It is well known that Paul A.M. Dirac (1902–1984) had a great impact on quantum mechanics. His celebrated wave equation won him the 1933 Nobel Prize, was dubbed "The Dirac Equation," and predicted the existence of antimatter particles 27 years before their experimental confirmation by Segre and Chamberlain in 1955.1 Lesser known is Dirac’s philosophy of science, particularly his aesthetic criterion of theory evaluation in physics. In "The Evolution of the Physicist’s Picture of Nature," Dirac argues that “it seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty.” In another place he says, “Physical laws should have mathematical beauty.” For Dirac, beauty is the mark of a good physical theory.2 And a good theory is more likely true than a worse alternative. So a beautiful theory is more likely true than some uglier alternative. In other words, beauty is a road to truth in physics. In the discussion that follows, I will set out the ways in which beauty may play a role in the sciences before briefly looking to how philosophers have tried to justify an aesthetic criterion of theory evaluation.
What is important to notice is that Dirac sees the role of beauty in science as central to theory evaluation, but this isn’t the only role it could have. Going back to Hans Reichenbach (1938), philosophers of science have distinguished between "the context of discovery" and "the context of justification."3 The context of discovery is the context in which scientists generate new hypotheses. It comprises whatever myth-making, guesswork, or methods of discovery scientists employ in creating hypotheses to test. In this area, beauty can play the role of a heuristic—a useful guide that we look to in brainstorming new ideas.
Opposite this is the context of justification: the context in which we test and evaluate those hypotheses. The context of justification is comprised of whatever criteria we use in accepting and rejecting hypotheses. Many believe that the evaluative criteria are exhausted by empirical and logical conditions alone, e.g., fit with observation and coherence. But contrary to this, Dirac insists that there is an aesthetic criterion of good physical theories. For Dirac, fit with experiment and coherence are important, but beauty is the sign of a likely theory. Beauty is a full-fledged evaluative criterion, operating in the context of justification. It is a kind of evidence.
There are two ways to spell out the evaluative role of beauty, or the thesis of an aesthetic criterion of good theories. We might hold that beauty is just one of several criteria for evaluating physical theories, while fit with experiment and observation remains paramount. This we can call the weak version of the thesis. If two theories are empirically equivalent, then the beautiful theory is more likely true than the uglier alternative. However, once a beautiful theory is tested and we find disconfirming evidence, we no longer find it probable. All else being equal, the beautiful theory is more likely true than not. But the aesthetic criterion plays a minor role compared to traditional criteria like fit with the empirical data.
On the other hand, we might believe that beauty is the feature of a good physical theory. Even in the face of disconfirming experimental results, the beautiful theory is more likely true than a better empirically supported, yet uglier, competitor. This we can call the strong version of the thesis. Not only did Dirac believe that beauty was a measure of the likelihood of a theory, he believed that it was more reliable a measure than experimental evidence. He claims “it is more important to have beauty in one’s equations than to have them fit experiment” (Op. cit., p. 50).
The consequences of the strong version are counterintuitive. First, we should retain a beautiful theory in the face of disconfirming experimental data. And second, we should reject theories that have already passed the tests, because they aren’t particularly beautiful. An example might make this seem more reasonable. Dirac cites Schrödinger’s wave theory as an example of a beautiful theory disagreeing with the results of an experiment. The first experiments into the behavior of the electron in the hydrogen atom produced results disagreeing with Schrödinger’s equation. According to Dirac, because the equation was particularly beautiful, it was nevertheless retained. It wasn’t until several years later that Schrödinger’s equation was corrected by accounting for the spin of the electron and the decision to retain the theory was vindicated. If we judged theories solely (or primarily) on the basis of fit with experiment, we would have rejected Schrödinger’s theory prematurely. However, if we accept the strong thesis of beauty’s evaluative role, how much disconfirming experimental evidence would it take for us to ever reject a beautiful theory?
The remainder of the discussion will briefly look to how philosophers of science have tried to justify an aesthetic criterion of theory evaluation. From the history of science we might glean a straightforward answer to the problem of justification: beauty is an evaluative criterion in science because nature is itself beautiful—the natural world has a mathematically beautiful form. But this doesn’t seem to be the right kind of justification. There is a distinction between warranting a belief in x based on the evidence and (something like) grounding the fact that x. Water on your windows may warrant the belief that it is raining, but it doesn’t ground the fact that it is raining. The water on the windows doesn’t cause it to rain and water can get on your windows without it having to rain (e.g. a garden hose). The fact that nature is beautiful grounds the fact that beauty is a mark of likelihood, but it doesn’t warrant our belief that beauty is such a mark. Or, it would only if we knew it already. This fact about the universe is something we are trying to find out through generating and evaluating theories in the first place. The straightforward answer—that nature is beautiful—presupposes what we want to show.
A popular author on the evaluative role of beauty is the Dutch philosopher Theo Kuipers. Kuipers believes that scientists are right in looking to aesthetic criteria as a guide to truth because what scientists in a certain field see as beautiful are the aesthetic features present in past successful theories. Experimental psychology has shown that repeated exposure to a stimulus will increase the emotional appreciation of the stimulus. The increase of aesthetic appreciation through repeated exposure is called the "mere-exposure effect." Kuipers applies the mere-exposure effect to the scientist’s context of justification. He finds that scientists are repeatedly exposed to the exemplary theories and experiments of their research paradigm and that this repeated exposure instills an appreciation of the exemplars’ aesthetic features. This might be simplicity, harmony, novelty, integration with other areas of science, etc. The scientist goes on to infer that, since the exemplar theories were successful in the past and had x, y, and z aesthetic features, when we find those same features in a new hypothesis we have reason to believe it is likely true. The aesthetic criterion of theory evaluation is justified by an induction on past successes in research areas.
The trouble for Kuipers’ account is that if it is correct, the thesis that beauty is a road to truth turns out to be trivial and uninteresting. Kuipers correlates beautiful* features and empirical success through induction, not beautiful features and empirical success. (Beautiful* features are the features that belong to the aesthetic canon of a research paradigm at a given time.) As we’ve seen, this can be any number of traits: simplicity, harmony, conserving our beliefs, novelty, etc. It can also include traits that we generally find ugly; for instance, in evolutionary biology we might value unrepeatable contingencies and irregularities. The unqualified notion of beauty being an evaluative criterion in science is an interesting and substantive thesis. The notion that beauty* (the non-empirical features of past successful theories relative to a research paradigm) is an evaluative criterion loses its bite. Of course, features of past successful theories are likely features of future successful theories. What we wanted to show was that beauty in its unqualified form is a feature of likely theories.
The philosopher of science Paul Thagard gives an alternative account of how our aesthetic criterion might be justified. Rather than being rooted in an aesthetic induction on past successful theories, beauty is a consequence of the coherence of a theory. “Beauty is the feeling that emerges to consciousness when a theory is very strongly coherent” (p. 368). Beauty is a measure of likelihood since it is a sign of a coherent theory. And "very strongly coherent" hypotheses are likely to be true. So beauty is a road to truth, but only in this derivative way.
The merit of Thagard’s account is that it attempts to answer the interesting version of the aesthetic criterion of theory evaluation; that is, one that appeals to the unqualified notion of beauty common to scientific and nonscientific domains. Coherence elicits feelings of beauty in mathematical equations as well as in paintings, films, and music. Harmony and coherence seem to be beautiful in general. So unqualified beauty can be a road to truth on Thagard’s account. However, it is unclear what we should make of Thagard’s account if we accept that incoherence can be beautiful in certain circumstances. He must assume that incoherence never produces feelings of beauty. However, if it can, then the feeling of beauty stems from either coherence or incoherence, while only the former is assumed evidence for the likelihood of a theory. We would be wrong to believe a beautiful theory where the feeling is had in the absence of coherence. So the question is: why should we mention beauty at all? Why doesn’t the scientist just look for coherence and let beauty fall by the wayside? Unfortunately, if we do this, we’re back to the empirico-logical criteria that Dirac’s criterion of beauty was meant to supplement.
Several scientists and philosophers throughout modern history have held the thesis that beauty is a road to truth—that beauty is a criterion of theory evaluation. Unfortunately, we are missing a justification of this thesis. What reason does Dirac or anyone else have for believing that a beautiful theory is more likely true than not? What does beauty have to do with truth? We can now see that the attempts to justify this thesis discussed above either presuppose what we’re trying to figure out, or they deflate the thesis to something trivial, uninteresting, or irrelevant. Whether beauty is evidence for the truth of a theory remains a question.
1. “The Dirac equation” is a wave equation that describes the behavior of fermions—subatomic particles with half-integer spins, like electrons—in a way consistent with special relativity. The prediction of antimatter particles stems from a realist interpretation of the equation’s negative solutions. For example, just as a quadratic equation has two solutions, Dirac’s equation has a positive solution describing an electron with positive energy and a negative solution describing an electron with negative energy (the antielectron). Of course, we can accept the equation without accepting the realist interpretation, but the results of Segre and Chamberlain and the ongoing experimental manipulation of antiprotons and antihydrogen are reason to believe antimatter does exist. Currently, CERN has five research projects that employ antimatter particles to other experimental ends, such as discovering the value of Earth’s gravitational acceleration (the AEGIS project).
2. There are several examples of scientists who judge theories based on aesthetic features. A popular example is Einstein, who believed that his General Theory of Relativity was most likely correct regardless of what Eddington’s experiments would find. Less popular examples are Steven Weinberg, Stephen Jay Gould, and Daniel Freedman at CERN.
3. Although Reichenbach understood this distinction in light of his other views in the philosophy of science, a crude distinction will serve our purpose in this article. It is important to note that some philosophers of science reject the distinction between a context of discovery and a context of justification, seeing it as stemming from the false idea that the elements of knowledge—theories, observations, etc.—are timeless elements, intelligible outside of their historical context. See Paul Feyerabend, Against Method, p. 105.
Thomas Morrison has an M.A. in philosophy from the University of Chicago. His main area of research is the philosophy of science, including the social sciences and psychology. He currently lives in Lawrence, Kansas, where he continues to write philosophy and work on his veggie korma recipe.
Thank you for this. I’ve often heard physicists talk about beauty, and it’s weird. Even eminent physicists have views on the question that seem poorly considered. Isn’t it presumptuous to speak of beauty as if it were objective – a controversial (probably minority) opinion. Beyond that, their standards of beauty seem to reduce to mere simplicity in an equation, which actually hides a great deal of “ugly” complexity via indirection to a half-dozen math textbooks. Although cosmologists will discuss observer selection effects with respect to fine-tuning, they never apply the same principle to question their standards of beauty: perhaps only students that found Riemann topology beautiful decided to pursue a career in theoretical physics. Would they accept a theologian’s standard of beauty as evidence of God’s existence as uncritically? They reflect on the unreasonable effectiveness of math, but never seem to contemplate the parallel notion of whether paint is unreasonably effective in portraiture, and their standard of ‘reasonableness’ again seems left unquestioned. Dirac’s naiveté is typical. It’s gotten so ridiculous that I recently heard a notable physicist declare something along the lines of “there has never been a theory in physics this beautiful that turned out not to be true.” (My apologies, I can’t recall who said this about string theory).
It’s a mystery why General Relativity, which failed to predict dark matter as well as dark energy, isn’t replaced with another more “ugly” theory, with more terms, that actually describes observed phenomena. I think the beauty of Relativity is something of a red-herring – what Einstein did was take an old fundamental force, Newton’s gravity, and made it more complicated, so that it fit observation. There should be more of that. Instead, an unconsidered notion of “beauty” often seems paramount. On the other hand, I’m glad physicists didn’t posit a new particle to explain the perturbation of the orbit of Mercury, (and further particles to explain additional inconsistencies the first particle raised or left out), as they are doing now with dark matter and dark energy. But I’m not sure that’s much different than adding new “beautiful” dimensions like a one-trick pony, just to avoid adding a new fundamental force – or, God forbid, a more complex (i.e. “ugly”) old fundamental force. On a related note, how useful is a theory if it requires tens of billions of dollars in machinery just to distinguish one functional model of the universe from another? Perhaps it could be better spent elsewhere. At the very least, let’s hope the LHC doesn’t unintentionally explain the Fermi paradox. Do physicists consider beauty when modeling risk?
Great piece. Beauty is tricky, but so is truth. Beauty seems to also mean simplicity and elegance in a theory, and while at times the beautiful theories might not account fully for the ugly reality, I think they do a good job at positing possibilities and routes for discovery. The aesthetic criteria of truth must be considered as a consequence.