Math Makers The Lives and Works of 50 Famous Mathematicians Alfred S Posamentier and Christian Spreitzer Guilford, Connecticut An imprint of The Rowman & Littlefield Publishing Group, Inc 4501 Forbes Blvd., Ste 200 Lanham, MD 20706 www.rowman.com Distributed by NATIONAL BOOK NETWORK Copyright © 2020 by Alfred S Posamentier and Christian Spreitzer Cover image of math symbols © Marina Sun Cover design by Liz Mills Cover design © Prometheus Books All rights reserved No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without written permission from the publisher, except by a reviewer who may quote passages in a review British Library Cataloguing in Publication Information available Library of Congress Cataloging-in-Publication Data Names: Posamentier, Alfred S., author | Spreitzer, Christian, 1979– author Title: Math makers: the lives and works of 50 famous mathematicians / Alfred S Posamentier and Christian Spreitzer Description: Amherst, New York: Prometheus Books, 2019 | Includes index Identifiers: LCCN 2018052509 (print) | LCCN 2018059061 (ebook) | ISBN 9781633885219 (ebook) | ISBN 9781633885202 (hardcover) Subjects: LCSH: Mathematicians—Biography | Mathematics—History Classification: LCC QA28 (ebook) | LCC QA28 P67 2019 (print) | DDC 510.92/2—dc23 LC record available at https://lccn.loc.gov/2018052509 The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ANSI/NISO Z39.48-1992 ALSO BY ALFRED S POSAMENTIER AND CHRISTIAN SPREITZER The Mathematics of Everyday Life ALSO BY ALFRED S POSAMENTIER, ROBERT GERETSCHLÄGER, CHARLES LI, AND CHRISTIAN SPREITZER The Joy of Mathematics ALSO BY ALFRED S POSAMENTIER AND ROBERT GERETSCHLÄGER The Circle ALSO BY ALFRED S POSAMENTIER AND INGMAR LEHMANN The Fabulous Fibonacci Numbers Pi: A Biography of the World’s Most Mysterious Number Mathematical Curiosities Magnificent Mistakes in Mathematics The Secrets of Triangles Mathematical Amazements and Surprises The Glorious Golden Ratio ALSO BY ALFRED S POSAMENTIER AND BERND THALLER Numbers ALSO BY ALFRED S POSAMENTIER The Pythagorean Theorem Math Charmers To my children and grandchildren, whose future is unbounded, Lisa, Daniel, David, Lauren, Max, Samuel, Jack, and Charles —Alfred S Posamentier To my mathematics instructors and mentors for fostering my love for mathematics —Christian Spreitzer Contents Introduction xi Chapter Thales of Miletus: Greek (ca 624–546 BCE) Chapter Pythagoras: Greek (575–500 BCE) Chapter Eudoxus of Cnidus : Greek (390–337 BCE) 15 Chapter Euclid: Greek (ca 300 BCE) 19 Chapter Archimedes: Greek (ca 287–ca 212 BCE) 26 Chapter Eratosthenes: Greek (276–194 BCE) 39 Chapter Claudius Ptolemy: Greco-Roman (100–170) 44 Chapter Diophantus of Alexandria: Hellenistic Greek (ca 201–285) 50 Chapter Brahmagupta: Indian (598–668) 56 Chapter 10 Leonardo Pisano Bigollo, “Fibonacci”: Italian (1170–1250) 61 Chapter 11 Gerolamo Cardano: Italian (1501–1576) 75 Chapter 12 John Napier: Scottish (1550–1617) 85 Chapter 13 Johannes Kepler: German (1571–1630) 96 Chapter 14 René Descartes: French (1596–1650) 106 vii viii M AT H M A K E R S Chapter 15 Pierre de Fermat: French (1607–1665) 116 Chapter 16 Blaise Pascal: French (1623–1662) 124 Chapter 17 Isaac Newton: English (1642–1727) 134 Chapter 18 Gottfried Wilhelm (von) Leibniz: German (1646–1716) 143 Chapter 19 Giovanni Ceva: Italian (1647–1734) 157 Chapter 20 Robert Simson: Scottish (1687–1768) 165 Chapter 21 Christian Goldbach: German (1690–1764) 173 Chapter 22 The Bernoullis: Swiss (1700–1782) 177 Chapter 23 Leonhard Euler: Swiss (1707–1783) 189 Chapter 24 Maria Gaetana Agnesi: Italian (1718–1799) 197 Chapter 25 Pierre Simon Laplace: French (1749–1827) 201 Chapter 26 Lorenzo Mascheroni: Italian (1750–1800) 209 Chapter 27 Joseph-Louis Lagrange: French/Italian (1736–1813) 229 Chapter 28 Sophie Germain: French (1776–1831) 236 Chapter 29 Carl Friedrich Gauss: German (1777–1855) 242 Chapter 30 Charles Babbage: English (1791–1871) 250 Chapter 31 Niels Henrik Abel: Norwegian (1802–1829) 256 Chapter 32 Évariste Galois: French (1811–1832) 263 Chapter 33 James Joseph Sylvester: English (1814–1897) 268 Chapter 34 Ada Lovelace: English (1815–1852) 272 Chapter 35 George Boole: English (1815–1864) 279 Chapter 36 Bernhard Riemann: German (1826–1866) 284 Chapter 37 Georg Cantor: German (1845–1918) 293 Contents ix Chapter 38 Sofia Kovalevskaya: Russian (1850–1891) 302 Chapter 39 Giuseppe Peano: Italian (1858–1932) 309 Chapter 40 David Hilbert: German (1862–1943) 313 Chapter 41 G H Hardy: English (1877–1947) 322 Chapter 42 Emmy Noether: German (1882–1935) 329 Chapter 43 Srinivasa Ramanujan: Indian (1887–1920) 336 Chapter 44 John von Neumann: Hungarian-American (1903–1957) 343 Chapter 45 Kurt Gödel: Austrian-American (1906–1978) 351 Chapter 46 Alan Turing: English (1912–1954) 358 Chapter 47 Paul Erdős: Hungarian (1913–1996) 366 Chapter 48 Herbert A Hauptman: American (1917–2011) 372 Chapter 49 Benoit Mandelbrot: Polish-American (1924–2010) 377 Chapter 50 Maryam Mirzakhani: Iranian (1977–2017) 389 Epilogue 399 Appendix: Hilbert’s Axioms 401 Notes 405 References 417 406 Notes statement can be found on page 67: Preus, Anthony, ed Essays in Ancient Greek Philosophy VI Before Plato ISBN13: 978-0-7914-4955-4 Aristotle wrote a monograph titled “On the Pythagoreans,” see https://www jstor.org/stable/283647?seq=1#page_scan_tab_contents Elisha S Loomis, The Pythagorean Proposition, 2nd ed (Reston, VA: National Council of Teachers of Mathematics, 1968) James A Garfield, “Pons Asinorum,” New England Journal of Education (1876): 116 Chapter 1 Lives of Eminent Philosophers, edited by Tiziano Dorandi, Cambridge: Cambridge University Press, 2013 (Cambridge Classical Texts and Commentaries, vol 50, new radically improved critical edition) Translation by R D Hicks (Eudoxus is in Book 8.) Chapter 1 “Classics of Mathematics,” Ronald Calinger, ed Oak Park, IL: Moore Publishing, 1982; Euclid’s Elements, Dana Densmore, ed Santa Fe, NM: Green Lion Press, 2003; A History of Mathematics, V J Katz, 3rd ed New York: Addison-Wesley/ Pearson, 2009 2 http://www.abrahamlincolnonline.org/lincoln/speeches/autobiog.htm https://www.nps.gov/liho/learn/historyculture/debate4.htm Chapter No authentic portraits of Archimedes have survived, so the Canadian sculptor R Tait McKenzie, who designed the medal, had to imagine Archimedes’s appearance, inspired by earlier portrayals of Archimedes by Renaissance artists Chisholm, Hugh, ed (1911). “Vitruvius,” Encyclopædia Britannica (11th ed.) Cambridge University Press His daughter (Marie Louis Sirieix was a man) knew where it was and tried to sell it Cambridge University only had one single page of palimpsest (the one von Tischendorf had excised), but the whole “book” was in Sirieix’s cellar The Archimedes Codex: How a Medieval Prayer Book Is Revealing the True Genius of Antiquity’s Greatest Scientist, by Reviel Netz and William Noel, Da Capo Press, 2007 The new owner of the book According to Simon Finch, who represented the anonymous buyer, stated that the buyer was “a private American” who worked in “the high-tech industry.” See https://en.wikipedia.org/wiki/Archimedes_ Palimpsest Notes 407 This quote is from the chapter “The Life of Marcellus” in the book The Parallel Lives by Plutarch A reproduction of The Parallel Lives as published in Vol V of the Loeb Classical Library edition, 1917, can be found on this webpage: http:// penelope.uchicago.edu/Thayer/E/Roman/Texts/Plutarch/Lives/Marcellus*.html The quote is from page 481 Chapter Cyrene was an ancient Greek and later Roman city near present-day Shahhat, Libya. A relatively reliable online source for this is http://www-groups.dcs.st -and.ac.uk/history/Biographies/Eratosthenes.html Chapter The chord of 60 degrees is the length of a line segment whose endpoints are on the unit circle and are separated by 60 degrees Chapter See https://en.wikipedia.org/wiki/Diophantus Chapter 10 Maxey Brooke, “Fibonacci Numbers and Their History through 1900,” Fibonacci Quarterly (April 1964): 149 Chapter 11 Gerolamo Cardano, “A Point of View: Are Tyrants Good for Art?” BBC, August 10, 2012, http://www.bbc.com/news/magazine-19202527 Victor J Katz, and Karen Hunger Parshall, Taming the Unknown: A History of Algebra from Antiquity to the Early (Princeton, NJ: Princeton University Press, 2014); MacTutor History of Mathematics archive, School of Mathematics and Statistics, University of St Andrews, Scotland, link: http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Cardan.html Encyclopedia Britannica: https://www.britannica com/biography/Girolamo-Cardano 3 “The Story of Mathematics,” website by Luke Mastin, link: http://www storyofmathematics.com/16th_tartaglia.html; MacTutor History of Mathematics archive, School of Mathematics and Statistics, University of St Andrews, Scotland, link: http://www-history.mcs.st-andrews.ac.uk/Biographies/Tartaglia.html 408 Notes MacTutor History of Mathematics archive, School of Mathematics and Statistics University of St Andrews, Scotland, link: http://www-history.mcs.st-and ac.uk/HistTopics/Tartaglia_v_Cardan.html; Benjamin Wardhaugh, How to Read Historical Mathematics (Princeton, NJ: Princeton University Press, 2010) 5 John Stillwell, Mathematics and Its History (Science & Business Media, 2013), Section 5.5, p 54 Chapter 12 The following paragraphs are derived from Alfred S Posamentier and Bernd Thaller, Numbers: Their Tales, Types, and Treasures (Amherst, NY: Prometheus Books, 2015), pp 212–20 Chapter 13 William J Broad, “After 400 Years, A Challenge to Kepler: He Fabricated His Data, Scholar Says,” New York Times, Science Section, January 23, 1990, p The English mathematician Thomas Simpson re-discovered Kepler’s rule one hundred years after Kepler However, he also developed more elaborate approximation formulas, generalizing the formula Kepler had found A frustum of a cone is the remaining part of the right circular cone, when the vertex portion is cut off by a plane perpendicular to the altitude of the cone Chapter 14 This is from the official press release of the Nobel Assembly at Karolinska Institutet: https://www.nobelprize.org/prizes/medicine/2017/press-release/ R E Langer, “Rene Descartes,” The American Mathematical Monthly Vol 44, No (October, 1937): pp 495–512 The quote is from page 497 See also: https:// www.jstor.org/stable/2301226?seq=3#metadata_info_tab_contents R E Langer, “Rene Descartes,” The American Mathematical Monthly Vol 44, No (October, 1937), pp 495–512 The quote is from page 498 See also: https:// www.jstor.org/stable/2301226?seq=3#metadata_info_tab_contents Descartes, Discourse on the Method (Duke Classics, 2012), p 34 Valentine Rodger Miller, René Descartes: Principles of Philosophy, translated, with explanatory notes, Collection des Travaux de L’Académie Internationale D’Histoire des Sciences No 30 (Netherlands: Springer), p xvii Rene Descartes, Discourse on the Method, translated by John Veitch (Cosimo, Inc., 2008), p 15 Notes 409 Chapter 15 A parliament was a provincial appellate court in the Ancien Régime of France In 1789, France had thirteen parliaments, the most important of which was the Parliament of Paris While the English word “parliament” derives from this French term, parliaments in this sense were not legislative bodies They consisted of a dozen or more appellate judges, or about 1,100 judges nationwide (see Wikipedia, s.v “Parliament,” last edited February 2, 2019, https://en.wikipedia.org/wiki/Parliament) André Weil, Zahlentheorie: Ein Gang durch die Geschichte von Hammurapi bis Legendre (Basel, Switzerland: Birkhäuser, 1992), p 40 Michael Sean Mahoney, The Mathematical Career of Pierre de Fermat, 1601– 1665, Second Edition (Princeton, NJ: Princeton University Press, 2018), p 192 George F Simmons, Calculus Gems: Brief Lives and Memorable Mathematics (Washington, DC: Mathematical Association of America, 2007), p 98 Michael Sean Mahoney, The Mathematical Career of Pierre de Fermat, 1601– 1665, 2nd ed (Princeton, NJ: Princeton University Press, 2018), p 61 The method of infinite descent is a special variant of a proof by contradiction To prove that a problem has no solution, one may be able to show that if a solution— which was in some sense related to one or more natural numbers—would exist, this would necessarily imply that another solution related to smaller natural numbers existed The existence of this solution would then automatically imply the existence of another solution, related to even smaller natural numbers, and so forth Since there cannot be an infinite sequence of smaller and smaller natural numbers (sooner or later one would encounter the smallest natural number with the desired property), the premise that the problem has a solution must be wrong For example, to show that is not a rational number, we may start a proof by contradiction by first assuming that is rational Then we would be able to write p 2= q with p and q some natural numbers We would then have 2q2 = p2, implying that p2 is even and thus p must be even as well (if p were odd, than p2 cannot be even) Now, if p is even, we can write p = 2k for some natural number k, which upon inserting in the last equation yields 2q2 = 4k2, that is, q2 = 2k2 and therefore q must also be even Thus, both p and q must be divisible by This means that if had a representation p 2= this fraction could always be reduced by dividing p and q as a rational number q by But this is impossible since we cannot reduce a fraction further and further, without end This contradiction tells us that cannot be rational Reinhard Laubenbacher and David Pengelley, Mathematical Expeditions: Chronicles by the Explorers (Springer Science & Business Media, 2013), p 165 John Tabak, Probability and Statistics: The Science of Uncertainty, The History of Mathematics Series (Infobase Publishing, 2014), p 27 Simon Singh, Fermat’s Last Theorem (Fourth Estate, 1997) 410 Notes Chapter 18 A vacuum tube is a device that controls electric currents between electrodes in an evacuated container Invented in 1904, vacuum tubes were a basic component for electronics throughout the first half of the twentieth century, which saw the diffusion of radio, television, large telephone networks, as well as analog and digital computers 2 Wikipedia, s.v “ENIAC,” last edited February 14, 2019, https://en.wikipedia org/wiki/ENIAC Caren L Diefenderfer and Roger B Nelsen, The Calculus Collection: A Resource for AP and Beyond (MAA, 2019) Richard T W Arthur, “The Remarkable Fecundity of Leibniz’s Work on Infinite Series,” Annals of Science Vol 63, Issue (2006) G W Leibniz, Interrelations between Mathematics and Philosophy, edited by Norma B Goethe, Philip Beeley, and David Rabouin (Springer, 2015), p 146 The translation of Leibniz’s text can be found on the website “Leibniz Translations” by Lloyd Strickland; here is the link to article about binary arithmetic: http://www.leibniz-translations.com/binary.htm The translation of Leibniz’s text can be found on the website “Leibniz Translations” by Lloyd Strickland; here is the link to article about binary arithmetic: http://www.leibniz-translations.com/binary.htm Richard C Brown, The Tangled Origins of the Leibnizian Calculus: A Case Study of a Mathematical Revolution (World Scientific, 2012), p 229 Chapter 19 The text that follows is derived from the appendix of Alfred S Posamentier, Robert Geretschläger, Charles Li, and Christian Spreitzer, The Joy of Mathematics: Marvels, Novelties, and Neglected Gems That Are Rarely Taught in Math Class (Amherst, NY: Prometheus Books, 2017), pp 289–91 This section is derived from Alfred S Posamentier and Ingmar Lehmann, The Secrets of Triangles: A Mathematical Journey (Amherst, NY: Prometheus Books, 2012), p 45 3 This is a “biconditional” statement that indicates that if the lines are concurrent, then the equation is true; and if the equation is true, then the lines are concurrent The following section is derived from Posamentier and Lehmann, Secrets of Triangles, pp 135–36 and 342 Chapter 20 One such example is Alfred S Posamentier and Robert L Bannister, Geometry: Its Elements and Structure, 2nd ed (New York: Dover, 2014) Notes 411 R A Rankin, “Robert Simson,” School of Mathematics and Statistics, University of St Andrews, Scotland, http://www-history.mcs.st-andrews.ac.uk/Biographies/ Simson.html Chapter 21 The following biographical information is derived from J J O’Connor and E. F Robertson, “Christian Goldbach,” August 2006, http://www-history.mcs.st -andrews.ac.uk/Biographies/Goldbach.html Tomas Oliveira e Silva, Siegfried Herzog, and Silvio Pardi, “Emperical Verification of the Even Goldbach Conjecture and computation of Prime Gaps up to · 1018,” Mathematics of Computation, Vol 83, No 288 (July 2014): pp 2033–60, S 0025-5718(2013)02787-1, article electronically published on November 18, 2013 H A Helfgott, “Major Arcs for Goldbach’s Theorem,” French National Centre for Scientific Research (May 2013) Chapter 22 1 Biographical Dictionary of Mathematicians, Vol (New York: Charles Scribner’s), p 221 Dirk Jan Struik, A Source Book in Mathematics, 1200–1800 (in the series Princeton Legacy Library) (Princeton, NJ: Princeton University Press, 2014), p 320 3 Biographical Dictionary of Mathematicians, Vol (New York: Scribner’s), p 228 Chapter 23 The following paragraphs are derived from Alfred S Posamentier and Christian Spreitzer, The Mathematics of Everyday Life (Amherst, NY: Prometheus Books, 2018), pp 237–41 Chapter 24 Clifford A Pickover, The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Milestones Series (Sterling Publishing Company, Inc., 2009), p 180 Chapter 25 Benjamin Libet, Mind Time: The Temporal Factor in Consciousness (Cambridge, MA: Harvard University Press, 2009) 412 Notes 2 Lev Vaidman, “Quantum Theory and Determinism,” Quantum Studies: Mathematics and Foundations, Vol 1, Issue 1–2 (September 2014): pp 5–38 The following biographical information is derived from J J O’Connor and E F Robertson, “Pierre-Simon Laplace,” January 1999, http://www-history.mcs.st -and.ac.uk/Biographies/Laplace.html This and the following biographical information is derived from Wikipedia, s.v “Pierre-Simon Laplace,” last edited March 15, 2019, https://en.wikipedia.org/ wiki/Pierre-Simon_Laplace Napier Shaw, Manual of Meteorology, Vol (Cambridge: Cambridge University Press, 2015), p 130 6 In fact, it is now understood that the solar system is not stable over very long periods of time Laplace’s methods were not sufficiently precise to prove stability, but they were essential steps in the development of a precise mathematical theory of celestial motion This famous quote has often been interpreted as evidence for Laplace’s atheism, but this conclusion might be wrong Physicist Stephen Hawking shared this view of Laplace, as evidenced by what he said in a 1999 public lecture: “I don’t think that Laplace was claiming that God does not exist It’s just that he doesn’t intervene, to break the laws of Science.” (Stephen Hawking, “Does God Play Dice?” lecture, 1999, https://web.archive.org/web/20000902184353/http://www.hawking org.uk:80/lectures/dice.html) 8 Napoleon”s Memoirs: Napoléon I, Emperor of the French Memoirs of the history of France during the reign of Napoleon 1823–1826, Vol I (H Colburn and Company, 1823), p 116 Laplace wrote this in the introduction to his work Théorie Analytique des Probabilitiés, the quote can be found in the MacTutor History of Mathematics archive, School of Mathematics and Statistics, University of St Andrews, Scotland Chapter 26 The following discussion of Mascheroni constructions is derived from Alfred S Posamentier and Robert Geretschläger, The Circle: A Mathematical Exploration beyond the Line (Amherst, NY: Prometheus Books, 2016), pp 199–215 2 For a proof of this theorem, see Alfred S Posamentier and Charles T Salkind, Challenging Problems in Geometry (New York: Dover, 1996), p 217 Chapter 27 1 The following biographical information is derived from Wikipedia, s.v “Joseph-Louis Lagrange: Biography,” last modified April 12, 2019, https://en.wikipedia org/wiki/Joseph-Louis_Lagrange Notes 413 2 Wikipedia, s.v “Tautochrone Curve,” last edited March 15, 2019, https:// en.wikipedia.org/wiki/Tautochrone_curve The three-body problem is the problem in physics of computing the trajectory of three bodies interacting with one another T S Blyth and E F Robertson, Further Linear Algebra (London: Springer, 2002), p 187 J J O’Connor and E F Robertson, “Joseph-Louis Lagrange,” January 1999, http://www.history.mcs.st-andrews.ac.uk/Biographies/Lagrange.html 6 Wikipedia, s.v “Joseph-Louis Lagrange: Prizes and Distinctions,” last modified April 12, 2019, https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange Chapter 28 Gina Kolata, “At Last, Shout of ‘Eureka!’ in Age-Old Math Mystery,” New York Times, June 24, 1993 2 The following biographical information is derived from Wikipedia, s.v “Sophie Germain,” last updated April 11, 2019, https://en.wikipedia.org/wiki/ Sophie_Germain J J O’Connor and E F Robertson, “Sophie Germain,” December 1996, http://www-history.mcs.st-andrews.ac.uk/Biographies/Germain.html The content of this paragraph is derived from Wikipedia, s.v., “Sophie Germain: Later Work in Elasticity,” last updated April 11, 2019, https://en.wikipedia org/wiki/Sophie_Germain The content of this paragraph is derived from Wikipedia, s.v “Sophie Germain: Final Years,” in ibid Chapter 29 This and the following biographical information is derived from J J O’Connor and E F Robertson, “Johann Carl Friedrich Gauss,” December 1996, http:// www-history.mcs.st-and.ac.uk/Biographies/Gauss.html https://thatsmaths.com/2014/10/09/triangular-numbers-eyphka/ https://www.scientificamerican.com/article/are-mathematicians-finall/ This and the following biographical information is derived from ibid Chapter 31 NBIM, Norges Bank, Statistics Norway; see, for example, “Factbox: Norway’s $960 Billion Sovereign Wealth Fund,” Reuters, June 2, 2017, https://www.reuters com/article/us-norway-swf-ceo-factbox/factbox-norways-960-billion-sovereign -wealth-fund-idUSKBN18T283 https://worldhappiness.report/ed/2017/ 414 Notes “Niels Henrik Abel,” Norsk Biografisk Leksikon, last updated February 13, 2009, https://nbl.snl.no/Niels_Henrik_Abel 4 Olav Arnfinn Laudal and Ragni Piene, The Legacy of Niels Henrik Abel: The Abel Bicentennial, Oslo, 2002 (Berlin: Springer, 2013) Arild Stubhaug, Called Too Soon by Flames: Niels Henrik Abel and His Times (Heidelberg: Springer, 2000) Arild Stubhaug, Niels Henrik Abel and his Times: Called Too Soon by Flames Afar, translated by R H Daly (Springer Science & Business Media, 2013), p 231 Krishnaswami Alladi, Ramanujan’s Place in the World of Mathematics (New Delhi: Springer, 2013), p 83 http://www.abelprize.no/c53680/artikkel/vis.html?tid=53897 Chapter 32 The content of this paragraph is derived from Wikipedia, s.v., “Évariste Galois: Final Days,” last updated April 25, 2019, https://en.wikipedia.org/wiki/%C3 %89variste_Galois Ibid Chapter 33 The content of this and the following paragraphs is derived from Wikipedia, s.v “James Joseph Sylvester: Biography,” last updated April 11, 2019, https:// en.wikipedia.org/wiki/James_Joseph_Sylvester 2 J D North, “James Joseph Sylvester,” Complete Dictionary of Scientific Biography (Charles Scribner’s Sons, 2008), available through MacTutor History of Mathematics at http://www-history.mcs.st-and.ac.uk/DSB/Sylvester.pdf, citing James Joseph Sylvester, Collected Mathematical Papers 4, no 53 (1888): 588 http://www-history.mcs.st-andrews.ac.uk/Quotations/Sylvester.html Chapter 34 Lord Byron, Childe Harold’s Pilgrimage (1812–1818), canto 3, ll 1–2 Betty Alexandra Toole, Ada, The Enchantress of Numbers (Mill Valley, CA: Strawberry), pp 240–61 Chapter 40 At these conferences the famous Fields Medals (i.e., equivalent in mathematics to the Nobel Prize) are awarded every four years to outstanding mathematicians not above age 40 2 The curious reader is referred to https://en.wikipedia.org/wiki/Hilbert %27s_problems Notes 415 Hajo G Meyer, Tragisches Schicksal Das deutsche Judentum und die Wirkung historischer Kräfte: Eine Übung in angewandter Geschichtsphilosophie (Berlin: Frank & Timme, 2008), 202 See http://www.storyofmathematics.com/20th_hilbert.html Chapter 41 Godfrey Harold Hardy, Collected Papers of G H Hardy (Oxford: Oxford University Press, 1979) Chapter 43 Warner Bros., 2016 Robert Kanigel, The Man Who New Infinity: A Life of the Genius Ramanujan (New York: Macmillan, 1991) “Quotations by Hardy,” archived from the original on July 16, 2012, accessed November 20, 2012, https://www-history.mcs.st-andrews.ac.uk/Quotations/Hardy html G S Carr, A Synopsis of Elementary Results in Pure and Applied Mathematics (Cambridge: Cambridge University Press, 2013) Chapter 44 William Poundstone, Prisoner’s Dilemma: John von Neumann, Game Theory, and the Puzzle of the Bomb (New York: Anchor, 1993) Claudia Dreifus, “Maria Konnikova Shows Her Cards,” New York Times, August 10, 2018 Chapter 45 See https://en.wikipedia.org/wiki/Hilbert%27s_program Chapter 46 A M Turing, “Intelligent Machinery” (manuscript) (Turing Archive, 1948), “von Neumann firmly emphasized to me, and to others I am sure, that the fundamental conception is owing to Turing—insofar as not anticipated by Babbage, Lovelace and others,” letter by Stanley Frankel to Brian Randell, 1972, quoted in Jack Copeland, The Essential Turing (New York: Oxford University Press, 2004), 22 See A S Posamentier and I Lehmann, The Fabulous Fibonacci Numbers (Amherst, NY: Prometheus Books, 2007) 416 Notes Chapter 47 To compute the Erdős number of an author, visit https://mathscinet.ams org/mathscinet/freeTools.html and go to the collaboration distance calculator The tool will automatically find a path in the MathSciNet database between any two people you wish (there is a special button for selecting Paul Erdős as one end of the path) According to “Facts about Erdős Numbers and the Collaboration Graph,” using the Mathematical Reviews database, the next highest article count is roughly 823, see http://oakland.edu/enp/trivia/ Paul Hoffman, The Man Who Loved Only Numbers (New York: Hyperion, 1998) ln(x) is the natural logarithm of the number, x, and is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459 Chapter 48 A S Posamentier and H A Hauptman, 101+ Great Ideas for Introducing Key Concepts in Mathematics, 2nd ed (Thousand Oaks, CA: Corwin Press, 2006) Chapter 49 Benoit B Mandelbrot, The Fractal Geometry of Nature (New York: W H Freeman, 1983), The Koch Snowflake was named in 1904 after the Swedish mathematician Helge von Koch (1870–1924) Named in 1915 after the Polish mathematician Waclaw Sierpiński (1882– 1969) The complex plane is the two-dimensional representation of complex numbers with a real axis and an imaginary axis In fact, figure 49.7 is only an approximation of the Mandelbrot set In actuality, we cannot know for sure whether a number c lies in the Mandelbrot set, because to determine that with absolute certainty we would need to iterate the “test” an infinite number of times But even with computers, we can obviously iterate anything only a finite number of times But it so happens that the sequence formed by iterating the rule to a certain value of c may behave differently only after a very large number of iterations So we can make our approximation better by iterating a great number of times Still, this will not lead to absolute accuracy A cardioid is a heart-shaped curve generated by a fixed point on a circle as it rolls around another circle of equal radius 7 The Economist, October 21, 2010 References Aaboe, Asger Episodes from the Early History of Mathematics New York: Random House, 1964 Anglin, W S., and J Lambek The Heritage of Thales New York: Springer, 1995 Artmann, Benno Euclid—The Creation of Mathematics New York: Springer, 1999 Ball, W W Rouse A Short Account of the History of Mathematics New York: Dover, 1960 Bell, E T Men of Mathematics New York: Simon & Schuster, 1937 Berlinghoff, William P., and Fernando Q Gouvea Math Through the Ages: A Gentle History for Teachers and Others Washington, DC: Mathematical Association of America, 2004 Boyer, Carl B The History of the Calculus and Its Conceptual Development New York: 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The First 50,000 Years Amherst, NY: Prometheus, 2007 Sanford, Vera A Short History of Mathematics Boston: Houghton Mifflin, 1958 Smith, David Eugene History of Mathematics Vol I and II New York: Dover, 1951 ——— A Source Book in Mathematics New York: McGraw-Hill, 1929 Smith, David Eugene, and Marcia L Latham The Geometry of René Descartes New York: Dover, 1925 Stein, Sherman Archimedes: What Did He Do Besides Cry Eureka? Washington, DC: Mathematical Association of America, 1999 Stillwell, John Mathematics and Its History New York: Springer, 1989 Struik, Dirk J A Concise History of Mathematics New York: Dover, 1948 420 References ——— A Source Book in Mathematics, 1200–1800 Princeton, NJ: Princeton University, 1986 Turnbull, Herbert Westren The Great Mathematicians New York: New York University, 1961 Van der Waerden, B L Geometry and Algebra in Ancient Civilizations New York: Springer, 1983 ——— A History of Algebra: From al-Khwarizmi to Emmy Noether New York: Springer, 1980 ——— Science Awakening: Egyptian, Babylonian and Greek Mathematics New York: Wiley, 1963 .. .Math Makers The Lives and Works of 50 Famous Mathematicians Alfred S Posamentier and Christian Spreitzer Guilford, Connecticut An imprint of The Rowman & Littlefield Publishing Group, Inc 4501 ... among them: they were often considered unable to blend into the social fabric of the culture of their times The brilliance and unusualness of these fifty mathematicians are revealed not only by the. .. fruits of their mathematical wonder and labor but also by the very lifestyles they led Some of their lives were rather sad, such as that of French mathematician Évariste Galois, the developer of