Can a math equation or a physical theory be beautiful?
By Gabriel Ferrero
The story goes that once somebody asked Einstein what he would think if it was discovered that the theory of relativity - one of his great contributions to science - was incorrect. Apparently, the great physicist answered that in this case “God lost the opportunity to do something really beautiful”. He obviously had no self-esteem troubles. But the adjective “beautiful” can be even more striking.
Many people, legions of high school students among them, often ask themselves how somebody can love and even consider math equations or physical theories beautiful. Curiously enough, though, it is quite common to hear a mathematician say that a theorem has a “very elegant” demonstration, or a physicist praise “the beauty” of a theory. I myself have often rejected the solution I have found for a physical-mathematical problem because I found it “ugly”.
What lies behind these statements? Perhaps an example could come in handy so that we can understand one another. I will copy Maxwell’s equations for the electromagnetic field. I know that many readers will have probably skipped this page at the mere sight of such formulas, but if you made it so far and didn’t study physics, I’ll let you into a little secret I learned reading Penrose: don’t worry about understanding them. Not even we physicists are sure about what they really represent.
Let’s take note of something which might not stand out at first sight: their simplicity. They are only four equations and contain but twelve symbols, eight of which represent math operations and four, physical magnitudes. For instance, “t” stands for time.
The amazing thing is that these equations are enough to describe, explain and predict a huge amount of phenomenon, among them: light, ultraviolet radiation, infrared radiation, X-rays, radio waves and the way each one of these spread through space, through time, throughout the entire universe. With some additional term, these equations also enable the invention, design and construction of many useful things: electric engines, antenna, loudspeakers, microphones, radios and radio receivers, tinted glass, microwave ovens, 3D movies, etc. The list is probably endless.
Using only twelve symbols, arranged in four equations, we understand countless phenomenon… Isn’t it marvelous? It is a magnificent synthesis! And if we think about the variety, the complexity and the extension of what they represent, we can partly understand why we say that equations are very simple.
Of course it is necessary to study a bit, perhaps more than a bit, in order to understand them, but we can all see that they are very brief, concise and have a limited and really small number of operations. Yet another example of their simplicity.
On the other hand, let’s look at the first and third equations. We can see that if we change symbol E, which represents the electric field, with B, which represents the magnetic field, the equations are the same. Likewise, the second and fourth are very similar. That’s why we say that there’s a certain symmetry in the equations. By interchanging some symbols, the equations become the other, totally or partially, but we always maintain, almost, the same group of equations.
This symmetry of the equations expresses something very deep about the nature of electricity and magnetism. In a way, each one of these phenomenon is the consequence of the other. One could say they express something, a fundamental property of matter which manifests itself in very different ways, yet symmetrical. That is also why we say that equations are beautiful. Simplicity and symmetry: two attributes that make what we call beauty.
Some months ago a colleague showed me a scientific article he had just written. In it he described the behavior of an electron inside a certain molecule under certain conditions in full detail and precision. The corresponding equations took up several pages crammed with symbols which were very hard to understand. I confess I couldn’t avoid saying: how beautiful!
There is also beauty in complexity, when simplicity or symmetries aren’t that obvious. In order to write that equation my friend had to find the way of expressing mathematically, in a clear and explicit way, the relation of each electron with several atomic nuclei, with other electrons, with other molecules, with the electric and magnetic fields present in the matter… in sum, a large number of relations. I think the beauty we perceive in complex phenomenon are related to the beauty of relationality, with something deeply constitutive of what is real, of that which exists.
Everything, from the most basic and elemental levels of matter, exists in relation to, is related to, further still, everything is itself only in relation to others. Perhaps this is the most typical beauty of our times.
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