The Universe in Zero Words

by Mackenzie, Dana

  On the other hand, not all of the signs are positive. Indeed, there are some reasons to think that the twentieth century may have been a high-water mark for interesting, consequential, and beautiful equations.

  First, although we have added 173 equations, formulas, and identities in the last three years, there is no guarantee that they meet the criteria I listed in the Introduction. Are they surprising, concise, consequential, and universal? Does the twenty-first century really have such monumental surprises in store as quantum physics, chaos, or Gödel’s Incompleteness Theorem? It’s impossible to predict, but to me those look like oceans that can only be crossed once. Geographers eventually ran out of new continents on Earth to discover, and it seems possible that mathematicians will face the same problem.

  Also, changes are afoot in the way mathematics is conducted. The computer has brought humans a new way of knowing. Forecasting the climate or mapping the human genome involve the collection of staggering amounts of data—amounts too great for the human mind to comprehend. Scientists have to devise new ways to sift through the data and identify what is important. The most important patterns may not be expressible any more in the form of an equation. Perhaps they will be encoded into an artificial intelligence and not even understandable by human brains at all.

  Let me give a specific example where this change has already happened. Over the last 25 years, chess players’ ways of knowing has been dramatically changed by the computer. For example, there are positions that a computer can solve but a human cannot. Perhaps the most technical endgame that humans can master is checkmate with a king, bishop, and knight against a lone king. The procedure is tricky, but it can be broken down into stage, and it takes at most 33 moves (according to the computer) if both sides play their best. However, computers have discovered other endgames, such as king, rook, and bishop against king and two knights, where it takes as long as 223 moves for the stronger side to win, assuming both players play perfectly. And the moves are completely incomprehensible to humans. When you compare the position after move 80 to the position after move 50, it is impossible to explain why the stronger side is 30 moves closer to victory.

  The point of this example is that some kinds of knowledge are too complicated for the human mind to grasp. They are not necessarily deep, just intricate. Equations have evolved as a powerful tool that enables us to grasp some ideas that cannot (or can only with great difficulty) be put into words. But the truths lurking in the databases of the twenty-first century may not be understandable even with equations. They may be the scientific equivalent of the 223-move checkmate.

  Finally, I expect some changes in the ways mathematics is used. Historically, mathematics has been tied closely to physics, but in the next century we are likely to see more applications to other disciplines, such as biology or social sciences. The idea of using math to cure cancer is tremendously exciting.

  But there is a catch. In order to say anything about the universe with mathematics, we have to construct a mathematical model. And models are always imperfect. First, they always oversimplify reality in some way; and second, every mathematical model begins with assumptions. Sometimes they seem so obvious or so well established by experiment that we forget they are assumptions. We fall in love with our models, and then a major trauma ensues when we have to modify or discard them. Think again of non-Euclidean geometry or the discovery of chaos in deterministic dynamical systems.

  For reasons that are not entirely understood, mathematical models historically have worked remarkably well in physics. However, in biology that is not likely to be the case. As a rule, any mathematical model that describes biological processes with any degree of fidelity will tend to fail the conciseness test. It will have many equations and it will be difficult to grasp the reasons for even the most fundamental behaviors. We start entering the domain of the 223-move checkmate. For example, mathematical biologists have developed computer simulations of the heart, which can reproduce such conditions as ventricular tachycardia and fibrillation. Nevertheless, there is not yet any agreement on why the most basic treatment, a defibrillator, actually works.

  In sum, I have no doubt that a sequel to this book written a hundred years from now will include six pretty wonderful equations from the twenty-first century. Whether they will be quite as wonderful as Einstein’s equations, or Dirac’s equation, or chaos, I’m not so sure. We will probably have major breakthroughs in mathematical biology that completely fail the conciseness test. There will be many discoveries analogous to the 223-move checkmate, which cannot be expressed either in words or equations but have to be encoded into an artificial intelligence. The whole idea of an equation might begin to look a little bit quaint.

  However, let’s not forget that mathematics has an extraordinarily long tradition. Certain things do not change rapidly. One hundred years from now, I predict that there will still be few things quite as satisfying as filling in both sides of an equals sign.

  acknowledgments

  I will always think of this book in the way that one thinks of a beloved pet that shows up on the doorstep one day, bedraggled and wagging its tail, not certain what it wants but absolutely certain that you are the person that can provide it. Elwin Street Productions had conceived the idea for a book about the history of mathematical equations and they started looking around for a writer. That’s when the synopsis landed on my doorstep.

  The proposed contents actually made me mad; I had a completely different view of what should be in the book. It also took a while to get used to writing a book that began as somebody else’s idea. But in truth, I had been waiting a long time to write a math book, and I knew that I could fix this one up.

  So I would first like to thank the people at Elwin Street who tolerated my wholesale changes to their concept, never lost faith in it, and eventually placed the book with the best co-publishers I could ever have asked for, Princeton University Press and NewSouth Publishing.

  I would also like to thank, in no particular order:

  John Wilkes, the founder of the Science Communication Program at the University of California at Santa Cruz, who has been a guiding light to so many scientists who wanted to become writers.

  Peter Radetsky, one of my teachers at UCSC, who told me, “I think you’re going to have a great adventure.”

  Peter Steinhart, who taught essay writing at UCSC, and Rosalind Reid, my first editor at American Scientist, for convincing me that the first person singular pronoun has a place (and a very important one) in science writing.

  Martin Gardner, the first popular math writer I ever read, who made it look so effortless.

  George Pólya, whose explanation of Euler’s Basel Formula (which I read in college) was like Lake Tahoe: so transparent and yet so deep.

  Nisaba, the Sumerian goddess of writing (and indirectly, of mathematics), who got the whole thing started.

  And Kay, my wife, who encouraged me to follow my dream of writing and then followed the same dream herself. I have learned from her that writing is much more than putting words on paper.

  bibliography

  Introduction: The Abacist Versus the Algorist

  Feynman, R. P. and R. Leighton. Surely You’re Joking, Mr. Feynman!, New York: W. W. Norton, 1997.

  Part One: Equations of Antiquity

  Boyer, C. A History of Mathematics, 2nd ed. New York: Wiley, 1991.

  Burnyeat, M. F. “Other lives,” London Review of Books, 22 February 2007.

  Burton, D. The History of Mathematics: An Introduction, Boston: Allyn and Bacon, 1985.

  Cipra, B. “Digits of Pi,” in What’s Happening in the Mathematical Sciences, Vol. 6. Providence, RI: American Mathematical Society, 2006.

  Dauben, J. “Ancient Chinese mathematics: The Jiu Zhang Suan Shu vs. Euclid’s Elements,” International Journal of Engineering Science 36 (1998), Nos. 12–14, pp. 1339–1359.

  Heath. T. L., ed. The Works of Archimedes, Cambridge, UK: Cambridge University Press, 1897.

  Huffman, C.
“Pythagoreanism,” in Stanford Encyclopedia of Philosophy, accessed online at http://plato.stanford.edu/entries/pythagoreanism.

  Iamblichus, “Life of Pythagoras,” in K. S. Guthrie, ed. The Complete Pythagoras (1921), edited for the Internet by P. Rousell and accessed online at http://www.completepythagoras.net.

  Kaplan, R. and E. Kaplan. Hidden Harmonies: The Lives and Times of the Pythagorean Theorem, New York: Bloomsbury Press, 2011.

  Katz, V. J., ed. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princteon, NJ: Princeton University Press, 2007.

  Lee, H. D. P., ed. Zeno of Elea: A Text, with Translation and Notes, Cambridge, UK: Cambridge University Press, 1936.

  Stillwell, J. Mathematics and its History, New York: Springer-Verlag, 1989.

  Part Two: Equations in an Age of Exploration

  Beer, A. and P. Beer, eds. “Kepler: Four Hundred Years,” Vistas in Astronomy, Vol. 18. Oxford: Pergamon, 1975.

  Bellhouse, D. “Decoding Cardano’s Liber de Ludo Aleae,” Historia Mathematica 32 (2005), pp. 180–202.

  Bradley, R. E., L. A. D’Antonio, and C. E. Sandifer, eds. Euler at 300: An Appreciation, Washington, DC: Mathematical Association of America, 2007.

  Chambers, R. “Sir Isaac Newton and the Apple,” in The Book of Days: A Miscellany of Popular Antiquities, pp. 757–8. London: W. & R. Chambers, 1832.

  Cipra, B. “Fermat’s Theorem—at Last!” in What’s Happening in the Mathematical Sciences, Vol. 3. Providence, RI: American Mathematical Society, 1996.

  Dunham, William. Journey Through Genius. New York: Penguin Books, 1991.

  Ekert, A. “Complex and unpredictable Cardano,” International Journal of Theoretical Physics 47 (2008): pp. 2101–2119.

  Grattan-Guinness, I., ed. Landmark Writings in Western Mathematics, Amsterdam: Elsevier, 2005.

  Gray, J. “A short life of Euler,” BSHM Bulletin 23 (2008), pp. 1–12.

  Montagu, A. “Newton’s Principia: Heroes, Legends, and Myths,” The New York Times, 18 April 1987.

  Ouellette, J. The Calculus Diaries, London: Penguin Books, 2010.

  Rickey, V. F. “Isaac Newton: Man, Myth, and Mathematics,” The College Mathematics Journal 18 (1987), pp. 362–389.

  Stedall, J. Mathematics Emerging: A Sourcebook 1540–1900, Oxford, UK: Oxford University Press, 2008.

  Weil, A. Number Theory: An Approach through History from Hammurapi to Legendre, Boston: Birkhauser, 1984.

  Wells, D. Are These the Most Beautiful? Mathematical Intelligencer 12 (1990), No. 3, 37-41.

  Westfall, R. Never at Rest: A Biography of Isaac Newton, Cambridge, UK: Cambridge University Press, 1983.

  Wikipedia. “Principia Mathematica,” article accessed online at http://en.wikipedia.org/wiki/Principia_Mathematica.

  Part Three: Equations in a Promethean Age

  Crease, R. P. “The Greatest Equations Ever,” Physics World, October 6, 2004. Accessed online at http://physicsworld.com/cws/article/print/20407.

  Greenberg, M. J. Euclidean and Non-Euclidean Geometries: Development and History, 4th Ed. New York: W. H. Freeman, 2007.

  Hankins, T. L. Sir William Rowan Hamilton, Baltimore: Johns Hopkins University Press, 1980.

  Harman, P. M. The Natural Philosophy of James Clerk Maxwell, Cambridge, UK: Cambridge University Press, 1998.

  Kaharl, V. A. “Sounding Out the Ocean’s Secrets,” in Beyond Discovery: The Path from Research to Human Benefits. Washington, DC: National Academy of Sciences, March 1999. Accessed online at http://www.beyonddiscovery.org.

  Keston, D. A. “Joseph Fourier—Politician and Scientist,” accessed online at http://www.todayinsci.com/F/Fourier_JBJ/FourierPoliticianScientistBio.htm.

  Lambek, J. “If Hamilton Had Prevailed: Quaternions in Physics,” Mathematical Intelligencer 17 (1995), No. 4, pp. 7–15.

  Laugwitz, D. Bernhard Riemann, 1826–1866: Turning Points in the Conception of Mathematics, Boston: Birkhauser, 1999.

  Rockmore, D. Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers, New York: Pantheon, 2005.

  Rothman, T. “Genius and Biographers: The Fictionalization of Évariste Galois,” American Mathematical Monthly 89 (1982), No. 2, pp. 84–106.

  Stubhuag, A. Niels Henrik Abel and his Times, Berlin, New York: Springer, 2000.

  Tolstoy, I. James Clerk Maxwell: A Biography, Edinburgh: Canongate, 1981.

  Part Four: Equations in Our Own Time

  Albers, D. J. and G. L. Alexanderson, eds. Mathematical People: Profiles and Interviews, Boston: Birkhauser, 1985.

  Albers, D. J., G. L. Alexanderson, and C. Reid, eds. More Mathematical People: Contemporary Conversations, Boston: Harcourt Brace Jovanovich, 1990.

  Anon. “Lights All Askew in the Heavens,” The New York Times, 10 November 1919.

  Aubin, D. and A. D. Dalmedico. “Writing the History of Dynamical Systems and Chaos: Longue Durée and Revolution, Disciplines, and Culture,” Historia Mathematica 29 (2002), pp. 273–339.

  Brown, K. “Reflections on Relativity,” accessed online at http://www.mathpages.com/rr/rrtoc.htm.

  Casti, J. L. and DePauli, W. Gödel: A Life of Logic, Cambridge, MA: Perseus, 2000.

  Courtault, J.-M., Y. Kabanov, B. Bru, P. Crépel, I Lebon, A. le Marchand. “Louis Bachelier on the Centenary of Theorie de la Speculation,”Math. Finance 10 (2000), No. 3, pp. 341–353.

  Dirac, P. A. M. “The Evolution of the Physicist’s Picture of Nature,” Scientific American, May 1963. Accessed online at http://bit.ly/dirac1963.

  Duffie, D. Black, “Merton and Scholes—Their Central Contributions to Economics,” in Legacy of Fischer Black, B. Lehmann, ed. Cary, NC: Oxford University Press, 2004, pp. 286–298.

  Einstein, A. “Does the Inertia of a Body Depend on its Energy Content?” Annalen der Physic 18 (1905), p. 639. Trans. by W. Perrett and G. B. Jeffery (1923), accessed online at http://www.fourmilab.ch/etexts/einstein/E_mc2/www.

  Friederich, B. and D. Herschbach. “Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics,” Physics Today, December 2003: pp. 53–59.

  Giulini, D. and N. Straumann. “Einstein’s Impact on the Physics of the Twentieth Century,” Stud. Hist. Phil. Modern Physics 37 (2006), pp. 115–173.

  Goddard, P., ed. Paul Dirac: The Man and His Work, Cambridge, UK: Cambridge University Press, 1998.

  Isaacson, W. Einstein: His Life and Universe, New York: Simon and Schuster, 2007.

  Janssen, M. “Of pots and holes: Einstein’s bumpy road to general relativity,”Ann. Phys. (Leipzig) 14 (2005), Supp. pp. 58–85.

  Kassouf, S. T. and Thorp, E. O. Beat the Market: A Scientific Stock Market System, New York: Random House, 1967.

  Lowenstein, R. When Genius Failed: The Rise and Fall of Long-Term Capital Management, New York: Random House, 2000.

  Ott, E. “Edward N. Lorenz, 1917–2008,” Nature 453, 15 May 2008, p. 300.

  Peitgen, H.-O., H. Jergens, D. Saupe. Chaos and Fractals: New Frontiers of Science, New York: Springer, 2004.

  Pogge, R. W. “Real-World Relativity: The GPS Navigation System,” accessed online at http://www.astronomy.ohio-state.edu/˜pogge/Ast162/Unit5/gps.html.

  Renn, J. and D. Hoffmann. “1905—A Miraculous Year,”Jour. Physics B: Atomic, Molecular, and Optical Physics 38 (2005), S437–S448.

  Schaeffer, S. M. “Robert Merton, Myron Scholes, and the Development of Derivative Pricing,” Scand. Jour. Of Economics 100 (1998), No. 2, pp. 425–445.

  Strogatz, S. Sync: The Emerging Science of Spontaneous Order, New York: Hyperion, 2003.

  Wikipedia. “The Photoelectric Effect,” accessed online at http://en.wikipedia.org/wiki/Photoelectric_effect.

  Wilczek, F. “The Dirac Equation,” in Proceedings of the Dirac Centennial Symposium, Singapore: World Scientific Publishing, 2003.

  Wilmott, P. Frequently Asked Questions in Quantitative Finance, Chichester, UK: Wiley, 2007.

  Woodin, W. H. “The Continuum Hypothesis, Part I,” Notices of the American Mathematic
al Society, June/July 2001, pp. 567–576.

  Wu, H. “Historical development of the Gauss-Bonnet theorem,” Science in China Series A: Mathematics 51 (2008), No. 3, pp. 1–8.

  Wu, H. “Shiing-Shen Chern: 1911–2004,” Bull. Amer. Math. Soc. 46 (2009), pp. 327–338.

  index

  Page numbers in italics refer to illustrations

  Abel, Niels Henrik 67, 114–16, 118–21

  Allendoerfer, Carl 178

  Anderson, Carl 170

  Arabic numerals 14–15

  Archimedes 40–3, 48–51, 52–5, 54

  Measurement of the Circle, The 40

  On Floating Bodies 53–4

  principle 53–4

  quadrature 82–3

  Quadrature of the Parabola 48–51

  Aristarchus of Samos 68

  Aristotle 70–71, 92

  Atiya, Michael 180

  atomic bomb 157, 158

  Bachelier, Louis 210

  Bacon, Francis 76

  Bailey, David 45

  Barrow, Isaac 83

  Bessel, Friedrich 124

  Biot, Jean–Francois 141

  Black, Fischer 204–13

  Black-Scholes formula 204–13

  Blaschke, Wilhelm 175

  Bolyai, Janos 124, 127, 174

  Bolyai, Wolfgang 123–4

  Bombelli, Rafael 66–7

  Bonnet, Pierre Ossian 176

  Borwein, Peter 45

  bosons 170, 172

  Brahe, Tycho 70–3

  Brahmagupta 27–9, 28, 61

  Brownian motion 210, 211

  Bryant, Robert 181

  butterfly effect 198

  calculus 80–9, 100–101

  Cantor, Georg 183–8

  Cantor’s Continuum Hypothesis 187, 192–3