The Cave and the Light

by Arthur Herman

  On the other hand, with his belief that statistics was “the mathematics of practical men,” Maxwell would have relished the example of horse racing.5 Of course, no one can say whether a particular horse will actually win a particular race on a particular day, even given the competing field. But someone knowledgeable about horses, and the field, can give you odds on your horse’s chances based on its probability of coming in win, place, or show.

  Still, that’s gambling, not science. But suppose you are running a thousand horses, all the same age and from the same bloodline and all raised on the same farm and fed the same diet—and that you have the same information about every horse in the field and its thousand siblings. You run them all in a series of races and plot the results on Gauss’s curve. Then it’s possible to learn a lot, not just about which bloodlines are likely to win and which to lose, but also about horses in general and how to raise a winner—as well as about the nature of the horse race itself. In fact, it may be possible to project the winning order of an entire field without mentioning a single particular horse—and without running a race at all.

  Maxwell was proposing to use statistical models to study phenomena, like atoms in a gas, where it wasn’t empirically possible to get an exact count; they pose the same problem as when we shake a box full of marbles and try to observe each marble’s speed and direction. Likewise, it could come in handy in studying things that won’t give us any useful sensory information, like the unseen world of the electromagnetic spectrum—or later, the atom.

  Maxwell used it in 1857 first to analyze the rings of Saturn. Although he couldn’t describe the motion or behavior of every particle that made up the rings, he could set up a description using his statistical model, showing that certain sizes of particles would end up in the different classes of orbits, and what their probable speeds might be.6 When he moved on to gases, the results were even more amazing. But when it came to studying light waves, Maxwell found that he had struck a conceptual reef. The question was, what do the waves move through? Without a fixed frame of reference for computing speed and distance, Maxwell’s theory looked incomplete. Some proposed “ether,” an “elastic solid” that provided the medium through which light could travel the way a lake provides the medium for waves from a speedboat or a dropped pebble. Surely if light moves, they argued, it must move through something. Once again, the materialist faith that all things must occupy some physical space—even light—remained unshakable.7

  So Maxwell’s breakthrough in the study and nature of light went nowhere for almost thirty years. Meanwhile, his insights into the explanatory power of statistics were about to be turned into an epoch-making theory about the very nature of matter, by an intense, rather alarming young man with a thick black beard named Ludwig Boltzmann.

  If Maxwell was the Da Vinci of nineteenth-century physics—introspective, insatiably curious, playful to the point of whimsy, and relishing the role of mysterious polymath—Boltzmann was its Michelangelo. Born the son of an Austrian civil servant in 1844, he was intense, physically imposing, moody, and explosive. His family had no intellectual pretensions; he showed no great aptitude for intellectual subjects, let alone physics or mathematics, until he met Josef Stefan at the University of Vienna in the early 1860s. Stefan was a physicist, and a good one: he was up for the directorship of the Institute of Physics previously held by Christian Doppler, discoverer of the Doppler effect. In 1866, Stefan beat out his competitor, another brilliant Austrian, the twenty-four-year-old Ernst Mach.

  Stefan took an immediate liking to the clever but awkward Boltzmann. He gave him articles by Maxwell and an English grammar and dictionary to help him read them; Stefan was one of the few scientists in central Europe who understood how revolutionary Maxwell’s theory of an invisible electromagnetic field really was. But above all, Stefan was a believer in the theory of the atom, and he passed belief in its existence on to his young disciple with the passion of an act of conversion.

  The idea of atoms, and their free motion in space, went back before Plato and Aristotle, of course, to the Eleatic philosophers Democritus and Leucippus, and then forward to their Roman admirer Lucretius (c.100–c. 55 BCE). For almost two thousand years, the idea that reality might be made of innumerable tiny unseen bodies that combined and recombined to form physical objects had seemed ludicrous, especially given the huge prestige of Aristotle’s own physics. Atomism also carried the taint of atheism, since it seemed to deny the existence of spiritual substances, including the soul—although Newton found himself occasionally drawn to the theory. Still, for centuries atomism seemed the last extremity of godless materialism—a strange irony, given its later fate.8

  Then in the mid-nineteenth century, it experienced a sudden resurgence, when the idea of the atom proved useful for isolating and studying the behavior of gases under different conditions and at different temperatures. Rudolf Clausius, for one, realized that heat is the result of motion, and that what is in motion could best be explained as tiny, as-yet-unseen molecules or atoms. Still, for believers like Stefan and his friend Josef Loschmidt, atoms remained an unproven hypothesis, nothing more.

  Maxwell’s theory of the electromagnetic field, however, gave them new hope. Atoms were invisible—but so was Maxwell’s electromagnetism. Maxwell’s electromagnetism explained certain phenomena by simplifying the mathematics; so did atomic theory. But it was thirty-year-old Boltzmann who saw that Maxwell’s work on statistical probability opened a whole new horizon for understanding the nature of the atom—and with it perhaps the fate of civilization itself. “Was it a God who wrote these signs?” he once exclaimed, while teaching Maxwell’s theories to his students, and meant it.9

  What Boltzmann did was connect Maxwell’s electromagnetism and Clausius’s atom directly to the hottest new issue in nineteenth-century physics, thermodynamics. In 1850, Clausius had discovered its second and latest law: For all the available energy in a system—the crankshaft of a moving train, the steam in a boiler, water flowing down a mill race or over a dam, a comet sweeping across the sky—some energy is lost and becomes unavailable. Over time, Clausius argued, the unavailable energy will tend to increase, as available energy gets lost—as when a spinning propeller gradually slows down or (on the issue of heat transfer) when very cold milk is poured into a very hot drink, and both tend to approach room temperature and become tepid.

  Clausius dubbed this unavailable lost energy entropy: and his famous second law stated that entropy is always increasing. To some, the idea was shocking. It suggested that all systems eventually run down, including (inevitably) the universe. To more lively imaginations in the later nineteenth century, the law of entropy seemed a signal of eventual doom—not only for the physical world but for civilization itself.*10

  Boltzmann, however, was able to give the law of entropy a more hopeful thrust, thanks to the laws of probability and the properties of the atom. Clausius and his colleagues were still thinking of heat as a fluid, a material that transferred from one place to another like water in a bath—which through entropy goes down the drain and is lost forever. But if heat was actually the random motion of atoms within a given system, then entropy was simply a measure of atoms reaching a disordered state after a more ordered one.11

  Imagine a deck of cards, arranged by suit and rank down to the last card—a highly ordered state. We shuffle the deck and find that some cards are now out of order, but others are still in their original sequence. We shuffle a second, even a third time: the number of hearts or diamonds still in sequence steadily diminishes. Finally we have a deck in which the distribution almost seems random: the final stage of entropy.12

  But Boltzmann pointed out that nothing says the process of shuffling the deck has to result in growing disorder. It is possible, but not probable, that hundreds or even thousands of reshufflings might restore the original sequence. In short, over time entropy tends to grow—but it is not inevitable. Disorderly states are more probable than orderly ones: but not every orderly state will
necessarily run down and dissolve into chaos. Ordered systems will take shape, and retain their shape, in certain places—in the solar system, in the human body, even in society—even as disorder takes over in others, by a formula Boltzmann drew up:

  S=K log W

  S, entropy, is proportional to the logarithm of W, the probability of a given state.13

  But Boltzmann’s theory was also saying something else, almost equally profound: the probability of any state increases as its energy decreases. Low-energy states are therefore more probable—while usable energy (like our train crankshaft, or water flowing over a dam) is energy in disequilibrium—and that includes electricity. Boltzmann’s statistical mechanics had suddenly opened the door to a new understanding of how electricity acts and flows from positive to negative—i.e., from high-energy states to low-energy ones—and how atoms directly affect the unseen properties of matter, from electrical conductivity to the vicosity of fluids.

  A whole new world was opening up, not only for physics but for chemistry, as well—even perhaps biology—that is, if atoms exist.14

  Others had their severe doubts. One was Boltzmann’s fellow countryman, Ernst Mach, his former mentor’s rival and now holder of a prestigious chair of experimental physics at the University of Prague.† Mach’s entire approach to science was that the only legitimate starting point for the investigation of nature had to be the classic one defined by the eighteenth-century empiricists: sense perception. “The aim of science,” Mach wrote, “is to obtain connections among phenomena” within that field of experience—and then describe them in the shortest and most economical way possible as scientific laws.15

  Boltzmann’s theorem flunked on both counts. No one had ever seen an atom. The theory of entropy worked just fine without them—and without Boltzmann’s elaborate statistical mechanics. Portraying heat as atomic motion seemed equally absurd; but nothing offended Mach and his empirically minded colleague Henri Poincaré at the University of Paris than Boltzmann’s introducing entities (i.e., atoms) whose existence could not be independently judged, solely on the grounds that they made the calculations clearer and more elegant.

  Where was the clear picture of reality? Mach wanted to know. Where were the facts? Stick to the verifiable facts, he told Boltzmann and everyone else, and leave these speculations to the metaphysicians. It’s a line John Locke himself might have taken—and one rooted in the firm confident faith in Newton’s universe.

  Boltzmann was devastated. With prestigious intellectual gatekeepers like Mach (who moved on to the University of Vienna in 1895) and Poincaré standing in his way, his theory of a vast and important unseen world of atoms determining the basic processes of nature, remained just a theory and an unproven one.

  At Vienna, meanwhile, Mach’s reputation as a philosopher of science would only grow. He drew to his classes and seminars three distinguished figures: the economist Otto Neurath, physicist Philipp Frank, and mathematician Hans Hahn—the core of what would become the famed Vienna Circle. They were delighted with Mach’s way of describing scientific laws as pictorial summaries of experimental facts or events, which we put together in order to make complex sense data comprehensible in terms of ordinary experience.

  Neurath, Frank, and Hahn were joined by other intellectual luminaries like Moritz Schlick and Rudolf Carnap to found an informal school of philosophy of science called Logical Positivism or Logical Empiricism. It would be the dominant school for scientists, thinkers, and mathematicians in eastern Europe for nearly half a century. Besides Vienna and Graz, its strongholds included Prague, Cracow, Budapest, and Lemberg (Lvov in Ukraine), with outposts as far away as Oxford, Chicago, and the University of Uppsala. At their first international meeting in Prague, they proudly drew up a list of their heroes from philosophy’s past.

  It was an impressive list. Auguste Comte, John Stuart Mill, David Hume, and the philosophers of the Enlightenment were joined by Epicurus, Jeremy Bentham, the physicist Albert Einstein, and the master of the logic of mathematics Bertrand Russell, as well as Mach and Poincaré. But one figure clearly held pride of place in their intellectual pantheon, namely Aristotle.16

  And no wonder. Aristotle was the founder not only of the logic to which they looked for their framework of truth, but also of the idea of science as a unified system based on human experience. Their ultimate goal, as one of them later put it, was to free the European mind from fuzzy metaphysics and dogma and to return “in some measure to the classical goal of philosophy as defined by Aristotle,” which was a way both of understanding nature and of living in the world.17

  In this, their chief foe was not the Catholic Church (many, like Mach, were raised and educated as Catholics) or Boltzmann—in the early days, they hardly gave him a thought. Rather, it was Georg Friedrich Hegel. In Hegel’s view, the laws of science were just one part of the unfolding of the great chain of dialectic, which would eventually integrate everything under the sun, from physics and biology and the output of industry to the meaning of the individual and the family and the State—into the ultimate objective reality of the Absolute.

  But underlying the Hegelian formulation of science, of course, was the great unspoken presence of Plato and his metaphysics.‡ For Hegel, the world of sense data—what he loftily dubbed subjectivity—is simply Plato’s original realm of the cave and the source of all unclear thinking. It confuses what is concrete (this is my bicycle, my spouse, my plans and dreams for the future) with what is real. Objectivity, the basis of all certain knowledge, can arise only when the individual disappears from view. This is true whether one is talking about the laws of physics and statistics, or about the laws of social and political progress. By thinking big and objectively, the Hegelian realizes how small and insignificant he—or at least most other people—must be.

  The metaphysical dogmas of Hegel were a dominant influence in the universities of the new German Empire, especially the University of Berlin, but they offended Mach and the Logical Empiricists. They wanted to lay out a more flexible, empirically based approach to science. Science should deal with perceptible individual facts and events, they argued, and nothing else. Our mind’s job is to match our words and mental images to those facts and nothing else.

  In short, Mach’s answer to Hegel was like Sergeant Friday’s: “The facts, ma’am, just the facts.” One of his disciples later put it almost the same way: “Something is real if it is a part of the system of symbols that denotes the world of facts.”18 All the rest, the whole weighty fabric of metaphysics from Plato to Hegel, needed to be shoved aside so that men could think clearly and accurately—and finally be free.

  The Logical Positivists’ goal, to free the European mind from the yoke of Hegel, might have been admirable. But in the process, their principal victim was Boltzmann.

  Henri Poincaré wrote a devastating attack on his kinetic theory of atoms, arguing that in any closed system, Boltzmann’s atoms, by his own formulation, must eventually return to their original condition, thus contradicting the law of entropy. (He conveniently ignored the fact that if everything is made up of atoms, there is no such thing as a closed system.)19

  Poincaré’s blast seemed the last word on the subject. Boltzmann’s bitterness grew, not only toward Mach and the scientific establishment but toward another rising physicist named Max Planck, who was using the same probability theory to study the release of energy—except by explicitly rejecting Boltzmann’s atoms and replacing them with electromagnetic waves.

  With the new century, Boltzmann seemed a forgotten, failed figure. In late 1905 he wrote to a friend, “I have reached 62 years of age and I have gained no peace of mind.”20 He spent that Christmas and New Year bedridden. In May 1906 the university decided he was too ill to teach any longer—the future of physics was passing into other hands. Then came news that his old antagonist Poincaré had been made president of the French Academy of Sciences. In a fit of despair, Boltzmann took his own life.

  The University of Vienna, according to newspaper
reports, plunged into “active mourning.”21 Ernst Mach made some brief remarks of regret about the death of the man whose theories he had done most to discredit, suggesting that perhaps Boltzmann’s nerves weren’t strong enough for the rough-and-tumble of classroom and laboratory. For others, however, the sense of grief would grow—not least because Boltzmann died at the very moment when his atomic theory was about to win, and when Mach himself would be denounced as a “false prophet” by the same physicist who had earlier rejected Boltzmann’s theory, Max Planck.

  The truly pivotal moment, however, had come even before that, when an obscure twenty-one-year employee at the Swiss Patent Office in Geneva merged Boltzmann’s theory with the next giant step in the demolition of Newton’s cosmos, relativity.

  His name was Albert Einstein. Born in Ulm in Germany in 1879 to Jewish parents, he had been a brilliant if underachieving student at the Zurich Polytechnic and, after marrying and having a child, decided a civil servant’s job and salary at the Swiss Patent Office suited his future well. Einstein, however, was one of those students who learn more outside the classroom than in it, and even at the Polytechnic, he had been fascinated by a series of experiments at the University of Berlin conducted by an American physicist, Albert Michelson, and their rather unexpected results.

  Michelson was a meticulous, conscientious man and a former science instructor at the U.S. Naval Academy in Annapolis. He had come to Berlin to work on a new kind of interferometer, a device made up of reflecting mirrors for measuring the speed of light. His had a longer optical path-length. This, he reasoned, would allow scientists to detect variations in light’s velocity with more precision than ever.