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A Passion for Mathematics  Numbers, Puzzles, Madness, Religion, and the Quest for Reality
A Passion for Mathematics  Numbers, Puzzles, Madness, Religion, and the Quest for Reality
Clifford A. Pickover
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A Passion for Mathematics is an educational, entertaining trip through the curiosities of the math world, blending an eclectic mix of history, biography, philosophy, number theory, geometry, probability, huge numbers, and mindbending problems into a delightfully compelling collection that is sure to please math buffs, students, and experienced mathematicians alike.
In each chapter, Clifford Pickover provides factoids, anecdotes, definitions, quotations, and captivating challenges that range from fun, quirky puzzles to insanely difficult problems. Readers will encounter mad mathematicians, strange number sequences, obstinate numbers, curious constants, magic squares, fractal geese, monkeys typing Hamlet, infinity, and much, much more. A Passion for Mathematics will feed readers’ fascination while giving them problemsolving skills a great workout!
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Year:
2005
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John Wiley & Sons
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411
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0471690988
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9780471690986
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A Passion for Mathematics Numbers, Puzzles, Madness, Religion, and the Quest for Reality CLIFFORD A. PICKOVER John Wiley & Sons, Inc. A Passion for Mathematics Works by Clifford A. Pickover The Alien IQ Test Black Holes: A Traveler’s Guide Calculus and Pizza Chaos and Fractals Chaos in Wonderland Computers, Pattern, Chaos, and Beauty Computers and the Imagination Cryptorunes: Codes and Secret Writing Dreaming the Future Egg Drop Soup Future Health Fractal Horizons: The Future Use of Fractals Frontiers of Scientific Visualization The Girl Who Gave Birth to Rabbits Keys to Infinity Liquid Earth The Lobotomy Club The Loom of God The Mathematics of Oz Mazes for the Mind: Computers and the Unexpected MindBending Visual Puzzles (calendars and card sets) The Paradox of God and the Science of Omniscience The Pattern Book: Fractals, Art, and Nature The Science of Aliens Sex, Drugs, Einstein, and Elves Spider Legs (with Piers Anthony) Spiral Symmetry (with Istvan Hargittai) Strange Brains and Genius Sushi Never Sleeps The Stars of Heaven Surfing through Hyperspace Time: A Traveler’s Guide Visions of the Future Visualizing Biological Information Wonders of Numbers The Zen of Magic Squares, Circles, and Stars A Passion for Mathematics Numbers, Puzzles, Madness, Religion, and the Quest for Reality CLIFFORD A. PICKOVER John Wiley & Sons, Inc. Copyright © 2005 by Clifford A. Pickover. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada Illustration credits: pages 91, 116, 137, 140, 142, 149, 150, 151, 157, 158, 159, 160, 162, 164, 167, 168, 169, 179, 199, 214, 215, 224, 225, 230, 274, 302, 336, 338, 341, 343, 345, and 348 by Brian C. Mansfield; 113, 114, 115, 145, 146, 332, 333, and 334 by Sam Loyd; 139 courtesy of Peter Hamburger and Edit Hepp; 141 by Stewart Raphael, Audrey Raphael, and Richard King; 155 by Patrick Grimm and Paul St. Denis; 165 and 166 from Magic Squares and Cubes by W. S. Andrews; 177 and 352 by Henry Ernest Dudeney; ; 200 by Bruce Patterson; 204 by Bruce Rawles; 206 by Jürgen Schmidhuber; 253 by Abram Hindle; 254, 255, 256, and 257 by Chris Coyne; 258 and 259 by Jock Cooper; 260 by Linda Bucklin; 261 by Sally Hunter; 262 and 263 by Jos Leys; 264 by Robert A. Johnston; and 266 by Cory and Catska Ench. Design and composition by Navta Associates, Inc. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate percopy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 7508400, fax (978) 6468600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 7486011, fax (201) 7486008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information about our other products and services, please contact our Customer Care Department within the United States at (800) 7622974, outside the United States at (317) 5723993 or fax (317) 5724002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress CataloginginPublication Data: Pickover, Clifford A. A passion for mathematics : numbers, puzzles, madness, religion, and the quest for reality / Clifford A. Pickover. p. cm. Includes bibliographical references and index. ISBN13 9780471690986 (paper) ISBN10 0471690988 (paper) 1. Mathematics. I. Title QA39.3.P53 2005 510—dc22 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 2004060622 Ramanujan said that he received his formulas from God. This book is dedicated to all those who find Ramanujan’s π formulas beautiful to look at: 1 π = 8 ∞ ∑ (1103 + 26390 n )( 2 n − 1 )!!( 4 n − 1 )!! 99 4 n + 2 32 n ( n!) 3 n =0 ( n 0 5 5 ∞ (11 n + 1 ) π= 2 3 n =0 ∑ () 1 2 Poch ( n ) () 1 6 Poch ( n ) ( n!) 3 ) () 5 6 Poch ( n ) 4 125 (where Poch(n) refers to the Pochhammer notation described in chapter 2) n −1 “Truly the gods have not from the beginning revealed all things to mortals, but by long seeking, mortals make progress in discovery.” —Xenophanes of Colophon (c. 500 B.C.) “Every blade of grass has its angel that bends over it and whispers, ‘grow, grow.’” —Talmudic commentary Midrash Bereishis Rabbah, 10:6 Contents Acknowledgments Introduction Acknowledgments xiii Introduction 1 1. Numbers, History, Society, and People 11 In which we encounter religious mathematicians, mad mathematicians, famous mathematicians, mathematical savants, quirky questions, fun trivia, brief biographies, mathematical gods, historical oddities, numbers and society, gossip, the history of mathematical notation, the genesis of numbers, and “What if?” questions. 2. Cool Numbers 45 In which we encounter fascinating numbers and strange number sequences. We’ll explore transcendental numbers, octonions, surreal numbers, obstinate numbers, cyclic numbers, Vibonacci numbers, perfect numbers, automorphic numbers, prime numbers, Wilson primes, palindromic primes, Fibonacci primes, Sophie Germain primes, BaxterHickerson primes, starcongruent primes, narcissistic numbers, amenable numbers, amicable numbers, padic numbers, large palindromes, factorions, hyperfactorials, primorials, ix palindions and hyperpalindions, exoticlooking formulas for π, the GolayRudinShapiro sequence, the wonderful Pochhammer notation, and famous and curious math constants (such as Liouville’s constant, the CopelandErdös constant, Brun’s constant, Champernowne’s number, Euler’s gamma, Chaitin’s constant, the LandauRamanujan constant, Mills’s constant, the golden ratio, Apéry’s constant, and constants even more bizarre). 3. Algebra, Percentages, Weird Puzzles, and Marvelous Mathematical Manipulations 111 In which we encounter treasure chests of zany and educational math problems that involve algebra, fractions, percentages, classic recreational puzzles, and various types of mathematical manipulation. Some are based on problems that are more than a thousand years old. Others are brand new. Get ready to sharpen your pencils and stretch your brains! 4. Geometry, Games, and Beyond 135 In which we explore tiles, patterns, position problems, arrays, Venn diagrams, tictactoe, other games played on boards, Königsberg bridges, catenaries, Loyd’s and Dudeney’s puzzles, Sherck’s surface, magic squares, lituuses, insideout Mandelbrot sets, the Quadratrix of Hippias, hyperspheres, fractal geese, Schmidhuber circles, Pappus’s Arbelos, Escher patterns, and chess knights. 5. Probability: Take Your Chances 209 In which we explore casinos, logic, guessing, decisions, combinations, permutations, competition, possibilities, games involving choice, monkeys typing Hamlet, Benford’s law, combinatorics, alien gambits, Rubik’s Cubes, card shuffles, marble mazes, nontransitive dice, dangerous movements of air molecules, the board game of the gods, and the tunnels of death and despair. x Contents 6. Big Numbers and Infinity 233 In which we explore very large numbers, the edges of comprehension, infinity, the Funnel of Zeus, the infinite gift, the omega crystal, the Skewes number, the Monster group, Göbel’s number, the awesome quattuordecillion, the ErdösMoser number, Archimedes’ famous “Cattle Problem,” classic paradoxes, Gauss’s “measurable infinity,” and Knuth’s arrow notation. 7. Mathematics and Beauty 251 In which we explore artistic forms generated from mathematics—delicate fungi, twentysecondcentury cityscapes, fractal necklaces and seed pods, alien devices, and a rich panoply of patterns that exhibit a cascade of detail with increasing magnifications. Answers 267 References 383 Index 387 Contents xi Contents Acknowledgments Introduction I thank Brian Mansfield for his wonderful cartoon diagrams that appear throughout the book. Over the years, Brian has been helpful beyond compare. Numerous people have provided useful feedback and information relating to the solutions to my puzzles; these individuals include Dennis Gordon, Robert Stong, Paul Moskowitz, Joseph Pe, Daniel Dockery, Mark Nandor, Mark Ganson, Nick Hobson, Chuck Gaydos, Graham Cleverley, Jeffrey Carr, Jon Anderson, “Jaymz” James Salter, Chris Meyers, Pete Barnes, Steve Brazzell, Steve Blattnig, Edith Rudy, Eric Baise, Martie Saxenmeyer, Bob Ewell, Teja Krasek, and many more. I discussed some of the original puzzles in this book at my Pickover Discussion Group, located on the Web at groups.yahoo.com/group/CliffordPickover, and I thank the group members for their wonderful discussions and comments. Many of the fancy formulas derived by the Indian mathematician Srinivasa Ramanujan come from Calvin Clawson’s Mathematical Mysteries, Bruce Berndt’s Ramanujan’s Notebooks (volumes 1 and 2), and various Internet sources. With respect to some of the elegant prime number formulas, Clawson cites Paulo Ribenboim’s The Book of Prime Number Records, second edition (New York: Springer, 1989), and The Little Book of Big Primes (New York: Springer, 1991). Calvin Clawson’s Mathematical Mysteries and David Wells’s various Penguin dictionaries provide a gold mine of mathematical concepts. A few problems in this book draw on, update, or revise problems in my earlier books, with information provided by countless readers who spend their lives tirelessly tackling mathematical conundrums. Other interesting sources and recommended reading are given in the reference section. Numerous Web sites are proliferating that give xiii comprehensive mathematical information, and my favorites include Wikipedia, the Free Encyclopedia (wikipedia.com), Ask Dr. Math (mathforum.org/dr.math/), The MacTutor History of Mathematics Archive (www.gapsystem.org/~history/), The Published Data of Robert Munafo (www.mrob.com/pub/index.html), and MathWorld (mathworld. wolfram.com). xiv Acknowledgments Acknowledgments Introduction “An equation means nothing to me unless it expresses a thought of God.” —Srinivasa Ramanujan (1887–1920) The Ramanujan Code “An intelligent observer seeing mathematicians at work might conclude that they are devotees of exotic sects, pursuers of esoteric keys to the universe.” —Philip Davis and Reuben Hersh, The Mathematical Experience, 1981 Readers of my popular mathematics books already know how I feel about numbers. Numbers are portals to other universes. Numbers help us glimpse a greater universe that’s normally shielded from our small brains, which have not evolved enough to fully comprehend the mathematical fabric of the universe. Higher mathematical discussions are a little like poetry. The Danish physicist Niels Bohr felt similarly about physics when he said, “We must be clear that, when it comes to atoms, language can be used only as in poetry.” When I think about the vast ocean of numbers that humans have scooped from the shoreless sea of reality, I get a little shiver. I hope you’ll shiver, too, as you glimpse numbers that range from integers, fractions, and radicals to stranger beasts like transcendental numbers, transfinite numbers, hyperreal numbers, surreal numbers, “nimbers,” quaternions, biquaternions, sedenions, and octonions. Of course, we have a hard time thinking of such queer entities, but from time to time, God places in our midst visionaries who function like the biblical prophets, those individuals 1 who touched a universe inches away that most of us can barely perceive. Srinivasa Ramanujan was such a prophet. He plucked mathematical ideas from the ether, out of his dreams. Ramanujan was one of India’s greatest mathematical geniuses, and he believed that the gods gave him insights. These came in a flash. He could read the codes in the mathematical matrix in the same way that Neo, the lead character in the movie The Matrix, could access mathematical symbols that formed the infrastructure of reality as they cascaded about him. I don’t know if God is a cryptographer, but codes are all around us waiting to be deciphered. Some may take a thousand years for us to understand. Some may always be shrouded in mystery. In The Matrix Reloaded, the wise Architect tells Neo that his life is “the sum of a remainder of an unbalanced equation inherent in the programming of the matrix.” Similarly, the great Swiss architect Le Corbusier (1887–1965) thought that gods played with numbers in a matrix beyond our ordinary reality: The chamois making a gigantic leap from rock to rock and alighting, with its full weight, on hooves supported by an ankle two centimeters in diameter: that is challenge and that is mathematics. The mathematical phenomenon always develops out of simple arithmetic, so useful in everyday life, out of numbers, those weapons of the gods: the gods are there, behind the wall, at play with numbers. (Le Corbusier, The Modulor, 1968) A century ago, Ramanujan was The Matrix’s Neo in our own reality. As a boy, Ramanujan was slow to learn to speak. He seemed to spend all of his time scribbling strange symbols on his slate board or writing equations in his personal notebooks. Later in life, while working in the Accounts Department of the Port Trust Office at Madras, he mailed some of his equations to the renowned British mathematician G. H. Hardy. Hardy immediately tossed these equations into the garbage—but later retrieved them for a second look. Of the formulas, Hardy said that he had “never seen anything in the least like them before,” and that some had completely “defeated” him. He quickly realized that the equations “could only be written down by a mathematician of the highest class.” Hardy wrote in Ramanujan: Twelve Lectures that the formulas “must be true because, if they were not true, no one would have had the imagination to invent them.” Indeed, Ramanujan often stated a result that had come from some sense of intuition out of the unconscious realm. He said that an Indian 2 A Passion for Mathematics goddess inspired him in his dreams. Not all of his formulas were perfect, but the avalanche of actual gems that he plucked from the mine of reality continues to boggle our modern minds. Ramanujan said that only in mathematics could one have a concrete realization of God. Blood Dreams and God’s Mathematicians Repeatedly, [Western mathematicians] have been reduced to inchoate expressions of wonder and awe in the face of Ramanujan’s powers—have stumbled about, groping for words, in trying to convey the mystery of Ramanujan.” —Robert Kanigel, The Man Who Knew Infinity, 1991 According to Ramanujan, the gods left drops of vivid blood in his dreams. After he saw the blood, scrolls containing complicated mathematics unfolded before him. When Ramanujan awakened in the morning, he scribbled only a fraction of what the gods had revealed to him. In The Man Who Knew Infinity, Robert Kanigel suggests that the ease with which Ramanujan’s spirituality and mathematics intertwined signified a “peculiar flexibility of mind, a special receptivity to loose conceptual linkages and tenuous associations. . . . ” Indeed, Ramanujan’s openness to mystical visions suggested “a mind endowed with slippery, flexible, and elastic notions of cause and effect that left him receptive to what those equipped with purely logical gifts could not see.” Before we leave Ramanujan, I should point out that many other mathematicians, such as Carl Friedrich Gauss, James Hopwood Jeans, Georg Cantor, Blaise Pascal, and John Littlewood, believed that inspiration had a divine aspect. Gauss said that he once proved a theorem “not by dint of painful effort but so to speak by the grace of God.” For these reasons, I have included a number of brief pointers to religious mathematicians in chapter 1. I hope these examples dispel the notion that mathematics and religion are totally separate realms of human endeavor. Our mathematical description of the universe forever grows, but our brains and language skills remain entrenched. New kinds of mathematics are being discovered or created all the time, but we need fresh ways to think and to understand. For example, in the last few years, mathematical proofs have been offered for famous problems in the history of mathematics, but the arguments have been far too long and complicated for experts Introduction 3 to be certain they are correct. The mathematician Thomas Hales had to wait five years before expert reviewers of his geometry paper—submitted to the journal Annals of Mathematics—finally decided that they could find no errors and that the journal should publish Hale’s proof, but only with a disclaimer saying they were not certain it was right! Moreover, mathematicians such as Keith Devlin have admitted (in the May 25, 2004, New York Times) that “the story of mathematics has reached a stage of such abstraction that many of its frontier problems cannot even be understood by the experts.” There is absolutely no hope of explaining these concepts to a popular audience. We can construct theories and do computations, but we may not be sufficiently smart to comprehend, explain, or communicate these ideas. A physics analogy is relevant here. When Werner Heisenberg worried that human beings might never truly understand atoms, Bohr was a bit more optimistic. He replied, “I think we may yet be able to do so, but in the process we may have to learn what the word understanding really means.” Today, we use computers to help us reason beyond the limitations of our own intuition. In fact, experiments with computers are leading mathematicians to discoveries and insights never dreamed of before the ubiquity of these devices. Computers and computer graphics allow mathematicians to discover results long before they can prove them formally, thus opening entirely new fields of mathematics. Even simple computer tools, such as spreadsheets, give modern mathematicians power that Heisenberg, Einstein, and Newton would have lusted after. As just one example, in the late 1990s, computer programs designed by David Bailey and Helaman Ferguson helped to produce new formulas that related pi to log 5 and two other constants. As Erica Klarreich reports in the April 24, 2004, edition of Science News, once the computer had produced the formula, proving that it was correct was extremely easy. Often, simply knowing the answer is the largest hurdle to overcome when formulating a proof. The Mathematical Smorgasbord As the island of knowledge grows, the surface that makes contact with mystery expands. When major theories are overturned, what we thought was certain knowledge gives way, and knowledge touches upon mystery differently. This newly uncovered mystery may be 4 A Passion for Mathematics humbling and unsettling, but it is the cost of truth. Creative scientists, philosophers, and poets thrive at this shoreline. —W. Mark Richardson, “A Skeptic’s Sense of Wonder,” Science, 1998 Despite all of my mystical talk about mathematics and the divine, mathematics is obviously practical. Mathematics has affected virtually every field of scientific endeavor and plays an invaluable role in fields ranging from science to sociology, from modeling ecological disasters and the spread of diseases to understanding the architecture of our brains. Thus, the fun and quirky facts, questions, anecdotes, equations, and puzzles in this book are metaphors for an amazing range of mathematical applications and notations. In fact, this book is a smorgasbord of puzzles, factoids, trivia, quotations, and serious problems to consider. You can pick and choose from the various delicacies as you explore the platter that’s set before you. The problems vary in scope, so you are free to browse quickly among concepts ranging from Champernowne’s number to the Göbel number, a number so big that it makes a trillion pale in comparison. Some of the puzzles are arranged randomly to enhance the sense of adventure and surprise. My brain is a runaway train, and these puzzles and factoids are the chunks of cerebrum scattered on the tracks. Occasionally, some of the puzzles in this book will seem simple or frivolous; for example, Why does a circle have 360 degrees? Or, Is zero an even number? Or, What’s the hardest license plate to remember? Or, Could Jesus calculate 30 × 24? However, these are questions that fans often pose to me, and I love some of these “quirkies” the best. I agree with the Austrian physicist Paul Ehrenfest, who said, “Ask questions. Don’t be afraid to appear stupid. The stupid questions are usually the best and hardest to answer. They force the speaker to think about the basic problem.” In contrast to the “quirkies,” some of the puzzles I pose in this book are so insanely difficult or require such an exhaustive search that only a computer hacker could hope to answer them, such as my problem “Triangle of the Gods” (see page 61). These are questions with which I have challenged my geeky colleagues, and on which they have labored for hours and sometimes days. I think you’ll enjoy seeing the results. Don’t be scared if you have no chance in hell of solving them. Just enjoy the fact that intense people will often respond to my odd challenges with apparent glee. Most of the problems in Introduction 5 the book are somewhere inbetween the extremes of simplicity and impossibility and can be solved with a pencil and paper. Chapter 3 contains most of the problems that teachers can enjoy with students. I will also tease you with fancy formulas, like those decorating the book from Ramanujan. Sometimes my goal is simply to delight you with wonderfullooking equations to ponder. Occasionally, a concept is repeated, just to see if you’ve learned your lesson and recognize a similar problem in a new guise. The different ways of getting to the same solution or concept reveal things that a single approach misses. I’ve been in love with recreational mathematics for many years because of its educational value. Contemplating even simple problems stretches the imagination. The usefulness of mathematics allows us to build spaceships and investigate the geometry of our universe. Numbers will be our first means of communication with intelligent alien races. Ancient peoples, like the Greeks, also had a deep fascination with numbers. Could it be that in difficult times numbers were the only constant thing in an evershifting world? To the Pythagoreans, an ancient Greek sect, numbers were tangible, immutable, comfortable, eternal—more reliable than friends, less threatening than Zeus. Explanation of Symbols NonEuclidean calculus and quantum physics are enough to stretch any brain, and when one mixes them with folklore, and tries to trace a strange background of multidimensional reality behind the ghoulish hints of the Gothic tales and the wild whispers of the chimneycorner, one can hardly expect to be wholly free from mental tension. —H. P. Lovecraft, “Dreams in the Witch House,” 1933 I use the following symbols to differentiate classes of entries in this book: signifies a thoughtprovoking quotation. signifies a mathematical definition that may come in handy throughout the book. signifies a mathematical factoid to stimulate your imagination. 6 A Passion for Mathematics signifies a problem to be solved. Answers are provided at the back of the book. These different classes of entries should cause even the most rightbrained readers to fall in love with mathematics. Some of the zanier problems will entertain people at all levels of mathematical sophistication. As I said, don’t worry if you cannot solve many of the puzzles in the book. Some of them still challenge seasoned mathematicians. One common characteristic of mathematicians is an obsession with completeness—an urge to go back to first principles to explain their works. As a result, readers must often wade through pages of background before getting to the essential ingredients. To avoid this, each problem in my book is short, at most only a few paragraphs in length. One advantage of this format is that you can jump right in to experiment or ponder and have fun, without having to sort through a lot of verbiage. The book is not intended for mathematicians looking for formal mathematical explanations. Of course, this approach has some disadvantages. In just a paragraph or two, I can’t go into any depth on a subject. You won’t find much historical context or many extended discussions. In the interest of brevity, even the answer section may require readers to research or ponder a particular puzzle further to truly understand it. To some extent, the choice of topics for inclusion in this book is arbitrary, although these topics give a nice introduction to some classic and original problems in number theory, algebra, geometry, probability, infinity, and so forth. These are also problems that I have personally enjoyed and are representative of a wider class of problems of interest to mathematicians today. Grab a pencil. Do not fear. Some of the topics in the book may appear to be curiosities, with little practical application or purpose. However, I have found these experiments to be useful and educational, as have the many students, educators, and scientists who have written to me. Throughout history, experiments, ideas, and conclusions that originate in the play of the mind have found striking and unexpected practical applications. A few puzzles come from Sam Loyd, the famous nineteenthcentury American puzzlemaster. Loyd (1841–1911) invented thousands of popular puzzles, which his son collected in a book titled Cyclopedia of Puzzles. I hope you enjoy the classics presented here. Introduction 7 Cultivating Perpetual Mystery “Pure mathematics is religion.” —Friedrich von Hardenberg, circa 1801 A wonderful panoply of relationships in nature can be expressed using integer numbers and their ratios. Simple numerical patterns describe spiral floret formations in sunflowers, scales on pinecones, branching patterns on trees, and the periodic life cycles of insect populations. Mathematical theories have predicted phenomena that were not confirmed until years later. Maxwell’s equations, for example, predicted radio waves. Einstein’s field equations suggested that gravity would bend light and that the universe is expanding. The physicist Paul Dirac once noted that the abstract mathematics we study now gives us a glimpse of physics in the future. In fact, his equations predicted the existence of antimatter, which was subsequently discovered. Similarly, the mathematician Nikolai Lobachevsky said that “there is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world.” A famous incident involving Murray GellMann and his colleagues demonstrates the predictive power of mathematics and symmetry regarding the existence of a subatomic particle known as the Omegaminus. GellMann had drawn a symmetric, geometric pattern in which each position in the pattern, except for one empty spot, contained a known particle. GellMann put his finger on the spot and said with almost mystical insight, “There is a particle.” His insight was correct, and experimentalists later found an actual particle corresponding to the empty spot. One of my favorite quotations describing the mystical side of science comes from Richard Power’s The Gold Bug Variations: “Science is not about control. It is about cultivating a perpetual condition of wonder in the face of something that forever grows one step richer and subtler than our latest theory about it. It is about reverence, not mastery.” Today, mathematics has permeated every field of scientific endeavor and plays an invaluable role in biology, physics, chemistry, economics, sociology, and engineering. Math can be used to help explain the structure of a rainbow, teach us how to make money in the stock market, guide a spacecraft, make weather forecasts, predict population growth, design buildings, quantify happiness, and analyze the spread of AIDS. 8 A Passion for Mathematics Mathematics has caused a revolution. It has shaped our thoughts. It has shaped the way we think. Mathematics has changed the way we look at the world. This introduction is dedicated to anyone who can decode the following secret message. .. /  .... .. . . /  .... .  / ... . .. . / ... . . /  ... / .  .. / . .. ... ... / ...   .... . .. /   / .. . ..  .. . /  .... .. ... / ... . .. .. .  /  . ... ... . . . ... “I am the thought you are now thinking.” —Douglas Hofstadter, Metamagical Themas, 1985 Introduction 9 1 2 Numbers, History,3Society, and People 4 5 I 6 N WHICH WE ENCOUNTER RELIGIOUS MATHEMATICIANS, MAD MATHEMATICIANS, famous mathematicians, mathematical savants, quirky questions, fun trivia, brief biographies, mathematical gods, historical oddities, numbers and society, gossip, the history of mathematical notation, the genesis of numbers, and “What if?” questions. Mathematics is the hammer that shatters the ice of our unconscious. 11 Ancient counting. Let’s start the book with a question. What is the earliest evidence we have of humans counting? If this question is too difficult, can you guess whether the evidence is before or after 10,000 B.C.— and what the evidence might be? (See Answer 1.1.) Mathematics and beauty. I’ve collected mathematical quotations since my teenage years. Here’s a favorite: “Mathematics, rightly viewed, possesses not only truth, but supreme beauty— a beauty cold and austere, like that of sculpture” (Bertrand Russell, Mysticism and Logic, 1918). The symbols of mathematics. Mathematical notation shapes humanity’s ability to efficiently contemplate mathematics. Here’s a cool factoid for you: The symbols + and –, referring to addition and subtraction, first appeared in 1456 in an unpublished manuscript by the mathematician Johann Regiomontanus (a.k.a. Johann Müller). The plus symbol, as an abbreviation for the Latin et (and), was found earlier in a manuscript dated 1417; however, the downward stroke was not quite vertical. Math beyond humanity. “We now know that there exist true propositions which we can never formally prove. What about propositions whose proofs require arguments beyond our capabilities? What about propositions whose proofs require millions of pages? Or a million, million pages? Are there proofs that are possible, but beyond us?” (Calvin Clawson, Mathematical Mysteries). The multiplication symbol. Mathematics and reality. Do humans invent mathematics or discover mathematics? (See Answer 1.2.) Mathematics and the universe. Here is a deep thought to start our mathematical journey. Do you think humanity’s longterm fascination with mathematics has arisen because the universe is constructed from a mathematical fabric? We’ll approach this question later in the chapter. For now, you may enjoy knowing that in 1623, Galileo Galilei echoed this belief in a mathematical universe by stating his credo: “Nature’s great book is written in mathematical symbols.” Plato’s doctrine was that God is a geometer, and Sir James Jeans believed that God experimented with arithmetic. Isaac Newton supposed that the planets were originally thrown into orbit by God, but even after God decreed the law of gravitation, the planets required continual adjustments to their orbits. In 1631, the multiplication symbol × was introduced by the English mathematician William Oughtred (1574–1660) in his book Keys to Mathematics, published in London. Incidentally, this Anglican minister is also famous for having invented the slide rule, which was used by generations of scientists and mathematicians. The slide rule’s doom in the mid1970s, due to the pervasive influx of inexpensive pocket calculators, was rapid and unexpected. Numbers, History, Society, and People 13 Math and madness. Many mathematicians throughout history have had a trace of madness or have been eccentric. Here’s a relevant quotation on the subject by the British mathematician John Edensor Littlewood (1885–1977), who suffered from depression for most of his life: “Mathematics is a dangerous profession; an appreciable proportion of us goes mad.” Mathematics and murder. What triple murderer was also a brilliant French mathematician who did his finest work while confined to a hospital for the criminally insane? (See Answer 1.3.) Creativity and madness. “There is a theory that creativity arises when individuals are out of sync with their environment. To put it simply, people who fit in with their communities have insufficient motivation to risk their psyches in creating something truly new, while those who are out of sync are driven by the constant need to prove their worth. 14 A Passion for Mathematics Mathematicians and religion. Over the years, many of my readers have assumed that famous mathematicians are not religious. In actuality, a number of important mathematicians were quite religious. As an interesting exercise, I conducted an Internet survey in which I asked respondents to name important mathematicians who were also religious. Isaac Newton and Blaise Pascal were the most commonly cited religious mathematicians. In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religion and mathematics struggle to express relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, as well as impenetrable language. Both exercise the deep recesses of our minds and stimulate our imagination. Mathematicians, like priests, seek “ideal,” immutable, nonmaterial truths and then often venture to apply these truths in the real world. Are mathematics and religion the most powerful evidence of the inventive genius of the human race? In “Reason and Faith, Eternally Bound” (December 20, 2003, New York Times, B7), Edward Rothstein notes that faith was the inspiration for Newton and Kepler, as well as for numerous scientific and mathematical triumphs. “The conviction that there is an order to things, that the mind can comprehend that order and that this order is not infinitely malleable, those scientific beliefs may include elements of faith.” In his Critique of Pure Reason, Immanuel Kant describes how “the light dove, cleaving the air in her free flight and feeling its resistance against her wings, might imagine that its flight would be freer still in empty space.” But if we were to remove the air, the bird would plummet. Is faith—or a cosmic sense of mystery—like the air that allows some seekers to soar? Whatever mathematical or scientific advances humans make, we will always continue to swim in a sea of mystery. They have less to lose and more to gain” (Gary Taubes, “Beyond the Soapsuds Universe,” 1977). Pascal’s mystery. “There is a Godshaped vacuum in every heart” (Blaise Pascal, Pensées, 1670). Leaving mathematics and approaching God. What famous French mathematician and teenage prodigy finally decided that religion was more to his liking and joined his sister in her convent, where he gave up mathematics and social life? (See Answer 1.4.) Ramanujan’s gods. As mentioned in this book’s introduction, the mathematician Srinivasa Ramanujan (1887–1920) was an ardent follower of several Hindu deities. After receiving visions from these gods in the form of blood droplets, Ramanujan saw scrolls that contained very complicated mathematics. When he woke from his dreams, he set down on paper only a fraction of what the gods showed him. Throughout history, creative geniuses have been open to dreams as a source of inspiration. Paul McCartney said that the melody for the famous Beatles’ song “Yesterday,” one of the most popular songs ever written, came to him in a dream. Apparently, the tune seemed so beautiful and haunting that for a while he was not certain it was original. The Danish physicist Niels Bohr conceived the model of an atom from a dream. Elias Howe received in a dream the image of the kind of needle design required for a lockstitch sewing machine. René Descartes was able to advance his geometrical methods after flashes of insight that came in dreams. The dreams of Dmitry Mendeleyev, Friedrich August Kekulé, and Otto Loewi inspired scientific breakthroughs. It is not an exaggeration to suggest that many scientific and mathematical advances arose from the stuff of dreams. Blaise Pascal (1623–1662), a Frenchman, was a geometer, a probabilist, a physicist, a philosopher, and a combinatorist. He was also deeply spiritual and a leader of the Jansenist sect, a Calvinistic quasiProtestant group within the Catholic Church. He believed that it made sense to become a Christian. If the person dies, and there is no God, the person loses nothing. If there is a God, then the person has gained heaven, while skeptics lose everything in hell. Legend has it that Pascal in his early childhood sought to prove the existence of God. Because Pascal could not simply command God to show Himself, he tried to prove the existence of a devil so that he could then infer the existence of God. He drew a pentagram on the ground, but the exercise scared him, and he ran away. Pascal said that this experience made him certain of God’s existence. One evening in 1654, he had a twohour mystical vision that he called a “night of fire,” in which he experienced fire and “the God of Abraham, Isaac, and Jacob . . . and of Jesus Christ.” Pascal recorded his vision in his work “Memorial.” A scrap of paper containing the “Memorial” was found in the lining of his coat after his death, for he carried this reminder about with him always. The three lines of “Memorial” are Complete submission to Jesus Christ and to my director. Eternally in joy for a day’s exercise on the earth. May I not forget your words. Amen. Numbers, History, Society, and People 15 Transcendence. “Much of the history of science, like the history of religion, is a history of struggles driven by power and money. And yet, this is not the whole story. Genuine saints occasionally play an important role, both in religion and science. For many scientists, the reward for being a scientist is not the power and the money but the chance of catching a glimpse of the transcendent beauty of nature” (Freeman Dyson, in the introduction to Nature’s Imagination). The value of eccentricity. “That so few now dare to be eccentric, marks the chief danger of our time” (John Stuart Mill, nineteenthcentury English philosopher). Counting and the mind. I quickly toss a number of marbles onto a pillow. You may stare at them for an instant to determine how many marbles are on the pillow. Obviously, if I were to toss just two marbles, you could easily determine that two marbles sit on the pillow. What is the largest number of marbles you can 16 A Passion for Mathematics quantify, at a glance, without having to individually count them? (See Answer 1.5.) Circles. Why are there 360 degrees in a circle? (See Answer 1.6.) Calculating π. Which nineteenthcentury British boarding school supervisor spent a significant portion of his life calculating π to 707 places and died a happy man, despite a sad error that was later found in his calculations? (See Answer 1.8.) The mystery of Ramanujan. After years of working through Ramanujan’s notebooks, the mathematician Bruce Berndt said, “I still don’t understand it all. I may be able to prove it, but I don’t know where it comes from and where it fits into the rest of mathematics. The enigma of Ramanujan’s creative process is still covered by a curtain that has barely been drawn” (Robert Kanigel, The Man Who Knew Infinity, 1991). The special number 7. In ancient days, the number 7 was thought of as just another way to signify “many.” Even in recent times, there have been tribes that used no numbers higher than 7. In the 1880s, the German ethnologist Karl von Steinen described how certain South American Indian tribes had very few words for numbers. As a test, he repeatedly asked them to count ten grains of corn. They counted “slowly The world’s most forgettable license plate? Today, mathematics affects society in the funniest of ways. I once read an article about someone who claimed to have devised the most forgettable license plate, but the article did not divulge the secret sequence. What is the most forgettable license plate? Is it a random sequence of eight letters and numbers—for example, 6AZL4QO9 (the maximum allowed in New York)? Or perhaps a set of visually confusing numbers or letters—for example, MWNNMWWM? Or maybe a binary number like 01001100. What do you think? What would a mathematician think? (See Answer 1.7.) but correctly to six, but when it came to the seventh grain and the eighth, they grew tense and uneasy, at first yawning and complaining of a headache, then finally avoided the question altogether or simply walked off.” Perhaps seven means “many” in such common phrases as “seven seas” and “seven deadly sins.” (These interesting facts come from Adrian Room, The Guinness Book of Numbers, 1989.) Carl Friedrich Gauss (1777–1855), a German, was a mathematician, an astronomer, and a physicist with a wide range of contributions. Like Ramanujan, after Gauss proved a theorem, he sometimes said that the insight did not come from “painful effort but, so to speak, by the grace of God.” He also once wrote, “There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example, touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.” Isaac Newton (1642–1727), an Englishman, was a mathematician, a physicist, an astronomer, a coinventor of calculus, and famous for his law of gravitation. He was also the author of many books on biblical subjects, especially prophecy. Perhaps less well known is the fact that Newton was a creationist who wanted to be known as much for his theological writings as for his scientific and mathematical texts. Newton believed in a Christian unity, as opposed to a trinity. He developed calculus as a means of describing motion, and perhaps for understanding the nature of God through a clearer understanding of nature and reality. He respected the Bible and accepted its account of Creation. Genius and eccentricity. James Hopwood Jeans “The amount of eccentricity in a society has been proportional to the amount of genius, material vigor and moral courage which it contains” (John Stuart Mill, On Liberty, 1869). (1877–1946) was an applied mathematician, a physicist, and an astronomer. He sometimes likened God to a mathematician and wrote in The Mysterious Universe (1930), “From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician.” He has also written, “Physics tries to discover the pattern of events which controls the phenomena we observe. But we can never know what this pattern means or how it originates; and even if some superior intelligence were to tell us, we should find the explanation unintelligible” (Physics and Philosophy, 1942). Mathematics and God. “The Christians know that the mathematical principles, according to which the corporeal world was to be created, are coeternal with God. Geometry has supplied God with the models for the creation of the world. Within the image of God it has passed into man, and was certainly not received within through the eyes” (Johannes Kepler, The Harmony of the World, 1619). Numbers, History, Society, and People 17 Leonhard Euler (1707–1783) was a prolific Swiss mathematician and the son of a vicar. Legends tell of Leonhard Euler’s distress at being unable to mathematically prove the existence of God. Many mathematicians of his time considered mathematics a tool to decipher God’s design and codes. Although he was a devout Christian all his life, he could not find the enthusiasm for the study of theology, compared to that of mathematics. He was completely blind for the last seventeen years of his life, during which time he produced roughly half of his total output. Euler is responsible for our common, modernday use of many famous mathematical notations—for example, f(x) for a function, e for the base of natural logs, i for the square root of –1, π for pi, Σ for summation. He tested Pierre de Fermat’s conjecture that numbers of the form 2n + 1 were always prime if n is a power of 2. Euler verified this for n = 1, 2, 4, 8, and 16, and showed that the next case 232 + 1 = 4,294,967, 297 = 641 × 6,700,417, and so is not prime. George Boole (1815–1864), an Englishman, was a logician and an algebraist. Like Ramanujan and other mystical mathematicians, Boole had “mystical” experiences. David Noble, in his book The Religion of Technology, notes, “The thought flashed upon him suddenly as he was walking across a field that his ambition in life was to explain the logic of human thought and to delve analytically into the spiritual aspects of man’s nature [through] the expression of logical relations in symbolic 18 A Passion for Mathematics or algebraic form. . . . It is impossible to separate Boole’s religious beliefs from his mathematics.” Boole often spoke of his almost photographic memory, describing it as “an arrangement of the mind for every fact and idea, which I can find at once, as if it were in a wellordered set of drawers.” Boole died at age fortynine, after his wife mistakenly thought that tossing buckets of water on him and his bed would cure his flu. Today, Boolean algebra has found wide applications in the design of computers. The value of puzzles. “It is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science” (Bertrand Russell, Mind, 1905). A mathematical nomad. What legendary mathematician, and one of the most prolific mathematicians in history, was so devoted to math that he lived as a nomad with no home and no job? Sexual contact revolted him; even an accidental touch by anyone made him feel uncomfortable. (See Answer 1.9.) Marin Mersenne (1588– 1648) was another mathematician who was deeply religious. Mersenne, a Frenchman, was a theologian, a philosopher, a number theorist, a priest, and a monk. He argued that God’s majesty would not be diminished had Mirror phobia. What brilliant, handsome mathematician so hated mirrors that he covered them wherever he went? (See Answer 1.10.) What is a mathematician? “A mathematician is a blind man in a dark room looking for a black cat which isn’t there” (Charles Darwin). Animal math. Can animals count? (See Answer 1.11.) He created just one world, instead of many, because the one world would be infinite in every part. His first publications were theological studies against atheism and skepticism. Mersenne was fascinated by prime numbers (numbers like 7 that were divisible only by themselves and 1), and he tried to find a formula that he could use to find all primes. Although he did not find such a formula, his work on “Mersenne numbers” of the form 2p – 1, where p is a prime number, continues to interest us today. Mersenne numbers are the easiest type of number to prove prime, so they are usually the largest primes of which humanity is aware. Mersenne himself found several prime numbers of the form 2p – 1, but he underesti mated the future of computing power by stating that all eternity would not be sufficient to decide if a 15 or 20digit number were prime. Unfortunately, the prime number values for p that make 2p – 1 a prime number seem to form no regular sequence. For example, the Mersenne number is prime when p = 2, 3, 5, 7, 13, 17, 19, . . . Notice that when p is equal to the prime number 11, M11 = 2,047, which is not prime because 2,047 = 23 × 89. The fortieth Mersenne prime was discovered in 2003, and it contained 6,320,430 digits! In particular, the Michigan State University graduate student Michael Shafer discovered that 220,996,011 – 1 is prime. The number is so large that it would require about fifteen hundred pages to write on paper using an ordinary font. Shafer, age twentysix, helped find the number as a volunteer on a project called the Great Internet Mersenne Prime Search. Tens of thousands of people volunteer the use of their personal computers in a worldwide project that harnesses the power of hundreds of thousands of computers, in effect creating a supercomputer capable of performing trillions of calculations per second. Shafer used an ordinary Dell computer in his office for nineteen days. What would Mersenne have thought of this large beast? In 2005, the German eye surgeon Martin Nowak, also part of the Great Internet Mersenne Prime Search, discovered the fortysecond Mersenne prime number, 225,964,951 – 1, which has over seven million digits. Nowak’s 2.4GHz Pentium4 computer spent roughly fifty days analyzing the number before reporting the find. The Electronic Frontier Foundation, a U.S. Internet campaign group, has promised to give $100,000 to whoever finds the first tenmilliondigit prime number. Numbers, History, Society, and People 19 Mathematics and God. “Before creation, God did just pure mathematics. Then He thought it would be a pleasant change to do some applied” (John Edensor Littlewood, A Mathematician’s Miscellany, 1953). The division symbol. The division symbol ÷ first appeared in print in Johann Heinrich Rahn’s Teutsche Algebra (1659). Donald Knuth (1938–) is a computer scientist and a mathematician. He is also a fine example of a mathematician who is interested in religion. For example, he has been an active Lutheran and a Sunday school teacher. His attractive book titled 3:16 consists entirely of commentary on chapter 3, verse 16, of each of the books in the Bible. Knuth also includes calligraphic renderings of the verses. Knuth himself has said, “It’s tragic that scientific advances have caused many people to imagine that they know it all, and that God is irrelevant or nonexistent. The fact is that everything we learn reveals more things that we do not understand. . . . Reverence for God comes naturally if we are honest about how little we know.” Mathematics and sex. “A wellknown mathematician once told me that the great thing about liking both math and sex was that he could do either one while thinking about the other” (Steven E. Landsburg, in a 1993 post to the newsgroup sci.math). Brain limitation. “Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions” (Ronald Graham, a prior director of Information Sciences Research at AT&T Research, quoted in Paul Hoffman’s “The Man Who Loves Only Numbers,” Atlantic Monthly, 1987). 20 A Passion for Mathematics Mystery mathematician. Around A.D. 500, the Greek philosopher Metrodorus gave us the following puzzle that describes the life of a famous mathematician: A certain man’s boyhood lasted 1⁄ 6 of his life; he married after 1⁄ 7 more; his beard grew after 1⁄12 more, and his son was born 5 years later; the son lived to half his father’s final age, and the father died 4 years after the son. Tell me the mystery man’s name or his age at death. (See Answer 1.12.) Mathematician starves. What famous mathematician deliberately starved himself to death in 1978? (Hint: He was perhaps the most brilliant logician since Aristotle.) (See Answer 1.13.) Understanding brilliance. “Maybe the brilliance of the brilliant can be understood only by the nearly brilliant” (Anthony Smith, The Mind, 1984). Georg Friedrich Bernhard Riemann (1826–1866) was a German mathematician who made important contributions to geometry, number theory, topology, mathematical physics, and the theory of complex variables. He also attempted to write a mathematical proof of the truth of the Book of Genesis, was a student of theology and bibli Calculating prodigy has plastic brain. Rüdiger Gamm is shocking the world with his calculating powers and is changing the way we think about the human brain. He did poorly at mathematics in school but is now a worldfamous human calculator, able to access regions of his brain that are off limits to most of us. He is not autistic but has been able to train his brain to perform lightning calculations. For example, he can calculate 53 to the ninth power in his head. He can divide prime numbers and calculate the answer to 60 decimal points and more. He can calculate fifth roots. Amazing calculating powers such as these were previously thought to be possible only by “autistic savants.” (Autistic savants often have severe developmental disabilities but, at the same time, have special skills and an incredible memory.) Gamm’s talent has attracted the curiosity of European researchers, who have imaged his brain with PET scans while he performed math problems. These breathtaking studies reveal that Gamm is now able to use areas of his brain that ordinary humans can use for other purposes. In particular, he can make use of the areas of his brain that are normally responsible for longterm memory, in order to perform his rapid calculations. Scientists hypothesize that Gamm temporarily uses these areas to “hold” digits in socalled “working memory,” the brain’s temporary holding area. Gamm is essentially doing what computers do when they extend their capabilities by using swap space on the hard drive to increase their capabilities. Scientists are not sure how Gamm acquired this ability, considering that he became interested in mathematical calculation only when he was in his twenties. (You can learn more in Steve Silberman’s “The Keys to Genius,” Wired, no. 11.12, December 2003.) cal Hebrew, and was the son of a Lutheran minister. The Riemann hypothesis, published by Riemann in 1859, deals with the zeros of a very wiggly function, and the hypothesis still resists modern mathematicians’ attempts to prove it. Chapter 3 describes the hypothesis further. A dislike for mathematics. “I’m sorry to say that the subject I most disliked was mathematics. I have thought about it. I think the reason was that mathematics leaves no room for argument. If you made a mistake, that was all there was to it” (Malcolm X, The Autobiography of Malcolm X, 1965). Mathematics and humanity. “Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house” (Robert A. Heinlein, Time Enough for Love, 1973). Numbers, History, Society, and People 21 Kurt Gödel (1906–1978) is an example of a mathematical genius obsessed with God and the afterlife. As discussed in Answer 1.13 about the mathematician who starved himself to death, Gödel was a logician, a mathematician, and a philosopher who was famous for having shown that in any axiomatic system for mathematics, there are propositions that cannot be proved or disproved within the axioms of the system. Gödel thought it was possible to show the logical necessity for life after death and the existence of God. In four long letters to his mother, Gödel gave reasons for believing in a next world. Math and madness. “Cantor’s work, though brilliant, seemed to move in halfsteps. The closer he came to the answers he sought, the further away they seemed. Eventually, it drove him mad, as it had mathematicians before him” (Amir D. Aczel, The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity, 2000). 22 A Passion for Mathematics Gottfried Wilhelm von Leibniz (1646–1716), a German, was an analyst, a combinatorist, a logician, and the coinventor of calculus who also passionately argued for the existence of God. According to Leibniz, God chooses to actualize this world out of an infinite number of possible worlds. In other words, limited only by contradiction, God first conceives of every possible world, and then God simply chooses which of them to create. Leibniz is also famous for the principle of “preestablished harmony,” which states that God constructed the universe in such a way that corresponding mental and physical events occur simultaneously. His “monad theory” states that the universe consists of an infinite number of substances called monads, each of which has its own individual identity but is an expression of the whole universe from a particular unique viewpoint. Greaterthan symbol. The greaterthan and lessthan symbols (> and <) were introduced by the British mathematician Thomas Harriot in his Artis Analyticae Praxis, published in 1631. Greaterthan or equalto symbol. The symbol ≥ (greater than or equal to) was first introduced by the French scientist Pierre Bouguer in 1734. Mathematician murdered. Why was the first woman mathematician murdered? (See Answer 1.14.) Going to the movies. What was the largest number ever used in the title of an Ameri can movie? Name the movie! (See Answer 1.15.) What is the largest number less than a billion ever used in a major, fulllength movie title? (Hint: The song was popular in the late 1920s and the early 1930s.) (See Answer 1.16.) Math and madness. Many mathematicians were depressed and religious at the same time. Which famous mathematician invented the concept of “transfinite numbers” (essentially, different “levels” of infinity), believed that God revealed mathematical ideas to him, and was a frequent guest of sanitariums? (See Answer 1.17.) Mathematics and diapering. “A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects” (Robert A. Heinlein, Time Enough for Love, 1973). omists and psychologists very seldom indeed. It is as their subject matter comes nearer to man himself that their antireligious bias hardens” (C. S. Lewis, The Grand Miracle: And Other Selected Essays on Theology and Ethics from God in the Dock, 1983). The Number Pope. As I write this book, I realize that a thousand years ago, the last Popemathematician died. Gerbert of Aurillac (c. 946– 1003) was fascinated by mathematics and was elected to be Pope Sylvester II in 999. His advanced knowledge of mathematics convinced some of his enemies that he was an evil magician. In Reims, he transformed the floor of the cathedral into a giant abacus. That must have been a sight to see! The “Number Pope” was also important because he adopted Arabic numerals (1, 2, 3, 4, 5, 6, 7, 8, 9) as a replacement for Roman numerals. He contributed to the invention of the pendulum clock, invented devices that tracked planetary orbits, and wrote on geometry. When he realized that he lacked knowledge of formal logic, he studied under German logicians. He said, “The just man lives by faith; but it is good that he should combine science with his faith.” Hardy’s six wishes. In the 1920s, the British mathematician G. H. Hardy wrote a postcard to his friend, listing six New Year’s wishes: 1. prove the Riemann hypothesis 2. score well at the end of an important game of cricket 3. find an argument for the nonexistence of God that convinces the general public 4. be the first man at the top of Mount Everest Mathematicians and God. “Mathematicians, astronomers, and physicists are often religious, even mystical; biologists much less often; econ 5. be the first president of the USSR, Great Britain, and Germany 6. murder Mussolini (The London Mathematical Society Newsletter, 1994) Numbers, History, Society, and People 23 Charles Babbage (1792–1871), an Englishman, was an analyst, a statistician, and an inventor who was also interested in religious miracles. He once wrote, “Miracles are not a breach of established laws, but . . . indicate the existence of far higher laws.” Babbage argued that miracles could occur in a mechanistic world. Just as Babbage could program strange behavior on his calculating machines, God could program similar irregularities in nature. While investigating biblical miracles, he assumed that the chance of a man rising from the dead is one in 1012. Babbage is famous for conceiving an enormous handcranked mechanical calculator, an early progenitor of our modern computers. Babbage thought the device would be most useful in producing mathematical tables, but he worried about mistakes that would be made by humans who transcribed the results from its thirtyone metal output wheels. Today, we realize that Babbage was a hundred years ahead of his time and that the politics and the technology of his era were inadequate for his lofty dreams. Tinkertoy computer. In the early 1980s, the computer geniuses Danny Hillis, Brian Silverman, and friends built a Tinkertoy computer that played tictactoe. The device was made from 10,000 Tinkertoy pieces. Fantasy meeting of Pythagoras, Cantor, and Gödel. I often fantasize about the outcome of placing mathematicians from different eras in the same room. For example, I would be intrigued to 24 A Passion for Mathematics gather Pythagoras, Cantor, and Gödel in a small room with a single blackboard to debate their various ideas on mathematics and God. What profound knowledge might we gain if we had the power to bring together great thinkers of various ages for a conference on mathematics? Would a roundtable discussion with Pythagoras, Cantor, and Gödel produce less interesting ideas than one with Newton and Einstein? Could ancient mathematicians contribute any useful ideas to modern mathemati cians? Would a meeting of timetraveling mathematicians offer more to humanity than a meeting of other scientists— for example, biologists or sociologists? These are all fascinating questions to which I don’t yet have answers. Mathematicians as God’s messengers. “Cantor felt a duty to keep on, in the face of adversity, to bring the insights he had been given as God’s messenger to mathematicians everywhere” (Joseph Dauben, Georg Cantor, 1990). Power notation. In 1637, the philosopher René Descartes was the first person to use the superscript notation for raising numbers and variables to powers—for example, as in x2. Numerical religion. What ancient mathematician established a numerical religion whose main tenets included the transmigration of souls and the sinfulness of eating beans? (See Answer 1.18.) Modern mathematical murderer. Which modern mathematician murdered and maimed the most people from a distance? (See Answer 1.19.) The ∞ symbol. Most high school students are familiar with the mathematical symbol for infinity (∞). Do you think this symbol was used a hundred years ago? Who first used this odd symbol? (See Answer 1.20.) Mathematics of tictactoe. In how many ways can you place Xs and Os on a standard tictactoe board? (See Answer 1.21.) The square root symbol. The Austrian mathematician Christoff Rudolff was the first to use the square root symbol √ in print; it was published in 1525 in Die Coss. Mathematics and poetry. “It is impossible to be a mathematician without being a poet in soul” (Sofia Kovalevskaya, quoted in Agnesi to Zeno by Sanderson Smith, 1996). Chickens and tictactoe. The mathematics of tictactoe have been discussed for decades, but can a chicken actually learn to play well? In 2001, an Atlantic City casino offered its patrons a tictactoe “chicken challenge” and offered cash prizes of up to $10,000. The chicken gets the first entry, usually by pecking at X or O on a video display inside a special henhouse set up in the casino’s main concourse. Gamblers standing outside the booth then get to make the next move by pressing buttons on a separate panel. There is no prize for a tie. A typical game lasts for about a minute, and the chicken seems to be trained to peck at an X or an O, depending on the human’s moves. Supposedly, the tictactoeplaying chickens work in shifts of one to two hours to avoid stressing the animals. Various animalrights advocates have protested the use of chickens in tictactoe games. Can a chicken actually learn to play tictactoe? (See Answer 1.22.) Mathematics and God. “God exists since mathematics is consistent, and the devil exists since we cannot prove the consistency” (Morris Kline, Mathematical Thought from Ancient to Modern Times, 1990). Lunatic scribbles and mathematics. “If a lunatic scribbles a jumble of mathematical symbols it does not follow that the writing means anything merely because to the inexpert eye it is indistinguishable from higher mathematics” (Eric Temple Bell, quoted in J. R. New man’s The World of Mathematics, 1956). God’s perspective. “When mathematicians think about algorithms, it is usually from the God’seye perspective. They are interested in proving, for instance, that there is some algorithm with some interesting property, or that there is no such algorithm, and in order to prove such things you needn’t actually locate the algorithm you are talking about . . . ” (Daniel Dennett, Darwin’s Dangerous Idea: Evolution and the Meaning of Life, 1996). Numbers, History, Society, and People 25 Erdös contemplates death. Once, while pondering his own death, the mathematician Paul Erdös (1913–1996) remarked, “My mother said, ‘Even you, Paul, can be in only one place at one time.’ Maybe soon I will be relieved of this disadvantage. Maybe, once I’ve left, I’ll be able to be in many places at the same time. Maybe then I’ll be able to collaborate with Archimedes and Euclid.” Creativity and madness. “Creativity and genius feed off mental turmoil. The ancient Greeks, for instance, believed in divine forms of madness that inspired mortals’ extraordinary creative acts” (Bruce Bower, Science News, 1995). Science, Einstein, and God. “The scientist’s religious feeling takes the form of a rapturous amazement at the harmony of natural law, which reveals an intelligence of such superiority that, compared with it, all the systematic thinking and acting of human beings is an utterly insignificant reflection. This feeling is the guiding principle of his life and work. . . . It is beyond question closely akin to that which has possessed the religious geniuses of all ages” (Albert Einstein, Mein Weltbild, 1934). Mathematics and the infinite. “Mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won’t ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite” (Paul Erdös, quoted in Paul Hoffman’s The Man Who Loved Only Numbers, 1998). First female doctorate. Who was the first woman to receive a doctorate in mathematics, and in what century do you think she received it? (See Answer 1.23.) 26 A Passion for Mathematics Math in the movies. In the movie A Beautiful Mind, Russell Crowe scrolls the following formulas on the blackboard in his MIT class: V = {F : 3 – X → 3smooth W = {F = ∇g} dim(V/W) = ? Movie directors were told by advisers that this set of formulas was subtle enough to be out of reach for most undergraduates, but accessible enough so that Jennifer Connelly’s character might be able to dream up a possible solution. Mathematics and homosexuality. Which brilliant mathematician was forced to become a human guinea pig and was subjected to drug experiments to reverse his homosexuality? (Hint: He was a 1950s computer theorist whose mandatory drug therapy made him impotent and caused his breasts to enlarge. He also helped to break the codes of the German Engima code machines during World War II.) (See Answer 1.24.) A famous female mathematician. Maria Agnesi (1718–1799) is one of the most famous female mathematicians of the last few centuries and is noted for her work in differential calculus. When she was seven years old, she mastered the Latin, the Greek, and the Hebrew languages, and at age nine she published a Latin discourse defending higher education for women. As an adult, her clearly written textbooks condensed the diverse research writings and methods of a number of mathematicians. They also contained many of her own original contributions to the field, including a discussion of the cubic curve that is now known as the “Witch of Agnesi.” However, after the death of her father, she stopped doing scientific work altogether and devoted the last fortyseven years of her life to caring for sick and dying women. Einstein’s God. “It was, of course, a lie what you read about my religious convictions, a lie which is being systematically repeated. I do not believe in a personal God and I have never denied this but have expressed it clearly. If something is in me which can be called religious then it is the unbounded admiration for the structure of the world so far as our science can reveal it” (Albert Einstein, personal letter to an atheist, 1954). Mathematician cooks. What eighteenthcentury French mathematician cooked himself to death? (See Answer 1.25.) Women and math. Despite horrible prejudice in earlier times, several women have fought against the establishment and persevered in mathematics. Emmy Amalie Noether (1882–1935) was described by Albert Einstein as “the most significant creative mathematical genius thus far produced since the higher education of women began.” She is best known for her contributions to abstract algebra and, in particular, for her study of “chain conditions on ideals of rings.” In 1933, her mathematical achievements counted for nothing when the Nazis caused her dismissal from the University of Göttingen because she was Jewish. Science and religion. “A contemporary has said, not unjustly, that in this materialistic age of ours the serious scientific workers are the only profoundly religious people” (Albert Einstein, New York Times Magazine, 1930). Mathematics and money. What effect would doubling the salary of every mathematics teacher have on education and the world at large? (See Answer 1.26.) A famous female mathematician. Sophie Germain (1776–1831) made major contributions to number theory, acoustics, and elasticity. At age thirteen, Sophie read an account of the death of Archimedes at the hands of a Roman soldier. She was so moved by this story that she decided to become a mathematician. Sadly, her parents felt that her interest in mathematics was inappropriate, so at night she secretly studied the works of Isaac Newton and the mathematician Leonhard Euler. Numbers, History, Society, and People 27 Mad mom tortures mathematician daughter. What brilliant, famous, and beautiful woman mathematician died in incredible pain because her mother withdrew all pain medication? (Hint: The woman is recognized for her contributions to computer programming. The mother wanted her daughter to die painfully so that her daughter’s soul would be cleansed.) (See Answer 1.27.) Mathematics and relationships. “‘No one really understood music unless he was a scientist,’ her father had declared, and not just a scientist, either, oh, no, only the real ones, the theoreticians, whose language is mathematics. She had not understood mathematics until he had explained to her that it was the symbolic language of relationships. ‘And relationships,’ he had told her, ‘contained the essential meaning of life’” (Pearl S. Buck, The Goddess Abides, 1972). Mathematician pretends. What important eleventhcentury mathematician pretended he was insane so that he would not be put to death? (Hint: He was born in Iraq and made contributions to mathematical optics.) (See Answer 1.28.) Mathematical greatness. “Each generation has its few great mathematicians, and mathematics would not even notice the absence of the others. They are useful as teachers, and their research harms 28 A Passion for Mathematics no one, but it is of no importance at all. A mathematician is great or he is nothing” (Alfred Adler, “Reflections: Mathematics and Creativity,” The New Yorker, 1972). Mathematics, mind, universe. “If we wish to understand the nature of the Universe we have an inner hidden advantage: we are ourselves little portions of the universe and so carry the answer within us” (Jacques Boivin, The Single Heart Field Theory, 1981). Christianity and mathematics. “The good Christian should beware of mathematicians, and all those who make empty prophesies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell” (St. Augustine, De Genesi Ad Litteram, Book II, c. 400). Mathematician believes in angels. What famous English mathematician had not the slightest interest in sex and was also a biblical fundamentalist, believing in the reality of angels, demons, and Satan? (Hint: According to most scholars, he is the most influential scientist and mathematician to have ever lived.) (See Answer 1.29.) History’s most prolific mathematician. Who was the most prolific mathematician in history? (If you are unable to answer this, can you guess in what century he lived?) (See Answer 1.30.) Suicidal mathematician. What mathematician accepted a duel, knowing that he would die? (Hint: He spent the night before the duel feverishly writing down his mathematical ideas, which have since had a great impact on mathematics.) (See Answer 1.31.) Marry a mathematician? Would you rather marry the best mathematician in the world or the best chess player? (See Answer 1.32.) Mathematics and truth. “We who are heirs to three recent centuries of scientific development can hardly imagine a state of mind in which many mathematical objects were regarded as symbols of spiri tual Truth” (Philip Davis and Reuben Hersh, The Mathematical Experience, 1981). Mathematics and lust. “I tell them that if they will occupy themselves with the study of mathematics they will find in it the best remedy against the lusts of the flesh” (Thomas Mann, The Magic Mountain, 1924). Newton’s magic. “Had Newton not been steeped in alchemical and other magical learning, he would never have proposed forces of attraction and repulsion between bodies as the major feature of his physical system” (John Henry, “Newton, Matter, and Magic,” in John Fauvel’s Let Newton Be!, 1988). Mathematical corpse. You come home and see a corpse on your foyer floor. Would you be more frightened if (1) scrawled on the floor is the Pythagorean theorem: a2 + b2 = c2, or (2) scrawled on the floor is the following complicated formula: 1 π = 2 2 9801 (See Answer 1.33.) ∞ ∑ k =0 ( 4 k !)(1103 + 26390 k ) ( k !) 4 396 4 k Earliest known symbols. The Egyptian Rhind Papyrus (c. 1650 B.C.) contains the earliest known symbols for mathematical operations. “Plus” is denoted by a pair of legs walking toward the number to be added. Mathematics and the divine. “Mathematical inquiry lifts the human mind into closer proximity with the divine than is attainable through any other medium” (Hermann Weyl, quoted in Philip Davis and Reuben Hersh, The Mathematical Experience, 1981). π and the law. In 1896, an Indiana physician promoted a legislative bill that made π equal to 3.2, exactly. The Indiana House of Representatives approved the bill unanimously, 67 to 0. The Senate, however, deferred debate about the bill “until a later date.” The mathematical life. “The mathematical life of a mathematician is short. Work rarely improves after the age of twentyfive or thirty. If little has been accomplished by Numbers, History, Society, and People 29 then, little will ever be accomplished” (Alfred Adler, “Mathematics and Creativity,” The New Yorker, 1972). Blind date. You are single and going on a blind date. The date knocks on your door and is extremely attractive. However, you note that the person has the following formulas tattooed on the right arm: dR 0 The genesis of x = 1. Ibn Yahya alMaghribi AlSamawal in 1175 was the first to publish dt dB dt = − k B B ( t ), R ( 0 ) = R0 = − k R R ( t ), B ( 0 ) = B0 x0 = 1 In other words, he realized and published the idea that any number raised to the power of 0 is 1. AlSamawal’s book was titled The Dazzling. His father was a Jewish scholar of religion and literature from Baghdad. Euler’s onestep proof of God’s existence. In mathematical books too numerous to mention, we have heard the story of the mathematician Leonhard Euler’s encounter with the French encyclopedist Denis Diderot. Diderot was a devout atheist, and he challenged the religious Euler to mathematically prove the existence of God. Euler replied, “Sir (a + bn)/n = x; hence, God exists. Please reply!” Supposedly, Euler said this in a public debate in St. 30 A Passion for Mathematics From this little information, do you think you would enjoy the evening with this person? (See Answer 1.34.) Fundamental Anagram of Calculus. Probably most of you have never heard of Newton’s “Fundamental Anagram of Calculus”: 6accdae13eff7i3l9n4o4qrr4s8t12ux Can you think of any possible reason Newton would want to code aspects of his calculus discoveries? (See Answer 1.35.) Petersburg and embarrassed the freethinking Diderot with this simple algebraic proof of God’s existence. Diderot was shocked and fled. Was Euler deliberately demonstrating how lame these kinds of arguments can be? Today we know that there is little evidence that the encounter ever took place. Dirk J. Struik, in his book A Concise History of Mathematics, third revised edition (New York: Dover, 1967, p. 129), says that Diderot was mathematically well versed and wouldn’t have been shocked by the formula. Moreover, Euler wasn’t the type of person to make such a zany comment. While people of the time did seek simple mathematical proofs of God, the “Euler versus Diderot” story was probably fabricated by the English mathematician De Morgan (1806–1871). Science and religion. “I have always thought it curious that, while most scientists claim to eschew religion, it actually dominates their thoughts more than it does the clergy” (Fred Hoyle, astrophysicist, “The Universe: Past and Present Reflections,” Annual Review of Astronomy and Astrophysics, 1982). Parallel universes and mathematics. In theory, it is possible to list or “enumerate” all rational numbers. How has this mathematical fact helped certain cosmologists to “prove” that there is an infinite number of universes alongside our own? (As you will learn in the next chapter, “rational numbers” are numbers like 1⁄ 2, which can be expressed as fractions.) (See Answer 1.36.) π savants. In 1844, Johann Dase (a.k.a., Zacharias Dahse) computed π to 200 decimal places in less than two months. He was said to be a calculating prodigy (or an “idiot savant”), hired for the task by the Hamburg Academy of Sciences on Gauss’s recommendation. To compute π = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196 Dase supposedly used π /4 = arctan(1/2) + arctan(1/5) + arctan(1/8) . . . with a series expansion for each arctangent. Dase ran the arctangent job in his brain for nearly sixty days. Not everyone believes the legend of Dase. For example, Arthur C. Clarke recently wrote to me that he simply doesn’t believe the story of Dase calculating pi to 200 places in his head. Clarke says, “Even though I’ve seen fairly well authenticated reports of other incredible feats of mental calculation, I think this is totally beyond credibility.” I would be interested in hearing from readers who can confirm or deny this story. discussion of the value of 00 is very old, and controversy raged throughout the nineteenth century. The mysterious 00. Students are taught that any number to the zero power is 1, and zero to any power is 0. But serious mathematicians often consider 00 undefined. If you try to make a graph of x y, you’ll see it has a discontinuity at the point (0,0). The Onepage proof of God’s existence. Which famous German mathematician “proved” God’s existence in a proof that fit on just one page of paper? (See Answer 1.37.) Greek numerals. Did you know that the ancient Greeks had two systems of numerals? The earlier of these was based on the initial letters of the names of numbers: the number 5 was indicated by the letter pi; 10 by the letter delta; 100 by the antique form of the letter H; 1,000 by the letter chi; and 10,000 by the letter mu. Numbers, History, Society, and People 31 Our mathematical perceptions. “The three of you stare at the school of fish and watch them move in synchrony, despite their lack of eyes. The resulting patterns are hypnotic, like the reflections from a hundred pieces of broken glass. You imagine that the senses place a filter on how much humans can perceive of the mathematical fabric of the universe. If the universe is a mathematical carpet, then all creatures are looking at it through imperfect glasses. How might humanity perfect those glasses? Through drugs, surgery, or electrical stimulation of the brain? Probably our best chance is through the use of computers” (Cliff Pickover, The Loom of God, 1997). Pierre de Fermat. In the early 1600s, Pierre de Fermat, a French lawyer, made brilliant discoveries in number theory. Although he was an “amateur” mathematician, he created mathematical challenges such as “Fermat’s Last Theorem,” which was not solved until 1994. Fermat’s Last Theorem states that xn + yn = zn has no nonzero integer solutions for x, y, and z when n > 2. Fermat was no ordinary lawyer indeed. He is considered, along with Blaise Pascal, a founder of probability theory. As the coinventor of analytic geometry, he is considered, along with René Descartes, one of the first modern mathematicians. The Beal reward. In the mid1990s, the Texas banker Andrew Beal posed a perplexing mathematical problem and offered $5,000 for the solution of this problem. In particular, Beal was curious about the equation Ax + B y = C z. The six letters represent integers, with x, y, and z greater than 2. (Fermat’s Last Theorem involves the special case in which the exponents x, y, and z are the same.) Oddly enough, Beal noticed that for any solution of this general equation he could find, A, B, and C have a common factor. For example, in the equation 36 + 183 = 38, the numbers 3, 18, and 3 all have the factor 3. Using computers at his bank, Beal checked equations with exponents up to 100 but could not discover a solution that didn’t involve a common factor. 32 A Passion for Mathematics Ancient number notation lets humans “think big.” The earliest forms of number notation, which used straight lines for grouping 1s, were inconvenient when dealing with large numbers. By 3400 B.C. in Egypt, and 3000 B.C. in Mesopotamia, a special symbol was adopted for the number 10. The addition of this second number symbol made it possible to express the number 11 with 2 symbols instead of 11, and the number 99 with 18 symbols instead of 99. Caterpillar vehicle. Many mathematicians were creative inventors, although not all of their inventions were practical. For example, Polishborn Josef HoënéWronski (1778–1853), an analyst, a philosopher, a combinatorialist, and a physicist, developed a fantastical design for caterpillarlike vehicles that he intended to replace railroad transportation. He also attempted to build a perpetual motion machine and to build a machine to predict the future (which he called the prognometre). Progress in mathematics. “In most sciences, one generation tears down what another has built and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure” (Hermann Hankel, 1839–1873, who contributed to the theory of functions, complex numbers, and the history of mathematics, quoted in Desmond MacHale, Comic Sections, 1993). Hilbert’s problems. In 1900, the mathematician David Hilbert submitted twentythree important mathematical problems to be targeted for solution in the twentieth century. These twentythree problems extend over all fields of mathematics. Because of Hilbert’s prestige, mathematicians spent a great deal of time tackling the problems, and many of the problems have been solved. Some, however, have been solved only very recently, and still others continue to daunt us. Hilbert’s twentythree wonderful problems were designed to lead to the furthering of various disciplines in mathematics. God’s math book. “God has a transfinite book with all the theorems and their best proofs. You don’t really have to believe in God as long as you believe in the book” (Paul Erdös, quoted in Bruce Schechter, My Brain Is Open: The Mathematical Journeys of Paul Erdös, 1998). Song lyrics. What is the largest number ever used in the lyrics to a popular song? (See Answer 1.38.) The magnificent Erdös. It is commonly agreed that Paul Erdös is the secondmost prolific mathematician of all times, being surpassed only by Leonhard Euler, the great eighteenthcentury mathematician whose name is spoken with awe in mathematical circles. In addition to Erdös’s roughly 1,500 published papers, many are yet to be published after his death. Erdös was still publishing a paper a week in his seventies. Erdös undoubtedly had the greatest number of coauthors (around 500) among mathematicians of all times. In 2004, an eBay auction offered buyers an opportunity to link their names, through five degrees of separation, with Paul Erdös. In particular, the mathematician William Tozier presented bidders with the chance to collaborate on a research paper. Tozier was linked to Erdös through a string of coauthors. In particular, he had collaborated with someone who had collaborated with someone who had collaborated with someone who had collaborated with Paul Erdös. A mathematician who has published a paper with Erdös has an Erdös number of 1. A mathematician who has published a paper with someone who has published a paper with Erdös has an Erdös number of 2, and so on. Tozier has an Erdös number of 4, quite a respectable ranking in the mathematical community. This means that the person working with Tozier would have an Erdös number of 5. During the auction, Tozier heard from more than a hundred wouldbe researchers. (For more information, see Erica Klarreich, “Theorems for Sale: An Online Auctioneer Offers Math Amateurs a Backdoor to Prestige,” Science News 165, no. 24 (2004): 376–77.) Numbers, History, Society, and People 33 Rope and lotus symbols. The Egyptian hieroglyphic system evolved special symbols (resembling ropes, lotus plants, etc.) for the numbers 10, 100, 1,000, and 10,000. Strange math title. I love collecting math papers with strange titles. These papers are published in serious math journals. For example, in 1992, A. Granville published an article with the strange title “Zaphod Beeblebrox’s Brain and the FiftyNinth Row of Pascal’s Triangle,” in the prestigious The American Mathematical Monthly (vol. 99, no. 4 [April]: 318–31). A calculating prodigy and 365,365,365,365,365,365. When he was ten years old, the calculating prodigy Truman Henry Safford (1836–1901) of Royalton, Vermont, was once asked to square, in his head, the number 365,365,365,365,365,365. His church leader reports, “He flew around the room like a top, pulled his pantaloons over the tops of his boots, bit his hands, rolled his eyes in their sockets, sometimes smiling and talking, and then seeming to be in agony, until in not more than a minute said he, 133,491,850,208,566,925,016,658,299,941, 583,225!” Incidentally, I had first mentioned this large number in my book Wonders of Numbers and had misprinted one of the digits. Bobby Jacobs, a tenyearold math whiz from Virginia, wrote to me with the corrected version that you see here. He was the only person to have discovered my earlier typographical error. that when we die, our souls survive. Incidentally, Jesus Christ is number 611,121 among more than 700,000 Creator Sons. Urantia religion and numbers. In the modernday Urantia religion, numbers have an almost divine quality. According to the sect, headquartered in Chicago, we live on the 606th planet in a system called Satania, which includes 619 flawed but evolving worlds. Urantia’s grand universe number is 5,342,482,337,666. Urantians believe that human minds are created at birth, but the soul does not develop until about age six. They also believe 34 A Passion for Mathematics [mathematical curves], spinning to their own music, is a wondrous, spiritual event” (Paul Rapp, in Kathleen McAuliffe, “Get Smart: Controlling Chaos,” Omni [1989]). Mathematics and God. “Philosophers and great religious thinkers of the last century saw evidence of God in the symmetries and harmonies around them— in the beautiful equations of classical physics that describe such phenomena as electricity and magnetism. I don’t see the simple patterns underlying nature’s complexity as evidence of God. I believe that is God. To behold The discovery of calculus. The English mathematician Isaac Newton (1642–1727) and the German mathematician Gottfried Wilhelm Leibniz (1646–1716) are generally credited with the invention of calculus, but various earlier mathematicians explored the concept of rates and limits, starting with the ancient Egyptians, who Newton’s giants. “If I have seen further than others, it is by standing upon the shoulders of giants” (Isaac Newton, personal letter to Robert Hooker, 1675 [see next quotation]). Abelson’s giants. “If I have not seen as far as others, it is because giants were standing on my shoulders” (Hal Abelson, MIT professor). Mathematical marvel. “Even stranger things have happened; and perhaps the strangest of all is the marvel that mathematics should be possible to a race akin to the apes” (Eric T. Bell, The Development of Mathematics, 1945). developed rules for calculating the volume of pyramids and approximating the areas of circles. In the 1600s, both Newton and Leibniz puzzled over problems of tangents, rates of change, minima, maxima, and infinitesimals (unimaginably tiny quantities that are almost but not quite zero). Both men understood that differentiation (finding tangents to curves) and integration (finding areas under curves) are inverse processes. Newton’s discovery (1665–1666) started with his interest in infinite sums; however, he was slow to publish his findings. Leibniz published his discovery of differential calculus in 1684 and of integral calculus in 1686. He said, “It is unworthy of excellent men, to lose hours like slaves in the labor of calculation. . . . My new calculus . . . offers truth by a kind of analysis and without any effort of imagination.” Newton was outraged. Debates raged for many years on how to divide the credit for the discovery of calculus, and, as a result, progress in calculus was delayed. The notations of calculus. Today we use Leibniz’s symdf __ bols in calculus, such as dx for the derivative and the ∫ symbol for integration. (This integral symbol was actually a long letter S for summa, the Latin word for “sum.”) The mathematician Joseph Louis Lagrange (1736–1813) was the first person to use the notation f ′(x) for the first derivative and f ″(x) for the second derivative. In 1696, Guillaume François de L’Hôpital, a French mathematician, published the first textbook on calculus. Jews, π, the movie. The 1998 cult movie titled π stars a mathematical genius who is fascinated by numbers and their role in the cosmos, the stock market, and Jewish mysticism. According to the movie, what is God’s number? (See Answer 1.39.) Mathematics and romance. What romantic comedy has the most complicated mathematics ever portrayed in a movie? (See Answer 1.40.) Greek death. Why did the ancient Greeks and other cultures believe 8 to be a symbol of death? (See Answer 1.41.) Numbers, History, Society, and People 35 The mechanical Pascaline. One example of an early computing machine is Blaise Pascal’s wheel computer called a Pascaline. In 1644, this French philosopher and mathematician built a calculating machine to help his father compute business accounts. Pascal was twenty years old at the time. The machine used a series of spinning numbered wheels to add large numbers. The wonderful Pascaline was about the size of a shoebox. About fifty models were made. The Matrix. What number is on Agent Smith’s license plate in the movie The Matrix Reloaded? Why? (See Answer 1.42.) At the movies. What famous book and movie title contains a number that is greater than 18,000 and less then 38,000? (See Answer 1.43.) The mathematical life. “The mathematician lives long and lives young; the wings of the soul do not early drop off, nor do its 36 A Passion for Mathematics pores become clogged with the earthly particles blown from the dusty highways of vulgar life” (James Joseph Sylvester, 1814–1897, a professor of mathematics at Johns Hopkins University, 1869 address to the British Mathematical Association). Mathematical progress. “More significant mathematical work has been done in the latter half of this century than in all previous centuries combined” (John Casti, Five Golden Rules, 1997). Simultaneity in science. The simultaneous discovery of calculus by Newton and Leibniz makes me wonder why so many discoveries in science were made at the same time by people working independently. For example, Charles Darwin (1809–1882) and Alfred Wallace (1823–1913) both independently developed the theory of evolution. In fact, in 1858, Darwin announced his theory in a paper presented at the same time as a paper by Wallace, a naturalist who had also developed the theory of natural selection. As another example of simultaneity, the mathematicians János Bolyai (1802–1860) and Nikolai Lobachevsky (1793–1856) developed hyperbolic geometry independently and at the same time (both perhaps stimulated indirectly by Carl Friedrich Gauss). Most likely, such simultaneous discoveries have occurred because the time was “ripe” for such discoveries, given humanity’s accumulated knowledge at the time the discoveries were made. On the other hand, mystics have suggested that there is a deeper meaning to such coincidences. The Austrian biologist Paul Kammerer (1880–1926) wrote, “We thus arrive at the image of a worldmosaic or cosmic kaleidoscope, which, in spite of constant shufflings and rearrangements, also takes care of bringing like and like together.” He compared events in our world to the tops of ocean waves that seem isolated and unrelated. According to his controversial theory, we notice the tops of the waves, but beneath the surface there may be some kind of synchronistic mechanism that mysteriously connects events in our world and causes them to cluster. First mathematician. What is the name of the first human who was identified as having made a contribution to mathematics? (See Answer 1.44.) Game show. Why was the 1950’s TV game show called The $64,000 Question? Why not a rounder number like $50,000? (See Answer 1.45.) God and the infinite. “Such as say that things infinite are past God’s knowledge may just as well leap headlong into this pit of impiety, and say that God knows not all numbers. . . . What madman would say so? . . . What are we mean wretches that dare presume to limit His knowledge?” (St. Augustine, The City of God, A.D. 412). Who was Pythagoras? You can tell from some of the following factoids that I love trivia that relates to the famous ancient Greek mathematician Pythagoras. His ideas continue to thrive after three millennia of mathematical science. The philosopher Bertrand Russell once wrote Mathematical scope. “Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer’s gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness, the life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud and cell, and is forever ready to burst forth into new forms of vegetable and animal existence” (James Joseph Sylvester, The Collected Mathematical Papers of James Joseph Sylvester, Volume III, address on Commemoration Day at Johns Hopkins University, February 22, 1877). that Pythagoras was intellectually one of the most important men who ever lived, both when he was wise and when he was unwise. Pythagoras was the most puzzling mathematician of history because he founded a numerical religion whose main tenets were the transmigration of souls and the sinfulness of eating beans, along with a host of other odd rules and regulations. To the Pythagoreans, mathematics was an ecstatic revelation. The Pythagoreans, like modern day fractalists, were akin to musicians. They created pattern and beauty as they discovered mathematical truths. Mathematical and theological blending began with Pythagoras and eventually affected all religious philosophy in Greece, played a role in religion of the Middle Ages, and extended to Kant in modern times. Bertrand Russell felt that if it were not for Pythagoras, theologians would not have sought logical proofs of God and immortality. If you want to read more about Pythagoras, see my book The Loom of God and Peter Gorman’s Pythagoras: A Life. Numbers, History, Society, and People 37 The secret life of numbers. To Pythagoras and his followers, numbers were like gods, pure and free from material change. The worship of numbers 1 through 10 was a kind of polytheism for the Pythagoreans. Pythagoreans believed that numbers were alive, independent of humans, but with a telepathic form of consciousness. Humans could relinquish their threedimensional lives and telepathize with these number beings by using various forms of meditation. Meditation upon numbers was communing with the gods, gods who desired nothing from humans but their sincere admiration and contemplation. Meditation upon numbers was a form of prayer that did not ask any favors from the gods. These kinds of thoughts are not foreign to modern mathematicians, who often debate whether mathematics is a creation of the human mind or is out there in the universe, independent of human thought. Opinions vary. A few mathematicians believe that mathematics is a form of human logic that is not necessarily valid in all parts of the universe. 38 A Passion for Mathematics Anamnesis and the number 216. Was the ancient Greek mathematician Pythagoras once a plant? This is a seemingly bizarre question, but Pythagoras claimed that he had been both a plant and an animal in his past lives, and, like Saint Francis, he preached to animals. Pythagoras and his followers believed in anamnesis, the recollection of one’s previous incarnations. During Pythagoras’s time, most philosophers believed that only men could be happy. Pythagoras, on the other hand, believed in the happiness of plants, animals, and women. In various ancient Greek writings, we are told the exact number of years between each of Pythagoras’s incarnations: 216. Interestingly, Pythagoreans considered 216 to be a mystical number, because it is 6 cubed (6 × 6 × 6). Six was also considered a “circular number” because its powers always ended in 6. The fetus was considered to have been formed after 216 days. The number 216 continues to pop up in the most unlikely of places in theological literature. In an obscure passage from The Republic (viii, 546 B–D), Plato notes that 216 = 63. It is also associated with auspicious signs on the Buddha’s footprint. Shades of ghosts. Pythagoras believed that even rocks possess a psychic existence. Mountains rose from the earth because of growing pains of the earth, and Pythagoras told his followers that earthquakes were caused by the shades of ghosts of the dead, which created disturbances beneath the earth. Pythagorean sacrifice. Although some historians report that Pythagoras joy fully sacrificed a hecatomb of oxen (a hundred animals) when he discovered his famous theorem about the rightangled triangle, this would have been scandalously unPythagorean and is probably not true. Pythagoras refused to sacrifice animals. Instead,