THE WORLD OF MATHEMATICS VOLUME IV Volume Four of THE WORLD OF MATHEMATICS A small library of the literature of mathematics from. A'h-mose the Scribe to Albert Einstein, presented with commentaries and notes by JAMES R. NEWMAN LONDON GEORGE ALLEN AND UNWIN LTD First Published in Great Britain in I960 This book is copyright under the Berne Convention, Apart from any fair dealing for the purpose of private study, research, atticism or review, as permitted under the Copyright Act, 1956, no portion maybe reproduced by any process without writ fen permission. Enquiry should be made to the publisher. © James R. Newman 1956 ACKNOWLEDGEMENTS The Editor wishes to express his gratitude for Mahomet", from What h Mathematics? by Richard permission to reprint ma le rial from the following Courant and Herbert Rabbins. sources: Philosophy of Sciertce, for "The Locus of Mathe- Messrs. John Lane. Ths Bodiey Head Ltd,, Tor matical Reality: An Anthropologics! Footnote", by "Common Sense and the Universe", by Stephen Leslie A. White, issue of October, 1947. Le acock. The Science Press for "Mathematical Creation", Professor R. L. Goodslein, and The Mathematics! from Foundations of Science, hy Henri Poincarc, Association for "Easy Mathematics and Lawn translated by George Bruce Haisted. Tennis" by T, J. LA. Bromwich, from Mathematical Scientific American, for "A Chess- Playing Gateite XI If, October, I92S, Machine", by Claude Shannon. Cambridge University Press for "Mathematics of Messrs. G. Bell & Sons Ltd., for "Pastimes of Music", from Science and Music, by Sir James Past and Present Times", from Mathematics and the Jeans; and for excerpt from A Mathematician's Imagination, by Edward Kasner and James R, Apology, by G. H. Hardy. Newman. Messrs- Constable & Co. Ltd., for "Mathematics Estate of A. M, Turing for "Can a Machine in Warfare", from Aircraft in Warfare, by Frederick Think?", from Mind, 1950. William Lan Chester. University of Chicago Press, for "The Mathe- Messrs. Chatfo & Windns Ltd., for Young Archi- matician", by John von Neumann, from TTie Works medes by Aldous Huxley, of the Mind, edited by Heywpod and Nef. Lane, The Bod ley Head Ltd., for Messrs. John Messrs, Chatto & Windus Ltd., for "Geometry "Mathematics for Golfers", by Stephen Lcacock. in the South Pacific", from Mr, Fortune's Maggot, Messrs, Alien &. XJnwin Ltd., for "Meaning of by Sylvia Townsertd Warner (with the assistance of Numbers", Tram The Decline vf the West, by Oswald Victor Butler). Spengler. John Wiley &. Sons Inc., for "The General and Macmillan & Company Ltd,, for "Arithmetical Logical Theory of Automata", by John von Res [orations", from Mathematical Recreations and Neumann, from Cerebral Mechanisms in Behavior, Essays, by W. W. Rouse Ball. reprinted with the permission of the Hi.soti louriiia- The New Yorker for "Inflexible Logic" by Russell tion and Dr. Lloyd A, JetTrcss. Maloney, © 1940 by The New Yorker Magazine Inc.; John Wiley & Sons Inc., and the Technology Press 4 and for "The Law", by Robert M, Coates. © 1 J47 of M.I.T. Tor "How to Hunt a Submarine" from The New Yorker Magazine Inc. Methods of Operations Research, by Phillip M. Morse Oxford University Press, for "The Lever of and George E. Kimball, Printed in Great Britain by Novella & Co. Ltd., Soho* London Table of Contents VOLUME ^2 FOUR part xviii : The Mathematician G, H. Hardy: Commentary 2024 1. A Mathematician's Apology by c. H. hardy 2027 The Etysivertess of hi vent ion: Commentary 2039 2. Mathematical Creation by kenri poincare 2041 The Use of a Top Hat as a Water Bucket: Commentary 2051 3. The Mathematician by john von neumann 2053 part xix : Mathematical Machines: Can a Machine Think? Automatic Computer.):: Commentary 2066 1. The General and Logical Theory of Automata 2070 by JOHN VON NEUMANN 2. Can a Machine Think? by a. m. Turing 2099 3. A Chess-Playing Machine by CLAUDE shannon 2124 part xx : Mathematics in Warfare Frederick William Lanchester: Commentary 2136 1. Mathematics in Warfare 2I3S by FREDERICK WILLIAM LA"NCHESTER Operations Research: Commentary 2158 2. How to Hunt a Submarine 2160 by phillip m. morse and GEORGE B, KIMBALL y Confetti:, part xxi : A Mathematical Theory of Art George Davit/ Birkhoff: Commentary 2182 1. Mathematics of Aesthetics by george david birkhoff 2185 part xxii : Mathematics of the Good 1. A Mathematical Approach to Ethics 2198 by GEORGE DAVID BIRKHOFF part xxin : Mathematics in Literature The Island of l.apula: Commentary 2210 1. Cycloid Pudding by Jonathan swift 2214 Aidoits Huxley: Commentary 2221 2. Young Archimedes by aldous huxley 2223 Mr, Famine; Commentary 2250 3. Geometry in the South Pacific 2252 by SYLVIA TOWNSEKD WARNER Statistics as a Literary Stiinirlns: Commentary 2261 4. Inflexible Logic by russell malone 2262 5. The Law by Robert m. coates 2268 part xxiv : Mathematics and Music Sir James Jeans: Commentary 2274 I. Mathematics of Music by sir JAMBS jeans 2278 part xxv: Mathematics as a Culture Clue Oswald Speng/er: Commentary 2312 1. Meaning of Numbers by Oswald spengler 2315 2. The Locus of Mathematical Reality: An Anthropological Footnote by Leslie a. white 2348 . t' anient s. part xxvi : Amusements, Puzzles, Fancies Augustus De Morgan, tin Estimable Man: Commentary 2366 1 Assorted Paradoxes by Augustus de morgan 2369 A Romance of Many Dimensions: Commentary 2383 2. Flatland by edwjn a. abbott 2385 Lewis Carroll: Commentary 2397 3. What the Tortoise Said to Achilles and Other Riddles 2402 by LEWIS CARROLL Continuity: Commentary 2410 4. The Lever of Mahomet 2412 by RICHARD COURANT and HERBERT ROBBINS Games and Puzzles: Commentary 2414 5. Pastimes of Past and Present Times 2416 by EDWARD KASNER and JAMES R. NEWMAN 6. Arithmetical Restorations by w. w, rouse ball 2439 7. The Seven Seven's by w. e. h. Berwick 2444 Thomas John I' Anson Bromu ich: Commentary 2449 8. Easy Mathematics and Lawn Tennis 2450 by t. j. i'a. bromwich Stephen Butler Lcacock: Commentary 2455 9. Mathematics for Golfers by Stephen leacock 2456 10. Common Sense and the Universe by Stephen leacock 2460 index 2471 . PART XVIII The Mathematician 1 A Mathematician's Apology by g. h. hardy 2. Mathematical Creation by henri poincare 3. The Mathematician by john von Neumann COMMENTARY ON G. H. HARDY GH. HARDY was a pure mathematician. The boundaries of this • subject cannot be precisely defined but for Hardy the word "pure" as applied to mathematics had a clear, though negative, meaning. To qualify as pure, Hardy said, a mathematical topic had to be useless; if useless, it was not only pure, but beautiful. If useful—which is to say opinions impure— it was ugly, and the more useful, the more ugly. These were not always well received. The noted chemist Frederick Soddy, re- viewing the book from which the following excerpts are taken, pro- nounced as scandalous Hardy's expressed contempt for useful mathe- matics or indeed for any applied science. "From such cloistral clowning," wrote Soddy, "the world sickens." n Hardy was a strange, original and enigmatic man. He was also a fine mathematician and a charming writer. Godfrey Harold Hardy was born in Surrey in February 1877. His parents were teachers and "mathematically minded." He was educated first at Winchester—which he hated—and then at Cambridge, where he taught the greater part of his life. From 1919 to 1931 he held the Savilian chair of geometry at Oxford; in 1931 he was elected to the Sadlerian chair of pure mathematics at Cambridge and resumed the Fellowship at Trinity College which he had held from 1898 to 1919. Hardy's main work was in analysis and arithmetic. He is known to students for his classic text, A Course of Pure Mathematics, which set a new standard for English mathematical education. But his reputation as the leader of pure mathematicians in Great Britain rests on his original and advanced researches. He wrote profound and masterly papers on such topics as the convergence and summability of series, inequalities and the analytic theory of numbers. The problems of number theory are often very easily stated (e.g., to prove that every even number is the sum of two prime numbers) "but all the resources of analysis are required to make any impression on them." - The problem quoted, and others of equally innocent appearance, are still unsolved "but they are not now—as 3 they were in 1910—unapproachable." This advance is due mainly to the joint work of Hardy and the British mathematician J. E. Littlewood. Their collaboration was exceptionally long and immensely fruitful; it is consid- ered the most remarkable of all mathematical partnerships. An equally brilliant but unhappily brief partnership existed between Hardy and the 1 Nature, Vol. 147, January 4. 1941. -Obituary of G. H. Hardy, Nature, Vol. 161, May 22, 1948, pp. 797-98. 'Ibid. 2024 G. H. Hardy 2025 self-taught Indian genius Ramanujan (see p. 368). It is hard to imagine two men further apart in training and background, yet Hardy was one of the first to discern what he termed Ramanujan's "profound and invincible originality." Ramanujan "called out Hardy's equal but quite different powers." "I owe more to him," Hardy said, "than to anyone else in the world with one exception, and my association with him is the one ro- mantic incident of my life." 4 I once encountered Hardy in the early 1930s at the subway entrance near Columbia University in New York City. It was a raw, wet winter day, but he was bareheaded, had no overcoat and wore a white cable- stitched turtle-necked sweater and a baggy pair of tennis slacks. I recall his delicately cut but strong features, his high coloring and the hair that fell in irregular bangs over his forehead.