Why Beauty is Truth
Page 33
Interestingly, Dirac’s concept of beauty in mathematics differed considerably from that of most mathematicians. It did not include logical rigor, and many steps in his work had logical gaps—the best-known example being his “delta function,” which has self-contradictory properties. Nevertheless, he made very effective use of this “function,” and eventually mathematicians reformulated the idea rigorously—at which point it was indeed a thing of beauty.
Still, as Dirac’s biographer Helge Kragh has remarked, “All of [Dirac’s] great discoveries were made before [the mid-1930s], and after 1935 he largely failed to produce physics of lasting value. It is not irrelevant to point out that the principle of mathematical beauty governed his thinking only during the later period.”
Not irrelevant, perhaps, but not correct either. Dirac may have made the principle explicit during his later period, but he was using it earlier. All of his best work is mathematically elegant, and he relied on elegance as a test of whether he was heading in a fruitful direction. What all this suggests is not that mathematical beauty is the same as physical truth but that it is necessary for physical truth. It is not sufficient. Many beautiful theories have turned out, once confronted with experiments, to be complete nonsense. As Thomas Huxley said, “Science is organized common sense, where many a beautiful theory was killed by an ugly fact.”
Yet there is much evidence that nature, at root, is beautiful. The mathematician Hermann Weyl, whose research linked group theory and physics, said, “My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful.” Werner Heisenberg, a founder of quantum mechanics, wrote to Einstein,
You may object that by speaking of simplicity and beauty I am introducing aesthetic criteria of truth, and I frankly admit that I am strongly attracted by the simplicity and beauty of the mathematical schemes which nature presents us. You must have felt this too: the almost frightening simplicity and wholeness of the relationship, which nature suddenly spreads out before us.
Einstein, in turn, felt that so many fundamental things are unknown—the nature of time, the sources of ordered behavior of matter, the shape of the universe—that we must remind ourselves how far we are from understanding anything “ultimate.” To the extent that it is useful, mathematical elegance provides us only local and temporary truths. Still, it is our best way forward.
Throughout history, mathematics has been enriched from two different sources. One is the natural world, the other the abstract world of logical thought. It is these two in combination that give mathematics its power to inform us about the universe. Dirac understood this relationship perfectly: “The mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which nature has chosen.” Pure and applied mathematics complement each other. They are not poles apart, but the two ends of a connected spectrum of thought.
The story of symmetry demonstrates how even a negative answer to a good question (“can we solve the quintic?”) can lead to deep and fundamental mathematics. What counts is why the answer is negative. The methods that reveal this can be used to solve many other problems—among them, deep questions in physics. But our story also shows that the health of mathematics depends on the infusion of new life from the physical world.
The true strength of mathematics lies precisely in this remarkable fusion of the human sense of pattern (“beauty”) with the physical world, which acts both as a reality check (“truth”) and as an inexhaustible source of inspiration. We cannot solve the problems posed by science without new mathematical ideas. But new ideas for their own sake, if carried to extremes, can degenerate into meaningless games. The demands of science keep mathematics running along fruitful lines, and frequently suggest new ones.
If mathematics were entirely demand-driven, a slave of science, you would get the work you expect from a slave—sullen, grudging, and slow. If the subject were entirely driven by internal concerns, you would get a spoilt, selfish brat—pampered, self-centered, and full of its own importance. The best mathematics balances its own needs against those of the outside world.
This is what its unreasonable effectiveness derives from. A balanced personality learns from its experiences, and transfers that learning to new circumstances. The real world inspired great mathematics, but great mathematics can transcend its origins.
The unknown Babylonian who discovered how to solve a quadratic equation could never have realized, in his wildest dreams, what his legacy would be more than three thousand years later. No one could have predicted that questions about the solvability of equations would lead to one of the core concepts of mathematics, that of a group, or that groups would prove to be the language of symmetry. Even less could anyone have known that symmetry would unlock the secrets of the physical world.
Being able to solve a quadratic has very limited utility in physics. Being able to solve a quintic is even less useful, if only because any solution must be numerical, not symbolic, or else employ symbols specially invented for the purpose, which do little more than cover the question with a fig leaf. But understanding why quintics cannot be solved, appreciating the crucial role of symmetry, and pushing the underlying idea as far as it could go—that opened up entire new physical realms.
The process continues. The implications of symmetry for physics, indeed for the whole of science, are still relatively unexplored. There is much that we do not yet understand. But we do understand that symmetry groups are our path through the wilderness—at least until a still more powerful concept (perhaps already waiting in some obscure thesis) comes along.
In physics, beauty does not automatically ensure truth, but it helps.
In mathematics, beauty must be true—because anything false is ugly.
FURTHER READING
John C. Baez, “The octonions,” Bulletin of the American Mathematical Society volume 39 (2002) 145–205.
E. T. Bell, Men of Mathematics (2 volumes), Pelican, Harmondsworth, 1953.
R. Bourgne and J.-P. Azra, Écrits et Mémoires Mathématiques d’Évariste Galois, Gauthier-Villars, Paris, 1962.
Carl B. Boyer, A History of Mathematics, Wiley, New York, 1968.
W. K. Bühler, Gauss: A Biographical Study, Springer, Berlin, 1981.
Jerome Cardan, The Book of My Life (translated by Jean Stoner), Dent, London, 1931.
Girolamo Cardano, The Great Art or the Rules of Algebra (translated T. Richard Witmer), MIT Press, Cambridge, MA, 1968.
A. J. Coleman, “The greatest mathematical paper of all time,” The Mathematical Intelligencer, volume 11 (1989) 29–38.
Julian Lowell Coolidge, The Mathematics of Great Amateurs, Dover, New York, 1963.
P. C. W. Davies and J. Brown, Superstrings, Cambridge University Press, Cambridge, 1988.
Underwood Dudley, A Budget of Trisections, Springer, New York, 1987.
Alexandre Dumas, Mes Mémoires (volume 4), Gallimard, Paris, 1967.
Euclid, The Thirteen Books of Euclid’s Elements (translated by Sir Thomas L. Heath), Dover, New York, 1956 (3 volumes).
Carl Friedrich Gauss, Disquisitiones Arithmeticae (translated by Arthur A. Clarke), Yale University Press, New Haven, 1966.
Jan Gullberg, Mathematics: From the Birth of Numbers, Norton, New York, 1997.
George Gheverghese Joseph, The Crest of the Peacock, Penguin, London, 2000.
Brian Greene, The Elegant Universe, Norton, New York, 1999.
Michio Kaku, Hyperspace, Oxford University Press, Oxford, 1994.
Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, Oxford, 1972.
Helge S. Kragh, Dirac—A Scientific Biography, Cambridge University Press, Cambridge, 1990.
Mario Livio, The Equation That Couldn’t Be Solved, Simon & Schuster, New York, 2005.
J.-P. Luminet, Black Holes, Cambridge University Press, Cambridge, 1992.
Oystein Ore, Niels Henrik Abel: Mathematician Extraordinary, University of Minnesota Press, Minneapolis, 1957.
Abraham Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein, Oxford University Press, Oxford, 1982.
Roger Penrose, The Road to Reality, BCA, London, 2004.
Lisa Randall, Warped Passages, Allen Lane, London, 2005.
Michael I. Rosen, “Niels Hendrik Abel and equations of the fifth degree,” American Mathematical Monthly volume 102 (1995) 495–505.
Tony Rothman, “The short life of Évariste Galois,” Scientific American (April 1982) 112–120. Collected in Tony Rothman, A Physicist on Madison Avenue, Princeton University Press, 1991.
H. F. W. Saggs, Everyday Life in Babylonia and Assyria, Putnam, New York, 1965.
Lee Smolin, Three Roads to Quantum Gravity, Basic Books, New York, 2000.
Paul J. Steinhardt and Neil Turok, “Why the cosmological constant is small and positive,” Science volume 312 (2006) 1180–1183.
Ian Stewart, Galois Theory (3rd edition), Chapman and Hall/CRC Press, Boca Raton 2004.
Jean-Pierre Tignol, Galois’s Theory of Algebraic Equations, Longman, London, 1980.
Edward Witten, “Magic, mystery, and matrix,” Notices of the American Mathematical Society volume 45 (1998) 1124–1129.
WEBSITES
A. Hulpke, Determining the Galois group of a rational polynomial: http://www.math.colosate.edu/hulpke/talks/galoistalk.pdf
The MacTutor History of Mathematics archive: http://www-history.mcs.st-andrews.ac.uk/index.html
A. Rothman, Genius and biographers: the fictionalization of Évariste Galois: http://godel.ph.utexas.edu/tonyr/galois.htm
INDEX
Abacus, 49
Abbott, Edwin, 227
Abel, Hans Mathias, 84, 88, 92–93, 275
Abel, Margaretha, 84–85
Abel, Niels Henrik, xi, 85, 86–88, 94–95, 97
Abel, Søren, 84–85, 86
Accelerations, 194
Aether, 175, 176, 191
Aghurmi, 17
Agriculture, 2
Airy, George, 153
Akkad, 1, 8
Alamut castle, 39
Alberti, Leone, 269, 270
Alcohol, 84
Alexander, King of Epirus, 18
Alexander the Great, 17–18, 19
Alexandria, 17, 19
Algebra, 14–15, 50, 143.
See also Division algebra; Lie algebras; Vector algebra
division, 157
questions, 161–162
symbolism in, 35
symmetries in, 121
Algebra (Bombelli), 60–61
Algebra (Khayyám), 41
Alternating groups, 115
Alvarez-Gaume, Luis, 252
Amun, 17
Analytical Dynamics (Whittaker), 213
Analytical Mechanics (Lagrange), 76–77
Angles
bisecting, 25–26
trisecting, 26, 27, 28, 125
Anomalies, 252
Antikythera, 22
Antimatter, 214, 245
Antiquarks, 237
Anti-Taurus Mountains, 1
Apollonius, 39
Apostolic Camera, 60
Applied mathematics, 22
Archimedes, 20, 22, 27, 126
Argand diagram, 148
Argand, Jean-Robert, 148
Aristotle, 21, 140, 174
Arithmetica, 35–36
Arslan, Alp, 37–38
Artillery of National Guard, 104, 105
Ashtekar, Abhay, 255
Asimov, Isaac, 223
Associative law, 151
Assyria, 1
Astrology, 46
Astronomy, 40–41, 70–71
Asymptotes, 40
Atiyah, Michael, 247–248
Atom, probing of, 235–236
August, Ernst, 73
Axioms of Euclid, 24–25
Azra, Jean-Pierre, 110
Babylon, 2
culture, 4–5
education in, 8–9
fractions in, 12–13
history of, 7
life in, 8–9
mathematics, 13–14
notation systems in, 11–12
time measurements in, 12
Bachelier, Louis, 186
Bader, Peter, 85
Baez, John, 260
Bahariya Oasis, 17
Bandarini, Lucia, 52
Bartels, Johann, 66
Bartolotti, E., 54
Bastille Day, 107
Bayly, Helen, 142
Beauty
mathematics and, 278, 280
nature and, 278–279
truth and, 188, 263, 275
Becquerel, Alexandre, 186
Bell, Eric Temple, 108, 138, 155
Benze, Dorothea, 64
Bernadotte, Jean Baptiste, 85–86
Bernoulli, Johann, 147, 148, 209
Bernstein, Carl, 243
Bertel, Annemarie, 204
Bianchi identities, 196
Bible, 6
Big Bang, 193, 242, 257–258
Big Crunch, 257–258
Bilson-Thompson, Sundance, 256
Bioctonions, 272
Bisection problem, 25
Black holes, 193
Blackbodies, 201
Bohr, Niels, 203, 205, 235
Bolyai, Wolfgang, 64, 67, 149
Bombelli, Rafaele, 60–61, 260
Book of My Life (Cardano, G.), 46–47
Born, Max, 208
Bose, Satyendranath, 236
Bosons, 236
Bourg-la-Reine, 98, 102
Bourgne, Robert, 110
Bousso, Raphael, 257
Brahe, Tycho, 40, 276
Brahmagupta, 37, 264
Braids, 255–256
Brenda, Georgine Emilia, 204
Brinkley, John, 140
Broom Bridge, 152
Brownian motion, 185–186, 186
Brunelleschi, Filippo, 269
Bruno, Giordano, 56, 188
Büttner, J.G., 64–66
Calabi-Yau manifold, 253, 254
Calculus, 5, 149
Cambyses II, 17
Candelas, Philip, 253
Cardano, Fazio, 50, 51
Cardano, Girolamo, 45–47, 57–60, 61, 78, 260, 267
cubic formula of, 145–146
Fontana, Niccolo, and, 53–55
gambling habits of, 51
imprisonment of, 56
Cardano Tower, 90
Cartan, Élie, 169, 171, 268–269
Cassiani, Paolo, 79
Cauchy, Augustin-Louis, 75, 80, 81, 101, 104–105, 117
Caussidière, 108
Cayley, Arthur, 122, 261, 262, 267
Cayley numbers, 262
Cayley-Dickson process, 265–266
Ceres, 71
Charles VIII, 51
Charles X, 103
Chevalier, Auguste, 109
Christianity, 48
Circles, squaring, 125, 129–130
City Philosophical Society, 177
Civilization, origins of, 2
Classifications, 167
Cleopatra, 18
Cloyne, Bishop of, 140
Coefficients, 57, 106
Colburn, Zerah, 138
Coleridge, Samuel Taylor, 139–140
Collège Charlemagne, 127
Commutative law, 151
Commutator, 164
Compass, 33, 43, 49, 63, 130, 134, 150
in Greek geometry, 25–30
problems solvable with, 126–129
Complex numbers, 69, 70, 142, 147, 156, 260, 264
as division algebras, 157
equations and, 68
Hamilton, William Rowan, and, 148–150
as pairs, 150
and triples, 150
Complex systems, 226
Conic sections, 27, 33, 4
1
Connes, Alain, 255
Conti, Vittoria, 76
Conway, John Horton, 136
Copenhagen interpretation, 205, 209
Copernicus, Nicolaus, 56
Cosa, 50
Coulomb, Charles Augustin, 77
Crelle, August, 92–93
Crick, Francis, 205
Crommelin, Andrew, 212
Cube roots, 48, 60–61
Cubes, duplicating, 125
Cubic equations, 14, 33, 56, 57, 78
of Cardano, 145–146
groups and, 116
solving, 42–43, 57–58, 90
types of, 41–42
Culture, Babylonian, 4–5
Cuneiform, 5
Curvature, space-time, 192, 193–194, 195, 197
intrinsic, 196
da Vinci, Leonardo, 51, 269
D’Alembert, Jean Le Rond, 76
Darboux, Gaston, 161
Darius III, 18
Darwin, Charles, 184
Davy, Humphry, 177
de Broglie, Louis, 203
de Laplace, Pierre-Simon, 71
de Saint-Venant, Adhémard Jean Claude Barré, 127
Decimal systems, 10
Decimals, 12–13
Decoherence, 206
Deep Throat, 243
Degen, Ferdinand, 85–86, 265
Degrees, 57
del Ferro, Scipione, 53, 54, 55–56
del Nave, Annibale, 54
Delambre, Jean, 80
Della Pittura (Alberti), 269
Desargues, Girard, 270–271
Descartes, René, 146, 179
D’Herbinville, Pescheux, 108, 110
Differential equations, 162
Differential fields, 164–165
Diffraction, 175–176
Digits, 10
Dimensions, 225–226
Diophantine equations, 36
Diophantus, 35–36, 41
Dirac, Charles, 211
Dirac effect, 212, 214
Dirac, Paul, 204, 219, 244, 273, 277, 278
education of, 211–212
Disney, Catherine, 139, 153