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11 Numbers,Numbers,2 History,History,3 Society,Society, andand PeoplePeople 4

N WHICH WE ENCOUNTER RELIGIOUS5 MATHEMATICIANS, MAD MATHEMATICIANS, Ifamous mathematicians, mathematical savants, quirky questions, fun trivia, brief biographies, mathematical gods, historical oddities, numbers and society, gossip, the history of mathematical6 notation, the genesis of numbers, and “What if?” questions.

COPYRIGHTEDMathematics is the hammer MATERIAL that shatters the ice of our unconscious.

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Ancient counting. Let’s The symbols of mathe- Math beyond humanity. start the book with a ques- matics. Mathematical notation “We now know that there tion. What is the earliest evi- shapes humanity’s ability to exist true propositions dence we have of humans efficiently contemplate math- which we can never formally counting? If this question is ematics. Here’s a cool factoid prove. What about proposi- too difficult, can you guess for you: The symbols + and tions whose proofs require whether the evidence is –, referring to addition and arguments beyond our before or after 10,000 B.C.— subtraction, first appeared in capabilities? What about and what the evidence might 1456 in an unpublished man- propositions whose proofs be? (See Answer 1.1.) uscript by the mathematician require millions of pages? Johann Regiomontanus Or a million, million (a.k.a. Johann Müller). The pages? Are there proofs plus symbol, as an abbrevia- that are possible, but beyond Mathematics and beauty. tion for the Latin et (and), us?” (Calvin Clawson, I’ve collected mathematical was found earlier in a manu- Mathematical Mysteries). quotations since my teenage script dated 1417; however, years. Here’s a favorite: the downward stroke was not “Mathematics, rightly quite vertical. viewed, possesses not only The multiplication symbol. truth, but supreme beauty— In 1631, the multiplication a beauty cold and austere, Mathematics and reality. symbol × was introduced by like that of sculpture” Do humans invent mathemat- the English mathematician (Bertrand Russell, Mysticism ics or discover mathematics? William Oughtred and Logic, 1918). (See Answer 1.2.) (1574–1660) in his book Keys to Mathematics, published in London. Inciden- Mathematics and the universe. Here is a deep thought to start tally, this Anglican minister our mathematical journey. Do you think humanity’s long-term is also famous for having fascination with mathematics has arisen because the universe invented the slide rule, is constructed from a mathematical fabric? We’ll approach which was used by genera- this question later in the chapter. For now, you may enjoy tions of scientists and knowing that in 1623, Galileo Galilei echoed this belief in a mathematicians. The slide mathematical universe by stating his credo: “Nature’s great rule’s doom in the mid- book is written in mathematical symbols.” Plato’s doctrine was 1970s, due to the pervasive that God is a geometer, and Sir James Jeans believed that God influx of inexpensive pocket experimented with arithmetic. Isaac Newton supposed that the calculators, was rapid and planets were originally thrown into orbit by God, but even unexpected. after God decreed the law of gravitation, the planets required continual adjustments to their orbits.

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Math and madness. Many Mathematicians and religion. Over the years, many of my mathematicians throughout readers have assumed that famous mathematicians are not reli- history have had a trace of gious. In actuality, a number of important mathematicians madness or have been eccen- were quite religious. As an interesting exercise, I conducted an tric. Here’s a relevant quota- Internet survey in which I asked respondents to name impor- tion on the subject by the tant mathematicians who were also religious. Isaac Newton British mathematician and Blaise Pascal were the most commonly cited religious John Edensor Littlewood mathematicians. (1885–1977), who suffered In many ways, the mathematical quest to understand infin- from depression for most of ity parallels mystical attempts to understand God. Both reli- his life: “Mathematics is a gion and mathematics struggle to express relationships dangerous profession; an between humans, the universe, and infinity. Both have arcane appreciable proportion of us symbols and rituals, as well as impenetrable language. Both goes mad.” 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 Mathematics and murder. the most powerful evidence of the inventive genius of the What triple murderer was human race? In “Reason and Faith, Eternally Bound” (Decem- also a brilliant French mathe- ber 20, 2003, New York Times, B7), Edward Rothstein notes matician who did his finest that faith was the inspiration for Newton and Kepler, as well work while confined to a as for numerous scientific and mathematical triumphs. “The hospital for the criminally conviction that there is an order to things, that the mind can insane? (See Answer 1.3.) comprehend that order and that this order is not infinitely mal- leable, those scientific beliefs may include elements of faith.” In his Critique of Pure Reason, Immanuel Kant describes Creativity and madness. how “the light dove, cleaving the air in her free flight and feel- “There is a theory that ing its resistance against her wings, might imagine that its creativity arises when indi- flight would be freer still in empty space.” But if we were to viduals are out of sync with remove the air, the bird would plummet. Is faith—or a cosmic their environment. To put it sense of mystery—like the air that allows some seekers to simply, people who fit in soar? Whatever mathematical or scientific advances humans with their communities have make, we will always continue to swim in a sea of mystery. insufficient motivation to risk their psyches in creating something truly new, while They have less to lose and Pascal’s mystery. “There those who are out of sync more to gain” (Gary Taubes, is a God-shaped vacuum in are driven by the constant “Beyond the Soapsuds Uni- every heart” (Blaise Pascal, need to prove their worth. verse,” 1977). Pensées, 1670).

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Leaving mathematics and he was not certain it was methods after flashes of approaching God. What famous original. The Danish physicist insight that came in dreams. French mathematician and Niels Bohr conceived the The dreams of Dmitry teenage prodigy finally model of an atom from a Mendeleyev, Friedrich decided that religion was dream. Elias Howe received August Kekulé, and Otto more to his liking and joined in a dream the image of the Loewi inspired scientific his sister in her convent, kind of needle design breakthroughs. It is not an where he gave up mathemat- required for a lock-stitch exaggeration to suggest that ics and social life? (See sewing machine. René many scientific and mathe- Answer 1.4.) Descartes was able to matical advances arose from advance his geometrical the stuff of dreams.

Ramanujan’s gods. As Blaise Pascal (1623–1662), a Frenchman, was a geometer, a mentioned in this book’s probabilist, a physicist, a philosopher, and a combinatorist. He introduction, the mathemati- was also deeply spiritual and a leader of the Jansenist sect, a cian Srinivasa Ramanujan Calvinistic quasi-Protestant group within the Catholic Church. (1887–1920) was an ardent He believed that it made sense to become a Christian. If the follower of several Hindu person dies, and there is no God, the person loses nothing. If deities. After receiving there is a God, then the person has gained heaven, while skep- visions from these gods in the tics lose everything in hell. form of blood droplets, Legend has it that Pascal in his early childhood sought to Ramanujan saw scrolls that prove the existence of God. Because Pascal could not simply contained very complicated command God to show Himself, he tried to prove the exis- mathematics. When he woke tence of a devil so that he could then infer the existence of from his dreams, he set down God. He drew a pentagram on the ground, but the exercise on paper only a fraction of scared him, and he ran away. Pascal said that this experience what the gods showed him. made him certain of God’s existence. Throughout history, cre- One evening in 1654, he had a two-hour mystical vision that ative geniuses have been open he called a “night of ﬁre,” in which he experienced ﬁre and “the to dreams as a source of God of Abraham, Isaac, and Jacob . . . and of Jesus Christ.” inspiration. Paul McCartney Pascal recorded his vision in his work “Memorial.” A scrap of said that the melody for the paper containing the “Memorial” was found in the lining of his famous Beatles’ song “Yester- coat after his death, for he carried this reminder about with him day,” one of the most popular always. The three lines of “Memorial” are songs ever written, came to him in a dream. Apparently, Complete submission to Jesus Christ and to my director. the tune seemed so beautiful Eternally in joy for a day’s exercise on the earth. and haunting that for a while May I not forget your words. Amen.

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Transcendence. “Much of quantify, at a glance, without Calculating π. Which the history of science, like having to individually count nineteenth-century British the history of religion, is a them? (See Answer 1.5.) boarding school supervisor history of struggles driven by spent a significant portion of power and money. And yet, his life calculating π to 707 this is not the whole story. Circles. Why are there 360 places and died a happy man, Genuine saints occasionally degrees in a circle? (See despite a sad error that was play an important role, both Answer 1.6.) later found in his calcula- in religion and science. For tions? (See Answer 1.8.) many scientists, the reward for being a scientist is not the The mystery of Ramanujan. power and the money but the After years of working The special number 7. In chance of catching a glimpse through Ramanujan’s note- ancient days, the number 7 of the transcendent beauty of books, the mathematician was thought of as just another nature” (Freeman Dyson, in Bruce Berndt said, “I still way to signify “many.” Even the introduction to Nature’s don’t understand it all. I may in recent times, there have Imagination). be able to prove it, but I don’t been tribes that used no num- know where it comes from bers higher than 7. and where it fits into the rest In the 1880s, the German The value of eccentricity. of mathematics. The enigma ethnologist Karl von Steinen “That so few now dare to be of Ramanujan’s creative described how certain South eccentric, marks the chief process is still covered by a American Indian tribes had danger of our time” (John curtain that has barely been very few words for numbers. Stuart Mill, nineteenth- drawn” (Robert Kanigel, The As a test, he repeatedly asked century English philosopher). Man Who Knew Infinity, them to count ten grains of 1991). corn. They counted “slowly

Counting and the mind. I The world’s most forgettable license plate? Today, mathematics quickly toss a number of mar- affects society in the funniest of ways. I once read an article bles onto a pillow. You may about someone who claimed to have devised the most forget- stare at them for an instant to table license plate, but the article did not divulge the secret determine how many marbles sequence. What is the most forgettable license plate? Is it a are on the pillow. Obviously, random sequence of eight letters and numbers—for example, if I were to toss just two mar- 6AZL4QO9 (the maximum allowed in New York)? Or perhaps bles, you could easily deter- a set of visually confusing numbers or letters—for example, mine that two marbles sit on MWNNMWWM? Or maybe a binary number like 01001100. the pillow. What is the largest What do you think? What would a mathematician think? (See number of marbles you can Answer 1.7.)

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but correctly to six, but when Isaac Newton (1642–1727), an Englishman, was a mathe- it came to the seventh grain matician, a physicist, an astronomer, a coinventor of calculus, and the eighth, they grew and famous for his law of gravitation. He was also the author tense and uneasy, at first of many books on biblical subjects, especially prophecy. yawning and complaining of a Perhaps less well known is the fact that Newton was a cre- headache, then finally avoided ationist who wanted to be known as much for his theological the question altogether or sim- writings as for his scientific and mathematical texts. Newton ply walked off.” Perhaps seven believed in a Christian unity, as opposed to a trinity. He devel- means “many” in such com- oped calculus as a means of describing motion, and perhaps mon phrases as “seven seas” for understanding the nature of God through a clearer under- and “seven deadly sins.” standing of nature and reality. He respected the Bible and (These interesting facts come accepted its account of Creation. from Adrian Room, The Guin- ness Book of Numbers, 1989.)

Genius and eccentricity. James Hopwood Jeans Carl Friedrich Gauss “The amount of eccentricity (1877–1946) was an applied (1777–1855), a German, was in a society has been propor- mathematician, a physicist, a mathematician, an tional to the amount of and an astronomer. He some- astronomer, and a physicist genius, material vigor and times likened God to a math- with a wide range of contri- moral courage which it con- ematician and wrote in The butions. Like Ramanujan, tains” (John Stuart Mill, On Mysterious Universe (1930), after Gauss proved a theorem, Liberty, 1869). “From the intrinsic evidence he sometimes said that the of his creation, the Great insight did not come from Architect of the Universe “painful effort but, so to Mathematics and God. “The now begins to appear as a speak, by the grace of God.” Christians know that the math- pure mathematician.” He has He also once wrote, “There ematical principles, according also written, “Physics tries to are problems to whose solu- to which the corporeal world discover the pattern of events tion I would attach an inﬁ- was to be created, are co- which controls the phenom- nitely greater importance eternal with God. Geometry ena we observe. But we can than to those of mathematics, has supplied God with the never know what this pattern for example, touching ethics, models for the creation of the means or how it originates; or our relation to God, or world. Within the image of and even if some superior concerning our destiny and God it has passed into man, intelligence were to tell us, our future; but their solution and was certainly not received we should ﬁnd the explana- lies wholly beyond us and within through the eyes” tion unintelligible” (Physics completely outside the (Johannes Kepler, The and Philosophy, 1942). province of science.” Harmony of the World, 1619).

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Leonhard Euler (1707–1783) was a prolific Swiss mathe- found wide applications in matician and the son of a vicar. Legends tell of Leonhard the design of computers. 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. The value of puzzles. “It is a Although he was a devout Christian all his life, he could not wholesome plan, in thinking find the enthusiasm for the study of theology, compared to that about logic, to stock the mind of mathematics. He was completely blind for the last seven- with as many puzzles as pos- teen years of his life, during which time he produced roughly sible, since these serve much half of his total output. the same purpose as is served Euler is responsible for our common, modern-day use of by experiments in physical many famous mathematical notations—for example, f(x) for a science” (Bertrand Russell, function, e for the base of natural logs, i for the square root of Mind, 1905). –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 = A mathematical nomad. 641 × 6,700,417, and so is not prime. What legendary mathematician, and one of the most prolific mathematicians in history, was so devoted to or algebraic form. . . . It is George Boole (1815–1864), math that he lived as a nomad impossible to separate an Englishman, was a logi- with no home and no job? Boole’s religious beliefs cian and an algebraist. Like Sexual contact revolted him; from his mathematics.” Ramanujan and other mysti- even an accidental touch by Boole often spoke of his cal mathematicians, Boole anyone made him feel uncom- almost photographic mem- had “mystical” experiences. fortable. (See Answer 1.9.) David Noble, in his book The ory, describing it as “an Religion of Technology, arrangement of the mind for notes, “The thought flashed every fact and idea, which I upon him suddenly as he was can find at once, as if it were Marin Mersenne (1588– walking across a field that his in a well-ordered set of 1648) was another mathe- ambition in life was to drawers.” matician who was deeply explain the logic of human Boole died at age forty- religious. Mersenne, a thought and to delve analyti- nine, after his wife mistak- Frenchman, was a theologian, cally into the spiritual enly thought that tossing a philosopher, a number theo- aspects of man’s nature buckets of water on him and rist, a priest, and a monk. He [through] the expression of his bed would cure his flu. argued that God’s majesty logical relations in symbolic Today, Boolean algebra has would not be diminished had

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Mirror phobia. What brilliant, handsome mathematician so hundred pages to write on hated mirrors that he covered them wherever he went? (See paper using an ordinary font. Answer 1.10.) Shafer, age twenty-six, helped find the number as a volunteer on a project called What is a mathematician? “A mathematician is a blind man the Great Internet Mersenne in a dark room looking for a black cat which isn’t there” Prime Search. Tens of thou- (Charles Darwin). sands of people volunteer the use of their personal computers in a worldwide project Animal math. Can animals count? (See Answer 1.11.) that harnesses the power of hundreds of thousands of He created just one world, mated the future of comput- computers, in effect creating instead of many, because the ing power by stating that all a supercomputer capable of one world would be infinite eternity would not be suffi- performing trillions of calcu- in every part. His first publi- cient to decide if a 15- or lations per second. Shafer cations were theological 20-digit number were prime. used an ordinary Dell com- studies against atheism and Unfortunately, the prime puter in his office for nine- skepticism. number values for p that teen days. What would Mersenne was fascinated make 2p – 1 a prime number Mersenne have thought of by prime numbers (numbers seem to form no regular this large beast? like 7 that were divisible only sequence. For example, the In 2005, the German eye by themselves and 1), and he Mersenne number is prime surgeon Martin Nowak, also tried to find a formula that he when p = 2, 3, 5, 7, 13, 17, part of the Great Internet could use to find all primes. 19, . . . Notice that when p Mersenne Prime Search, dis- Although he did not find such is equal to the prime number covered the forty-second

a formula, his work on 11, M11 = 2,047, which is Mersenne prime number, “Mersenne numbers” of the not prime because 2,047 = 225,964,951 – 1, which has over form 2p – 1, where p is a 23 × 89. seven million digits. Nowak’s prime number, continues to The fortieth Mersenne 2.4-GHz Pentium-4 computer interest us today. Mersenne prime was discovered in spent roughly fifty days ana- numbers are the easiest type 2003, and it contained lyzing the number before of number to prove prime, so 6,320,430 digits! In particu- reporting the find. The Elec- they are usually the largest lar, the Michigan State Uni- tronic Frontier Foundation, a primes of which humanity is versity graduate student U.S. Internet campaign aware. Michael Shafer discovered group, has promised to give Mersenne himself found that 220,996,011 – 1 is prime. $100,000 to whoever finds several prime numbers of the The number is so large that it the first ten-million-digit form 2p – 1, but he underesti- would require about fifteen prime number.

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Mathematics and God. each of the books in the Mystery mathematician. “Before creation, God did Bible. Knuth also includes Around A.D. 500, the Greek just pure mathematics. Then calligraphic renderings of the philosopher Metrodorus gave He thought it would be a verses. Knuth himself has us the following puzzle that pleasant change to do some said, “It’s tragic that scientific describes the life of a famous applied” (John Edensor Lit- advances have caused many mathematician: tlewood, A Mathematician’s people to imagine that they A certain man’s boyhood Miscellany, 1953). know it all, and that God is 1 lasted ⁄6 of his life; he irrelevant or nonexistent. The 1 married after ⁄ 7 more; his fact is that everything we 1 beard grew after ⁄12 more, learn reveals more things that The division symbol. The and his son was born 5 we do not understand. . . . division symbol ÷ first years later; the son lived to Reverence for God comes appeared in print in Johann half his father’s final age, naturally if we are honest Heinrich Rahn’s Teutsche and the father died 4 years about how little we know.” Algebra (1659). after the son. Tell me the mystery man’s name or his age at death. (See Donald Knuth (1938–) is a Mathematics and sex. “A Answer 1.12.) computer scientist and a well-known mathematician mathematician. He is also a once told me that the great fine example of a mathemati- thing about liking both math Mathematician starves. cian who is interested in reli- and sex was that he could do What famous mathematician gion. For example, he has either one while thinking deliberately starved himself been an active Lutheran and a about the other” (Steven E. to death in 1978? (Hint: He Sunday school teacher. His Landsburg, in a 1993 post to was perhaps the most brilliant attractive book titled 3:16 the newsgroup sci.math). logician since Aristotle.) consists entirely of commen- (See Answer 1.13.) tary on chapter 3, verse 16, of

Brain limitation. “Our brains have evolved to get us out of Understanding brilliance. the rain, ﬁnd where the berries are, and keep us from getting “Maybe the brilliance of the killed. Our brains did not evolve to help us grasp really brilliant can be understood large numbers or to look at things in a hundred thousand only by the nearly brilliant” dimensions” (Ronald Graham, a prior director of Information (Anthony Smith, The Mind, Sciences Research at AT&T Research, quoted in Paul 1984). Hoffman’s “The Man Who Loves Only Numbers,” Atlantic Monthly, 1987).

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Georg Friedrich Bernhard physics, and the theory of cal Hebrew, and was the son Riemann (1826–1866) was a complex variables. He also of a Lutheran minister. German mathematician who attempted to write a mathe- The Riemann hypothesis, made important contributions matical proof of the truth of published by Riemann in to geometry, number theory, the Book of Genesis, was a 1859, deals with the zeros of topology, mathematical student of theology and bibli- a very wiggly function, and the hypothesis still resists modern mathematicians’ Calculating prodigy has plastic brain. Rüdiger Gamm is shock- attempts to prove it. Chapter ing the world with his calculating powers and is changing the 3 describes the hypothesis way we think about the human brain. He did poorly at mathe- further. matics in school but is now a world-famous 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 A dislike for mathematics. to the ninth power in his head. He can divide prime numbers “I’m sorry to say that the and calculate the answer to 60 decimal points and more. He subject I most disliked was can calculate ﬁfth roots. mathematics. I have thought Amazing calculating powers such as these were previously about it. I think the reason thought to be possible only by “autistic savants.” (Autistic was that mathematics leaves savants often have severe developmental disabilities but, at the no room for argument. If you same time, have special skills and an incredible memory.) made a mistake, that was all Gamm’s talent has attracted the curiosity of European there was to it” (Malcolm X, researchers, who have imaged his brain with PET scans while The Autobiography of Mal- he performed math problems. These breathtaking studies colm X, 1965). 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 Mathematics and humanity. responsible for long-term memory, in order to perform his “Anyone who cannot cope rapid calculations. Scientists hypothesize that Gamm tem- with mathematics is not porarily uses these areas to “hold” digits in so-called “working fully human. At best he is a memory,” the brain’s temporary holding area. Gamm is essen- tolerable subhuman who has tially doing what computers do when they extend their capa- learned to wear shoes, bathe, bilities by using swap space on the hard drive to increase their and not make messes in the capabilities. Scientists are not sure how Gamm acquired this house” (Robert A. Heinlein, ability, considering that he became interested in mathematical Time Enough for Love, calculation only when he was in his twenties. (You can learn 1973). more in Steve Silberman’s “The Keys to Genius,” Wired, no. 11.12, December 2003.)

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Kurt Gödel (1906–1978) is Gottfried Wilhelm von Leibniz (1646–1716), a German, was an example of a mathematical an analyst, a combinatorist, a logician, and the coinventor of genius obsessed with God calculus who also passionately argued for the existence of and the afterlife. As discussed God. According to Leibniz, God chooses to actualize this in Answer 1.13 about the world out of an infinite number of possible worlds. In other mathematician who starved words, limited only by contradiction, God first conceives of himself to death, Gödel was every possible world, and then God simply chooses which of a logician, a mathematician, them to create. and a philosopher who was Leibniz is also famous for the principle of “preestablished famous for having shown harmony,” which states that God constructed the universe in that in any axiomatic system such a way that corresponding mental and physical events for mathematics, there are occur simultaneously. His “monad theory” states that the uni- propositions that cannot be verse consists of an infinite number of substances called mon- proved or disproved within ads, each of which has its own individual identity but is an the axioms of the system. expression of the whole universe from a particular unique Gödel thought it was viewpoint. possible to show the logical necessity for life after death and the existence of God. Greater-than symbol. The greater-than and less-than In four long letters to his symbols (> and <) were introduced by the British mathemati- mother, Gödel gave reasons cian Thomas Harriot in his Artis Analyticae Praxis, published for believing in a next in 1631. world.

Greater-than or equal-to symbol. The symbol ≥ (greater than or equal to) was first introduced by the French scientist Pierre Math and madness. Bouguer in 1734. “Cantor’s work, though brilliant, seemed to move in half- steps. The closer he came to Mathematician murdered. can movie? Name the movie! the answers he sought, the Why was the first woman (See Answer 1.15.) What is further away they seemed. mathematician murdered? the largest number less than Eventually, it drove him mad, (See Answer 1.14.) a billion ever used in a as it had mathematicians major, full-length movie title? before him” (Amir D. Aczel, (Hint: The song was popular The Mystery of the Aleph: in the late 1920s and the Mathematics, the Kabbalah, Going to the movies. What early 1930s.) (See Answer and the Search for Infinity, was the largest number ever 1.16.) 2000). used in the title of an Ameri-

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Math and madness. Many omists and psychologists some of his enemies that he mathematicians were very seldom indeed. It is as was an evil magician. depressed and religious at the their subject matter comes In Reims, he transformed same time. Which famous nearer to man himself that the floor of the cathedral into mathematician invented the their antireligious bias hard- a giant abacus. That must concept of “transfinite num- ens” (C. S. Lewis, The Grand have been a sight to see! The bers” (essentially, different Miracle: And Other Selected “Number Pope” was also “levels” of infinity), believed Essays on Theology and important because he adopted that God revealed mathemati- Ethics from God in the Dock, Arabic numerals (1, 2, 3, 4, cal ideas to him, and was a 1983). 5, 6, 7, 8, 9) as a replacement frequent guest of sanitari- for Roman numerals. He ums? (See Answer 1.17.) contributed to the invention of the pendulum clock, The Number Pope. As I invented devices that tracked Mathematics and diapering. write this book, I realize that planetary orbits, and wrote on “A human being should be a thousand years ago, the last geometry. When he realized able to change a diaper, Pope-mathematician died. that he lacked knowledge of plan an invasion, butcher a Gerbert of Aurillac (c. 946– formal logic, he studied hog, conn a ship, design a 1003) was fascinated by under German logicians. He building, write a sonnet, bal- mathematics and was elected said, “The just man lives by ance accounts, build a wall, to be Pope Sylvester II in faith; but it is good that he set a bone, comfort the dying, 999. His advanced knowledge should combine science with take orders, give orders, of mathematics convinced his faith.” cooperate, act alone, solve equations, analyze a new Hardy’s six wishes. In the 1920s, the British mathematician problem, pitch manure, pro- G. H. Hardy wrote a postcard to his friend, listing six New gram a computer, cook a Year’s wishes: tasty meal, fight efficiently, die gallantly. Specialization 1. prove the Riemann hypothesis is for insects” (Robert A. 2. score well at the end of an important game of cricket Heinlein, Time Enough for 3. find an argument for the nonexistence of God that con- Love, 1973). vinces the general public 4. be the first man at the top of Mount Everest Mathematicians and God. 5. be the first president of the USSR, Great Britain, and “Mathematicians, astrono- Germany mers, and physicists are often religious, even mystical; biol- 6. murder Mussolini ogists much less often; econ- (The London Mathematical Society Newsletter, 1994)

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Charles Babbage (1792–1871), an Englishman, was an ana- cians? Would a meeting of lyst, a statistician, and an inventor who was also interested in time-traveling mathematicians religious miracles. He once wrote, “Miracles are not a breach offer more to humanity than a of established laws, but . . . indicate the existence of far higher meeting of other scientists— laws.” Babbage argued that miracles could occur in a mecha- for example, biologists or nistic world. Just as Babbage could program strange behavior sociologists? These are all fas- on his calculating machines, God could program similar irreg- cinating questions to which I ularities in nature. While investigating biblical miracles, he don’t yet have answers. assumed that the chance of a man rising from the dead is one in 1012. Babbage is famous for conceiving an enormous hand- Mathematicians as God’s cranked mechanical calculator, an early progenitor of our messengers. “Cantor felt a modern computers. Babbage thought the device would be duty to keep on, in the face most useful in producing mathematical tables, but he worried of adversity, to bring the about mistakes that would be made by humans who tran- insights he had been given as scribed the results from its thirty-one metal output wheels. God’s messenger to mathe- Today, we realize that Babbage was a hundred years ahead of maticians everywhere” his time and that the politics and the technology of his era (Joseph Dauben, Georg were inadequate for his lofty dreams. Cantor, 1990).

Tinkertoy computer. In the gather Pythagoras, Cantor, early 1980s, the computer and Gödel in a small room Power notation. In 1637, geniuses Danny Hillis, Brian with a single blackboard to the philosopher René Silverman, and friends built debate their various ideas on Descartes was the ﬁrst person a Tinkertoy computer that mathematics and God. What to use the superscript notation played tic-tac-toe. The device profound knowledge might for raising numbers and vari- was made from 10,000 we gain if we had the power ables to powers—for exam- 2 Tinkertoy pieces. to bring together great ple, as in x . thinkers of various ages for a conference on mathematics? Would a roundtable discus- Numerical religion. What Fantasy meeting of sion with Pythagoras, Cantor, ancient mathematician Pythagoras, Cantor, and Gödel. and Gödel produce less inter- established a numerical reli- I often fantasize about the esting ideas than one with gion whose main tenets outcome of placing mathe- Newton and Einstein? included the transmigration maticians from different eras Could ancient mathemati- of souls and the sinfulness of in the same room. For exam- cians contribute any useful eating beans? (See Answer ple, I would be intrigued to ideas to modern mathemati- 1.18.)

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Modern mathematical Chickens and tic-tac-toe. The mathematics of tic-tac-toe murderer. Which modern have been discussed for decades, but can a chicken actually mathematician murdered and learn to play well? In 2001, an Atlantic City casino offered its maimed the most people from patrons a tic-tac-toe “chicken challenge” and offered cash a distance? (See Answer prizes of up to $10,000. The chicken gets the first entry, usu- 1.19.) ally 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 The ∞ symbol. Most high pressing buttons on a separate panel. There is no prize for a school students are familiar tie. A typical game lasts for about a minute, and the chicken with the mathematical sym- seems to be trained to peck at an X or an O, depending on the bol for infinity (∞). Do you human’s moves. think this symbol was used a Supposedly, the tic-tac-toe-playing chickens work in shifts hundred years ago? Who first of one to two hours to avoid stressing the animals. Various used this odd symbol? (See animal-rights advocates have protested the use of chickens in Answer 1.20.) tic-tac-toe games. Can a chicken actually learn to play tic-tac- toe? (See Answer 1.22.)

Mathematics of tic-tac-toe. In how many ways can you Mathematics and God. “God man’s The World of Mathe- place Xs and Os on a stan- exists since mathematics is matics, 1956). dard tic-tac-toe board? (See consistent, and the devil Answer 1.21.) exists since we cannot prove the consistency” (Morris God’s perspective. “When Kline, Mathematical Thought mathematicians think about The square root symbol. The from Ancient to Modern algorithms, it is usually from Austrian mathematician Times, 1990). the God’s-eye perspective. Christoff Rudolff was the ﬁrst They are interested in prov- to use the square root symbol ing, for instance, that there is √ in print; it was published Lunatic scribbles and some algorithm with some in 1525 in Die Coss. mathematics. “If a lunatic interesting property, or that scribbles a jumble of mathe- there is no such algorithm, matical symbols it does not and in order to prove such Mathematics and poetry. “It follow that the writing means things you needn’t actually is impossible to be a mathe- anything merely because locate the algorithm you are matician without being a poet to the inexpert eye it is talking about . . . ” (Daniel in soul” (Soﬁa Kovalevskaya, indistinguishable from higher Dennett, Darwin’s Dangerous quoted in Agnesi to Zeno by mathematics” (Eric Temple Idea: Evolution and the Sanderson Smith, 1996). Bell, quoted in J. R. New- Meaning of Life, 1996).

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Erdös contemplates death. Creativity and madness. harmony of natural law, Once, while pondering his “Creativity and genius feed which reveals an intelligence own death, the mathematician off mental turmoil. The of such superiority that, com- Paul Erdös (1913–1996) ancient Greeks, for instance, pared with it, all the system- remarked, “My mother said, believed in divine forms of atic thinking and acting of ‘Even you, Paul, can be in madness that inspired mor- human beings is an utterly only one place at one time.’ tals’ extraordinary creative insigniﬁcant reﬂection. This Maybe soon I will be relieved acts” (Bruce Bower, Science feeling is the guiding princi- of this disadvantage. Maybe, News, 1995). ple of his life and work. . . . once I’ve left, I’ll be able to It is beyond question closely be in many places at the same akin to that which has pos- time. Maybe then I’ll be able Science, Einstein, and God. sessed the religious geniuses to collaborate with “The scientist’s religious feel- of all ages” (Albert Einstein, Archimedes and Euclid.” ing takes the form of a rap- Mein Weltbild, 1934). turous amazement at the

Mathematics and the infinite. “Mathematics is the only infi- Math in the movies. In the movie A Beautiful Mind, Russell nite human activity. It is con- Crowe scrolls the following formulas on the blackboard in his ceivable that humanity could MIT class: eventually learn everything in V = {F : 3 – X → 3smooth physics or biology. But W = {F = ∇g} humanity certainly won’t ever dim(V/W) = ? be able to find out everything in mathematics, because the Movie directors were told by advisers that this set of formulas subject is infinite. Numbers was subtle enough to be out of reach for most undergraduates, themselves are infinite” (Paul but accessible enough so that Jennifer Connelly’s character Erdös, quoted in Paul Hoff- might be able to dream up a possible solution. man’s The Man Who Loved Only Numbers, 1998). Mathematics and homosexuality. Which brilliant mathematician was forced to become a human guinea pig and was First female doctorate. Who subjected to drug experiments to reverse his homosexuality? was the first woman to (Hint: He was a 1950s computer theorist whose mandatory receive a doctorate in mathe- drug therapy made him impotent and caused his breasts matics, and in what century to enlarge. He also helped to break the codes of the German do you think she received it? Engima code machines during World War II.) (See Answer (See Answer 1.23.) 1.24.)

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A famous female mathematician. Maria Agnesi (1718–1799) Science and religion. “A is one of the most famous female mathematicians of the last contemporary has said, not few centuries and is noted for her work in differential calcu- unjustly, that in this material- lus. When she was seven years old, she mastered the Latin, the istic age of ours the serious Greek, and the Hebrew languages, and at age nine she pub- scientific workers are the only lished a Latin discourse defending higher education for profoundly religious people” women. As an adult, her clearly written textbooks condensed (Albert Einstein, New York the diverse research writings and methods of a number of Times Magazine, 1930). 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.” How- Mathematics and money. ever, after the death of her father, she stopped doing scientific What effect would doubling work altogether and devoted the last forty-seven years of her the salary of every mathemat- life to caring for sick and dying women. ics teacher have on education and the world at large? (See Einstein’s God. “It was, of Women and math. Despite Answer 1.26.) course, a lie what you read horrible prejudice in earlier about my religious convic- times, several women have tions, a lie which is being fought against the establish- A famous female mathe- systematically repeated. I do ment and persevered in matician. Sophie Germain not believe in a personal God mathematics. Emmy Amalie (1776–1831) made major and I have never denied this Noether (1882–1935) was contributions to number but have expressed it clearly. described by Albert Einstein theory, acoustics, and elastic- If something is in me which as “the most significant cre- ity. At age thirteen, Sophie can be called religious then it ative mathematical genius read an account of the death is the unbounded admiration thus far produced since the of Archimedes at the hands for the structure of the world higher education of women of a Roman soldier. She was so far as our science can began.” She is best known for so moved by this story that reveal it” (Albert Einstein, her contributions to abstract she decided to become a personal letter to an atheist, algebra and, in particular, for mathematician. Sadly, her 1954). her study of “chain condi- parents felt that her interest in tions on ideals of rings.” In mathematics was inappropri- 1933, her mathematical ate, so at night she secretly Mathematician cooks. achievements counted for studied the works of Isaac What eighteenth-century nothing when the Nazis Newton and the mathemati- French mathematician caused her dismissal from the cian Leonhard Euler. cooked himself to death? University of Göttingen (See Answer 1.25.) because she was Jewish.

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Mad mom tortures mathematician daughter. What brilliant, Christianity and mathe- famous, and beautiful woman mathematician died in incredi- matics. “The good Christian ble pain because her mother withdrew all pain medication? should beware of mathemati- (Hint: The woman is recognized for her contributions to com- cians, and all those who make puter programming. The mother wanted her daughter to die empty prophesies. The dan- painfully so that her daughter’s soul would be cleansed.) ger already exists that the (See Answer 1.27.) mathematicians have made a covenant with the devil to darken the spirit and to con- fine man in the bonds of Mathematics and relationships. “‘No one really understood Hell” (St. Augustine, De music unless he was a scientist,’ her father had declared, and Genesi Ad Litteram, Book II, not just a scientist, either, oh, no, only the real ones, the theo- c. 400). reticians, 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 Mathematician believes had told her, ‘contained the essential meaning of life’” (Pearl in angels. What famous S. Buck, The Goddess Abides, 1972). English mathematician had not the slightest interest in sex and was also a biblical fundamentalist, believing in Mathematician pretends. no one, but it is of no impor- the reality of angels, demons, What important eleventh- tance at all. A mathematician and Satan? (Hint: According century mathematician pre- is great or he is nothing” to most scholars, he is the tended he was insane so that (Alfred Adler, “Reflections: most influential scientist he would not be put to death? Mathematics and Creativity,” and mathematician to have (Hint: He was born in Iraq The New Yorker, 1972). ever lived.) (See Answer and made contributions to 1.29.) mathematical optics.) (See Answer 1.28.) Mathematics, mind, universe. “If we wish to understand the nature of the History’s most prolific math- Mathematical greatness. Universe we have an inner ematician. Who was the most “Each generation has its few hidden advantage: we are prolific mathematician in his- great mathematicians, and ourselves little portions of the tory? (If you are unable to mathematics would not even universe and so carry the answer this, can you guess in notice the absence of the oth- answer within us” (Jacques what century he lived?) ers. They are useful as teach- Boivin, The Single Heart (See Answer 1.30.) ers, and their research harms Field Theory, 1981).

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Suicidal mathematician. tual Truth” (Philip Davis and Earliest known symbols. What mathematician Reuben Hersh, The Mathe- The Egyptian Rhind Papyrus accepted a duel, knowing that matical Experience, 1981). (c. 1650 B.C.) contains the he would die? (Hint: He spent earliest known symbols for the night before the duel mathematical operations. feverishly writing down his Mathematics and lust. “I “Plus” is denoted by a pair of mathematical ideas, which tell them that if they will legs walking toward the num- have since had a great impact occupy themselves with the ber to be added. on mathematics.) (See study of mathematics they Answer 1.31.) will find in it the best remedy against the lusts of the flesh” Mathematics and the divine. (Thomas Mann, The Magic “Mathematical inquiry lifts Marry a mathematician? Mountain, 1924). the human mind into closer Would you rather marry the proximity with the divine than best mathematician in the is attainable through any other world or the best chess Newton’s magic. “Had medium” (Hermann Weyl, player? (See Answer 1.32.) Newton not been steeped in quoted in Philip Davis and alchemical and other magical Reuben Hersh, The Mathe- learning, he would never have matical Experience, 1981). Mathematics and truth. “We proposed forces of attraction who are heirs to three recent and repulsion between bodies π centuries of scientific devel- as the major feature of his and the law. In 1896, an opment can hardly imagine a physical system” (John Indiana physician promoted a π state of mind in which many Henry, “Newton, Matter, and legislative bill that made mathematical objects were Magic,” in John Fauvel’s Let equal to 3.2, exactly. The regarded as symbols of spiri- Newton Be!, 1988). Indiana House of Representa- tives approved the bill unani- mously, 67 to 0. The Senate, however, deferred debate Mathematical corpse. You come home and see a corpse about the bill “until a later on your foyer floor. Would you be more frightened if (1) date.” scrawled on the floor is the Pythagorean theorem: a2 + b2 = c2, or (2) scrawled on the floor is the following complicated formula: The mathematical life. “The mathematical life of a mathe- 122∞ (!)(4kk1103+ 26390 ) = ∑ matician is short. Work rarely π 44k 9801 k =0 (!)k 396 improves after the age of twenty-five or thirty. If little (See Answer 1.33.) has been accomplished by

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then, little will ever be Blind date. You are single and going on a blind date. The accomplished” (Alfred Adler, date knocks on your door and is extremely attractive. How- “Mathematics and Creativ- ever, you note that the person has the following formulas ity,” The New Yorker, 1972). tattooed on the right arm:

dR =− = 0 kBtRB (),(0 ) R0 The genesis of x = 1. dt Ibn Yahya al-Maghribi dB Al-Samawal in 1175 was =− = kRtBR (),(0 ) B0 the ﬁrst to publish dt x0 = 1 From this little information, do you think you would enjoy the In other words, he realized evening with this person? (See Answer 1.34.) and published the idea that any number raised to the power of 0 is 1. Al-Samawal’s book was titled The Dazzling. Fundamental Anagram of Calculus. Probably most of you have His father was a Jewish never heard of Newton’s “Fundamental Anagram of Calculus”: scholar of religion and literature from Baghdad. 6accdae13eff7i3l9n4o4qrr4s8t12ux Can you think of any possible reason Newton would want to code aspects of his calculus discoveries? (See Answer 1.35.) Euler’s one-step proof of God’s existence. In mathematical books too numerous to Petersburg and embarrassed 129), says that Diderot was mention, we have heard the the freethinking Diderot with mathematically well versed story of the mathematician this simple algebraic proof of and wouldn’t have been Leonhard Euler’s encounter God’s existence. Diderot was shocked by the formula. with the French encyclopedist shocked and ﬂed. Was Euler Moreover, Euler wasn’t Denis Diderot. Diderot was a deliberately demonstrating the type of person to make devout atheist, and he chal- how lame these kinds of such a zany comment. While lenged the religious Euler to arguments can be? people of the time did seek mathematically prove the Today we know that there simple mathematical proofs existence of God. Euler is little evidence that the of God, the “Euler versus replied, “Sir (a + bn)/n = x; encounter ever took place. Diderot” story was probably hence, God exists. Please Dirk J. Struik, in his book A fabricated by the English reply!” Concise History of Mathe- mathematician De Morgan Supposedly, Euler said this matics, third revised edition (1806–1871). in a public debate in St. (New York: Dover, 1967, p.

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Science and religion. π savants. In 1844, Johann Dase (a.k.a., Zacharias Dahse) “I have always thought it computed π to 200 decimal places in less than two months. He curious that, while most was said to be a calculating prodigy (or an “idiot savant”), scientists claim to eschew hired for the task by the Hamburg Academy of Sciences on religion, it actually dominates Gauss’s recommendation. To compute their thoughts more than it π does the clergy” (Fred Hoyle, = 3.14159 26535 89793 23846 26433 83279 astrophysicist, “The Uni- 50288 41971 69399 37510 58209 74944 59230 verse: Past and Present 78164 06286 20899 86280 34825 34211 70679 Reflections,” Annual Review 82148 08651 32823 06647 09384 46095 50582 of Astronomy and Astro- 23172 53594 08128 48111 74502 84102 70193 physics, 1982). 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. Parallel universes and math- Dase ran the arctangent job in his brain for nearly sixty days. ematics. In theory, it is possi- Not everyone believes the legend of Dase. For example, ble to list or “enumerate” all Arthur C. Clarke recently wrote to me that he simply doesn’t rational numbers. How has believe the story of Dase calculating pi to 200 places in his this mathematical fact helped head. Clarke says, “Even though I’ve seen fairly well authenti- certain cosmologists to cated reports of other incredible feats of mental calculation, I “prove” that there is an infi- think this is totally beyond credibility.” nite number of universes I would be interested in hearing from readers who can con- alongside our own? (As you firm or deny this story. will learn in the next chapter, “rational numbers” are num- 1 bers like ⁄ 2, which can be expressed as fractions.) (See discussion of the value of 00 Greek numerals. Did Answer 1.36.) is very old, and controversy you know that the ancient raged throughout the nine- Greeks had two systems of teenth century. numerals? The earlier of The mysterious 00. these was based on the Students are taught that any initial letters of the names number to the zero power is One-page proof of God’s of numbers: the number 5 1, and zero to any power is 0. existence. Which famous Ger- was indicated by the letter But serious mathematicians man mathematician “proved” pi; 10 by the letter delta; often consider 00 undefined. God’s existence in a proof 100 by the antique form of If you try to make a graph of that fit on just one page of the letter H; 1,000 by the xy, you’ll see it has a disconti- paper? (See Answer 1.37.) letter chi; and 10,000 by the nuity at the point (0,0). The letter mu.

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Our mathematical percep- Pierre de Fermat. In the Ancient number notation lets tions. “The three of you stare early 1600s, Pierre de Fer- humans “think big.” The earliest at the school of fish and watch mat, a French lawyer, made forms of number notation, them move in synchrony, brilliant discoveries in num- which used straight lines for despite their lack of eyes. The ber theory. Although he was grouping 1s, were inconven- resulting patterns are hyp- an “amateur” mathematician, ient when dealing with large notic, like the reflections from he created mathematical numbers. By 3400 B.C.in a hundred pieces of broken challenges such as “Fermat’s Egypt, and 3000 B.C.in glass. You imagine that the Last Theorem,” which was Mesopotamia, a special sym- senses place a filter on how not solved until 1994. Fer- bol was adopted for the num- much humans can perceive of mat’s Last Theorem states ber 10. The addition of this the mathematical fabric of the that xn + yn = zn has no second number symbol made universe. If the universe is a nonzero integer solutions for it possible to express the mathematical carpet, then all x, y, and z when n > 2. number 11 with 2 symbols creatures are looking at it Fermat was no ordinary instead of 11, and the number through imperfect glasses. lawyer indeed. He is consid- 99 with 18 symbols instead How might humanity perfect ered, along with Blaise of 99. those glasses? Through drugs, Pascal, a founder of surgery, or electrical stimula- probability theory. As the tion of the brain? Probably coinventor of analytic Caterpillar vehicle. Many our best chance is through the geometry, he is considered, mathematicians were creative use of computers” (Cliff Pick- along with René Descartes, inventors, although not all of over, The Loom of God, one of the first modern math- their inventions were practi- 1997). ematicians. cal. For example, Polish-born Josef Hoëné-Wronski (1778–1853), an analyst, a The Beal reward. In the mid-1990s, the Texas banker philosopher, a combinatorial- Andrew Beal posed a perplexing mathematical problem and ist, and a physicist, developed offered $5,000 for the solution of this problem. In particular, a fantastical design for Beal was curious about the equation Ax + By = C z. The six caterpillar-like vehicles that letters represent integers, with x, y, and z greater than 2. he intended to replace rail- (Fermat’s Last Theorem involves the special case in which road transportation. He also the exponents x, y, and z are the same.) Oddly enough, Beal attempted to build a perpetual noticed that for any solution of this general equation he could motion machine and to build find, A, B, and C have a common factor. For example, in the a machine to predict the equation 36 + 183 = 38, the numbers 3, 18, and 3 all have the future (which he called the factor 3. Using computers at his bank, Beal checked equations prognometre). with exponents up to 100 but could not discover a solution that didn’t involve a common factor.

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Progress in mathematics. God’s math book. “God has The Mathematical Journeys “In most sciences, one gener- a transfinite book with all the of Paul Erdös, 1998). ation tears down what another theorems and their best has built and what one has proofs. You don’t really have established another undoes. to believe in God as long as Song lyrics. What is the In mathematics alone, each you believe in the book” largest number ever used in generation adds a new story (Paul Erdös, quoted in Bruce the lyrics to a popular song? to the old structure” (Her- Schechter, My Brain Is Open: (See Answer 1.38.) mann Hankel, 1839–1873, who contributed to the theory of functions, complex num- The magnificent Erdös. It is commonly agreed that Paul bers, and the history of math- Erdös is the second-most prolific mathematician of all times, ematics, quoted in Desmond being surpassed only by Leonhard Euler, the great eigh- MacHale, Comic Sections, teenth-century mathematician whose name is spoken with 1993). 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 Hilbert’s problems. In 1900, in his seventies. Erdös undoubtedly had the greatest number the mathematician David of coauthors (around 500) among mathematicians of all Hilbert submitted twenty- times. three important mathematical In 2004, an eBay auction offered buyers an opportunity to problems to be targeted for link their names, through five degrees of separation, with Paul solution in the twentieth cen- Erdös. In particular, the mathematician William Tozier pre- tury. These twenty-three sented bidders with the chance to collaborate on a research problems extend over all paper. Tozier was linked to Erdös through a string of coau- fields of mathematics. thors. In particular, he had collaborated with someone who Because of Hilbert’s prestige, had collaborated with someone who had collaborated with mathematicians spent a great someone who had collaborated with Paul Erdös. A mathemati- deal of time tackling the cian who has published a paper with Erdös has an Erdös num- problems, and many of the ber of 1. A mathematician who has published a paper with problems have been solved. someone who has published a paper with Erdös has an Erdös Some, however, have been number of 2, and so on. Tozier has an Erdös number of 4, solved only very recently, and quite a respectable ranking in the mathematical community. still others continue to daunt This means that the person working with Tozier would have an us. Hilbert’s twenty-three Erdös number of 5. During the auction, Tozier heard from wonderful problems were more than a hundred would-be researchers. (For more infor- designed to lead to the fur- mation, see Erica Klarreich, “Theorems for Sale: An Online thering of various disciplines Auctioneer Offers Math Amateurs a Backdoor to Prestige,” in mathematics. Science News 165, no. 24 (2004): 376–77.)

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Rope and lotus symbols. A calculating prodigy and 365,365,365,365,365,365. When The Egyptian hieroglyphic he was ten years old, the calculating prodigy Truman Henry system evolved special sym- Safford (1836–1901) of Royalton, Vermont, was once asked to bols (resembling ropes, lotus square, in his head, the number 365,365,365,365,365,365. His plants, etc.) for the numbers church leader reports, “He flew around the room like a top, 10, 100, 1,000, and 10,000. 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 Strange math title. I love minute said he, 133,491,850,208,566,925,016,658,299,941, collecting math papers with 583,225!” strange titles. These papers Incidentally, I had first mentioned this large number in my are published in serious math book Wonders of Numbers and had misprinted one of the dig- journals. For example, in its. Bobby Jacobs, a ten-year-old math whiz from Virginia, 1992, A. Granville published wrote to me with the corrected version that you see here. He an article with the strange was the only person to have discovered my earlier typographi- title “Zaphod Beeblebrox’s cal error. Brain and the Fifty-Ninth Row of Pascal’s Triangle,” in the prestigious The American that when we die, our souls [mathematical curves], spin- Mathematical Monthly (vol. survive. Incidentally, Jesus ning to their own music, is a 99, no. 4 [April]: 318–31). Christ is number 611,121 wondrous, spiritual event” among more than 700,000 (Paul Rapp, in Kathleen Creator Sons. McAuliffe, “Get Smart: Urantia religion and num- Controlling Chaos,” Omni bers. In the modern-day Uran- [1989]). tia religion, numbers have an Mathematics and God. almost divine quality. “Philosophers and great According to the sect, head- religious thinkers of the last The discovery of calculus. quartered in Chicago, we live century saw evidence of God The English mathematician on the 606th planet in a sys- in the symmetries and Isaac Newton (1642–1727) tem called Satania, which harmonies around them— and the German mathemati- includes 619 flawed but in the beautiful equations of cian Gottfried Wilhelm Leib- evolving worlds. Urantia’s classical physics that describe niz (1646–1716) are grand universe number is such phenomena as electric- generally credited with the 5,342,482,337,666. Urantians ity and magnetism. I don’t invention of calculus, but var- believe that human minds are see the simple patterns under- ious earlier mathematicians created at birth, but the soul lying nature’s complexity explored the concept of rates does not develop until about as evidence of God. I believe and limits, starting with the age six. They also believe that is God. To behold ancient Egyptians, who

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Newton’s giants. “If I have seen further than others, it is by Latin word for “sum.”) The standing upon the shoulders of giants” (Isaac Newton, per- mathematician Joseph Louis sonal letter to Robert Hooker, 1675 [see next quotation]). Lagrange (1736–1813) was the first person to use the notation f ′(x) for the first ″ Abelson’s giants. “If I have not seen as far as others, it is derivative and f (x) for the because giants were standing on my shoulders” (Hal Abelson, second derivative. In 1696, MIT professor). Guillaume François de L’Hôpital, a French mathematician, published the first textbook on calculus. 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). Jews, π, the movie. The 1998 cult movie titled π stars a mathematical genius who is developed rules for calculat- integral calculus in 1686. He fascinated by numbers and ing the volume of pyramids said, “It is unworthy of excel- their role in the cosmos, the and approximating the areas lent men, to lose hours like stock market, and Jewish of circles. slaves in the labor of calcula- mysticism. According to the In the 1600s, both Newton tion. . . . My new calculus . . . movie, what is God’s num- and Leibniz puzzled over offers truth by a kind of ber? (See Answer 1.39.) problems of tangents, rates of analysis and without any change, minima, maxima, effort of imagination.” New- and infinitesimals (unimagin- ton was outraged. Debates ably tiny quantities that are raged for many years on how Mathematics and romance. almost but not quite zero). to divide the credit for the What romantic comedy has Both men understood that discovery of calculus, and, as the most complicated mathe- differentiation (finding tan- a result, progress in calculus matics ever portrayed in a gents to curves) and integra- was delayed. movie? (See Answer 1.40.) tion (finding areas under curves) are inverse processes. Newton’s discovery The notations of calculus. (1665–1666) started with his Today we use Leibniz’s sym- Greek death. Why did __df the ancient Greeks and interest in infinite sums; how- bols in calculus, such as dx for ever, he was slow to publish the derivative and the ∫ sym- other cultures believe 8 to his findings. Leibniz pub- bol for integration. (This inte- be a symbol of death? (See lished his discovery of differ- gral symbol was actually a Answer 1.41.) ential calculus in 1684 and of long letter S for summa, the

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The mechanical Pascaline. pores become clogged with Mathematical progress. One example of an early the earthly particles blown “More significant mathemati- computing machine is Blaise from the dusty highways of cal work has been done in Pascal’s wheel computer vulgar life” (James Joseph the latter half of this century called a Pascaline. In 1644, Sylvester, 1814–1897, a than in all previous centuries this French philosopher and professor of mathematics combined” (John Casti, Five mathematician built a calcu- at Johns Hopkins University, Golden Rules, 1997). lating machine to help his 1869 address to the British father compute business Mathematical Association). accounts. Pascal was twenty years old at the time. The machine used a series of Simultaneity in science. The simultaneous discovery of cal- spinning numbered wheels to culus by Newton and Leibniz makes me wonder why so many add large numbers. discoveries in science were made at the same time by people The wonderful Pascaline working independently. For example, Charles Darwin was about the size of a shoe- (1809–1882) and Alfred Wallace (1823–1913) both independ- box. About fifty models were ently developed the theory of evolution. In fact, in 1858, Dar- made. win 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. What number The Matrix. As another example of simultaneity, the mathematicians is on Agent Smith’s license János Bolyai (1802–1860) and Nikolai Lobachevsky plate in the movie The Matrix (1793–1856) developed hyperbolic geometry independently Reloaded? Why? (See and at the same time (both perhaps stimulated indirectly by Answer 1.42.) Carl Friedrich Gauss). Most likely, such simultaneous discoveries have occurred because the time was “ripe” for such dis- At the movies. What coveries, given humanity’s accumulated knowledge at the time famous book and movie title the discoveries were made. On the other hand, mystics have contains a number that is suggested that there is a deeper meaning to such coincidences. greater than 18,000 and less The Austrian biologist Paul Kammerer (1880–1926) wrote, then 38,000? (See Answer “We thus arrive at the image of a world-mosaic or cosmic 1.43.) 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 The mathematical life. ocean waves that seem isolated and unrelated. According to “The mathematician lives his controversial theory, we notice the tops of the waves, but long and lives young; the beneath the surface there may be some kind of synchronistic wings of the soul do not mechanism that mysteriously connects events in our world and early drop off, nor do its causes them to cluster.

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First mathematician. What say that God knows not all that Pythagoras was intellec- is the name of the first human numbers. . . . What madman tually one of the most impor- who was identified as having would say so? . . . What are tant men who ever lived, both made a contribution to math- we mean wretches that dare when he was wise and when ematics? (See Answer 1.44.) presume to limit His knowl- he was unwise. Pythagoras edge?” (St. Augustine, was the most puzzling mathe- The City of God, A.D. 412). matician of history because Game show. Why was the he founded a numerical reli- 1950’s TV game show called gion whose main tenets were The $64,000 Question? Why Who was Pythagoras? the transmigration of souls not a rounder number like You can tell from some of the and the sinfulness of eating $50,000? (See Answer 1.45.) following factoids that I beans, along with a host of love trivia that relates to the other odd rules and regula- famous ancient Greek mathe- tions. To the Pythagoreans, God and the infinite. “Such matician Pythagoras. His mathematics was an ecstatic as say that things infinite are ideas continue to thrive after revelation. past God’s knowledge may three millennia of mathemati- The Pythagoreans, like just as well leap headlong cal science. The philosopher modern day fractalists, were into this pit of impiety, and Bertrand Russell once wrote akin to musicians. They created pattern and beauty as Mathematical scope. “Mathematics is not a book confined they discovered mathematical within a cover and bound between brazen clasps, whose con- truths. Mathematical and the- tents it needs only patience to ransack; it is not a mine, whose ological blending began with treasures may take long to reduce into possession, but which Pythagoras and eventually fill only a limited number of veins and lodes; it is not a soil, affected all religious philoso- whose fertility can be exhausted by the yield of successive phy in Greece, played a role harvests; it is not a continent or an ocean, whose area can be in religion of the Middle mapped out and its contour defined: it is limitless as that space Ages, and extended to Kant which it finds too narrow for its aspirations; its possibilities in modern times. Bertrand are as infinite as the worlds which are forever crowding in and Russell felt that if it were multiplying upon the astronomer’s gaze; it is as incapable of not for Pythagoras, theolo- being restricted within assigned boundaries or being reduced gians would not have sought to definitions of permanent validity, as the consciousness, the logical proofs of God and life, which seems to slumber in each monad, in every atom of immortality. matter, in each leaf and bud and cell, and is forever ready to If you want to read more burst forth into new forms of vegetable and animal existence” about Pythagoras, see my (James Joseph Sylvester, The Collected Mathematical Papers book The Loom of God and of James Joseph Sylvester, Volume III, address on Commemo- Peter Gorman’s Pythagoras: ration Day at Johns Hopkins University, February 22, 1877). A Life.

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The secret life of numbers. Anamnesis and the number 216. Was the ancient Greek math- To Pythagoras and his follow- ematician Pythagoras once a plant? This is a seemingly bizarre ers, numbers were like gods, question, but Pythagoras claimed that he had been both a plant pure and free from material and an animal in his past lives, and, like Saint Francis, he change. The worship of num- preached to animals. Pythagoras and his followers believed in bers 1 through 10 was a kind anamnesis, the recollection of one’s previous incarnations. of polytheism for the During Pythagoras’s time, most philosophers believed that Pythagoreans. only men could be happy. Pythagoras, on the other hand, Pythagoreans believed that believed in the happiness of plants, animals, and women. numbers were alive, inde- In various ancient Greek writings, we are told the exact pendent of humans, but with a number of years between each of Pythagoras’s incarnations: telepathic form of conscious- 216. Interestingly, Pythagoreans considered 216 to be a mysti- ness. Humans could relin- cal number, because it is 6 cubed (6 × 6 × 6). Six was also quish their three-dimensional considered a “circular number” because its powers always lives and telepathize with ended in 6. The fetus was considered to have been formed these number beings by using after 216 days. various forms of meditation. The number 216 continues to pop up in the most unlikely of Meditation upon numbers places in theological literature. In an obscure passage from was communing with the The Republic (viii, 546 B–D), Plato notes that 216 = 63. It is gods, gods who desired noth- also associated with auspicious signs on the Buddha’s footprint. ing from humans but their sincere admiration and con- templation. Meditation upon numbers was a form of prayer Shades of ghosts. Pythago- fully sacriﬁced a hecatomb of that did not ask any favors ras believed that even rocks oxen (a hundred animals) from the gods. possess a psychic existence. when he discovered his These kinds of thoughts Mountains rose from the famous theorem about the are not foreign to modern earth because of growing right-angled triangle, this mathematicians, who often pains of the earth, and would have been scandalously debate whether mathematics Pythagoras told his followers un-Pythagorean and is proba- is a creation of the human that earthquakes were caused bly not true. Pythagoras mind or is out there in the by the shades of ghosts of the refused to sacriﬁce animals. universe, independent of dead, which created distur- Instead, the Pythagoreans human thought. Opinions bances beneath the earth. believed in the theurgic vary. A few mathematicians construction of agalmata— believe that mathematics is a statues of gods consisting of form of human logic that is Pythagorean sacrifice. herbs, incense, and metals to not necessarily valid in all Although some historians attract the cosmic forces. parts of the universe. report that Pythagoras joy-

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Pythagoras and aliens. Famous epitaphs. Replace x and y with the names of two UFOs and extraterrestrial famous mathematicians in these two puzzles. In Puzzle 1, the life are hot topics today. But following couplet of Alexander Pope is the engraved epitaph who would believe that these on the mystery person’s sarcophagus in Westminster Abbey, in same ideas enthralled the London: Pythagoreans a few millennia ago? In fact, Pythagore- Nature and Nature’s laws lay hid in the night; ans believed that all the God said, “Let x be” and all was light. planets in the solar system —Alexander Pope (1688–1744) are inhabited, and humans In Puzzle 2, the epitaph “S = klnW ” is engraved on y’s tomb- dwelling on Earth were less stone. Who are x and y? (See Answer 1.46.) advanced than these other inhabitants were. (The idea of advanced extraterrestrial neighbors curiously has continued, for some, to this Matrix prayers. Some of prayer that he succinctly day.) According to later you will be interested in writes as Underwood Dudley’s Pythagoreans, as one travels P < R{S} → {U}, r > 0 farther from Earth, the Mathematical Cranks beings on other planets and (Washington, D.C.: Mathe- He says, “This prayer should in other solar systems matical Association of be sufficiently concise to be become less flawed. America, 1992). The book acceptable to Christ, yet Pythagoras went as far as contains musings of slightly every single Christian inhabi- to suggest that disembodied mad mathematicians. My tant of Northern Ireland has intelligences existed in the favorite chapter is on the been separately included.” universe. These mind- topic of “Matrix Prayers,” The chapter concludes with creatures had very tenuous designed by a priest of the information regarding the physical bodies. Many of Church of England. The geometry of heaven. Pythagoras’s followers priest regularly prayed to God in mathematical terms believed that Pythagoras Secret mathematician. using matrices, and he taught himself had once been a Who really was the famous the children in his church to superior being who inhabited and secretive French mathe- pray and think of God in the Moon or the Sun. matician “N. Bourbaki”? matrices. Mathematical (See Answer 1.47.) Cranks goes into great detail Gods and sets. “The null regarding “revelation matri- set is also a set; the absence ces,” “Polite Request Opera- How much? How much of a god is also a god” (A. tors,” and the like. The priest mathematics can we know? Moreira). finally derives a beautiful (See Answer 1.48.)

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Mathematics and reality. From string theory to quantum the- Einstein on mathematics ory, the deeper one goes in the study of physics, the closer one and reality. “At this point an gets to pure mathematics. Mathematics is the fabric of reality. enigma presents itself which Some might even say that mathematics “runs” reality in the in all ages has agitated same way that Microsoft’s Windows runs your computer and inquiring minds. How can it shapes your interactions with the vast network beyond. be that mathematics, being Schrödinger’s wave equation—which describes basic reality after all a product of human and events in terms of wave functions and probabilities—is thought which is independent the evanescent substrate on which we all exist: of experience, is so admirably appropriate to the objects of  h 2  ∂ψ reality? Is human reason, −∇+2 Vr()→  ψ (,) rt→ = ih (,)rt→  2m  ∂t then, without experience, merely by taking thought, Freeman Dyson, in the introduction to Nature’s Imagination, able to fathom the properties speaks highly of this formula: “Sometimes the understanding of real things? In my opinion of a whole field of science is suddenly advanced by the dis- the answer to this question is covery of a single basic equation. Thus it happened that the briefly this: As far as the laws Schrödinger equation in 1926 and the Dirac equation in 1927 of mathematics refer to real- brought a miraculous order into the previously mysterious ity, they are not certain; and processes of atomic physics. Bewildering complexities of as far as they are certain, they chemistry and physics were reduced to two lines of algebraic do not refer to reality” symbols.” (Albert Einstein’s address to the Prussian Academy of Science in Berlin, 1921). Factorial symbol. In 1808 Insights and analysis. Christian Kramp (1760– “Einstein’s fundamental 1826) introduced the “!” as insights of space/matter Ramanujan’s tongue. In the the factorial symbol as a con- relations came out of philo- spring of 2003, the Aurora venience to the printer. sophical musings about the Theater Company of Berke- nature of the universe, not ley performed Ira Hauptman’s from rational analysis of play Partition, which focuses Nobel Prize. Why is there observational data—the on the collaboration of the no Nobel Prize for mathemat- logical analysis, prediction, mathematicians Ramanujan ics? (See Answer 1.49.) and testing coming only and Hardy. In the play, Nama- after the formation of the giri, Ramanujan’s personal creative hypotheses” deity and inspiration in real Roman numerals. Why (R. H. Davis, The Skeptical life, is seen literally writing don’t we use Roman numerals Inquirer, 1995). equations on Ramanujan’s anymore? (See Answer 1.50.) tongue with her finger.

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Mathematics and reality. of higher mathematics, mation coming in, new “My complete answer to the how close are we to a real worlds to explore, a con- late 19th century question understanding?” (Susan stantly expanding domain of ‘what is electrodynamics Kruglinski, “When Even life, consciousness, and trying to tell us’ would sim- Mathematicians Don’t memory” (Freeman Dyson, ply be this: Fields in empty Understand the Math,” New “Time without End: Physics space have physical reality; York Times, May 25, 2004). and Biology in an Open Uni- the medium that supports verse,” Reviews of Modern them does not. Having thus Physics, 1979). removed the mystery from Dyson on the infinite reser- electrodynamics, let me voir of mathematics. “Gödel immediately do the same for proved that the world of pure More mathematics and quantum mechanics: mathematics is inexhaustible; reality. “It is difficult to Correlations have physical no finite set of axioms and explain what math is, let reality; that which they corre- rules of inference can ever alone what it says. Math may late does not” (N. David encompass the whole of be seen as the vigorous struc- Mermin, “What Is Quantum mathematics; given any finite ture supporting the physical Mechanics Trying to Tell set of axioms, we can find world or as a human idea in Us?” American Journal of meaningful mathematical development. [Dr. John Casti Physics 66, [1998]: 753–67). questions which the axioms says that] ‘the criteria that leave unanswered. I hope that mathematicians use for what an analogous situation exists constitutes good versus bad More mathematics and in the physical world. If my mathematics is much more reality. “[Much of frontier view of the future is correct, close to that of a poet or a mathematics] confounds even it means that the world of sculptor or a musician than it mathematicians and physi- physics and astronomy is also is to a chemist’” (Susan cists, as they use math to cal- inexhaustible; no matter how Kruglinski, “When Even culate the inconceivable, far we go into the future, Mathematicians Don’t undetectable, nonexistent and there will always be new Understand the Math,” New impossible. So what does it things happening, new infor- York Times, May 25, 2004). mean when mainstream explanations of our physical reality are based on stuff that Creativity and travel. “One way of goosing the brain is even scientists cannot com- traveling, particularly internationally. It helps shake up prehend? When nonscientists perspective and offers new experiences. Interviews with 40 read about the strings and MacArthur ‘genius’ award winners found 10 lived overseas branes of the latest physics permanently or temporarily, three traveled at least a few theories, or the Riemann months each year, and at least two have a ‘horror of a home’” surfaces and Galois fields (Sharon McDonnell, “Innovation Electrified,” American Way).

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Ramanujan redux. How The Fractal Murders. In 2002, Mark Cohen, a lawyer and a would mathematics have been judge, published The Fractal Murders, a novel in which three advanced if Ramanujan had mathematicians, all of whom are experts in fractals, have died. developed in a more nurtur- Two were murdered. The third was an apparent suicide. In the ing early environment? novel, the math professor Jayne Smyers hires a private eye, Although he would have been Pepper Keane, to look into the three deaths, which seem to be a better-trained mathemati- related only because each victim was researching fractals. cian, would he have become such a unique thinker? Could he have discovered so many Mathematical universe. understand. One may say ‘the wonderful formulas if he Why does the universe seem eternal mystery of the world had been taught the rules of to operate according to is its comprehensibility.’ It is mathematics early on and mathematical laws? (See one of the great realizations pushed to publish his results Answer 1.51.) of Immanuel Kant that this with rigorous proofs? Perhaps setting up of a real external his relative isolation and world would be senseless poverty enhanced the great- A universe of blind mathe- without this comprehensi- ness of his mathematical maticians. Sighted mathemati- bility” (Albert Einstein, thought. For Ramanujan, cians generally work by “Physics and Reality,” 1936). equations were not just the studying vast assemblages of means for proofs or calcula- numbers and symbols scrib- tions. The beauty of the equa- bled on paper. It would seem Wigner on mathematics and tion was of paramount value extremely difﬁcult to do reality. “The miracle of appro- for Ramanujan. mathematics without being priateness of the language of able to see, and to be forced mathematics for the formula- to keep the information “all tion of the laws of physics is The secret life of formulas. in one’s head.” Can a great a wonderful gift which we “We cannot help but think mathematician be totally neither understand nor that mathematical formulae blind? (See Answer 1.52.) deserve. We should be grate- have a life of their own, that ful for it, and hope that it will they know more than their remain valid for future discoverers do and that they Einstein on comprehensibil- research, and that it will return more to us than we ity and reality. “The very fact extend, for better or for have invested in them” that the totality of our sense worse, to our pleasure even (Heinrich Hertz, German experiences is such that by though perhaps also to our physicist, quoted in Eric means of thinking . . . it can bafﬂement, to wide branches Bell, Men of Mathematics, be put in order, this fact is of learning” (Eugene Wigner, 1937). one which leaves us in awe, “The Unreasonable Effective- but which we shall never ness of Mathematics,” 1960).

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Close to reality. We’ve better word—the math of the Why learn mathematics? talked quite a bit about math- Well of Souls. We feel the It has been estimated that ematics and reality. Who do energy ﬂow, the ties and much greater than 99.99 you think is in more direct bands, in each and every percent of all Americans contact with reality, a mathe- particle of matter and energy. will never use the quadratic matician or a physicist? What All reality is mathematics, all formula or most of the other do famous twentieth-century existence—past, present, algebraic or geometrical mathematicians say on this and future—is equations’” relations they learn in school. subject? (See Answer 1.53.) (Jack Chalker, Quest for the Why teach or learn mathe- Well of Souls, 1985). matics beyond the basic operations of addition, All reality is mathematics. subtraction, multiplication, “The Gedemondan chuckles. Pythagoras on mathematics and division? (See Answer ‘We read probabilities. You and reality. “All things are 1.54.) see, we see—perceive is a numbers.”

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