does not develop directly from them. Thales had no established postulates, yet his arguments are as persuasive as any that could be drawn from a rigorous logical sequence of inferences. Indeed, his arguments are simply convincing and persuasive, with little room for doubt. Ideas for theorems seem to come from intuition that follows experience. Experience enough drawings of right triangles inscribed in a circle, and you will discover that the sum of the angles of a triangle is 180°. Experience drawing enough triangles, and you will discover that you can split any triangle into two right triangles. That’s the way you would demonstrate a theorem: persuasion by experience. Later, when you are sure that your intuition is right because you have persuasive arguments to show that it is right, you might want to find your theorem’s position in the web of others that are ultimately built on self-evident assumptions.
* * *
Jesus and I developed an insatiable appetite for mathematics. At the time, I was studying architecture and had no real training in mathematics. The solution to the truck winching problem came to me by accident, without thought. It not only impressed Jesus, but it inspired both of us to consider studying more mathematics. Jesus decided that he would go to university to study engineering.
I started thinking about proof. Pushing the limits of my knowledge at the time, I explained a simple proof of a proposition found in Book I of Euclid’s Elements. It says that an isosceles triangle has its base angles equal. Thales proved this proposition sometime back in the fifth century B.C. Later, Pappus, a Greek mathematician who lived around A.D. 300, gave a different proof—a curious proof: Flip the given isosceles triangle around its vertical axis. The triangle is unchanged. Since the angle on the left of the base coincides with the angle on the right, the two base angles must be equal.
If the sides of such an isosceles triangle are extended equal distances beyond the base, the angles under the base will also be equal. This was proven by the same “flip” argument.
In medieval times, this theorem was the end of the road for courses in mathematics. It was known as “the bridge of fools” because the figure looks like the truss of a bridge and also because only a fool couldn’t make it across.
This kind of heuristic proof was acceptable in Thales’s time, but not in Euclid’s. The same proposition appears as Proposition 5 in Book I of Euclid’s Elements, with a proof based on far stricter standards. Although heuristics might have played a central role in guiding discovery and acceptance of the theorem, rigor would inevitably challenge any argument not based on strict rules of proof.
I confess that at the time of meeting Jesus, my knowledge of mathematics did not extend beyond this bridge. But later that bridge was moved closer to that wonderful theorem in mathematics called the Pythagorean theorem, which states that the sum of the squares of the lengths of the sides of a right triangle equals the square of the length of the hypotenuse.6 According to the Guinness Book of World Records, the Pythagorean theorem has the most known proofs: 520 different proofs.7
When Jesus heard about Pythagoras’s theorem, he tested it on all the triangles he knew, making light sketches of right triangles and measuring sides and hypotenuses. If it looked like it was true for a few triangles, he would not be satisfied until he sketched others and tested the statement again and again on squat, fat, and tall triangles.
How could he have convinced himself of the truth? He would have had to draw some fairly precise right triangles and take reasonably precise measurements of the sides and hypotenuses. Although this method might persuade some people, it wouldn’t be enough to convince Jesus, and it certainly wouldn’t be enough to convince mathematicians.
However, it doesn’t take much evidence to form opinions about mathematical statements. There was good evidence to suggest that the theorem is true. So Roger gave Jesus one of the standard arguments in favor of the truth.
“Draw any right triangle and three squares, one on each side and the third on the hypotenuse,” he said.
“You can place the contents of the lower square into the tilted square on the hypotenuse with room to spare. Next, you see that the remaining square on the left will fit into the remaining space of the square on the hypotenuse without room to spare. This should convince you that the sum of the areas of the squares on the sides is equal to the area of the square on the hypotenuse.”
“It doesn’t convince me,” Jesus said.
“Repeat the testing by redrawing right triangles,” continued Roger. “You will see that every time you draw a right triangle, no matter how squat or tall, the theorem holds. You should become more accepting, but not entirely persuaded. What you need is some more ‘evidence,’ not a formal proof in the mathematical sense, but a ‘somewhat credible’ argument.” He sketched several triangles and said, “Examine these triangles and you will become persuaded that the theorem is true.”
“I’m not persuaded,” I said. “Suppose the triangle has two equal sides. Does it work for such a triangle?” I drew a picture.
Form two squares on the sides of the isosceles right triangle and one square on its hypotenuse.
Divide the smaller squares in half to make four isosceles right triangles.
Fit the triangles into the larger square to fill it exactly.
“Okay, I’m persuaded that it works for the isosceles right triangle,” I said. “Now, what others can we be sure of?” I drew a triangle and played with it, but I could not get anywhere.
Frustrated, I began to doodle.
Jesus took one look at my doodling and shouted, “You proved it! All you have to do is rearrange the four triangles in the big square to make the two smaller squares!” And he made the following sketch.
The four shaded triangles can be repositioned from one figure to the other to show that the two squares in the figure on the right must equal the square C in the figure on the left.
It was brilliant. My sketch was nothing more than doodling, yet Jesus saw something I did not. This was by no means a formal proof through a string of logical arguments starting from axioms; nevertheless, it seemed to me to be an airtight argument.
The next day, Jesus told us that he had had a dream that the theorem was not specifically about squares. And when he awoke, he realized that his dream was right.
“It seems that the Pythagorean theorem is about numbers—that is, the squares of the lengths of the sides,” Jesus said. “But the square of a length is also the area of a square whose side is that length. So, is the theorem about numbers or geometry?”
“Perhaps it’s about both,” I said, amazed by the thought and innocent of the remarkable possibility that geometry and arithmetic could reflect each other.
A few days into our trip, we came to the little town of San Mauricio. What a great surprise. The sidewalks on both sides of its short main street were paved in the same mosaic tesseral pattern as those of Caracas. In the center was a bronze statue of Simón Bolivar. The mayor was Jesus’s uncle, who spoke English well enough for me to understand him without Roger’s help. He was a short man with a bushy moustache, a round face, and curly black hair who constantly chewed tobacco and never removed his sunglasses, even as the evening turned dark. He arranged a feast in our honor. Most of the town was visible from our outdoor feast on a little hill under the shade of tall mahogany trees, with nearby monkeys playfully jumping in all directions. In the distance, I could see long, low buildings that might have been army barracks; three gray water towers; and two rows of thatched roofed mud houses, churuatas. I could see the town communal vegetable garden sprawling down to a wide section of the Manapire River that zigzagged through lush, receding valleys with birds of all colors tunefully sharing fruit and berries of their heavenly habitat.
A magnificently embroidered tablecloth of geometric design was laid out on three long boards supported by tree stumps. Exotic fruits of every kind were placed around the ends to keep the cloth from the welcome evening breezes. Apricots, mangos, and guava were piled high in the center, along with several cooked chickens bathing in thick coconut milk. Two large goldfish bowls of lemonade and cut lemons balanced the boards. My thirst was perpetual, and although I recall the lemonade more vividly than anything else on the table, I still remember the flowers and rainbow-colored berries strategically placed to make an impressive presentation. It was quite a surprise that such a remote village would salute unimportant, uninvited, undeserving strangers with such generous display.
Our host was very interested in America, especially President Kennedy. Conversation was a battle. Although he seemed to be an educated man, he had the charmingly innocent background of someone with only Caracas newspaper and radio versions of the bigger world outside of his remote village.
Jesus, meanwhile, couldn’t get enough mathematics. He returned to his thoughts about the Pythagorean theorem and felt that somehow he had to be more convinced. I had no real idea how to prove the theorem in any conventional sense and could only try to persuade him by virtue of experience, intuition, and compelling forces suggesting that it couldn’t not be true. Parmenides was referring to this kind of persuasion in his poem The Way of Truth: “The only ways of enquiry that can be thought of: the one way, that it is and cannot not-be, is the path of Persuasion, for it attends upon Truth.”
I was amazed to find that even small schoolboys in San Mauricio knew the Pythagorean theorem and could give reasonably persuasive arguments in favor of belief. The mayor called for a boy, no older than ten, to give credible evidence that the theorem is true and to show how well his town’s school prepared schoolchildren. The mayor pretended to be choosing a boy at random from a small group of curious children standing at a comfortable distance behind an imaginary dividing line. The boy was dark-skinned, shirtless, and smiling, and had hair combed in an attempt to cover a recent wound to the left side of his head. With a great deal of help from the mayor, the boy allegedly proved the theorem while drawing a picture on the back of an old movie poster of Clark Gable hovering over the exposed cleavage of Vivien Leigh.
The schoolboy had given a well-known Hindu “proof” of the Pythagorean theorem. The Hindu version simply gives a picture like the one the boy drew, except that it includes the word BEHOLD! It is a very persuasive picture. From it, you can deduce the Pythagorean theorem by comparing the areas of the triangles, rectangles, and squares in the figure.
The problem with this “Hindu proof” is that it is not really a proof, but merely a picture. The proof is in the ability to rearrange the triangles, rectangles, and squares in such a way that the two small (shaded) squares fit inside the tilted square.
This is essentially what the boy said: “Draw a right triangle. (I). Then draw a square on the hypotenuse (II). You can fit both your triangle and the square on the hypotenuse into a larger square (III). From two corners of the original triangle, draw two lines parallel to the sides (IV), and color or shade the four rectangles defined by the large square and the two lines that you just drew. Your picture should look something like V.”
In Figure V, there are four triangles that have the same area as the original triangle. There is a tilted square, which is really the square on the hypotenuse of the original triangle. And there is a big square that borders the entire figure. Next, notice that there are two small squares that are the squares on each of the sides of the original triangle. Let A represent the area of the original triangle. Let B represent the area of the big square.
Check that B – 4A equals the area of the tilted square. Then check that B – 4A also equals the sum of the areas of the squares on the sides of the original triangle. Now use the fact that two things that are equal to a third must be equal to each other to convince yourself that the theorem is true for your original triangle.
The next morning, the mayor, encouraged by the schoolboy’s performance, took us to the village school that was established in 1902 and appropriately named, in translation, The Search for Knowledge School. It was impressively active; students were seated at desks working on what appeared to be math projects. The teacher seemed to know the history of mathematics better than mathematics itself, but he was clearly attached to his students. He was a tall man, whiter than his fellow villagers, with a small black beard and a nervous habit of using one hand to pick at the fingernails of the other. He claimed to have spent some time in Texas doing migrant farm work. His English was clear. When I asked about textbooks, he exchanged glances with the mayor and said, “I am the only one with a textbook. We don’t have enough textbooks; so I give the lessons from my copy.” But the mayor interrupted.
“We have a fine library,” he said and immediately led us out of the schoolhouse to proudly lead us to one more place.
CHAPTER 2
How to Persuade Jesus
Is the Pythagorean Theorem True?
“As the scale of the balance must necessarily be depressed when weights are put in it, so the mind must yield to clear demonstrations.” The more the mind is empty and without counterpoise, the more easily it yields under the weight of the first persuasion.
—Montaigne quoting Cicero, Essays
The mayor was right to be proud. The town library was a one-room wooden bungalow, painted on the inside from floor to very low ceiling in lively colors. Two windows and the front door welcomed the daylight that reflected off a mural mimicking the real windows and door. It was painted with such skill that the brightness of the painted light suggested that we were looking outside at the huge ferns that really were just on the other side of the wall. The books were old and in bad physical shape, leaning at various angles against milk crates on low bookcases. Two metal wire chairs with round bottoms faced the windows. They were the uncomfortable kind, popular in American ice cream parlors at the time. A five-pointed star inscribed in a circle was faintly painted on the floor and could barely be seen under heaps of scuff markings.
Humidity and mildew had stained many books. Others were in worse shape, having been unfortunately stored more directly under leaks in the roof. But pick up any book on any shelf, and it would hold your interest until you began to think that you should be browsing rather than reading. I spotted an 1889 math book written in English. Its title summoned me—something containing the name Euclid. The musty book, with its curled hard cover, cracked open to a page containing the same diagram that the schoolboy drew on the back of the Gone with the Wind poster. A neatly folded note fell from the book as I opened it to where it wanted to open: to an illustration almost identical to the one I doodled, the one Jesus claimed to be a proof. It seemed that someone had tried to work out the proof from the illustration.
I borrowed the book to study it further. Jesus did not know that the Egyptians had discovered his proof long ago. The book claimed that there was little doubt that the Egyptians were dissecting figures that enclosed areas and playing with general ideas in geometry long before Greek travelers came to the North African continent.
The mayor left us back at the table under the mahogany trees, where I rested and read my borrowed book while the villagers went about and a few preschool children watched from a distance. I was happy to have one more book written in English. I was surprised to learn that the Egyptians had very persuasive arguments supporting general theorems, although I felt sure that such arguments fell far short of Greek standards of rigor. Unlike the Greeks, who went by cult names such as Pythagoreans and personal names such as Thales, these Egyptians remained nameless. Histories refer to them as simply “the Egyptians” and give no time period for their contributions. They could have made contributions at any time from the building of the Great Pyramid of Gizeh to the end of the Trojan War. Quite possibly this is because in Egypt and more ancient civilizations, special knowledge of writing and mathematics was limited to the priesthood, which monopolized and preserved learning as collected sacred dogma.
The book told tales of Thales as if the author personally had known him. It painted a picture of a man sailing back and forth from his hometown of Miletus to Egypt. He was in the business of buying and selling papyrus, farming tools, boats, olive oil, and olive presses. While in Egypt, he met with a priest who took him to see the pyramids at an inspiring moment when the sun cast a sharp shadow along the golden desert floor. At that moment, Thales had an idea. Could he determine the height of the pyramid just by measuring shadows? He poked a rod into the sand and waited for the moment in the day when the length of the rod’s shadow cast by the sun would also be equal to the rod’s height. That would be the moment when the length of the pyramid’s shadow would also equal the pyramid’s height. He was quick to turn particular observations into general truths.
Aristotle tells a tale to demonstrate Thales’s shrewd business sense. Eating fresh figs under a barren tree in an olive grove, Thales realized that a tree’s abundance of fruit depended on the weather and that weather follows cycles. He then thought about how to use astronomical observations to predict favorable weather conditions for olives and bought up almost all the olive presses in Miletus to corner the market for substantial financial gain. We learn charming anecdotes about Thales from Proclus, Plutarch, Aesop, Plato, and many others, but we still know very little about the man himself. Aesop tells a tale of Thales’s donkey: By accidentally rolling over one day in a stream, it learned how to lighten its load of sacks of salt. Thales’s clever solution was to load the donkey’s sacks with sponges. The stereotype of the absent-minded professor may be attributed to Plato’s tale of Thales walking into a ditch. One of his attendants called out, “You know what’s happening in the sky but can’t see what’s happening at your feet.”
When Amasis, King of Egypt, heard about Thales’s clever technique for measuring the heights of pyramids, he invited him to the palace. An artist was working on a mural in one of the great halls. He had prepared the wall with square markings to line up and orient his figures. Those markings inspired Thales’s next idea. He noticed that he could prove a special case of the Pythagorean theorem (in which the right triangle has two equal sides) by simply drawing a picture similar to the palace artist’s markings. (This is the same as the proof described on page 16.)