Book Review The Road to Reality: A Complete Guide to the Laws of the Universe Reviewed by Brian Blank The Road to Reality: A Complete Guide to the and a mathematical formulation of one of Nature’s Laws of the Universe interactions, gravity. Roger Penrose Many of the hallmarks of progress along the road Alfred A. Knopf, 2005 to reality can already be discerned in the first Sci- ISBN 0-679-45443-8 entific Revolution. Because Nature so deftly hides $40.00, 1,099 pages her secrets, a crazy theory is often a prerequisite for making any headway. Copernicus, for example, For the title of his latest epic, The Road to Re- defied not only established authority but also the ality, Roger Penrose has selected a metaphor that common sense of every observer who, under the appears frequently in popular expositions of illusion of being at rest, watched the sun move physics. It is no wonder that the phrase has become across the sky. Other necessities for progress— a favorite among physicists, for it suggests a mathematics for formulating and developing a the- single-minded pursuit of the ultimate destination: ory and a physical apparatus for testing it—were an understanding of all the underlying principles also essential components of the revolution. The that govern the behavior of our universe. Perhaps improved instruments for measurement devised by that may seem to be an ambitious program. After Tycho Brahe permitted Kepler to refute the orbits all, it was not so very long ago that Eugene Wigner of Copernicus’s system. Newton’s calculus allowed asserted, “The great success of physics is due to a him to extend Galilean dynamics and explain the restriction of its objectives.” Since that sober as- laws that Kepler had observed. The road to reality sessment, however, stunning progress has changed had taken its now familiar course of revolution fol- the outlook of physics so greatly that several of its lowed by successive approximation. leading proponents have been emboldened to sug- The nineteenth century witnessed the second gest that a complete grasp of the laws of nature lies Scientific Revolution. Between 1850 and 1865, fun- just ahead of us. damental notions such as energy and entropy were As Penrose asserts, the voyage of discovery has introduced. At first, many scientists deprecated en- lasted more than two and a half millennia and has ergy as a mathematical abstraction. By the end of been profoundly difficult. At the start of the jour- the century, however, energy was replacing force ney, around 500 B.C.E., Heraclitus identified the as the preferred attribute of reality around which major stumbling block: Nature is wont to conceal her- to organize physical theories. Several new branches self. Mathematical advances aside, the first signifi- of physics—thermodynamics, statistical mechan- cant steps on the road to reality were achieved in ics, the kinetic theory of gases—arose accordingly. the period between 1543, the year Copernicus pub- The second revolution in physics culminated in a lished his heliocentric theory of planetary motion, working awareness of our solar system’s place in and 1687, the year Newtonian mechanics was in- the Milky Way, the concept of a “disembodied” troduced. This era, the first Scientific Revolution, field, the mathematical description of a second culminated in a working awareness of the solar fundamental interaction, namely electromagnetism, system, a basic framework for studying dynamics, the discovery of an elementary particle (the elec- tron), and, by virtue of the second law of thermo- Brian Blank is professor of mathematics at Washington dynamics, a stark new aspect of reality: the ther- University. His email address is [email protected]. modynamic arrow of time. JUNE/JULY 2006 NOTICES OF THE AMS 661 For all the successes of the second revolution, a theory that experimental physicists have re- physics faced several challenges at the beginning peatedly confirmed to exacting standards. Particles of the twentieth century. The debate over the wave continue to be discovered—notably the top quark versus particle nature of light, which had erupted in 1995 and the τ-neutrino in 2000—but they fit during the first revolution, was not definitively into the theory the way the man-made synthetic el- settled by the second, Maxwell’s characterization ements fit into the periodic table. of light as electromagnetic radiation notwith- During the same time span in which high energy standing. All experiments to detect a medium physicists probed the smallest bits of reality, as- through which light propagated, the hypothetical tronomers and astrophysicists revolutionized our luminiferous aether, failed. Newton’s law of grav- understanding of the largest objects of reality, in- itation remained a useful scorecard of gravity, but cluding our universe itself. By 1923 astronomers it neither explained the mechanism by which grav- had confirmed the existence of galaxies beyond the ity is effected nor permitted time to play any role Milky Way. Observations of distant celestial bod- in gravity’s action. The new theories of the second ies coupled with general relativity gave rise to a new revolution presented even more troublesome para- branch of physics, cosmology, that tells us much doxes, chief among which was the prediction of about how our universe came to be and how it will blackbody radiation having arbitrarily large en- cease to be. Though elementary particle physics and ergy. From an evolutionary point of view, our un- cosmology deal with objects at diametrically op- derstanding of reality seems to have advanced not posite ends of reality, the two fields have come to by a march down an orderly road but by an alter- be intricately intertwined. Knowledge gained from nating sequence of leaps between the frying pan the study of subatomic processes is the basis for and the fire. understanding the physics of stars and the syn- Historians of science often state that the twen- thesis of heavy elements in the universe. In return, tieth century witnessed two revolutions in physics: the exotic constituents of the universe provide im- quantum theory and general relativity. The first res- portant tests of particle theory. cued physics from the ultraviolet catastrophe of The whirlwind tour we have just concluded rep- blackbody radiation and resolved the dilemmas resents only a tiny fraction of what The Road to Re- posed by the properties of light. By blurring the dis- ality covers in its 1,100 pages. Anyone who casu- tinction between wave and particle, quantum the- ally flips through a few of those pages will recognize ory presented counterintuitive insights into the immediately that more than length distinguishes nature of matter and energy. The second revolu- The Road to Reality from other expositions that tar- tion, general relativity, combined space and time get a roughly similar audience. Here, uniquely so to provide a theory of gravity that is deeper than far as I am aware, we find an author presenting so- a mere bookkeeping formula. Both revolutions pro- phisticated concepts of physics by invoking so- foundly changed our conceptions of physical re- phisticated concepts of mathematics. Even an ex- ality. perienced mathematician who happens upon a The twentieth century was, indeed, a productive page illustrated with diagrammatic tensor nota- one for physicists; two revolutions may not give tion might shy away from Penrose’s Road. As the them their due. A second elementary particle, the author explains in his preface, “What I have to say photon, was detected in 1923. By 1932 both the pro- cannot be reasonably conveyed without a certain ton and the neutron had also been discovered. amount of mathematical notation and the explo- These nucleons led physicists to an additional two ration of genuine mathematical concepts.” Do not interactions: the strong and weak nuclear forces. take this declaration to be a contemporary version Within a few decades, a large menagerie of sub- of Copernicus’s Mathemata mathematicis scribun- atomic particles had been assembled: positrons tur: Penrose’s idea is that mathematics should be and muons in the 1930s, pions and kaons in the written not only for mathematicians but also for 1940s, Pauli’s long-conjectured neutrino in the anyone willing to learn. To that end, remedial 1950s, and a great many others. The ever increas- lessons begin in the preface, where rational num- ing particle zoo became ever more perplexing. Once, bers are defined as equivalence classes. after having given a speculative lecture at Colum- The first sixteen chapters of The Road to Real- bia, Wolfgang Pauli admitted, “This is a crazy the- ity are primarily devoted to the mathematics needed ory.” From the audience Niels Bohr called out, “Un- to express modern physical theory. By the time fortunately, it is not crazy enough!” In 1963 Murray page 383 is reached, the intrepid reader will have Gell-Mann and, independently, George Zweig pro- been introduced to a large number of topics in posed a theory of fractionally charged elementary analysis, algebra, and geometry. Of these, the particles (christened quarks by Gell-Mann) that demands of analysis are comparatively modest: cal- proved to be just crazy enough. In the next decade culus, Fourier series, hyperfunctions, Riemann sur- and a half, the so-called Standard Model of ele- faces, and enough complex function theory to state mentary particles and their interactions arose. It is the Riemann mapping theorem. The necessities 662 NOTICES OF THE AMS VOLUME 53, NUMBER 6 from algebra include quaternions, Clifford and find the backward references helpful. The forward Grassmann algebras, linear algebra, transforma- references may strengthen the incentives of some tion groups, and enough Lie theory to discuss the readers to slog through seemingly abstract math- classical groups and their Lie algebras and repre- ematics.