BOLTZMANN WAS INVIGORATED BY his lively engagement with the English physicists in Oxford over the meaning of kinetic theory. In September 1895, the German Society of Natural Scientists was scheduled to meet in the Baltic port city of Lübeck, and in June of that year Boltzmann wrote to his friend the chemist Wilhelm Ostwald that he “would like to bring about, if possible, a debate a la british association [sic], mainly for my own education. It is essential that the chief leaders of opinion attend. I hardly need tell you how important to me in particular your own attendance would be.”
Boltzmann had met Ostwald in 1887 when the younger man, then 34 years old, had come to Graz for a few months to study. Ostwald quickly made a name for himself as one of the greatest of German chemists, a prominent leader and organizer, as well as an original scientist. He largely founded the discipline that is now called physical chemistry, in essence a marriage of chemistry and thermodynamics. The aim of physical chemistry is to understand the exchanges of energy involved in chemical reactions, the influence of temperature and other external conditions on reaction rates, and in general to elucidate the dependence of chemical changes on physical circumstances.
Boltzmann had been impressed by Ostwald, who had in turn enjoyed the friendly reception he found in Graz. At the time of his brief sojourn in Graz, Ostwald was in the process of moving from the University of Riga (now the capital of Latvia), which was his birthplace, to the old and important University of Leipzig, 100 miles southwest of Berlin in the German state of Saxony. There he stayed for almost two decades, building up an enormously influential department. He was the founder in 1889 of the Zeitschrift für Physikalische Chemie, the first scientific journal for his discipline.
Despite his early encounter with Boltzmann and kinetic theory, Ostwald fell under the influence of Mach and his antipathy toward theorizing. It was possible at that time to be a chemist without believing that atoms and molecules were real objects. Rather, they could be viewed as convenient but abstract accounting devices, notional divisions of matter, whose essential purpose was to allow a chemist to keep track of the bookkeeping in chemical reactions. Hydrogen and oxygen were well known to combine in two-to-one proportions to form water, but that did not necessarily mean that two bona fide bits of hydrogen linked up to a single bit of oxygen. In chemistry, even toward the end of the 19th century, belief in the reality of atoms was optional. Many chemists found it not even an interesting question, too abstract for their tastes.
Ostwald, however, had a strong philosophical bent, and this kind of agnosticism proved unsatisfactory. His knowledge of physics, moreover, made him hanker after a version of chemistry based on some fundamental principles. Atoms wouldn’t do, because they constituted a form of abstract theorizing—metaphysics, in Mach’s dismissive vocabulary. Instead, Ostwald became enamored of an idea called energetics or energeticism, which was built on the notion that energy, since it is an observable and tangible thing, ought to constitute the elemental stuff of scientific explanation. In this view, heat was undoubtedly a form of energy, but as to the nature of heat, no more could or should be said. The law of energy conservation was the prime rule, and other laws, including other laws of thermodynamics and even Newtonian mechanics, ought to flow from that essential principle.
To some energeticists, atoms were a needless but not necessarily sinful hypothesis. But in the quest for philosophical purity, the leaders of this new movement developed a positive antipathy toward atomism. In 1887, the chemist Georg Helm published a book called The Theory of Energy, which argued for the recognition of energy as the fundamental stuff of the physical world. Helm’s book had little immediate impact, but its message was seized on by Ostwald: this was the foundation he needed for his interpretation of physics and chemistry. During the early 1890s, Ostwald energetically took up the cause of energeticism, hoping that somehow all the known laws of physics could be shown to derive from rules governing the transformation of energy. There would then be no talk of atoms. In 1892, Ostwald visited Boltzmann briefly in Munich, and Boltzmann wrote shortly afterward asking that if he and Helm were ever to get around to formulating their ideas in a more systematic way, they should please keep him informed of progress.
A little later Ostwald indeed sent a manuscript, which he described as a sketch of the foundations of energeticism. He displayed a certain awkwardness and trepidation, admitting himself to be a “bungler” in mathematics, asking for Boltzmann’s critical analysis but also asking for any comments to be kept between them. Later, thanking Boltzmann for his “friendly opinion” of the manuscript, he added, “you can hardly know how valuable this is to me. In such matters it’s easy to mistake a great stupidity for a great discovery.”
Boltzmann’s replies also show a delicacy of phrasing unusual for him. He emphasizes his admiration for the effort Ostwald is making, and cautions him against taking his criticisms too badly. He warns against excessive rigidity of thinking: “against the dogma that nature can be explained only mechanically (through the motion of atoms) I would not like to set the opposite view, that it cannot be explained in that way.” Ostwald admired and liked Boltzmann, but was also fearful of submitting his ideas to what he acknowledged were the other man’s great powers of physical and mathematical analysis. Boltzmann in turn liked and admired Ostwald, but at heart thought his crusade for energeticism philosophically dubious and scientifically just plain wrong.
Against this background, the Lübeck debate unfolded. Ostwald and Helm took the field for energeticism, and Boltzmann, seconded by a young mathematician named Felix Klein, argued the case for atoms. The event was, despite Boltzmann’s original intent, nothing like the debate at Oxford. There, interested physicists had gathered to discuss kinetic theory and hammer out, if they could, the meaning of the H-theorem, the nature of reversibility, the probabilistic nature of the laws of thermodynamics, and so on. It was a scientific debate devoted to the elucidation of subtle issues implicit in a profound but still unfinished theory.
At Lübeck, on the other hand, Boltzmann and Klein had to defend the very essence of kinetic theory against opponents who simply did not believe in the existence of atoms, who saw Boltzmann’s work as elaborate mathematical speculation founded on pure assumption, and who would not even allow Boltzmann the privilege of thinking that his theories constituted a respectable kind of scientific investigation. Ostwald and Helm were there not to debate the merits of kinetic theory but to deny them altogether.
Ostwald, moreover, was something of an orator, where Boltzmann was generally not; he could be a fierce and impressive speaker, but he was not always articulate. Ostwald was fluent, and adept at putting his ideas into easily assimilable form. Over the course of his life he wrote many books, both technical and nontechnical, concerning general issues of philosophy and intellectual development as well as science itself, and he capped it off with a three-volume autobiography.
The debate took place on September 16, 1895, in a large hall filled with hundreds of eager listeners, and it occupied most of the day. Helm and then Ostwald sketched their view of energetics, arguing that although it was still far from complete, it held the promise of all-encompassing explanatory power based on elementary principles. They were, they claimed, offering a program of research, not a finished body of knowledge. Both chemists, they tripped up when they tried to explain how the well-known laws of mechanics and thermodynamics were supposed to follow from the conservation of energy alone.
Boltzmann began his reply amicably enough, making some general remarks about the need to explore a range of scientific hypotheses in order to move science forward. He professed himself eager to avoid hostility: “I hope I can count among my closest friends the scholars whose names I will mention later, and I reckon countless of their achievements among the most outstanding scientific works; I set myself only against their specific publications on energeticism. This observation will suffice to prevent the subsequent attacks directed at any of their conclusions or mathematical formulas from being imbued with the slightest personal character.”
Having made this lofty declaration of neutrality, Boltzmann then tore deliberately into the theoretical pretensions of his opponents. What followed was a lengthy and fairly technical analysis, but the gist of it was simple. He explained what any physicist knew to be true. Newtonian mechanics was based on more than just energy conservation. The second law of thermodynamics was distinct from the first and not, as some physicists had originally thought, derivable from it. These were not questions of philosophical preference but matters of physical reasoning and mathematical proof. Energeticism simply could not achieve the things it set out to achieve. He may have included no personal remarks, but Boltzmann made no attempt to sugarcoat his lack of respect for his opponents’ scientific aims. He explained in unrelenting detail all the reasons why he thought the cherished hopes of the energeticists were utterly at odds with physics as it was then understood.
Helm, writing to his wife the next day, said that “things went hard.” Boltzmann and Klein “touched on things which during my preparation and in my correspondence I had not anticipated, and which seemed inappropriate, and it was hard for me to put in even a few words of clarification.” Ostwald, in his autobiography, recalled that he felt surrounded by “closed antagonism,” and that the Lübeck debate was “the first time I personally found myself confronted by such a unanimous band of downright adversaries.” Svante Arrhenius, the Swedish chemist who had come under Boltzmann’s influence at Graz, also attended the debate and later wrote that “the energeticists were thoroughly defeated at every point, above all by Boltzmann, who brilliantly expounded the elements of kinetic theory. . . . Ostwald was quite exhausted when the discussion ended, and Helm spoke of having been lured into an ambush.”
Many years later, at a meeting held in wartime Vienna to mark the 1944 centenary of Boltzmann’s birth, another physicist recollected the contest. Arnold Sommerfeld, who succeeded Boltzmann in Munich and became, in the early part of the 20th century, a champion of the new quantum theory, had attended the Lübeck meeting as a young man. He compared the debate between the dogged Boltzmann and the nimble Ostwald to “the fight of the bull with the lithe swordsman. But this time, in spite of all his swordsmanship, the toreador was defeated by the bull. The arguments of Boltzmann broke through. At the time, we mathematicians all stood on Boltzmann’s side.”
By all accounts, in other words, even according to Ostwald and Helm, the much ballyhooed debate in Lübeck was a rout, with Boltzmann on the winning side. The friendship and correspondence between Boltzmann and Ostwald cooled for some years.
Still, though Boltzmann must presumably have felt some sense of victory, he had not succeeded in convincing Helm or Ostwald of the error of their ways. It was a perpetual annoyance to him in his university lecturing if, despite his best efforts, he could not get all the students in the room to grasp what he was saying. Likewise, the fact that he could not persuade his opponents in Lübeck to alter their thinking would have struck Boltzmann as a failure on his part. Boltzmann did not find, in the Lübeck debate, anything like the lively exchange of views he had encountered the previous year in Oxford. He and his opponents had gone at each other stubbornly, stating and restating their opinions, but in the end everyone was left standing and none of the principals had switched sides. If his aim had been not just to explain but to persuade and convert, Boltzmann had failed.
Back in Vienna, moreover, he now found himself face to face with the very symbol of continuing opposition to atomism. Ernst Mach had finally escaped from Prague, and was now a professor in Vienna—and a professor, no less, of philosophy.
The two men had been in Vienna at the same time once before. When Boltzmann was there as an undergraduate, Mach had already graduated and was offering a variety of lecture courses in general physics—none of which Boltzmann attended. When Stefan had taken over the Institute of Physics, Mach had gone off to Graz; Boltzmann went there a couple of years later, but by then Mach had left “cheerful, friendly Graz for beautiful, gloomy Prague.”
Frequently riven by nationalistic conflicts between its German and Czech inhabitants, Prague was not always a pleasant home for Mach. His family was in fact of Czech origin, and Mach, to the inhabitants of Prague and to the students and officials of the university there, was a recognizably Czech name. He spoke only German, however. The mere contrast between his name and his language was enough, in Prague, to raise suspicions.
Over the years, Mach made a number of attempts to return from Prague closer to the center of Austria-Hungary. On a few occasions he found himself in competition with Boltzmann, and coming up short. Mach had some desire for the position that Boltzmann took in Graz after Toepler’s departure, but his cause had too little support. In the middle 1880s, the university in Munich was looking for an experimental physicist, and Mach’s name came up, but at that time he was unable to obtain a release from Prague. A few years after that, it was Boltzmann who landed up in Munich. After Boltzmann’s first refusal of the position made vacant in Vienna by Stefan’s death in 1893, Mach’s name was on the list again, but instead of going to him, the Viennese authorities waited a few months, by which time Boltzmann’s renewed availability became known. Boltzmann returned to Vienna in 1894, while Mach, after 27 years, remained in Prague.
Like Boltzmann in Graz, Mach became university rector for a time in Prague and similarly had to deal with fractious students. Prague was a divided city, with German and Czech factions. As elsewhere in the Habsburg Empire, Slavic populations were beginning to coalesce into factions that opposed the rule of Vienna. Count Edward von Taaffe, an Austrian of Irish ancestry, became chief minister to Emperor Franz-Josef in 1879, the same year that Mach became rector in Prague. It was von Taaffe, in one account, who popularized the notion of fortwürsteln, muddling through. He was pragmatic and not at all ideological, and he tried to deal with simmering and perpetual nationalistic disputes by creating a system in which all parties felt they had some power and participation. This was a generally well-meant philosophy, but it turned inevitably into a succession of stopgap changes that left no one happy for very long.
In Prague, as a result of von Taaffe’s initiatives, the university split into two supposedly equal parts, one Czech and the other German. Neither side wanted to abandon its claims to the ancient buildings of Charles-Ferdinand University, so instead the old institution was divided up, leaving the Czech and German schools sharing the same buildings as reluctant and untrusting neighbors.
Mach, as rector, tried to deal evenhandedly with the factions and was for that reason mistrusted equally by both sides. At a meeting of the German students’ association in 1880, he spoke for tolerance and moderation, but those words were forgotten in comparison to strident noises from other speakers, and when riots between the student groups developed over the next few days, Mach was lumped by the Czech press into the pro-German movement. A few years later Mach was rector for a second term, and agitation between the Czech and German halves broke out once again. This time Mach more actively opposed Czech-inspired attempts to reorganize parts of the university and was branded a pro-German. On the other hand, as overt anti-Semitism began to infect the German faction, Mach distinguished himself by defending the Jews and opposing August Rohling, who had been named professor of Hebrew Antiquities and used his position to broadcast ancient blood libels against the Jews. Mach, meanwhile, was attacked with allegations of atheism, which were more or less true, except that Mach remained formally a Catholic and was not one to advertise his irreligious views. In 1884, beset by endless squabbles over university administration, ethnic factionalism, and religion, Mach threw in the towel and resigned his rectorship, hoping to keep his head down and return to his scientific and philosophical writings. As the 1890s approached, the number of German students in physics fell and financial support for his researches waned. Irked by skirmishing between his old, alcoholic technical assistant and a new, incompetent one, Mach put his oldest son, Ludwig, in charge of his lab, which only added to the resentments.
Nationalistic discontent was rising too in the heart of Austria, but still, Vienna was calmer than the fringes of the empire. And the University of Vienna was still the apex of the Austrian intellectual world. Mach had lost out to Boltzmann once again after Stefan’s death, but his yearning to be in Vienna was powerful. On top of all his other troubles, Mach’s younger son, Heinrich, had killed himself, at the age of 20, a few days after receiving his doctorate in chemistry from the University of Göttingen in Germany. The reasons for his suicide were obscure. Just a couple of weeks later, Mach was in Vienna giving a philosophical lecture on the meaninglessness of cause and effect. His argument, that while one may often observe a certain phenomenon happening after some other, one should resist the temptation to infer any causative link between them, impressed the philosophical faculty and others at the university. A movement began to draw Mach to Vienna as a professor of philosophy rather than physics. In May 1895, he was formally appointed to the chair of the History and Philosophy of the Inductive Sciences. Before leaving Prague he wrote to Boltzmann, saying that he looked forward to a collegial relationship despite their obvious differences of opinion. Boltzmann wrote back in a similar vein, expressing the hope that he would be able to learn from his new faculty colleague.
Mach’s philosophy, no matter how elaborate it became, always rested on a simple principle: science should be based on observable facts, not hypotheses or theories. This may seem an innocent enough statement, and may even seem self-evident, as if science were by definition a matter of dealing with facts about the world, but it soon runs into trouble. The exertion of pressure by gas on the walls of a container is, by Mach’s standards, an acceptable fact, since it is something that all can agree on. To explain that pressure in terms of underlying atomic motions is an unacceptable hypothesis, since it relies on entities—atoms—that cannot be seen.
Similarly, in Mach’s view, heat is a primary phenomenon, a fact, since anyone can feel the difference between a hot dinner plate and a cold doorknob on an icy day. Portraying heat as atomic motion is unacceptable.
Even the most ardent atomists had to accept that the objects of their affections and desires could not be directly observed. The philosophical question, at its simplest, therefore boiled down to a debate over the extent to which it was permissible in science to make hypotheses in order to achieve a fuller or more unified understanding. The atomists claimed that they could explain both heat and pressure in similar terms—atomic motions—so that the hypothesis of atoms led to a deeper understanding.
In Mach’s estimation, this supposed increase in understanding was an illusion, purchased at the expense of dreaming into existence particles whose reality could not independently be judged. This argument, conceived by Mach at a time when theoretical physics was young, reverberates still today. During the 20th century, physicists have predicted the existence of many a subatomic particle that was only later proved experimentally to exist, and now some physicists argue for the existence of superstrings and other curious entities that will never be seen directly. It remains, even now, a profound question whether the cost of proposing such very hypothetical objects as superstrings is sufficiently compensated by the benefit in understanding that the hypothesis brings. Mach’s critical attitude retains merit.
In his own day, however, Mach sought to deal with his concerns by stifling theory altogether. He insisted that the atomic hypothesis was beyond the bounds of true science, and that the physicist should instead deal with temperature and pressure as fundamental entities in themselves. A scientific understanding of the behavior of gases then amounted to describing empirical relationships between temperature and pressure. Mach’s stringent philosophy makes science, in essence, a matter of depiction rather than anything that Boltzmann or his sympathizers would call understanding. As he pursued in his single-minded fashion the consequences of what he took to be an unarguable assumption about the nature of science, Mach came to disparage theories altogether as mere props to understanding. If the kinetic theory of gases was of some assistance in puzzling out new aspects of the measured behavior of gases, then it was not altogether useless or inadmissible, but the point was to eventually disencumber oneself of the theory once new relationships among measured properties were found and established.
Mach began to think of himself as an “anti-philosopher.” Any philosophy, any systematization of knowledge, was usually founded on some assumption, whereas Mach, so he claimed, began with no assumptions at all. Stick to verifiable facts, he declared, make the goal of science the finding of mathematical relationships between those facts, and what would result would be an utterly reliable account of the workings of the world. Theories had meaning only insofar as they were useful and practical; the intellectual content of theory had no meaning at all. Or rather, any meaning it had was put there by theorists, not derived from the world itself. Atoms were a fiction, Mach maintained, possibly a useful device but nothing more.
But Mach’s own philosophy, whatever he may have thought, had assumptions of its own—principally that facts were unarguable, and that everyone could agree on what the facts were. The stringent belief that one must take every observation at face value is the central unexplained element of Mach’s view of the world, and he seems to have held it since childhood. As a young boy, he later recollected, he found great difficulty in understanding why a long table looks wider at the near end than the far end, and he seems in a sense never to have got over this difficulty. He objected to the way artists used tricks of perspective to portray three-dimensional objects on a two-dimensional canvas, as if they were somehow fooling the viewer into accepting a distorted view of reality as the real thing.
His later philosophy amounted to an insistence, in all areas of science, that a long table really is wider at the near end, because that’s how a viewer sees it. But therein lies the flaw in Mach’s thinking. An observer who walks around a table sees its appearance change. The conventional explanation is that there exists an independent object called a table, endowed with certain fixed properties, and its changing appearance is the result of looking at it from different perspectives. Reality, in other words, is the table itself, not the way the table happens to look from this or that angle. Similarly, in science, true reality is not what is seen directly but what is consistently inferred from a variety of observations. This is why Boltzmann and the atomists believed in the value of what they were doing. It might not be possible to see an atom directly, just as it is not possible to see the real table, free of the distortions of perspective, in any single view. But the table exists, and so do atoms.
Mach clung to his ideas fiercely throughout his life. Unlike Boltzmann, who had been an eager but docile schoolboy, Mach had resented instruction where it disagreed with his own notions. He liked to memorize the features of geographical maps or lists of historical events, but he struggled against conjugations and declensions in Greek and Latin, those being arbitrary and therefore unreal constructions to him. His obstreporousness may have lain behind the judgment of the Benedictine monks at one of his schools, who had found him “sehr talentlos.” Mach’s father, himself a schoolteacher, took his son away from the care of the monks and taught him at home for a while, and succeeded in getting the classical languages into his head. Later, Mach went to another public school, and once again had difficulty. Formal instruction again triggered rebellion in him. Like many children who know themselves to be intelligent but find themselves in difficulty at school, he attributed the success of others to a dubious kind of “school cleverness and slyness” that he himself lacked.
Nevertheless, he graduated from this school, and in 1855, at the age of 17, arrived in Vienna to study mathematics and physics at the university. Although he did well as an undergraduate, supporting himself by tutoring, he continued to complain. “Kaiser Franz had let the Austrian universities go to the dogs, and I didn’t have enough money to go to a German university,” he recalled much later. He remained “a stranger with respect to all of my professors, an outsider, someone they mistrusted and against whom they visibly tried to excite mistrust.” Boltzmann, who arrived in Vienna a few years after Mach and studied mainly under the congenial Josef Stefan, remembered his undergraduate days as a period of encouragement and accomplishment compared to which his later academic experiences seemed less golden. Mach was there before Stefan arrived, and studied with the older, less distinguished, and considerably less modern physicist Andreas von Ettingshausen. Even so, the difference between Mach’s recollections of the University of Vienna and Boltzmann’s reflects their own personalities as much as any great change in the nature of the institution after Stefan took over. Boltzmann soaked up knowledge gleefully; Mach took it in bit by bit, critically, judging each idea against his own notions, undeveloped as they were at that stage, of what knowledge ought to consist of.
After graduating, Mach began to deliver lectures in elementary physics, earning some money so that he could also do experimental research in a variety of subjects. His lectures were popular, but printed booklets of his teaching failed to find many readers. At the same time—and quite unlike Boltzmann—he fell in with a varied group of writers, journalists, and critics who used to meet at the Café Elefant in Vienna. Especially toward the end of the 19th century, the numerous coffeehouses of Vienna served as the foci for various earnest intellectual groups: the Freudians at one establishment, Mahler and his group at another, Trotsky presiding at a third. Coffee came into the city’s life in a characteristically Viennese way, arriving by the agency of a would-be invader. The Ottoman Turks were, over the centuries, frequently at the gates of the city, and only after their final defeat at the end of a siege in 1683 was Vienna finally free of the Turkish threat. Still, the Viennese abstracted a number of choice elements of Turkish culture, coffee being first among them. The traditional story is that the fleeing Turks left behind bags of unroasted beans along with brewing equipment, and that a Polish spy working for the Austrians took charge of the mysterious items, since he was the only person who knew their purpose, and founded the city’s first coffeehouse.
Mach owed his introduction to café society to his knowledge of the tone perception theories of Helmholtz. Overhearing a conversation led by a newspaper music critic, Mach was invited into the discussion and impressed the crowd with his ability to explain scientific aspects of the way music is heard. The classical education dinned into him by his father now yielded benefits, and he joined a diverse group of writers, musicians, and social philosophers. From the latter, in particular, he gained some understanding of broader issues that would in time come to inform his own thinking.
During his tenure in Prague, Mach developed from a versatile if not brilliant physicist into a more focused and purposeful philosopher. His scientific achievements included the invention of a laboratory demonstration of the Doppler effect, which became a staple of physics classes for years to come. He studied acoustics and fluid flow, made attempts at microphotography, worked at improving a medical device to measure blood flow and pulse rate, and began to think about human perception of shapes, colors, and sounds with the idea of coming up with physical explanations for physiological responses. He had even dabbled in atomic theory—not very successfully, as it turned out—in order to understand the flow of liquids through tubes. These wide-ranging activities attested to a fertile mind and an ingenious pair of hands, but they also betrayed an attitude that the job of a physicist was primarily experimental and observational. Even at this early stage, Mach distrusted theorizing and mathematizing except insofar as it might provide simple explanations for the quantities a physicist could measure in the lab.
From the beginning he had interests beyond the world of physics alone. In attempting to explore scientifically the means of human perception, Mach had come into contact with a long seam of philosophical thought. This in turn had led him to start thinking about the development of physics from a philosophical standpoint, and he began to attract a little—at first, a very little—attention as a commentator on scientific history and philosophy. In 1872, after he had been in Prague for five years, his book The Conservation of Energy appeared. In it, among other things, Mach made some attacks on the kinetic theory of heat and atomism in general. But the book was perhaps too philosophical for most scientists, too scientific for most philosophers, and generally too poorly argued for both camps. It quickly disappeared.
Mach wrote prolifically, both books and scientific articles, describing his experimental achievements, but for many years his writings slipped from the presses and sank out of sight with hardly a ripple. Indefatigable, he kept at it and gradually began to make an impression. As the years in Prague wore on, he turned away from experimental work and increasingly to writings of an expository and philosophical manner. After The Conservation of Energy, his 1883 monograph The Science of Mechanics was a much greater success and began to exert an influence on a younger generation of physicists.
Mach tried to draw a distinction between “mathematical physics,” which was simply the elucidation of mathematical relationships between measurable physical quantities (and which was a good thing), and “theoretical physics,” which connoted the attachment of deeper meaning, the attribution of some sort of reality, to mathematically defined quantities (and which was, by Mach’s lights, a bad thing). At the end of The Conservation of Energy, Mach had declared that “the object of natural science is the connection of phenomena, but theories are like dry leaves which fall away when they have ceased to be the lungs of the tree of science.” In much of his writing, Mach was at pains to scrutinize large areas of science, especially physics, and judge what were leaves and what was solid wood.
Even Newton did not come up to Mach’s standards. He found that in Newtonian mechanics the concepts of “mass” and “force” are not defined independently from directly measurable quantities, but achieve definition only through the very laws they enter into. In other words, a mass can be defined from the force needed to move it, but forces are themselves defined according to their ability to move mass. This, Mach thought, was circular and unacceptable. He was half-right: Newton’s laws do contain a degree of circularity, but the success of these laws is not that they are somehow self-evident, or can be derived from some more fundamental laws, but that they in effect define the subject they aim to describe. This is not a weakness but a strength.
To put it another way, any new scientific law has to rest on some kind of theoretical assumption. Newton showed not that mass and force could be independently defined in some unarguable way, but that the mass and force implied by his laws had universal meaning and applicability. This is indeed somewhat circular, but it has to be: Newton is erecting a theoretical edifice where none previously existed.
This was an aspect of scientific theory whose necessity and inevitability Mach could never grasp. He wanted all laws to rest only on definitions that had some obvious independent meaning; he would not, or could not, accept that science must devise qualities and characteristics whose usefulness can be proved only within the system that defines them. For scientific theories, the proof of the pudding is in the eating.
Mach’s antipathy to theorizing and to the invocation of “metaphysical” and therefore unprovable notions led him to some extreme opinions. In The Conservation of Energy he remarks: “We say now that water consists of hydrogen and oxygen, but this hydrogen and oxygen are merely thoughts or names which, at the sight of water, we keep ready to describe phenomena which are not present but which will appear again whenever, as we say, we decompose water.”
Mach is saying that it is acceptable to speak of hydrogen and oxygen when they are individually present, but that any suggestion that hydrogen and oxygen constitute water, or that water contains hydrogen and oxygen, is going beyond the bounds of reason. This is, clearly, a restrictive philosophy. It means that a scientist may only say “there was hydrogen and oxygen, but now there is water” or “there was water, but now there is hydrogen and oxygen.” To hint that water is actually made of hydrogen and oxygen is to suggest a metaphysical connection that goes beyond the certifiable facts.
Mach’s argument here is oddly reminiscent of a stage that very young children go through. Infants do not understand how a teddy bear can disappear behind a screen and then reappear on the other side; they tend to think that the bear has gone, do not know to look for it behind the screen, and regard the newly revealed bear as a quite new object. But very early, in an automatic developmental stage, infants realize that the bear is still there, but out of sight. They know it is behind the screen even though they can’t see it. Evidently, the urge to attribute reality to objects we cannot see is something we all learn at an early age, and which is essential to our being able to navigate in the real world. But Mach, as with his difficulties in grasping the notion of perspective in art, seemed to put his strict conception of reason above simple common sense. He could not accept that hydrogen and oxygen continued to exist while they were hidden from him in the form of water.
Nevertheless, Mach’s views began to win a following. To be fair, he preached an adherence to experimental facts and a caution against unfounded theoretical speculation, which were, and remain, important elements of scientific style. He emphasized the importance of simplicity, in the traditional sense of trying to find the simplest account of observed phenomena but also in a larger sense, that scientific explanation as a whole should constitute as simple and coherent a system as could be obtained. But his devotion to these reasonable causes was overzealous to the point of fanaticism, and in the end his philosophy, as far as it concerned the practice of science, amounted to a list of things that scientists ought not to do. Theorizing was prime among those sins.
The second half of the 19th century was the period when theoretical physics first began to establish itself as a subject in its own right. Max Planck recalled that when he was an undergraduate in Munich in the 1870s, he could not study theoretical physics because no such course was offered; he learned experimental physics and mathematics separately. But that was changing. Boltzmann went to Munich in 1890 specifically as a professor of theoretical physics, and he continued teaching in Vienna under that same banner. Kinetic theory and Maxwell’s electromagnetic theory were the first great theoretical constructs, harmonizing large subjects by means of a physical model expressed in explicitly mathematical terms. Not everyone in physics saw this as a laudable development, however, and Mach’s ideology became a flag for those who found the methods and ideas espoused by Clausius, Maxwell, and Boltzmann too abstract, too removed from the empirical world, to qualify as acceptable science.
Mach’s return to Vienna, at the age of 57, became a triumph. His lectures, which ranged across history and philosophy, physics and psychology, drew large, rapt audiences. He was saying nothing that he hadn’t been saying, and writing down in numerous dense volumes, for many years, but in Vienna he finally gained a wide following among intellectuals of all stripes. Mach had become increasingly enamored of an idea borrowed from the fledgling science of economics, according to which there was a marketplace of ideas that favored simplicity as a kind of efficiency: the most explanatory power for the least investment of hypothesis. This principle certainly should apply to physics, he argued, where it ruled against elaborate theorizing and in favor of mere description and observation, but it could also be extended into the arena of moral and ethical thought. Mach had a kind of Panglossian belief, coupled with an interpretation of Darwin’s thinking on evolution, that from the general ferment of ideas and behaviors moral actions would emerge as those that benefited most of the people most of the time. In a Vienna fragmented by nationalistic and political divisions of growing vehemence, this may have seemed a soothing philosophy, suggesting that one should refrain from trying to explain what was going on in terms of mysterious and subterranean social forces, and trust instead that all would be well eventually. In the mid-1890s, Ernst Mach became a public intellectual of great repute, attracting some young physicists to his cause but also influencing poets and writers, musicians and artists.
All this must have been galling to Boltzmann. He had come to Vienna on the death of his mentor Josef Stefan, and very soon afterward, in July 1895, his old colleague Josef Loschmidt had died. “Did I return to Vienna as the gravedigger for those who had been so dear to me?” Boltzmann lamented in a memorial address. Loschmidt had sometimes been critical of kinetic theory—it was he who first voiced the reversibility objection clearly—but like the British physicists, he fundamentally believed in atoms and wanted above all to find out how they worked. Boltzmann never became close to the younger (and generally less talented) physicists who were in Vienna when he returned, many of whom came under Mach’s influence. Boltzmann had hoped that in Vienna he would find serious-minded colleagues ready to debate atomism and kinetic theory; he found, instead, a university in the grip of a philosophy he could not understand, or if he did, thought it foolish.
ADDING TO Boltzmann’s woes, the old reversibility objection came up again, in somewhat different form, and from a new corner. The great French mathematician Henri Poincaré had in 1893 proved a theorem showing that any closed mechanical system must, in the course of time, return to its starting point. This conclusion was relevant to a point that both Boltzmann and Maxwell had addressed but not resolved. In thinking about the way a set of atoms constantly moves from one possible dynamical state to another, they had assumed, without proof, a degree of randomness, so that a gas would visit all possible states in an essentially statistical manner. Poincaré’s theorem proved that in at least one respect this randomness was not absolute. It stated with mathematical certainty that the system would at some point come back to its starting point and therefore begin repeating itself. Poincaré noted at the time that his result might prove troublesome for what he called “the English kinetic theories.”
A couple of years later, a student of Max Planck’s by the name of Ernst Zermelo made the attack specific. If, as Poincaré’s theorem demanded, the atoms in a gas must sooner or later return to the exact configuration they began in, then Boltzmann’s H-theorem could not always be obeyed. If the system evolved at first so that H decreased and entropy increased, then eventually it must go back the other way, with H increasing and entropy decreasing. Therefore, Zermelo said, the idea of kinetic theory that a gas of atoms would inevitably evolve toward equilibrium—maximum entropy—and stay there was simply wrong.
Although he had a powerful new theorem to back him, Zermelo was saying nothing that Loschmidt and then the English critics had not said before. Yes, Boltzmann agreed, a system might sometimes move in a way that would decrease entropy. Now Poincaré had proved that such things indeed must happen. But the question, as always, was how likely such events were. To say, even with mathematical certainty, that an event must happen is not to say that it will happen often, or even in a humanly imaginable period of time. Boltzmann took to the fray again, no doubt with some weariness. His published reply to Zermelo displays a mixture of sarcasm and petulance.
“Herr Zermelo’s paper indeed shows that the relevant works of mine have not been understood; even so I am bound to be pleased by this paper as the first indication that these works have been given any attention at all in Germany,” he declared in the introduction, and after going through all the technical matters concluded: “All objections raised against the mechanical view of nature are thus void of substance and based on error. He however who finds himself unable to overcome the difficulties that a clear exposition of gas-theory principles offers should in that case follow Herr Zermelo’s advice, and resolve to give the matter up.”
To make his point specific, Boltzmann estimated the approximate time that a simple system of about one trillion atoms, constituting one cubic centimeter of gas at room temperature, would take to return, as Poincaré said it must, to some precise dynamical state it had already passed through. The recurrence interval he came up with was a number of seconds containing trillions of digits—an unimaginable length of time. For comparison, he observed that if every star in the sky had the same number of planets as the Sun, and if every planet had the same number of people on it as the Earth, and if every one of those people lived a trillion years, the sum total of their combined lifespans would even so amount to a number of seconds with less than 50 digits. Poincaré’s recurrence theorem might be mathematically unarguable, he concluded, but it was not of any practical concern.
He summarized Zermelo’s objection in a more easily grasped way. From a strictly mathematical point of view, a set of 1,000 dice thrown enough times must eventually come up all ones. But such an outcome is fantastically unlikely. Zermelo, Boltzmann said, “is like a dice-player who . . . concludes that something is wrong with his dice because such an occurrence has not yet presented itself to him.”
More subtly, the reconciliation between Poincaré’s recurrence theorem and Boltzmann’s H-theorem hangs on probabilities and timescales. Given a cosmic perspective, in which one is prepared to watch for limitless eons, Poincaré and Zermelo are correct: a system must eventually come back to its starting point. But on human timescales (and even on timescales of trillions and trillions of years), the likelihood of recurrence is negligible. In practical terms, therefore, the assumption that a gas explores all the available dynamical states in a random fashion may not be strictly true, but it’s pragmatically close enough to true as makes no difference. Once again, Boltzmann’s sense of the physics of the matter held true.
In both his Nature communication of 1895 and the response to Zermelo in 1896, Boltzmann enlarged on this point a little. In thinking of the universe as a whole, which was generally presumed at that time to be eternal, it might seem that everything would have to settle down into a perfectly uniform, perfectly stable equilibrium—clearly not the heterogeneous universe of stars and planets and empty space that astronomers were beginning to map out. The notion of an inexorable winding down of the universe into a featureless stasis had been pointed out by Clausius, who called it the “heat death.” Boltzmann now suggested that even in such a state, there would be pockets that, strictly for reasons of chance, ran temporarily away from the general equilibrium and then fell back again. The corner of the universe currently occupied by humanity, he suggested, must be just such a place, where entropy happened to have hit a temporary low and was increasing again. Elsewhere there would be pockets of the universe where entropy was running down, and in such places, Boltzmann speculated, it might appear that time itself was running backward.
To Boltzmann, this may have seemed like a plausible speculation adding to the depth and interest of his kinetic theory. To his critics it seemed like a farfetched piece of backpedaling, indicative of the lengths Boltzmann had to resort to in order to defend his theory. Far from being a true and definitive result, the H-theorem, Boltzmann now seemed to be admitting, was true only some of the time, in certain special places in the universe. There remained, on the question of reversibility, no consensus. Few physicists had yet accustomed themselves to arguments involving matters of probability. Poincaré’s theorem was perfectly true: a system must eventually come back to its starting point. Boltzmann accepted this, yet argued in his baffling way that somehow it wasn’t important.
Zermelo made a further brief reply, notable in that he expressed considerable surprise that Boltzmann blithely admitted the second law of thermodynamics to be a probabilistic and not an absolute rule. Zermelo was Planck’s student, after all, and even in the mid-1890s this notion shocked him. Still, he was not alone in his puzzlement. In England, William Thomson (who had become Lord Kelvin in 1892) had run up against the idea of theoretical predictions that amounted to calculations of probability, not assertions of certainty, and the seeming contradiction had brought him to a halt. In 1895, he wrote to Boltzmann (the two had corresponded occasionally for a few years) that “whenever other occupations allow me I return to it, but alas! I make absolutely no progress towards comfort of happiness in regard to it. This is very sad, as on it the whole of Thermodynamics hangs.”
Zermelo’s objection struck many physicists as acute, and Boltzmann’s answers as evasive. Planck and Kelvin were respected and influential figures. Boltzmann began to feel once more unhappy, unappreciated, and alone. Mach’s disciples began to refer to Boltzmann as “the last pillar” of atomism. Other young physicists recollected that “with few exceptions, the leaders in Germany and France were persuaded that the atomistic kinetic theory had already played out its role” and that the atomists “in those days fought somewhat on the defensive.”
Maxwell was long dead, Clausius had died in 1888, and Stefan and Loschmidt were more recently gone. In Vienna in the 1890s, Boltzmann felt himself surrounded by his intellectual enemies, with no young physicists to support him. His letters to a journal editor in connection with the publication of Zermelo’s objection and his rebuttal reveal how much Boltzmann felt himself isolated and vulnerable. “Now I come to a delicate point,” he wrote. Observing that Planck was an adviser to the journal and Zermelo his student, he went on, “I believe I am entitled to demand: 1, that Herr Planck does not delay the appearance of my note; 2, that no word of it should be changed; 3, that no reply [from Zermelo] should appear in the same issue, but that they should answer later as they wish and are able to.”
In the same letter Boltzmann portrays his beleaguered position: “Since I am, so it appears, now that Maxwell, Clausius, Helmholtz, etc., are dead, the last Epigone for the view that nature can be explained mechanically rather than energetically, I would say that in the interests of science I am duty-bound to take care that at least my voice does not go unheard.” (In Greek mythology the Epigoni were seven warriors who sacked Thebes to avenge the deaths of their fathers in an earlier failed attack on the same city.) In another letter to the same editor Boltzmann muses, “Whether I will soon be alone in opposing the present direction of German science I cannot say.”
Boltzmann’s enemies were calling him “the last pillar” and he himself seemed to concur at times in that judgment. Even so, he strove to stay upright.