The Beauty of Physics

The greatest poet of the English language

Whenever physicists confer, they are likely to declare their belief in the beauty of the laws of nature. It must also be acknowledged that passion for beauty and search for harmony are part of the very essence of being human. Now, after these initial assertions, two things should be highlighted. The first is a question: Isn’t an “act of faith” – such as that in the beauty of natural laws – exactly what a scientist should avoid when engaging in research? The second is a statement: Passion for harmony could distort objectivity and cause cognitive distortions, from which scientific discourse should be free.

Leaving this issue aside for a moment, we might ask ourselves who is the greatest English-language poet of all time.  Poets are often  counted among those who seek beauty: Is it William Shakespeare, Samuel Taylor Coleridge, or George Gordon, Lord Byron? It is very probable that this question will never have an uncontested and satisfactory answer. It may be impossible to reach a general consensus.

For Graham Farmelo, however,[1] the greatest English-speaking poet of all time was Paul Adrien Maurice Dirac.[2] Dirac was an English theoretical physicist who, together with Erwin Schrödinger, received the Nobel Prize in 1933 “for the discovery of new forms of atomic theory,” which later became quantum mechanics. Perhaps Farmelo lacks the literary ability to discern who is “the greatest of the English-speaking poets.” Perhaps his statement is meant to be highly provocative and an invitation to reflect both on the similarities between physics and poetry – and the fact that they need each other – and on how the subtle power of beauty manifests itself in both fields.  

In this article we will consider how scientific knowledge – in a similar fashion to poetic discourse – makes ample use of analogical language, and how both science and poetry tend – albeit at different levels – to summarize their concepts, preferring a terse form of expression to long and excessively detailed descriptions. Above all we will see how poetry and physics have an eye for aesthetic taste, albeit with the necessary differences.

A hidden aesthetic taste

To the mind  of many physicists, nature expresses itself paradoxically through a loud and clear silence; and at the same time it is simply “beautiful.” However, the taste for beauty among physicists is not uniformly shared, just as the case with literary passion. Dirac claimed that he did not understand poetry at all and that he did not understand how some of his illustrious colleagues – including Robert Oppenheimer[3] – could write sonnets. He even went so far as to say that he did not understand “how someone can work on the frontiers of physics and at the same time compose poetry. The two things are contradictory. In physics one wants to say something that no one knew before in terms that everyone can understand. In poetry you are forced to say things that everyone already knows in terms that no one understands.” This is a most provocative statement.

Richard Feynman,[4] several years later, reiterated this statement, claiming that “Poets say science takes away from the beauty of the stars – mere globs of gas atoms. I too can see the stars on a desert night, and feel them. But do I see less or more? […] A vast pattern, of which I am a part. Perhaps my stuff was belched from some forgotten star, as one is belching there. […] What is the pattern, or the meaning, or the why? It does not do harm to the mystery to know a little about it. For far more marvelous is the truth than any artists of the past imagined it. Why do the poets of the present not speak of it”?

What kind of beauty are we talking about, then, when we even go so far as to count physicists as potentially “great poets” and regard them as artists capable of creating sublime works? Is this a terrible and unforgivable error of judgement, or is there a certain aesthetic sense even in science? If so, in what does the beauty of physics consist?

The elegance of mathematics and the gift of analogy

Eugene Wigner[5] wondered why we are able to describe the world in  mathematical terms, and answered that essentially mathematical language is wonderful and exemplary not only because it is the only one which is universal but also because it is the most appropriate one in which to describe nature. At this point the Hungarian physicist spoke of a real “miracle” and a “gift”: the miracle of the appropriateness of mathematical language in the formulation of the laws of physics, and the wonderful gift that we neither understand nor deserve.[6] Although Wigner declared that he did not believe in God, in his position there is a glimmer of sublime spirituality.

One could say that the most famous equation in physics is E = mc2, formulated by Albert Einstein[7] in the theory of special relativity. It puts an equality between the energy (E) of a body and the mass of the body (m), multiplied by the speed of light squared (c2).

Now, with Farmelo,[8] we would propose as the most beautiful equation one proposed by Paul Dirac: i∂ = m. We will not enter here into the details of this formula, nor in the physical-mathematical meaning of the symbols it contains. Suffice to note that it describes the behavior of electrons or quarks, in particular the motion and conservation of the total energy of the reaction, unifying in a consistent formulation the principles of quantum mechanics and those of special relativity. This equation even appears on the floor of Westminster Abbey, to commemorate the life of the physicist who formulated it.[9]

What can we say about these mathematical symbols that express one of the highest achievements of human knowledge? We perceive in them a beautiful analogy and a poetic form of great refinement and elegance. We do not think we are exaggerating if we see in the mathematical formula aspects in common with poetry. The equation shares with poetic language one of the most powerful tools of human knowledge, analogy.  It tells us that “this” is equal to “that,” that the term on the left is equal to the term on the right and vice versa. It is indeed an analogy.

As in spoken language, where direct or indirect analogies (metaphors) abound, so the mathematical formulations of physics open  up extraordinary scenarios, connecting hitherto unexplored notions to concepts with which we are – some more, some less – familiar. All this pushes us beyond the known, beyond the banal. With its equations, physics transforms statements that would risk being tautological – obvious and adding nothing to our knowledge – into an expansion of our knowledge toward unexplored horizons. If engineering stops at what is possible and what is “materially” feasible, physics pushes us beyond, moving the limits of knowledge a little further, to the extremities of the imagination.[10]

Considering mathematical language on the one hand, and analogical language on the other, we observe how in an equation the evolution of a quantity is determined starting from the knowledge of another quantity; in the same way an analogy is nothing other than an abstraction that allows us to know concepts without the need of a direct proof. So there are two ways – equations and analogies – to indicate that starting from a known image we can visualize others, both  unknown and new, creating bridges between something known and other elements that we do not control directly. Moreover, analogies create bridges and relationships between themselves: the same tools we develop to understand one bridge are also used to build new ones. And the most surprising bridge seems to be the one that leads to the connection between poetry and physics. Both are tools at our disposal to describe and understand the world, as if they were in close, communicating vessels. And these two tools can express a passion for beauty. Who other than a poet is attracted by beauty, and who other than a physicist can investigate the beauty of nature in a profound way? But, when it comes to  Dirac, is this really the case?

Follow your heart…

As far as analogies are concerned, the English physicist certainly understood, or at least intuited, their profound nature and the informative mysterious links that animate them. It is said that Dirac was particularly bizarre character, and that in his spare time he loved to climb mountains. But for training he had found a perfect analogy in climbing the apple trees that surrounded the Cambridge University campus. His ultimate goal, as far as leisure was concerned (“climbing mountains”), was prepared, supported and accompanied by an activity that was not the same, but analogous (“climbing apple trees”). The more exciting the activity and the connection that exists between the two, the more effective and lasting the knowledge becomes.

For Feynman, scientific research draws its lifeblood from an intimate source – the deep heart of the scientist – which, without forcing the terms too much, we could define as “biblical” in nature. The scientist is “someone in whom thoughts spring,” to quote the words of Dante.[11] The American genius has explained: “The same thrill, the same reverent awe and the same mystery still return when we delve into any problem. As knowledge expands and deepens, the mystery becomes more and more attractive and invites us to go further. […] Few non-scientists live this particular religious experience. Our poets do not write about it; our artists do not attempt to represent it. I do not understand why.”[12]

Both the sublimity of Dante’s analogy and the intensity of the heartfelt sentiment of the American physicist are undeniable. But it must be recognized, if we are intellectually honest , that there is a clear distinction between the beauty present in science and the beauty that moves us in art or literature. In this regard, one can read Dirac’s views on poetry and Feynman’s views on artists. Perhaps the heart leads to aesthetic perceptions that are not immediately transferable into one another and to polychrome forms that authorize scientists to see the “beauty of physics” and to be, at the same time, practically insensitive to the “beauty of art.”

Fundamental to  this field is the type of language that is used. Most probably, the most beautiful scientific work, from a literary point of view, is the De rerum natura (“The nature of things”) by Titus Lucretius Carus, philosopher and poet. The work, written in the first century B.C., is a Latin poem of the epic-philosophical genre, composed of six books in hexameters, in which are set out the Epicurean theories on the reality of the world, regulated by a “natural order” and totally independent from the gods. The universe, according to its atomistic, materialistic and mechanistic vision, is composed of tiny invisible and indivisible elements, the “atoms.”

What is surprising is the fact that the philosophical system of De rerum natura, while ranging from physics to ethics, is presented in the form of a poem. Feynman’s words come to mind: “Poets say that science takes away the beauty of the stars. […] Because the truth is much more wonderful than any artist of the past has imagined. Why do poets of the present not speak of it?” It really seems that we cannot speak of a shared sense of beauty between art and poetry, on the one hand, and science, on the other. So, what is the beauty to which science refers?

For those who do not know mathematics it is difficult to perceive, or rather to feel , the profound beauty of Nature. If you want to know Nature and appreciate it, you have to understand the language it speaks: that is, mathematical language. According to Gian Francesco Giudice,[13] “we cannot write exact mathematical rules that establish whether a theory is fascinating or not. However, it is surprising that the beauty and elegance of a theory are universally recognized by people of different cultures.” In most cases it is a matter of instinct, of intuition, which encompasses in a correct combination the justification of empirical results and the use of fundamental principles: what makes a theory beautiful lies in its harmony and coherence with fundamental principles and successful in its agreement with experimental data. The dimension beyond reason makes physics exciting and thrilling and leads us to believe that probably the meaning of beauty of a physical theory must be something inherent in our brain. The bottom line is that science is not art, and theories are not sought to elicit emotional reactions, but rather to function as explanations for what is observed in the natural world. Science is an endeavor organized so as to overcome the weaknesses of the human intellect and avoid the fallacies of intuition. Science does not deal with emotions, but with numbers and equations, data and graphs, with facts and with logic and mathematics.[14]

Werner Heisenberg, one of the founding fathers of quantum mechanics, firmly believed in the beauty of physical models and in the fact that it is beauty that leads us to truth. He said: “If nature leads us to mathematical forms of great simplicity and beauty, we cannot help but believe that they are true, and that they describe an authentic feature of the real world.”[15] The wife of the great physicist also recalls how, during a night walk under the starry sky, he, in an ecstatic and romantic transport, affirmed that what is symmetrical, beautiful and harmonious cannot but be an original archetype of creation, and therefore intrinsically and necessarily true.[16] Probably one should not trust too much the sentimental effusions of physicists, who thereby risk becoming more mediocre poets than rigorous scientists.

The Beautiful is not what is beautiful, but what is natural

According to Frank Wilczek,[17] explanations that seem successful become beautiful, and so we learn to recognize their beauty. Perhaps the time has come for the aesthetic ideals of the past to be abandoned in physics. It seems inevitable and reasonable to base ourselves on the experience of those who have preceded us: Bernard of Chartres[18] said that we are like dwarfs on the shoulders of giants, so that we can see more things than they could and things further away, certainly not because of the acumen of our sight or the height of our body, but because we are lifted up and carried high by the stature of the giants, by tradition in fact. However, we should add that it would be foolish to stop there, and that such an attitude would not be in agreement with the forward movement of scientific knowledge, in its contemporary sense. A criterion of beauty derived from the past risks being mendacious and treacherous, and risks leading physicists astray in many circumstances.

It is in fact interesting to note how the advent of quantum mechanics – the greatest scientific revolution of the contemporary era – was, in many ways, one of the many failures of the aesthetic criterion. The disappearance of the notions of position and velocity of the “quantum objects,” and the rejection of the criterion of the reality of the objects themselves, which are, so to speak, dissolved in a probabilistic and indeterminate reality, even on the level of being,[19] assumed a drastic distance from the aesthetic purity of the entities in which the universe was organized up and including  the cosmological models of Kepler and Newton.

It should also be added that Heisenberg’s own ideas that have been successful – and that have survived criticism from the scientific community – are not exactly classifiable as wonders of beauty. So, the aesthetic criterion is most likely not a determining factor in science at all. Or at least it is not rigidly and immutably so.

According to Steven Weinberg,[20] any new theory suggests aesthetic criteria and, at the same time, demands a comparison with real data. In this process of experimental confirmation the sense and perception of scientific beauty change and adapt in accordance with our experimental experiences.

In the evolution of science there has been a shift from a representation of nature in holistic terms – no doubt aesthetically very beautiful, but pseudoscientific: think of astrology – to a quest for truth free from having beauty as the ultimate criterion. Thus, according to physicists, knowledge of the physical world has been freed from the anthropocentric and “anthropomorphic prison” of the past. Max Planck[21] is very clear and insistent on this process of de-anthropomorphizing scientific knowledge.[22]  A theory should not refer to human beings, and beauty could be a mere anthropic concept that, by  becoming rigid – as it could according to some aesthetic canons of classical Greece – would risk hindering scientific progress. Planck himself argued that new scientific truths do not become established because their opponents are convinced of their correctness (and even less of their beauty), but rather because eventually the opponents of the theory die and a new generation is born for whom the concepts have become familiar. Probably a similar thing happens with the sense of beauty.

Physics must be consistent in its mathematical formulation and consistent with experimental results. Physics is not mathematics, but, from the time of Isaac Newton onward, it finds in mathematical formalism its most effective expression, since the quantitative formulation is the most economical and least ambiguous of human forms of communication. One can be wrong in mathematics, but one cannot lie using it. Requiring consistency in a theory may not be enough, however. There are many things that are mathematically very beautiful, but have nothing to do with reality. One solution would be to require that theories be mathematically consistent and, on the other hand, that they allow the understanding – description and prediction – of a very large number of phenomena. However, it remains firm, as a nodal point – again according to Weinberg, who takes up an idea of Thomas Kuhn[23] – that every scientific revolution must overthrow the concept of beauty.

Beauty in physics

According to Anthony Zee,[24] in the eyes of a physicist “beauty” means “symmetry.”[25] In physics, the concept of symmetry refers to a property that is repeated essentially identically in time and space during physical processes. Therefore a scientific law that is true both in Rome and in Beijing – which is the case for all physical formulations – and is therefore “invariant” for spatial displacements, satisfies the aesthetic canons of physics.[26] The point is that with symmetries we can say a lot with very little, and derive more knowledge than what has been entered as input. How can we not think of Dirac’s statement that “in physics you want to say something that no one knew before in terms that everyone can understand; in poetry you are forced to say things that everyone already knows in terms that no one understands?” For a physicist, beauty is economy and simplicity.

We can then try to identify three main criteria to get an idea of “beauty in physics.” 1) Simplicity. This means being able to do with less: Ockham’s famous razor.[27] However, we understand how this concept has a purely relative value and is not immediately quantifiable or free from a certain subjectivism. 2) Naturalness. This means that we do not use ad hoc hypotheses, that is hypotheses that work only and exclusively for the specific case considered. In a natural vision every assumption should have a justification and not be intruded. This aspect links mathematical consistency, offered as an indication, to the compatibility of the mathematical model with the experimental data. Yet this remains a criterion of aesthetic origin and theory, and not a purely scientific one. 3) Elegance. This is the most elusive criterion. It is a kind of combination of simplicity and astonishment, which opens to a new awareness and converges in an “unexpected explanatory closure,” to use an expression of Richard Dawid.[28] Elegance emerges unexpectedly from the economy of means and is neither formalized nor systematically aimed at. Besides, how could it be, if it is practically the manifestation of scientific genius? Elegance therefore remains a subjective criterion.

Beauty in physics is therefore a combination of simplicity, naturalness and a certain amount of unexpectedness.[29]

The miraculous mystery of the beauty of physics

In conclusion, we could say that no matter how beautiful a theory is, no matter how clever the person who devised it is: if it does not agree with the data of the experiment, it is wrong. Physics appears to us to involve a delicate and mysterious balance between aesthetic and ingenious intuition on the one hand, and rigorous logical and experimental verification on the other. And what is the practical effect of this combination? Paraphrasing Feynman, we can say that, if human curiosity represents a need, then studies have a practical sense,  that of satisfying such curiosity.[30]

DOI: La Civiltà Cattolica, En. Ed. Vol. 5, no.11 art. 11, 1121: 10.32009/22072446.1121.11

[1].      Graham Paul Farmelo (May 18, 1953) is a biographer and science writer, lecturer at Churchill College,  Cambridge, and adjunct professor of physics at Northeastern University, Boston. He wrote a highly successful biography of the theoretical physicist Paul Dirac, entitled: The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom, New York, Faber and Faber, 2009.

[2].      Paul Adrien Maurice Dirac (August 8, 1902 – October 20, 1984), English theoretical physicist, is considered one of the most important scientists of the 20th century.

[3].      Robert Oppenheimer (April 22, 1904 – February 18, 1967) was an American theoretical physicist and professor of physics at the University of California, Berkeley. He was the head of Los Alamos National Laboratory, and is among those who can be called the “fathers of the atomic bomb,” for his role in the Manhattan Project.

[4].      Richard Phillips Feynman (May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics and the theory of quantum electrodynamics. He assisted in the development of the atomic bomb during World War II and received the Nobel Prize in 1965, along with Julian Schwinger and Shin’ichir? Tomonaga.

[5].      Eugene Paul Wigner (November 17, 1902 – January 1, 1995) was a Hungarian theoretical physicist who also contributed to mathematical physics. He became an American citizen in 1937, and won the Nobel Prize in 1963.

[6].       Cf. E. P. Wigner, L’irragionevole efficacia della matematica nelle scienze naturali, Milan, Adelphi, 2017, 39

[7].      Albert Einstein (March 14, 1879 – April 18, 1955), German-born theoretical physicist, widely recognized as one of the greatest physicists of all time. He received the Nobel Prize in 1921, and both his intellectual achievements and his originality has  made “Einstein” synonymous with “genius.”

[8].      We also point out how several people – more or less young, and not exactly experts in physics, mainly active on the web – consider Dirac’s equation as the “equation of love.”  We do not think that this superficial mixture of science and love deserves to be rigorously rejected, as it lacks the necessary honest rigor, a necessary condition for a dialogue that is enriching.

[9].      The memorial plaque in Westminster Abbey, which features the equation, was unveiled on November 13, 1995.

[10].     Cf. M. Malvaldi, L’infinito tra parentesi. Storia sentimentale della scienza da Omero a Borges, Milan, Rizzoli, 2016, 42.

[11].     Dante Alighieri, Divine Comedy. Purgatory, canto V, v. 16.

[12].     R. Feynman, Il piacere di scoprire, Milan, Adelphi, 2020, 154.

[13].    Gian Francesco Giudice (January 25, 1961) is an Italian theoretical physicist working at the European Organization for Nuclear Research (CERN) in Geneva, in the field of particle physics and cosmology.

[14].     Cf. S. Hossenfelder, Sedotti dalla matematica. Come la bellezza ha portato i fisici fuori strada, Milan, Raffaello Cortina, 2019, 21f.

[15].     W. K. Heisenberg, Physics and beyond: encounters and conversations, New York, HarperCollins, 1971, 68.

[16].    See Id., Inner Exile: Recollections of a Life with Werner Heisenberg, Basel, Birkhäuser, 1984, 143.

[17].    Frank Anthony Wilczek (May 15, 1951) is an American theoretical physicist, and mathematician and urrently professor of physics at the Massachusetts Institute of Technology (MIT). Together with David Gross and David Politzer, he received the Nobel Prize in Physics in 2004 “for the discovery of asymptotic freedom in the theory of strong interaction between particles.”

[18].    Bernard of Chartres (died after 1124) was a 12th-century French neo-Platonic philosopher, master of rhetoric in the cathedral school of Chartres.

[19].    One thinks of Heisenberg’s “uncertainty principle,” according to which in the sphere of reality the conditions are formulated by quantum theory; natural laws, therefore, do not lead to a complete determination of what happens in space and time, and the  understanding of what is happening is rather left to the play of chance.

[20].    Steven Weinberg (May 3, 1933 – July 23, 2021) was an American physicist who won the Nobel Prize in 1979.

[21].    Max Karl Ernst Ludwig Planck (April 23, 1858 – October 4, 1947) was a German theoretical physicist; his discovery of energy quanta earned him the Nobel Prize in 1918.

[22].     Cf. M. Planck, La conoscenza del mondo fisico, Turin, Bollati Boringhieri, 1965.

[23].    Thomas Samuel Kuhn (July 18, 1922 – June 17, 1996) was an American philosopher of science. His book The Structure of Scientific Revolutions introduced the term “paradigm shift.”

[24].    Anthony Zee (1945) is a Chinese-American physicist, professor at the Kavli Institute for Theoretical Physics and the Department of Physics at the University of California, Santa Barbara.

[25].     Cf. S. Hossenfelder, Sedotti dalla matematica…, op. cit., 43.

[26].    A similar argument can be made for temporal changes.

[27]. Ockham’s razor, or the principle of economics, states that when one has several hypotheses for solving a problem, one should choose, given equal results, the simplest way that involves the lowest possible number of assumptions and variables. William of Ockham (1288 – April 10, 1347), was an English Franciscan theologian and philosopher.

[28].    Richard Dawid is professor of philosophy of science at Stockholm University.

[29].     S. Hossenfelder, Sedotti dalla matematica…, op. cit., 120.

[30].     R. Feynman, Il piacere di scoprire…, op. cit., 253.