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 earthqu...
