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In 1819, the British Romantic poet John Keats composed a poem with the title “Ode on a Greek Urn,” which was published six years later. The poem is famous, not least because of its last lines:
“Beauty is truth, truth beauty—that is all
Ye know on earth, and all ye need to know.”
Keats’ enigmatic equation between two apparently unrelated concepts—truth and beauty—has its scientific counterpart in physicists’ frequent allusions to beautiful equations. They sometimes state that certain theories or equations are particularly beautiful and for this reason supposedly are also fundamental, revealing deep truths about nature. Maxwell’s equations of electromagnetism fall into this category. They are fundamental, have a high degree of symmetry and harmony, and they are condensed enough to fit on a T-shirt.
Whatever the meaning of Keats’ poem, it is a fact that scientists—and theoretical physicists in particular—often associate their ideas and theories with aesthetic qualities. More than any other modern scientist, Dirac became a prophet of beauty in science, the belief that a truly fundamental theory must be beautiful. But what is beauty? To Dirac, the essential element was one that was missing in Keats’ equation (and which Keats probably did not have in mind), namely mathematics. What mattered was not just any kind of beauty, but mathematical beauty. Since the mid-1930s, the idea that beautiful mathematics is all-important in fundamental physics had increasingly dominated Dirac’s thinking. “A physical law must possess mathematical beauty,” Dirac wrote on the blackboard when he visited the University of Moscow in 1956 and was asked to write an inscription summarizing his basic view of physics.
The general idea of an intimate connection between mathematics and physics, sometimes in the more extreme version that nature is, in essence, mathematical, can be found as far back as in ancient Greece. In the early years of the 20th century, it was revived in the form of the influential doctrine of a “pre-established harmony between nature and mathematics,” as Hilbert called it. According to Hilbert and his school in Göttingen, the pre-established harmony implied that mathematics was the royal road to progress in theoretical physics. Hilbert and his mathematician colleague, Hermann Minkowski, were deeply involved in the development of the theory of relativity, which they tended to see as a branch of mathematics rather than physics. Max Born and other Göttingen physicists who founded quantum mechanics subscribed happily to Hilbert’s doctrine. Most likely Dirac was exposed to a heavy dose of the same doctrine during his visits to Göttingen. By linking it to aesthetical considerations, he developed it in his own way.
In 1960, Eugene Wigner, Dirac’s brother-in-law and a former Göttingen student (and Hilbert’s assistant), famously discussed the “unreasonable effectiveness of mathematics in the natural sciences.” He suggested that the pervasive usefulness of mathematics in physics bordered on the miraculous. This was not a new observation, for in a 1921 lecture Einstein had referred to the same question: “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?” And 18 years later, Dirac repeated: “There is no logical reason why [the method of mathematical reasoning] should be possible at all, but one has found in practice that it does work and meets with considerable success.” If there was no logical reason, what reason could there be?
Dirac’s answer, offered in his 1939 Edinburgh lecture, was that nature possessed a “mathematical quality” or that its fundamental laws were written in the language of mathematics. “One may describe the situation,” he explained, “by saying that the mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on, it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen.” At a later occasion, he phrased the same message in terms similar to those which Galileo had used more than 300 years earlier: “God is a mathematician of a very high order, and he used very advanced mathematics in constructing the universe.” Contrary to Galileo, a faithful Catholic, Dirac’s rare reference to God was just conventional and not an indication of belief in divine creation. Had Dirac stopped at this point, his idea of a perfect marriage between mathematics and physics would have been rather unoriginal. It would have been just one more version of a line of thought cultivated since antiquity by many other physicists, mathematicians, philosophers, and even theologians.
To claim that nature is constructed in accordance with mathematics, or that the laws of physics are in essence mathematical laws, is not very informative. Mathematics is so rich that it can cover almost any imaginable universe (and a good deal more), so why precisely this universe? There are immensely more mathematical structures and equations than those which govern the fundamental laws of physics, so why precisely this mathematics? The traditional answer, again one with long historical roots, was that nature was constructed in the simplest possible way and therefore described by the simplest possible mathematical structures. Simplicity was the answer. Like practically all physicists, Dirac valued simplicity highly, meaning that he preferred simple equations over more complicated ones with the same empirical power. But he thought that simplicity was not enough. Sometimes the endeavor for mathematically simple equations might even lead the researcher astray from the ultimate goal of physics, to find the truths about nature.
In his Edinburgh lecture, Dirac argued that what characterized the deepest and most successful theories of modern physics was mathematical beauty, a concept that was not, in general, the same as simplicity. Newton’s classical law of gravitation is about as simple as a fundamental law of physics can be, involving only ordinary numbers and simple mathematical operations such as multiplication, division, and squaring. On the other hand, Einstein’s law, as given by the equations of general relativity, is expressed by a complicated system of equations that involve mathematical quantities called tensors. No doubt Newton’s law is simpler than Einstein’s; but Einstein’s theory is better, deeper, more general, and nearer to the truth than Newton’s. Dirac’s advice to his fellow theoretical physicists was to strive towards the element that, more than anything else, characterized Einstein’s theory, namely mathematical beauty. “The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty,” he said. Simplicity was worth considering too, but only as subordinate to beauty. “It often happens that the requirements and beauty are the same, but where they clash the latter must take precedence.”
Dirac was not always fascinated by mathematical beauty, a concept he first dealt with explicitly and at any length in his Edinburgh address. However, in a more embryonic form, it was in his mind at an earlier date. Without relating specifically to mathematical beauty, in his 1931 paper on the magnetic monopole, he advocated the mathematical route to physical discovery in a way that pointed towards his later, aesthetically formulated view. The surprising physical consequences of his essentially mathematical approach to the electron’s wave equation in 1928 were instrumental in turning him into an apostle of mathematically beautiful equations. Although he usually referred to “beauty,” sometimes Dirac used other words, which he apparently considered to be synonymous; among them “elegance” and “prettiness.” One of his last articles, published on the occasion of his 80th birthday, carried the title “Pretty Mathematics.” His favored term for those theories he found to be sorely lacking in beauty was “ugly.” Renormalization quantum electrodynamics was his paradigm of a mathematically ugly theory.
Why should the physicist strive toward beautiful mathematics in his equations? Dirac was not aiming at beauty either for artistic or purely mathematical reasons. He was a physicist, neither an artist nor a mathematician, and his main reason for giving such priority to mathematical beauty was that it would lead to better theories of physics. As far as fundamental physics was concerned, he believed that the aesthetic criterion was a safer guide to progress than just agreement with experiment. “A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data,” he claimed in 1970, referring to quantum electrodynamics.
For the principle of mathematical beauty to be scientifically valuable, one needs to know more precisely what it is. After all, “beauty” is a nebulous concept, and it does not become less nebulous just because “mathematical” is placed in front of it. Dirac was characteristically vague when he spoke, as he often did, on the meaning of mathematical beauty and its role in physical theory. A beautiful theory, he said on one occasion, “is a theory based on simple mathematical concepts that fit together in an elegant way, so that one has pleasure in working with it.” Pleasure? Yes, there was an emotional element involved and not just the cold austerity of mathematics and logic. “The beauty of the equations provided by nature … gives one a strong emotional reaction,” he stated in a 1979 interview.
The emotional element is also what Dirac indicated in a moving obituary he wrote for his friend and colleague Schrödinger in 1961. He confided that the Austrian physicist was “the one that I felt to be most closely similar to myself.” The reason, he believed, was that “Schrödinger and I both had a very strong appreciation of mathematical beauty, and this appreciation of mathematical beauty dominated all our work.” Dirac described it “as a sort of faith with us that any equations which describe fundamental laws of nature must have great beauty in them. It was like a religious faith with us.”
There is little doubt that Dirac relied on his own intuitions and basic beliefs, and also on his personal experiences as a physicist when he decided which mathematical methods and concepts were beautiful and which were ugly. Nonetheless, he suggested that his own intuitions were largely the same as those of other mathematicians and physicists. “Mathematical beauty,” he wrote, “is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating.” And yet, contrary to the subjective nature of beauty in painting, sculpture, poetry, and theater, he believed mathematical beauty was objective and absolute, almost as it existed in a Platonic heaven. This is what Dirac said in a talk given in Florida in 1972. “It is of a completely different kind. It is the same in all countries and at all periods of time.” No justification was given, and no further explication followed.
What if a mathematically beautiful theory flatly disagrees with evidence from experiment and observation? This happened to Dirac’s theory of a varying gravitational constant based on the beautiful hypothesis of interconnected constants of nature. Aware of the dilemma, Dirac proposed the radical idea that in some cases considerations of mathematical aesthetics should be given greater weight than experimental facts. In effect, the principle of mathematical beauty would then become a criterion of truth, overruling the traditional criterion of experimental testing. Dirac spelled out his radical proposal in no uncertain terms: “There are occasions when mathematical beauty should take priority over agreement with experiment.”
Just as Dirac believed that a beautiful theory should not be rejected just because it contradicted experiments, so he believed that empirically successful theories might nonetheless be wrong if they were aesthetically displeasing. Empirical virtues and aesthetic shortcomings may well go together, as was the case with the “ugly and incomplete” theory of quantum electrodynamics. In Dirac’s view, aesthetic criteria could and in some cases should preserve a theory from refutation in the face of falsifying experiments; the same criteria could also prompt the abandonment, or at least mistrust of, a theory that is as yet empirically satisfactory.
Did Dirac seriously mean that the physicist, convinced of the sublime beauty of some theory, should obstinately stick to that theory and disregard any kind of conflicting evidence, however strong? Not really, for Dirac realized that, in the long run, a scientist could not ignore the verdict of experiment and still be credible as a scientist. What he did mean and was quite serious about, was that the physicist should not care too much if a beautiful theory were apparently disproved by experiments. He liked to illustrate the point by referring to Einstein’s admired theory of general relativity. Let us imagine that a discrepancy, well confirmed and substantiated, had turned up shortly after Einstein had completed his theory. Should one then conclude that the theory had been falsified, or proved wrong? Not according to Dirac and also, for that matter, not according to Einstein. “Anyone who appreciates the fundamental harmony connecting the way nature runs and general mathematical principles must feel that a theory with the beauty and elegance of Einstein’s theory has to be substantially correct,” Dirac stated.
Apart from Einstein and general relativity, Dirac often referred to Heisenberg’s original quantum mechanics as a beautiful theory, despite its lack of incorporating relativity. Remarkably, he never discussed his own contributions in this context. Many physicists of today would single out Dirac’s linear wave equation of 1928 as a masterpiece of beauty, harmony, and elegance, but Dirac never mentioned it. He was too modest to call attention to his own work.