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It may sound strange to refer to Dirac as a natural philosopher, a term with a curiously archaic ring to it. But until the beginning of the 20th century, British physicists often preferred their field to be called “natural philosophy” rather than “physics.” Dirac’s predecessor in the Lucasian chair, Joseph Larmor, was of the old school and thought of himself as a natural philosopher rather than a physicist. When Max Born became professor of physics at the University of Edinburgh in 1936, his title was Professor of Natural Philosophy. One of Dirac’s first papers, a note on “The Relativity Dynamics of a Particle,” was published in 1924 in the Philosophical Magazine, a distinguished journal of physics founded in 1788. So Paul Dirac could rightly be called a natural philosopher, at least of a kind. But he was definitely not a philosopher in the conventional sense, and he never fancied himself as one. In fact, he was plainly uninterested in philosophy, a subject of which he had only the most superficial knowledge and was never tempted to take up for serious study.
As an engineering student in Bristol, 18-year-old Dirac listened to Broad’s lectures on the philosophical aspects of relativity theory, which induced him to do some reading in philosophy. What he read did not appeal to him, and he soon came to the conclusion that philosophy was merely an unimportant play with words. He felt as he stated later in life, “that all the things that philosophers said were rather indefinite, and [I] came to the conclusion eventually that I did not think philosophy could contribute anything to the advance of physics.” This was a view that stayed with Dirac throughout his life. In a 1963 interview, he put it this way: “I feel that philosophy will never lead to important discoveries. It’s just a way of talking about discoveries which have already been made.” Notice that for Dirac philosophy could only be of value if it served physicists in their theories and experiments. He did not appreciate that there might be other reasons to take philosophy seriously.
Dirac’s attitude to philosophy, more indifferent than hostile, was not shared by many of his continental colleagues. Physicists such as Schrödinger and Heisenberg, not to mention Einstein and Bohr, were deeply interested in philosophical questions and not only in those that were considered scientifically useful. On the other hand, Dirac’s attitude was far from exceptional. “Philosophy of science is as useful to scientists as ornithology is to birds,” Feynman is supposed to have said. Never mind if the quotation is apocryphal (which it is), it expresses a view shared by many physicists, including Dirac. Nevertheless, philosophical considerations—in the vague meaning of considerations that go beyond measurement and calculation—were not absent from his science. There is, in fact, a good deal of implicit or spontaneous philosophy in Dirac’s writings, ideas of a philosophical kind that grew out of his physics. For example, the Large Numbers Hypothesis (LNH) was partly of a philosophical nature. Although certain consequences of the LNH can be tested, such as a varying gravitational constant, the LNH itself can neither be tested nor derived from the laws of physics. It is a belief beyond the range of science.
Recall Dirac’s prediction of magnetic monopoles in 1931. Dirac proved that, under certain circumstances, monopoles could exist according to the laws of electrodynamics and quantum mechanics. But do they actually exist? Dirac thought they did, his sole argument being that “one would be surprised if Nature had made no use of it,” namely, turned the possibility into a reality. Without reflecting on the nature of his argument, in effect, he postulated that there is a close correlation between potential and actual existence. And more than that, because the former kind of existence requires the latter. If an entity is allowed to exist, meaning that it does not violate any of the fundamental laws of nature, then it does exist. This kind of argument, or rather belief, was of a philosophical nature since it could not be justified by either experiment or theory. Dirac may not have been aware that he argued as a philosopher, but in fact, he did.
When the monopole began to attract attention in the physics community and experimentalists started to hunt for the mysterious magnetic counterpart to the electron, the argument was sometimes stated in more explicit and elaborate forms. Thus, in a popular 1963 paper on monopoles, we read: “One of the elementary rules of nature is that, in the absence of law prohibiting an event or phenomenon, it is bound to occur with some degree of probability. To put it simply and crudely: Anything that can happen does happen. Hence physicists must assume that the magnetic monopole exists unless they can find a law barring its existence.” Another and shorter version that physicists are fond of is, “anything which is not prohibited is compulsory.”
The argument used rather casually by Dirac in 1931 has a long and dignified past in the history of ideas where it was extensively discussed by philosophers and theologians centuries before it entered the physical sciences. Known as the principle of plenitude, in its modern version, it goes back principally to the great German philosopher and mathematician Gottfried Wilhelm Leibniz. When physicists today believe in the existence of black holes, gravitational waves, supersymmetric elementary particles, other universes, and superstrings, it may be because there is some observational evidence for the entities; but a stronger reason for this belief is that they can be described consistently by the laws of physics. High-energy physicists were convinced that the Higgs particle existed many years before it was detected in 2012. It just had to exist. The principle of plenitude is well and alive.
At about the same time that Dirac suggested the monopole, he applied similar reasoning in his interpretation of the negative-energy solutions appearing in his relativistic wave equation of 1928. The two extra variables that turned up in his wave function had to correspond to something physical. They couldn’t be just mathematical symbols without physical referents. Dirac was inclined to think that the potential existence of positive electrons implied that they actually existed, meaning that they were inhabitants of the real world that physicists investigated. In all probability, he had never heard of the philosophical principle of plenitude. But in his 1931 paper, he referred to a somewhat similar principle suggested by Eddington as a means of establishing relations between mathematical and physical quantities appearing in general relativity theory. Dirac adopted Eddington’s principle in a wider form, as a realist interpretation of mathematical quantities in terms of physical entities.
Apart from his work on monopoles and positrons, Dirac also applied a kind of plenitude reasoning in 1941 when he proposed a new version of quantum theory based on the counterintuitive notion of negative transition probabilities. Rather than dismissing such transitions between quantum states as non-existent, he wanted to interpret them physically. “Negative energies and probabilities should not be considered as nonsense,” he said, pointing out that “they are well-defined concepts mathematically, like a negative sum of money.” On the other hand, Dirac did not believe that just because a concept was mathematically well-defined it would be present in nature in some form or another. The negative probabilities “should be considered simply as things which do not appear in experimental results,” he noted. If they did not appear in experiments, then where did they appear? According to Dirac, they belonged to a “hypothetical world” with properties that could tell the physicist something about the real world to which experiments were limited.
Dirac was aware that not all mathematical terms that appeared in a physical theory could be ascribed a physical meaning. After all, mathematics is immensely greater than physics, including an infinity of structures and symbols that cannot possibly have any counterpart in nature. What about a space with an infinite number of dimensions? There is a lot of difference between the claim that “the whole of the description of the universe has its mathematical counterpart,” as Dirac said in 1939, and the claim that the whole of mathematics has its physical counterpart. Dirac never went as far as some modern physicists who have seriously suggested that there was a one-to-one correspondence between mathematics and physics—in other words, everything that exists mathematically also exists physically. This kind of mathematical metaphysics did not appeal to Dirac. But he thought that mathematical relations with a high degree of beauty were likely also to represent something real in nature. His implicit use of the principle of plenitude was guided by an even more powerful principle, what he sometimes referred to as the principle of mathematical beauty.
It is not surprising that Dirac was a believer in unity, namely, that in the last resort, nature consists of very few basis entities, which can be understood in terms of very few basic laws. This is a credo common to the large majority of physicists. The belief that nature, despite its confusing diversity, is made up of just a single entity in different disguises has been a powerful—if sometimes deceitful—guiding theme throughout history, from the atoms of the ancient Greeks to the superstrings of modern mathematical physics.
This belief influenced Dirac’s thinking when he was looking for a candidate for the hypothetical positive electron and at first proposed the proton. In this way, he would “have all matter built up from one fundamental kind of particle,” a result that appealed greatly to him. It would have been much nicer with just one elementary particle than two of them. Two is also a small number, but it is not aesthetically pleasing in the same way as one. In 1933, Dirac and most other physicists had accepted the neutron as a new constituent of the atomic nucleus. At the Solvay Congress that year, he suggested that the nucleus, in addition to protons and neutrons, also contained electrons. He had no problem with this more crowded picture of the nucleus. His curious argument was that “this number [three] may appear to be large, but, from this viewpoint, two is already a large number.”
The unity of nature may refer to its constituents, but it may also refer to the methods, laws, and equations by means of which nature is understood. Dirac firmly believed that all of nature should be subject to the same mathematical treatment. In 1939, he used this version of nature’s unity to provide an argument against the determinism of classical mechanics that was quite independent of quantum mechanics. Classical or Newtonian mechanics was a powerful theory that Dirac much admired, but it was “very unsatisfactory philosophically, as it goes against all ideas of the Unity of Nature,” he said. Dirac’s reasoning was that in a problem of classical mechanics the equations of motion were in a mathematical form, while the initial conditions were not. A freely falling body was governed by Galileo’s law, but we could only tell what the speed of the body had been at some point in space if we knew where it started and what its initial velocity was. And the initial conditions were not subject to any law. In this way, Dirac suggested, an unpleasant asymmetry arose in the classical description of nature, separating it from one sphere ruled by mathematics and another sphere where mathematics did not apply. This he found intolerable because it ran contrary to the expectation of unity in nature.
Dirac’s dream, as he exposed it in his 1939 lecture, was that ultimately all of nature, including initial conditions, elementary particles, and constants of nature, should be amenable to mathematical analysis. In Dirac’s philosophical vision, the physicist of the future would be able to deduce everything in the universe by pure mathematical reasoning. He would be omniscient, as God: “A person with a complete knowledge of mathematics could deduce, not only astronomical data, but also all the historical events that take place in the world, even the most trivial ones.”
More realistically, Dirac’s concern for the unity of nature—or rather the unity of physics—showed itself in his response to the so-called S-matrix theory, which in the 1960s was a popular alternative to quantum field theory. However, it was primarily a theory of nuclear forces and did not describe electrons, for example. Although S-matrix theory built on a foundation that Dirac in many ways agreed with, he nonetheless found the theory to be unattractive. His principal objection was that it destroyed the unity of physics. “We have reason to think of physics as a whole and we need to have the same underlying basis for the whole of physical theory,” he wrote in 1969. And three years earlier: “In physics one should aim at a comprehensive scheme for the description of the whole of Nature. … It is necessary that quantum field theory be based on concepts and methods that can be unified with those used in the rest of physics.”
While Dirac was in favor of unity in terms of concepts and methods, he remained skeptical concerning ambitious attempts at constructing a final and grand unified synthesis of all of physics. The general idea of this kind of theory is to find an equation or a system of equations that encompass and unify all the laws of physics, including quantum mechanics and general relativity. Such an ultimate theory cannot be derived from a still deeper theory—if it could, it would not be ultimate. The dream still attracts considerable interest among modern physicists who refer to it as the Theory of Everything (TOE), a name that was coined in 1985. The idea, however, is old. During Dirac’s career, it was pursued by, among others, Eddington, Heisenberg, and the great German mathematician David Hilbert. Dirac did not believe in a theory of everything, but thought that the secrets of nature would only be revealed by gradually improving and criticizing existing theories. He likened the process to biological evolution, although one where mutations played an important role. First of all, fundamental changes would have to follow equally fundamental changes in the mathematical framework of physics. These changes would mutate indefinitely.
In the fourth, 1958 edition of Principles of Quantum Mechanics, Dirac criticized “from general philosophical grounds” the underlying reductionist attempts at constructing an ultimate theory of matter. Ordinary matter consists of atoms, which are built up of electrons and nuclei; the nuclei are combinations of protons and neutrons, and both of these nuclear particles consist of tightly bounds quarks. But why shouldn’t electrons and quarks consist of even more fundamental particles? As Dirac argued, this kind of classical reductionist thinking was hopeless because it led to an infinite regress: “It becomes necessary to postulate that each constituent part is itself made up of smaller parts, in terms of which its behaviour is to be explained,” he wrote. “There is clearly no end to this procedure, so that one can never arrive at the ultimate structure of matter on these lines.”
Since the 1920s, the interpretation of quantum mechanics has attracted an enormous amount of attention among physicists as well as philosophers. Publications on the subject can easily fill a medium-sized library. As noted in Chapter 2, Dirac was not much involved in the debate over the quantum world that occurred in the 1920s and 1930s, but he largely agreed with the mainstream views of Bohr, Heisenberg, and Pauli. When Heisenberg created the basic framework of quantum mechanics in 1925, he was inspired by the semi-philosophical idea that the new theory should contain only relations between observable quantities. Impressed by what he sometimes referred to as “Heisenberg’s principle,” Dirac praised it on several occasions.
One version of Heisenberg’s observability principle implies that quantities that are not observable do not belong to the physical world. In a few cases, Dirac used this crude version to rule out the existence of entities, arguing that they could not be observed either in principle or in practice. For example, in 1936, he dismissed the neutrino on this basis. Dirac was no less off target in 1962 when he used similar reasoning to exclude the interior of black holes from the realm of physics. At the time, he worked on problems of general relativity, and at a meeting of the Royal Society, he discussed some of his results. The mysterious black holes—the name “black hole” was only coined five years later—attracted much interest among physicists trying to understand and further develop Einstein’s theory of gravitation. Dirac knew that the interior of a black hole could be described mathematically, but also that a signal emitted from the interior would never escape the black hole. Hence, the interior was unobservable and for this reason, Dirac concluded that it “must belong to a different universe and not be taken into account in any physical theory.” He was as wrong in this case as he was in the case of the neutrino. Although black holes have not been detected in the same direct way as neutrinos, in a more indirect way they have been observed. Even the interior of a black hole is no longer considered unobservable in principle.
In most other cases, Dirac did not apply the observability principle in this rigorous way. On the contrary, he often proposed quantities that had only the slightest connection to experimental results, such as quantum states with negative energy. Despite his rhetoric, Dirac realized that the division between observables and non-observables was not given by nature alone, but also by theory. “There must be unobservable quantities coming into the theory and the hard thing is to find what these unobservable quantities are,” he concluded in a paper of 1973.
In his later years, Dirac often expressed views about the quantum world, which not only differed from those he had held earlier, but also were at odds with the ideas entertained by the majority of physicists. One of the cornerstones of quantum physics is Heisenberg’s uncertainty principle, as given by the formula ΔqΔp ∼ h. The validity of Heisenberg’s principle, admitted even by Einstein, presupposes that Planck’s constant h is absolute and fundamental. Yet in 1963, Dirac questioned the unquestionable, proposing that h might in some way be derived from the elementary charge e and the speed of light c. There is little doubt that the radical proposal was related to his cosmological ideas based on the constants of nature. At any rate, if h is not fundamental, it seems to follow that the uncertainty relations are not fundamental either. “I think one can make a safe guess that uncertainty relations in their present form will not survive in the physics of the future,” he suggested. It was not, perhaps, a safe guess, but it was daring. Dirac continued to doubt the fundamental nature of Heisenberg’s principle. In a 1976 lecture, he denied that it was really a cornerstone of quantum mechanics, arguing that “it is not a precise equation, but only a statement about indeterminacies.”
While Dirac had almost nothing to say about the Bohr-Einstein debate in the 1930s, later on, he did refer to it in a surprising way, given his earlier allegiance to the ideas of the Copenhagen school. “My own belief was that Einstein was basically right, but he did not have a sufficiently general mathematical basis,” he wrote in unpublished notes from 1974, apparently referring to his view in the 1930s. Five years later, he made it clear that his allegiance was to Einstein rather than Bohr. According to Dirac, quantum mechanics was not in its final form, such as indicated by the infinity problems of quantum electrodynamics. “The new quantum mechanics will have determinism in the way Einstein wanted,” he noted. Then he added: “I think it is very likely, or at any rate quite possible, that in the long run Einstein will turn out to be correct.” When he suggested that determinism would return in fundamental physics, Dirac did not have in mind the determinism of classical mechanics, but some new form of determinism, the nature of which was still unknown.
Dirac never elaborated in a systematic manner his half-baked ideas concerning the interpretation of quantum mechanics. His interest in the issue was sporadic and on a level quite different from the detailed arguments of, for example, Bohr, Heisenberg, and Schrödinger. It seems that the subject did not really interest him. In one of his last papers, he said, revealingly, “I don’t want to discuss this question of the interpretation of quantum mechanics … I want to deal with more fundamental things.” Clearly, Dirac was not a quantum philosopher.
He did share with Einstein (and several other physicists) the view that there are no final theories in physics. Quantum mechanics and general relativity were no exceptions. This had not always been his view; in his younger days, impressed by the apparently endless victories of quantum mechanics, he tended to believe that physics was approaching completeness, at least in principle. In the introduction to a paper he wrote in 1929 dealing with the application of quantum mechanics to atoms with many electrons, Dirac noted that “The general theory of quantum mechanics is now almost complete.” He then proceeded to say: “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known.” The whole of chemistry! Dirac’s reductionist declaration was not kindly received by chemists of the old school, who considered it one more proof of the theoretical physicists’ arrogance and disciplinary imperialism. But it soon became part of the view adopted by the young generation of chemists who eagerly and successfully sought to turn quantum mechanics into a resource for their own science. By the mid-1930s, quantum chemistry was a reality.
The second edition of Newton’s masterwork, Principia, stated that “to treat of God from phenomena is certainly a part of natural philosophy.” Recall that “natural philosophy” means physics. Newton’s 20th-century successor, Dirac was far from inspired by religious feelings just vaguely similar to those held by the former Lucasian professor. He was not a religious man and had little understanding of the cause and nature of religion. At least in his younger days, he seemed to have favored an atheistic or perhaps agnostic view, if he felt religion to be a worthy subject at all. In his memoirs, Heisenberg mentioned an informal discussion between Dirac and a couple of other physicists during the 1927 Solvay Congress. On this occasion, Heisenberg wrote, Dirac showed himself to be a “fanatic of rationalism,” who considered religion to be merely a system of myths and God just an invention of the clergy. Religious ideas could not possibly appeal to the man of science. Religion and science were opposites and necessarily in conflict. Having listened to Dirac’s militant atheism, Pauli is said to have commented: “But yes, our friend Dirac has a religion, and the basic postulate of this religion is: ‘There is no God, and Dirac is his prophet.”
Whatever the authenticity of Heisenberg’s story, the gist of it was supported by Dirac’s 1933 notes. There, Dirac unambiguously rejected religion as a superstitious belief suitable for only “a primitive community.” Any belief in God, he wrote, is “inadmissible from the point of view of modern science … [and] a well-organized society.” Later in life, Dirac expressed some interest in religion, though in his own somewhat naïve and restricted way, he never abandoned his dislike of organized religion.
Dirac may have discussed the relationship between science and religion with his friend Lemaître, the physicist-priest, as he recalled in an insightful memorial article written in 1968. Dirac had first become acquainted with Lemaître in 1933 when the latter gave a talk to the Kapitza Club on his new exploding-universe hypothesis, and they met again in Princeton two years later. In a letter to Margit, Paul Dirac described the Belgian cosmologist as a pleasant man who was “not strictly religious as one might expect from an Abbé.” Dirac was pleased with Lemaître’s model of an ever-expanding universe, which he thought made room not only for cosmic evolution but also for perpetual biological and human evolution. One might expect, he wrote, that the continuing evolution would lead “to a better and brighter future for all mankind.” Dirac suggested to Lemaître that of all the branches of science, cosmology was the one closest to religion: after all, God supposedly created the universe and the laws governing its structure and evolution. He naturally thought that the eminent cosmologist Lemaître would concur, but “Lemaître did not agree with me. After thinking it over he suggested psychology as lying closest to religion.”
A few years later, Dirac discussed in public for the first and only time a religious issue, none other than the existence of God. He insisted on approaching this question, not from the perspective of theology or philosophy—which is “really just sort of guessing or expressing one’s feelings”—but from the perspective of physics. In short, Dirac argued that the existence of God was intimately connected with the origin of intelligent life. If life could come into being relatively easily and be explained by the laws of physics, there would be no need for God. On the other hand, “if physical laws are such that to start off life involves an excessively small chance, so that it will not be reasonable to suppose that life would have started just by blind chance, then there must be a god.” As usual, Dirac did not commit himself. He did not reveal which of the two possibilities he found the most likely or most appealing.
When it was proposed shortly after Dirac’s death to commemorate him in Westminster Abbey, at first, the request was turned down. The reason was not, of course, that the Dean of Westminster doubted Dirac’s greatness as a scientist; rather, he questioned, not without reason, whether Dirac had ever been a sincere Christian. Nonetheless, after 11 years of consideration, a commemoration ceremony did take place in the church. Stephen Hawking gave the final address: “It has taken eleven years for the nation to recognise that he was probably the greatest British theoretical physicist since Newton, and belatedly to erect a plaque for him in Westminster Abbey. It is my task to explain why. That is, why he was so great, not why it took so long.”