Mathematics Is the Music of Reason

Pythagoras, according to ancient accounts, discovered that a plucked string sounds an octave higher when its length is halved and a fifth higher when it is shortened by one third. Pleasing harmonies, he found, correspond to simple ratios of whole numbers. The discovery was one of the first links between mathematics and the senses. The phrase of the topic, that mathematics is the music of reason, is attributed to the nineteenth century mathematician James Joseph Sylvester, and it carries the same intuition, that reason too has a harmony which can be heard by those trained to listen.

The comparison begins with structure. Music is made of a small number of elements, notes, rhythms and intervals, combined by rules into patterns that can be varied endlessly. Mathematics is the same. A mathematician starts with a few axioms and builds a vast edifice, as a composer builds a symphony from a theme. In both, the pleasure lies in seeing a simple idea develop and return in new forms. A fugue by Bach and a proof by Euler share the quality of inevitability, where each step feels both surprising and necessary.

The two also share a language of abstraction. Neither refers directly to the everyday world. A melody tells no story, and a theorem about prime numbers describes no object that one can touch. Yet both have a precise notation, the staff and the equation, that allows a stranger in another century to reproduce the work exactly. The Indian grammarian Pingala, in his treatise on poetic metre composed around the second century BCE, described patterns of long and short syllables in a way that anticipated binary numbers, so that mathematics and rhythm were bound together in the Sanskrit tradition from the start.

The link is more than metaphor. In the early nineteenth century Joseph Fourier showed that any periodic sound can be broken into a sum of simple sine waves, which are the pure tones of which every instrument's voice is made. The mathematics he introduced is now the basis of digital audio, so that when a phone plays a song it is performing the calculation that he proposed. Kepler, searching in 1619 for the order in the heavens, wrote The Harmony of the World and believed that the planets sang in ratios. His astronomy was right where his music was wrong, but the hope had been fruitful.

His astronomy was right where his music was wrong, but the hope had been fruitful.

Mathematicians themselves insist on the aesthetic. G. H. Hardy wrote in A Mathematician's Apology in 1940 that the patterns of a mathematician, like those of a painter or poet, must be beautiful, and that there is no permanent place in the world for ugly mathematics. Srinivasa Ramanujan, who sent Hardy pages of formulas from Madras in 1913, produced results of such strange elegance that Hardy compared them to nothing he had seen. Euler's identity, which links five of the most important numbers in a single short equation, is often called the most beautiful formula in mathematics, and people who understand it speak of it as others speak of a piece of music.

Another reason for the analogy lies in the way both are learnt. A student of music practises scales for years before the pleasure of playing arrives, and a student of mathematics works through exercises before the pleasure of proof. In each case the beginner experiences drudgery, and the expert experiences flow. Poor teaching fails in the same way in both fields, by presenting rules to be memorised and not patterns to be heard. A teacher who plays a piece to the class before explaining its rules often finds that interest arrives first and understanding follows, and the same order works for a theorem. People who say they are not musical or not mathematical often mean that nobody showed them the point at which the practice becomes joy.

Yet the analogy has limits, and they are instructive. Music is felt, and mathematics is proved. A piece can move a listener who knows nothing of its construction, while a theorem moves only those who can follow it. Music may be wrong in no sense, since a dissonance can be deliberate, but a false theorem is simply false. Music is also tied to emotion in a way mathematics is not, as it can stir grief, joy and longing at once. Reason, in the topic, is supposed to be cool. The word music suggests that it need not be.

A harder question is raised by Eugene Wigner, who wrote in 1960 of the unreasonable effectiveness of mathematics in the natural sciences. Mathematical structures invented for their beauty, such as complex numbers and non Euclidean geometry, turned out to describe quantum mechanics and relativity. Music has no comparable power over the world. A symphony does not predict the motion of a planet. If mathematics is music, it is a music that the universe seems to be playing, and it is not clear whether we discovered the score or composed it, or why the two match.

This puzzle makes the topic more than a pretty phrase. It suggests that reason is not merely a tool for getting things done, but a faculty with its own delights, to which education should introduce every child. A school that teaches mathematics only as a gate to jobs will produce clever calculators and few lovers of the subject. A school that shows its patterns, with its rhythms and surprises, may find that students who thought themselves incapable begin to hear what others have long heard. The effect of music on the soul has never needed a justification, and mathematics may deserve the same freedom.

What the analogy cannot settle is whether the harmony is in the world or in the listener. Pythagoras believed the cosmos was built on number, and many physicists today feel the same. Others argue that the human mind, shaped by evolution, finds order wherever it looks and calls it mathematics. If so, the music of reason is the sound of our own thinking echoing back from nature, and it would be impossible to distinguish this from a real harmony except by the pleasure each one gives.