A Miscellany

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AMS / MAA SPECTRUM VOL 58

Is Mathematics Inevitable? A Miscellany

ISBN 978-0-88385-566-9

Library of Congress number 2007940798

Printed in the United States of America

Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 10.1090/spec/058

'ls Ji1atliematics 'lnevitab{e? 3l. :Nfisce((any

edited by

Underwood Dudley Professor Emeritus, DePauw University

Published and Distributed by The Mathematical Association of America SPECTRUM SERIES The Spectrum Series of the Mathematical Association ofAmerica was so named to reflect its purpose: to publish a broad range of books including biographies, accessible expositions of old or new mathematical ideas, reprints and revisions of excellent out-of-print books, popular works, and other monographs of high interest that will appeal to a broad range of readers, including students and teachers of mathematics, mathematical amateurs, and researchers.

Coordinating Council on Publications James Daniel, Chair Spectrum Editorial Board Gerald L. Alexanderson, Editor Robert Beezer Jeffrey L. Nunemacher William Dunham J. D. Phillips Michael Filaseta Ken Ross Erica Flapan Marvin Schaefer Michael A. Jones Sanford Segal Keith Kendig Franklin Sheehan

777 Mathematical Conversation Starters, by John de Pillis 99 Points of Intersection: Examples-Pictures-Proofs, by Hans Walser. Translated from the original German by Peter Hilton and Jean Pedersen. All the Math Thats Fit to Print, by Keith Devlin Calculus Gems: Brief Lives and Memorable Mathematics, by George F. Simmons Carl Friedrich Gauss: Titan of Science., by G. Waldo Dunnington, with additional material by Jeremy Gray and Fritz-Egbert Dohse The Changing Space of Geometry, edited by Chris Pritchard Circles: A Mathematical View, by Dan Pedoe Complex Numbers and Geometry, by Liang-shin Hahn Cryptology, by Albrecht Beutelspacher The Early Mathematics ofLeonhard Euler, by C. Edward Sandifer The Edge of the Universe: Celebrating JO Years of Math Horizons, edited by Deanna Haunsperger and Stephen Kennedy Euler and Modern Science, edited by N. N. Bogolyubov, G. K. Mikhailov, and A. P. / Yushk:evich. Translated from Russian by Robert Burns. Euler at 300: An Appreciation, edited by Robert E. Bradley, Lawrence A. D'Antonio., and C. Edward Sandifer Five Hundred Mathematical Challenges, Edward J. Barbeau, Murray S. Klamkin, and William 0. J. Moser The Genius ofEuler: Reflections on his Life and Work, edited by William Dunham The Golden Section, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. The Harmony of the World: 75 Years of Mathematics Magazine, edited by Gerald L. Alexanderson with the assistance of Peter Ross How Euler Did It, by C. Edward Sandifer Is Mathematics Inevitable? A Miscellany, edited by Underwood Dudley I Want to Be a Mathematician, by Paul R. Halmos Journey into Geometries, by Marta Sved JULIA: a life in mathematics, by Constance Reid The Lighter Side ofMathematics: Proceedings of the Eugene Strens Memorial Confer ence on Recreational Mathematics & Its History, edited by Richard K. Guy and Robert E. Woodrow Lure ofthe Integers, by Joe Roberts Magic Numbers of the Professor, by Owen O'Shea and Underwood Dudley Magic Tricks, Card Shuffling, and Dynamic Computer Memories: The Mathematics of the Perfect Shuffle, by S. Brent Morris Martin Gardner s Mathematical Games: The entire collection of his Scientific Ameri can columns The Math Chat Book, by Frank Morgan Mathematical Adventures for Students and Amateurs, edited by David Hayes and Ta- tiana Shubin. With the assistance of Gerald L. Alexanderson and Peter Ross. Mathematical Apocrypha, by Steven G. Krantz Mathematical Apocrypha Redux, by Steven G. Krantz Mathematical Carnival, by Martin Gardner Mathematical Circles Vol I: In Mathematical Circles Quadrants I, II, Ill, IV, by Howard W. Eves Mathematical Circles Vo l JI: Mathematical Circles Revisited and Mathematical Circles Squared, by Howard W. Eves Mathematical Circles Vol Ill: Mathematical Circles Adieu and Return to Mathematical Circles, by Howard W. Eves Mathematical Circus, by Martin Gardner Mathematical Cranks, by Underwood Dudley Mathematical Evolutions, edited by Abe Shenitzer and John Stillwell Mathematical Fallacies, Flaws, and Flimflam, by Edward J. Barbeau Mathematical Magic Show, by Martin Gardner Mathematical Reminiscences, by Howard Eves Mathematical Treks: From Surreal Numbers to Magic Circles, by !vars Peterson Mathematics: Queen and Servant ofScience, by E.T. Bell Memorabilia Mathematica, by Robert Edouard Moritz Musings ofthe Masters: An Anthology ofMathematical Reflections, edited by Raymond G.Ayoub New Mathematical Diversions, by Martin Gardner Non-Euclidean Geometry, by H. S. M. Coxeter Numerical Methods That Work, by Forman Acton Numerology or What Pythagoras Wrought, by Underwood Dudley Out ofthe Mouths ofMathematicians, by Rosemary Schmalz Penrose Tiles to Trapdoor Ciphers .. . and the Return ofDr. Matrix, by Martin Gardner Polyominoes, by George Martin Power Play, by Edward J. Barbeau The Random Walks of George P6lya, by Gerald L. Alexanderson Remarkable Mathematicians,from Euler to von Neumann, loan James The Search for E.T. Bell, also known as John Taine, by Constance Reid Shaping Space, edited by Marjorie Senechal and George Fleck Sherlock Holmes in Babylon and Other Tales ofMathematical History, edited by Mar low Anderson, Victor Katz, and Robin Wilson Student Research Projects in Calculus, by Marcus Cohen, Arthur Knoebel, Edward D. Gaughan, Douglas S. Kurtz, and David Pengelley Symmetry, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. The Trisectors, by Underwood Dudley Twenty Years Before the Blackboard, by Michael Stueben with Diane Sandford The Words of Mathematics, by Steven Schwartzman

MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 800-331-1622 FAX 301-206-9789 Sliort Preface

A traditional function of a preface is to give excuses for the book that follows it, no matter how inexcusable it is, and, since mathematics is a subject with a long tradition, it would not do to depart from custom. I should first apologize for the disingenuous title. It was meant to catch your . eye and, if you are reading this, it has succeeded. It does not, however, describe the contents of the book, which is a collection of more or less unrelated pieces. One of them considers the question of the title and gives the answer, "yes." A more accurate title for the book, though less appealing, would be "Some Mathematical Stuff." I think that the stuff that follows has some interesting things in it that the reader would probably not encounter otherwise. The book is not an assemblage of Mathematics' Greatest Hits, nor of classics. It does not have a theme. Nevertheless, I think that it deserves to exist. Writing on mathematics has more of a claim to be preserved and reprinted than does most writing. What appears in newspapers is notoriously ephemeral, and magazines are much the same. Not many people would want to read issues of, say, Popular Science in the 1950s for their content. Even books can quickly become dated and irrelevant. Mathematics, however, is permanent. There are gems to be found in the literature of mathematics, periodical or otherwise, that shine as brightly today as when they first appeared in print, and they deserve to be seen and admired. This is not to say that everything that follows is a flawless jewel, nor that there are not other items that deserve equal or greater exposure. I have been buying mathematics books and subscribing to mathematics journals (and even reading some of them) for many decades and what is included here was, for the most part, taken from my shelves. I hope the reader will find some of it interesting, entertaining, enlightening, or all three at once. That's probably enough excuses. 'Underwood 'Dud(ey

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Contents

Sliort Preface ...... vii 1 'Dieudonne on 9'tfatliematics ...... 1 Mathematics and Mathematicians, Jean Dieudonne ...... 1 2 'Wliy 'ls 9'tfatliematics? ...... 15 Mathematics and the Physical World, Morris Kline ...... 16 3 'ls 9'tfatliematics 'lnevita6fe? ...... 31 Mathematics in Fun and in Earnest, Nathan Altshiller Court ...... 32 4 .:JI. 'Defense ef Quadratic 'Equations ...... 41 Quadratic Equations ...... 43 5 Obtuse Triang{es ...... 55 There Are Three Times as Many Obtuse-Angled Triangles as There Are Acute-Angled Ones, Richard K. Guy ...... 57 6 .:Jl.Sma{{Paradox ...... 63 Why Your Classes Are Larger than "Average", David Hemenway .... 63 Why is a Restaurant's Business Worse in the Owner's Eyes Than in the Customers'?, Wong Ngai Ying ...... 66 7 .:Jtyyfied 9'tfatliematics ...... 69 Assigning Driver's License Numbers, Joseph A. Gallian ...... 69 8 'The ,(aw ef Sma{{'J'.-0:,mbers ...... 83 The Strong Law of Small Numbers, Richard K. Guy ...... 83 9 'The Para{{e{ Postu{ate...... 103 The Problem With Postulate 5, Richard J Trudeau ...... 105 10 .:Jl.ritlimetic in tlie 'United' States ...... l19 A Calculating People, Patricia Cline Cohen ...... 120 l1 Tlie Moore Metliod ...... 141 I Want to be a Mathematician, Paul Ha/mos ...... 141

ix X 'Js 7vCatliematics 'lnevitaGre?

12 'Early [afcu{us ...... 157 13 Prob fems ...... 173 On the Origin of Certain Typical Problems, David Eugene Smith . . . 17 5 14 Jl 'Tan9{ed' 'Tafe ...... 181 A Tangled Tale, Lewis Carroll ...... 182 15 Jl 'Brief £Jfe ...... 189 Brief Lives, John Aubry ...... 190 16 [ardano...... 197 The Book of My Life, Jerome Cardan ...... 199 17 'Boo{e and'Finite

20 1:!9isfatin!J :,r ...... 261 Indiana's Squared Circle, Arthur E. Hallerberg ...... 261 21 7vCatliematics and .'.Music ...... 271 Emblems of Mind, Edward Rothstein ...... 272 22 .'.Matliematics 'Books ...... 285 On the Value of Mathematics (Books), G. L. Alexanderson and L. F. Klosinski ...... 285 23 'lrrationa{ Square 'R.pots ...... 293 Theodorus' Irrationality Proofs, Robert L. McCabe ...... 295 24 Tlie Eu{er-

1. Jean Dieudonne, Mathematics-The Music ofR eason, translated by H. G. and J. C. Dales, Springer-Verlag, 1992, pp. 7- 17. With kind permission of Springer Science and Business Media. 2. Morris Kline, Mathematics and the Physical World, Dover Publications, 1981 , pp vii- ix, 464- 475. With kind permission of Dover Publications. 3. Nathan Altshiller Court, Mathematics in Fun and in Earnest, Dial Press, 1958, pp. 96-107. 4 Hansard (House of Commons Daily Debates), 26 June 2003, vol. 407, part 117, columns 1260- 1268. 5. Richard K. Guy, There are three times as many obtuse-angled triangles as there are acute-angled ones, Mathematics Magazine 66 #3 (1993), pp. 175- 179. 6. David Hemenway, Why your classes are larger than "average", Mathemat ics Magazine, 55 #3 (1982), pp. 162- 164. Wong Ngai Ying, Why is a restaurant's business worse in the owner's eyes than in the customers'?, College Mathematics Journal, 18 #4 (1987), 315- 316. 7. Joseph Gallian, Assigning driver's license numbers, Mathematics Maga zine, 64 #1 (1991), 13- 22. 8. Richard K. Guy, The strong law of small numbers, American Mathemati cal Monthly, 95 #8 (1988), pp. 697- 712. 9. Richard J. Trudeau, The Non-Euclidean Revolution, Birkhauser, 1987, pp. 118- 131. With kind permission of Springer Science and Business Media. 10. Patricia Cline Cohen, A Calculating People: The Spread ofNum eracy in Early America, Routledge Publishing Inc., 1999, pp. 116- 138. Used with permission of the author and the Copyright Clearance Center. 11. Paul R. Halmos, / Want to be a Mathematician, Mathematical Association of America, 1988, pp. 255- 268.

323 324 'ls :Jll(arfiematics '1nevita6re?

12. Encyclopcedia Britannica, A. Bell and C. Macfarquhar, Edinburgh, 1771, vol. 2, pp. 607-611, plate LXXXI. 13. David Eugene Smith, On the origin of certain typical problems, American Mathematical Monthly, 24 #2 ( 1917), pp. 64-71 . 14. Lewis Carroll, A Tangled Tale, Dover, 1958, pp. 4- 12, 84-89. (First edi tion 1885.) 15. John Aubrey, Brief Lives, The University of Michigan Press, 1957, pp. 229-233. 16. Jerome Cardan, The Book of My Life, translated from the Latin De Vita Propria Liber by Jean Stoner, Dover, 1962, pp. 19- 25, 73-74, 97- 118. Original edition, Dutton, 1930. 17. J. L. Synge, George Boole and the calculus of finite differences in George Boole, A Miscellany, pp. 4- 18, Copyright 1969 J. L. Synge, by permis sion of Cork University Press, Youngline Industrial Estate, PouladuffRod, Togher, Cork Ireland. 18. Steven B. Smith, The Great Mental Calculators, Copyright 1983 by Co lumbia University Press, pp. 221-226, 289-298. Reprinted with permis sion of the publisher. 19. James Smith, The Quadrature of the Circle, Simpkin, Marshall, 1861 , pp. vii-xxv. 20. Arthur E. Hallerberg, Indiana's squared circle, Mathematics Magazine 50 #3 (1977), pp. 136-140. 21. Edward Rothstein, Emblems of Mind, University of Chicago Press, 2006, pp. 135- 140, 191- 201. 22. G. L. Alexanderson and L. F. Klosinski, On the value of mathematics (books), Mathematics Magazine 55 #2 (1982), pp. 98- 103. 23. Robert L. McCabe, Theodorus' irrationality proofs, Mathematics Maga zine, 49 #4 (1976), 201-203. 24. R. J. Gillings, The so-called Euler-Diderot incident, American Mathemati cal Monthly, 61 #2 (1954), pp. 77- 80. 25. Carl E. Linderholm, Mathematics Made Difficult, World Scientific, 1972, pp. 154- 155, 89-99. 26. Marlow Sholander, On the set oflegs of a horse, Pi Mu Epsilon Journal, l #3 (1950), pp. 103-106. Despite every effort to contact copyright holders and obtain permission prior to publication, in some cases this has not been possible. If notified, we will undertake to rectify any errors or omissions at the earliest opportunity. .Jlbout tlie 'Editor

Underwood Dudley earned his B.S. and M.S. degrees from the Carnegie Institute of Tech nology and his doctorate (in number theory) from the University of Michigan. He taught I briefly at the Ohio State University and then at DePauw University from 1967- 2004. Woody has written six books and many papers, re views, and commentaries. He has served in many editing positions, including editor of The Pi Mu Epsilon Journal, 1993- 96 and The College Mathematics Journal, 1999-2003. He is widely known and admired for his speaking ability-especially his ability to find humor in mathematics. He was the PME J. Sutherland Frame lecturer in 1992 and the MAA P6lya lecturer in 1995- 96. Woody's contributions to mathematics have earned him many awards, includ ing the Trevor Evans award, from the MAA in 1996, the Distinguished Service Award, from the Indiana Section of the MAA in 2000, and the Meritorious Service Award, from the MAA in 2004.

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AMS / MAA SPECTRUM

Is Mathematics Inevitable? A Miscellany

Underwood Dudley, Editor

This is a collection of gems from the literature of mathematics that shine as brightly today as when they first appeared in print. They deserve to be seen and admired. The selections include two opposing views on the purpose of mathematics, The Strong Law of Small Numbers, the treatment of calculus in the 1771 Encyclopaedia Britannica, several proofs that the number of legs on a horse is infinite, a deserved refutation of the ridiculous Euler-Diderot anecdote, the real story of pi and the Indiana Legislature, the reason why Theodorus stopped proving that square roots were irrational when he got to the square root of 17, an excerpt from Mathematics Made Difficult; a glimpse into the mind of a calculating prodigy. There will be something of interest here for almost anyone interested in mathematics. A more accurate question to appear as the title would be “Is All Mathematics Inevitable?” It is clear that given the development of human-level intelligence some mathematics is inevitable. For example, counting is clearly inevitable and there have been experiments that demon- strate that human babies can count at a very early age and that many animals can perform rudimentary counting. However, it is uncertain whether some of the more abstract areas of mathematics were inevitable, it is a very interesting point of philosophical debate, being rooted in the Greek mathematics of Plato. “Is Mathematics Inevitable?” by Nathan Altshiller Court is just one of the articles. The book is a collection of articles about mathematics, the people that pushed it forward and the context in which they lived their lives. All of the papers are ex- pository, while some of the topics are philosophically deep; the level of mathematics never gets to the point where it would overwhelm an intelligent undergraduate that is beyond calculus. ...Some mathematics books are fun to read, this one is fun, nearly always interesting and could be useful as a text in survey or capstone courses. —Charles Ashbacher, Journal of Recreational Mathematics