On Mathematical Beauty in Physics
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like a sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.”
— Bertrand Russell
Physics, the most fundamental and abstract of the sciences, is inextricably linked to mathematics, the language it is expressed in. Central to physics is the concept of mathematical beauty. This aesthetic quality, whilst not easily definable, is not subject to any doubt to those who can appreciate it. A study of the brain scans of a group of mathematicians at University College London reveals that beautiful equations have an emotional effect that is not unlike great art and music; observed here was an increase in activity in the medial orbito-frontal cortex—the emotional part of the brain—when gazing at equations whose beauty they rated highly.
The lack of a precise definition of mathematical beauty, and beauty itself being a subjective concept, may lead one to dismiss the subject as not worthy of further discussion. However, a naïve survey of the field reveals a general agreement on the underlying characteristics of what makes an equation “beautiful”. Amongst the not-too-tangible qualities often cited to constitute mathematical beauty are simplicity, symmetry, and elegance. Euler’s equation, highly regarded among mathematicians, is one such example where the union of five constants is made in a rather simple and neat fashion.
The expression of the fundamental laws of nature in simple mathematical equations thus carries in it an element of poetic beauty. The rather succinct Schrödinger’s equation below, for instance, describes the behaviour of atoms and their constituents down to the finest detail, which if solved, gives the probability of finding a particle at a given position.
A personal favourite of mine is the set of Maxwell’s equations. In this elegant formulation we find the marriage of two previously unrelated realms of electricity and magnetism. As an undergraduate, a charismatic lecturer took us through a long journey towards the finale of the four neat equations, whereupon he exclaimed with certain gravity in his voice, “And that’s how Maxwell came up with these equations of electromagnetism, right here in this building”. It sent a chill down my spine which I have never quite forgotten.
“A physical law must possess mathematical beauty.”
There was perhaps no one amongst the 20th century physicists more obsessed with mathematical beauty than Paul Dirac. A trained engineer, he was spurred to take up physics by Einstein’s work on relativity and later became a pioneer of quantum field theory. To Dirac, it was precisely this aesthetic quality that makes the general theory of relativity great, despite the greater mathematical complexities and its going against the principle of simplicity.
So obsessed was Dirac with mathematical beauty it led him to state “It is more important to have beauty in one’s equation than to have them fit experiment”. Quite notably, in the James Scott Prize lecture Dirac delivered in 1939, he offered a devastating insight in a just few short lines: “…the mathematician plays a game in which he himself invents the rule while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen.”
Few physicists would today contest the beauty of his eponymous equation. Deceptively simple, the equation united the then distinctly separate special theory of relativity and the quantum theory to describe the electron, and remarkably predicted the existence of antimatter—an outlandish claim at the time, but is now taken for granted to constitute half of the early universe, as per the Big Bang theory.
This aestheticism will regrettably remain inscrutable to many. In the words of Dirac, mathematical beauty is “a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics have no difficulty in appreciating”. It is presumably this which led Bertrand Russell to call it cold and austere.