A few months ago the BBC conducted a survey to find the most beautiful mathematical equation in the world. There is of course a long history of the relationship between the idea of beauty and mathematics. The ancients certainly pondered that relationship as well as connecting it to the concepts of truth and of goodness. In Plato’s mind certainly, as well as in the mind of others, the true, the good and the beautiful were objectively one in the transcendental realm, whilst they could only be perceived as pale reflections in physical reality. To what degree can beauty and mathematics be associated? What do we mean by beauty? Are beautiful equations always true, and in what sense? Is mathematics the best conceptual as well as concrete expression of beauty? These questions should be a typical part of a TOK exploration of Mathematics, let’s see what answers to some of them might look like.

The first step for TOK of course would be an attempt to understand the concept of beauty in the first place. It has become commonplace to quote this tired old cliché that “beauty is in the eye of the beholder.” This is to say that the range of things described as beautiful is so vast that one cannot possibly agree on a set of criteria or qualities (tone, symmetry or form, the presence of the Golden Ratio…) by which all things can then be judged. If this is so, then beauty is indeed an entirely subjective concept and as long as one prefaces any statement about individual things having some of the elements of beauty with “In my humble opinion…”, then one is perfectly safe from the beauty-objectivists out there. However, Plato has found some support from modern philosopher Charles Sanders Peirce, who concluded that “Logic (truth) follows ethics (goodness) and both follow aesthetics (beauty)” (my italics). In fact Peirce, a logician and mathematician came to the view that beauty should govern all human conduct and thought. Would all or most Mathematicians agree with this?

In an effort to answer this question, a few years ago, British researchers strapped a number of Mathematicians to a functional magnetic resonance imaging scanner (fMRI), and showed them a number of famous equations. It turned out that many of these equations triggered activity in the brain usually associated with the experience of visual and musical beauty  (in field A1 of the medial orbito-frontal cortex, apparently). This happened so consistently over the number of subjects that the researchers concluded that not only was beauty recognised in mathematical equations but, taking other research into consideration, they also triggered activity connected with the beauty of understanding and of moral goodness. Plato’s view that the most perfect expression of beauty was in mathematics, and that it is intimately connected to truth and goodness, has therefore been vindicated in a way few people would have anticipated. Sadly, for the poor numerically illiterates out there, they’ll have to be content with expressions of beauty in less elevated forms.

But what of the winning equation in the BBC survey? Out of twelve chosen equations,  Paul Dirac’s equation, which describes how particles behave when they travel close to the speed of light, came out as the clear winner. And what of the equation which so moved those Mathematicians’ brains in the fMRI scanner? Well, Euler’s Identity garnered 50% fewer votes than Dirac’s, so it may well be that, after all, beauty is indeed in the eye of the mathematical beholder.