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Beauty & Science
Marilyn Kane wonders what scientists mean when they say nature is beautiful.
The advancement of scientific knowledge doesn’t result from any single research method, but is the combined effect of several different approaches, which have been summarized as follows:1
1) “careful refinements in measurements to uncover anomalous behaviour,
2) mathematical extrapolation of existing theory,
3) critical re-examination of apparently obvious but untested presuppositions,
4) argument by symmetry or analogy
5) aesthetic judgement,
6) accident,
7) hunch.”
The first four look reasonable; the last two are serendipitous and fortuitous effects. But what of approach 5, aesthetic judgement? This seems out of place, perhaps somehow in the wrong list. Can it really be that scientists rely on aesthetic judgements when analyzing data or developing hypotheses, and that beauty plays a role in their work?
Voltaire wrote: “Ask a toad what beauty is… He will answer that it is his female, with two large round eyes sticking out of her little head, a large and flat snout, a yellow belly, a brown back. Question the devil: he will tell you that the beautiful is a pair of horns, four claws, and a tail. Finally, consult the philosophers: their answer will be… grandiloquent nonsense…”
Note that Voltaire didn’t specify what scientists would say about beauty, presumably because in his day scientists were lumped in with philosophers. But now that scientists are a distinct group, what do they say about beauty? Surprisingly, scientists have a great deal to say about it, and they particularly like to speak of beauty in science. Are the things that they say about beauty wise and sensible… or is it still grandiloquent nonsense?
The mathematician and astronomer Henri Poincaré said: “The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it; and he takes pleasure in it because it is beautiful. If nature were not beautiful, it would not be worth knowing and life would not be worth living… I mean the intimate beauty which comes from the harmonious order of its parts and which a pure intelligence can grasp.”2
By Hercules! What extravagant claims this fellow made.
Is nature beautiful? There is much obscurity here, beginning with the word beautiful, which seems incongruous when used as a central word in science. Does it have a scientific definition? Can its presence be observed and measured, as can length or velocity? Or is the meaning of beauty, as it is allegedly sensed by the scientist studying nature, the same as when sensed by other people? In that case the same philosophical problems that arise in aesthetics will now arise in science too, or at least in philosophy of science. What is beauty? Artists, presumably more familiar than scientists with the concept of beauty, are less certain of its meaning or even its existence. The Dadaist Tristan Tzara, echoing Nietzsche, said: “Beauty is dead.” And Pablo Picasso remarked “I hate that aesthetic game of the eye and the mind, played by these connoisseurs, these mandarins who ‘appreciate’ beauty. What is beauty, anyway? There’s no such thing.”
The vagueness of ‘Nature is beautiful’ is even greater in the scientific context. We might know what to expect if the words came from the mouth of a nature-worshipper: the adoration of the sun and moon, of trees, rivers, rain, breezes, fire, etc. But anthropologists tell us that nonliterate or primitive peoples seem to have lacked the concept of nature as a totality.3 When Poincaré says that nature is beautiful, does he mean nature as the grand unified sum of stuff, or that sum minus human artifice and perhaps God? Or does he mean just those particular aspects of nature that a scientist focuses on, so that for a biologist the structure of organic systems is beautiful, stars for an astronomer, and perhaps sub-atomic particles for a physicist? Does Poincaré mean that all of nature is beautiful, or only some of it?
If his claim is that all of nature is beautiful, we may ask whether he means to imply that all the parts of nature are beautiful. We often say something is beautiful without meaning that all its component parts are beautiful: someone may find a face beautiful but not the nose, for example. It would be interesting to get an answer to this question, because without one Poincaré’s comments are very unclear, and so do not help us to understand what is meant by ‘beauty’ in science. If all of nature is said to be beautiful when looked at by the scientist, then we know that the meaning of beauty has undergone a metamorphosis from its more usual one. Ordinarily the word beautiful is applied to some things and not to others, but not to all things. To say that all of nature is beautiful is contrary to the more common opinion that some of nature is beautiful and some isn’t. As Parmenides pointed out to the young Socrates, there is something problematic about having a Form for mud and a Form for dirt; so too, there would be something peculiar about calling everything beautiful. Also, to ascribe beauty to everything is to render beauty an almost meaningless term. As George Santayana points out: “If we could so transform our taste as to find beauty everywhere, because, perhaps, the ultimate nature of things is as truly exemplified in one thing as in another, we should, in fact, have abolished taste altogether. For the ascending series of aesthetic satisfactions we should have substituted a monotonous judgement of identity.”
Next Poincaré states that nature wouldn’t be worth knowing were it not beautiful. But why should we believe this? There are plenty of things worth knowing about that seem not-sobeautiful. The mathematician and physicist Paul Dirac held that “A physical law must possess mathematical beauty.” He called the 1929 quantum electrodynamic (QED) theory of Heisenberg and Pauli “ugly” and spent much time searching for a better theory. He said: “I really spent my life mainly trying to find better equations for quantum electrodynamics, and so far without success…” Later, in 1947 and 1948, Dirac judged a new theory of QED by Richard Feynman, Freeman Dyson and others also to be “ugly”. If these theories were indeed ugly, they would not be worth knowing, according to Poincaré’s standards. Yet Heisenberg, Pauli and Feynman were all Nobel Prize winners and they thought the theories were worth knowing.
Furthermore, the belief that only the beautiful is worth knowing promises problems akin to those discussed by Socrates and dubbed the learning paradox. Must the scientist have sensed or otherwise come to believe in the beauty of nature in advance of doing any research on it, in order to justify the worth of that research? Or does the scientist reach the conclusion that nature is beautiful through experimentation and theorizing, in which case one might say the scientist started off investigating matters without being sure whether the investigations would yield results worth knowing. Or, if the scientist only investigates areas where she or he has already sensed beauty, then if s/he finds beauty, s/he can’t reasonably claim that all of nature is beautiful, nor that non-beautiful aspects of nature are not worth knowing, since s/he purposefully avoids such aspects, and so never learns enough about them to tell whether or not they are worth knowing.
Poincaré tells us that the beauty of nature comes from the ‘harmonious order of its parts’. Nature is beautiful because of the order of its parts, but are the parts themselves beautiful?
If only the harmonious order of nature’s parts is beautiful, then nature itself is not beautiful, nor are its parts, but only the harmonious order. Then stars and rainbows are not beautiful, nor sunsets. Beauty is synonymous with ‘harmonious order’; where there is no harmonious order of parts, there can be no beauty. Of course this is in contradiction to other statements about beauty, such as Bacon’s dictum that “There is no excellent beauty that hath not some strangeness in the proportion.” In fact, on occasion one sees a rainbow in a part of the sky, illuminated by a bit of sun, while the rest of the sky is grey, and bleak and stormy. Is this a harmonious order of parts?
Much of modern music and art accentuates discord and dissonance rather than harmony, such as the paintings of Jackson Pollock, aka ‘Jack the Dripper’, and the music of Pierre Boulez. Even in Beethoven’s highly praised last quartets there is much that is harsh, jarring, and inharmonious and grates the ear. Is Hamlet harmonious? Weinberg describes it thus: “…Shakespeare’s plays are not spare, perfect structures like general relativity or Oedipus Rex; they are big messy compositions whose messiness mirrors the complexity of life. That is part of the beauty of the plays, a beauty that to my taste is of a higher order than the beauty of a play by Sophocles or the beauty of general relativity.”4
Harmony is considered to be related to beauty, but it isn’t usually thought that there can be beauty if and only if harmony is present. And perhaps, as Weinberg suggests, the beauty of messy complexity is of a higher order than the beauty of harmony, and Poincaré worships a lesser god.
“This beauty can be grasped by a pure intelligence.” What is pure intelligence? A scientist? Not likely. It sounds more like a definition of a deity. Pure intelligence, devoid of human feelings and exempt from the limitations of human perception may be able to grasp beauty, but perhaps a pure intelligence might not recognize beauty at all, if beauty is not something recognized by intelligence alone, but is more an intermediary characteristic seen only by a combination of human perceptions, emotions, and intelligence. As physicist and philosopher of science Sir Arthur Stanley Eddington wrote: “…let us consider humour. I suppose that humour can be analyzed to some extent and the essential ingredients of the different kinds of wit classified. Suppose that we are offered an alleged joke. We subject it to scientific analysis as we would a chemical salt of doubtful nature, and perhaps after careful consideration of all its aspects we are able to confirm that it really and truly is a joke. Logically, I suppose our next procedure would be to laugh. But it may be… that as a result of this scrutiny we shall have lost all inclination we may ever have had to laugh at it. The classification… preserves all the characteristics of a joke except its laughableness.”
Might it not be so also for beauty? And, if so, then perhaps Poincaré’s ‘pure intelligence’ might not grasp the beauty after all.
Finally, “If nature were not beautiful, it would not be worth knowing and life would not be worth living.” Whether or not life is worth living has been a subject of contemplation for many people. Schopenhauer, ever cheerful, said that if life is a gift, “…everyone would have declined such a gift if he could have seen it and tested it beforehand”, and that if life is a lesson, “…I wish I had been left in the peace of the allsufficient nothing, where I would have no need of lessons, or of anything else.” Tolstoy added that life is a “stupid fraud” and “not worthwhile.” Others have taken a more optimistic view, that life is worthwhile, while a third group remain in doubt, now leaning one way, now the other. Poincaré’s position seems to be that life is worthwhile, but only because nature is beautiful and so gives pleasure. The beauty and/or the pleasure make life worthwhile for Poincaré. His position seems quite dogmatic in that the situation, as he sees it, is the same for everyone. The rather insulting inference would be that one who lacked sufficient ‘pure intelligence’ to grasp the ‘intimate beauty which comes from the harmonious order of nature’s parts’, would lead a life that is not worth living.
If Voltaire had listened to Poincaré’s statement concerning science, beauty, pleasure and the value of life, would he (Voltaire) not have exclaimed: “Grandiloquent nonsense!”? Perhaps not… the ‘nonsense’ seems too harsh, but the ‘grandiloquent’ quite appropriate.
Poincaré’s vision of beauty in science is one way that scientists use the concept. That is, beauty is a quality that is present in nature, and is discoverable through scientific effort. A second way that some scientists are concerned with beauty is that when they detect beauty in their work, they see it also as a signpost pointing them in the right direction.
Dirac said: “…it is more important to have beauty in one’s equations that to have them fit experiment… the discrepancy may well be due to minor features… that will get cleared up with further developments of the theory.” Dirac would become legendary among physicists for at least one of his ‘beautiful’ equations. He developed wave equations describing energy, in which the energy of an atomic particle worked out to be the plus or minus square root of something. If the plus sign denotes the energy of an electron, what does the negative sign denote? Does anything in nature correspond to it? Why should a term in a mathematical equation have a counterpart in nature? Dirac had faith in the beauty of his equation, and put forward the hypothesis that for every charged atomic particle there exists also a corresponding particle with identical mass but opposite electric charge, an ‘antiparticle’. Dirac’s hypothesis was later extensively corroborated, and the existence of anti-matter accepted.
Dirac’s faith that a beautiful theory would probably turn out to be right was rewarded in this case, but is faith justified in these situations?
The mathematician Hermann Weyl said: “My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.” Thinking along Einsteinian lines, Weyl guessed that just as gravity distorts space, altering directions, so too dimensions of space and of objects in space may also be subject to distortion. Equations describing such effects turned out to be identical with equations describing electromagnetic effects. Weyl thought that he might be onto a theory unifying gravity with electromagnetism, and he wrote to Einstein to share his ideas. Although Einstein called Weyl’s theory a stroke of genius, and a beautiful theory, he also explained that, “in my opinion it is impossible that your theory corresponds to nature.”
Although Weyl was wrong about how his theory fitted into explanations of nature, the passing of years has shown that Weyl’s theory does ‘correspond to nature’, albeit not in the way he had thought. It has found a place in quantum theory, fibre bundle theory and in describing the strong nuclear forces that bind together protons and neutrons.
It is puzzling that what begins as mathematics ends as physics, that mathematical equations become theories of reality. ‘Beautiful’ equations and theories, ones that appear to the educated intuition of mathematicians and physicists to have a special grandeur, seem also to be the ones that have stayingpower and lasting significance. This suggests complex relations between mathematics, aesthetics, mind and world. Of course, no doubt many equations and theories have been discredited, and some of those may have once been considered beautiful. Credibility in such matters arises when one predicts things, or one’s theories do, and there is subsequent corroboration. In Shakespeare’s Henry IV there is a similar situation. The Welsh leader and so-called magician Owain Glyndwr boasts that he can “call forth spirits from the vasty deep.” Henry Hotspur, always sceptical, retorts:
“Why, so can I, or so can any man;
But will they come when you do call them?”
Anyone can make predictions, but one whose predictions consistently come true, has methods worth looking into. Glyndwr summons musicians, who presently “hang in the air a thousand leagues from hence, and straight they shall be here.” Then music plays, and even doubting Hotspur must admit that Glyndwr’s ‘prediction’ came true.
If beautiful hypotheses and theories, and elegant mathematical equations, seem to lead to scientific truth significantly more often than ‘ugly’ or nonbeautiful hypotheses and theories, then it should be worthwhile to ask why.
What a scientist means when s/he says ‘beauty’ in reference to scientific matters may not be exactly what a non-scientist means by beauty. Richard Feynman said that physicists delight in using ordinary words for something else.5 And another physicist, Steven Weinberg, wrote: “A physicist who says that a theory is beautiful does not mean quite the same thing that would be meant in saying that a particular painting or a piece of music or poetry is beautiful. It is not merely a personal expression of aesthetic pleasure; it is much closer to what a horse-trainer means when he looks at a racehorse and says that it is a beautiful horse. The horse-trainer is of course expressing a personal opinion, but it is an opinion about an objective fact: that, on the basis of judgements that the trainer could not easily put into words, this is the kind of horse that wins races… The physicist’s sense of beauty is also supposed to serve a purpose – it is supposed to help the physicist select ideas that help us to explain nature.”4
So, according to Feynman and Weinberg, scientists (and horse-trainers) use some words in other than their ordinary public sense, and especially the words beauty and beautiful. How then do they use it? It is rare to find a scientist who’ll give a precise explanation, and even rarer is an attempt to justify that explanation, but there are several concepts that appear frequently in discussions of scientific beauty: harmony, simplicity (for physicists), complexity (for chemists), elegance, symmetry, proportion, unity, completeness, inevitability, and others. In one chapter of Weinberg’s book Dreams of a Final Theory, he names seventeen varieties of scientific beauty. However, all these aspects of beauty, and beauty itself, are most often discussed in conjunction with, and generally subordinate to, scientific usefulness and truth.
The strange cases of Dirac’s equation and Weyl’s speculations are two successful examples of the chief way that scientists use the word beautiful. That is the way of which Weinberg wrote, and which could be called the ‘aesthetic hypothesis’6. The ‘aesthetic hypothesis’ suggests that there is a close connection between beauty and truth, or more modestly, between beauty and well-corroborated, non-refuted theories. Beauty is viewed as a sign of the true. This means that when a scientist declares a theory beautiful, s/ he means more than just surface or formal beauty, (though that may be appreciated too) but rather senses some truth about the theory, senses that the theory may be a winner, to follow the racehorse analogy. So when a scientist expresses aesthetic admiration for some theory, s/he is often expressing intellectual satisfaction with the content of the theory, as one that seems likely to hold true under testing.
If the presence of aesthetic qualities signifies the presence of truth, then the scientist is justified in expecting to find truth nearby once s/he has located the aesthetic aspects of some theory or equation.
However, aesthetics is an unusual criterion in this regard as a case may be made that what a scientist refers to as aesthetic may really be an aspect of truth. This is because scientists use terms to denote aesthetic properties in undefined ways, and then impute to the scientific meaning of aesthetics various attributes that really are attributes of scientific truth or laws of nature. For example, scientists hold that laws of nature have a regularity that can be stated in mathematical form. So when a scientist collects some data that shows regular relations between objects, this could well be a sign that some law of nature applies there. If the scientist says: “Hmmm… I believe I sense beauty”, s/he is, perhaps, assuming the existence of some extraneous component. The scientist could have left aesthetics out completely, and said: “Hmmm… this regular relationship may indicate a law of nature.”
Or one might call this a circular argument: “This seems aesthetic, and I know from past experience that aesthetics oft implies some working of nature.” But in reality the aesthetic something that was detected was some working of nature. So perhaps the scientist is saying: “I detect the regular pattern of nature at work here; therefore I detect the regular pattern of nature at work here.” Which is redundant to say the least. The circularity here conjures up Nietzsche’s critique of science: “If someone hides an object behind a bush, then seeks and finds it there, that seeking and finding is not very laudable…”
© Marilyn Kane 1997
1 Encyclopaedia Britannica, article on Physics
2 J.H.Poincaré, quoted in The Aesthetic Dimension of Science, 1980 Nobel Conference.
3 Encyclopaedia Britannica, article on Nature Worship
4 Steven Weinberg, Dreams of a Final Theory
5 Richard Feynman The Character of Physical Law
6 I am borrowing the phrase ‘aesthetic hypothesis’ from Clive Bell, but have given it a totally different meaning.
Marilyn Kane received her MA from St David’s College, (University of Wales, Lampeter). She is now working toward a PhD in Philosophy at the University of British Columbia.
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