Transforming Mathematics into a Means of Comprehending Reality: The Legacy of Pythagoras

The phrase “the music of the spheres” may not immediately bring mathematics to mind, but in its historical origin, it was all about math. Back in ancient Egypt and Mesopotamia, math served mainly as a practical tool, used to facilitate human interactions. It was employed to calculate the area of a farmer's land, keep track of workers' wages and determine the right amount of ingredients when making bread or beer. Nobody used math to investigate the nature of physical reality until ancient Greek philosophers began to seek scientific explanations for natural phenomena without recourse to myths. The first of these Greeks to put math to use for this purpose was Pythagoras, the enigmatic religious cult leader of Samos.

Pythagoras was born in about 575 B.C. and traveled widely to Egypt and Babylonia, and perhaps even Persia, to learn how math was used in ancient cultures. After studying with the aging Thales, the founder of the ancient Greek effort to explain the world rationally, Pythagoras headed west to southern Italy and settled in the Greek colony town of Croton. There he initiated a new phase of ancient science, mixing religion, music and math in a cult devoted to living in harmony with nature. For Pythagoras, philosophy ceased to be primarily about explaining nature but instead “becomes the search for a way of life whereby a right relationship might be established between the philosopher and the universe.”

Though Pythagoras established a religious cult, he and his followers expanded the Greek quest to explain the cosmos. Pythagoras believed that reality was made from numbers. To him, numbers were the seeds from which all reality grew. Members of the Pythagorean cult had to recite an oath crediting him with identifying the numbers “which contain the fount and root of ever-flowing nature.”

Pythagoras identified reality’s root in the tetractys, consisting of the first four integers: 1, 2, 3 and 4. Added together, they equal 10, which Pythagoras believed was the “perfect” number and the key to understanding nature. Pythagoras realised that these harmony-producing ratios all involved the numbers 1, 2, 3 and 4. Therefore, he concluded, their sum, 10, was the key number for developing a theory of the universe.

Pythagoras’ belief that reality was rooted in numbers may sound crazy to modern minds, but at the time, major philosophers had their own ideas about what sort substance served as reality’s foundation. For instance, Thales thought everything derived from water, while his student Anaximander argued that reality consisted of featureless material called the apeiron. Later philosophers, such as Anaximenes and Heraclitus, claimed that air and fire were reality's foundations, respectively.

The Pythagorean cult's use of math and music to explain the cosmos marks a significant moment in the history of math’s relationship with science. Pythagoras turned math from a mere tool for practical purposes into the key for unlocking the mysteries of the universe. Geoffrey Lloyd, the historian, observed that “Pythagoreans were … the first theorists to have attempted deliberately to give the knowledge of nature a quantitative, mathematical foundation.”

Pythagoras himself may not have developed that theory fully, but his later followers produced a vision of the universe consisting of heavenly bodies revolving around a “central fire.” That fire was NOT the sun, which was just one of the other heavenly bodies. The sun shone brightly because it reflected the light from the central fire back to the Earth (and the inhabited part of the Earth always faced away from the fire).

The Pythagoreans surmised that the motions of the heavenly bodies generated pleasant music. As Aristotle later explained it, those bodies move rapidly and therefore they must make sound, because anything moving quickly on Earth makes sound (think arrows whizzing through the air). Proper ratios of the planets’ speeds (which depended on their distances from the central fire) guaranteed that the sounds would be harmonious. Hence the moving planets created a “harmony of the heavens.” Because later Greek writers supposed that each planet is carried on its orbit by a rotating sphere, eventually that harmony became known as “the music of the spheres.”

So in a sense, the Pythagoreans believed that the universe itself could be regarded as a gigantic musical instrument. As Aristotle put it, the Pythagoreans thought “the whole heaven to be a musical scale and number.”

But this picture had a problem. The Pythagoreans knew of only eight heavenly bodies: Earth, moon, sun, Mercury, Venus, Mars, Jupiter and Saturn. A ninth, outermost sphere transported the fixed stars. But to be perfect, the cosmos needed a 10th body. So the Pythagoreans proposed the existence of another planet, a “counter-Earth,” orbiting the central fire inside the orbit of the Earth. Nobody could see that planet because it was always on the other side of the fire. Rather than limit their description of the cosmos to what could be observed, the Pythagoreans resorted to mathematical theory to infer the existence of an unseen reality.

This theory was wrong, of course. But it nevertheless foreshadowed the modern use of mathematics to predict unseen phenomena. In the 19th century, for instance, James Clerk Maxwell used equations to predict the existence of radio waves. In the 20th century, Paul Dirac used math to predict the existence of antimatter. And in the 21st century, astronomers detected gravitational waves, vibrating space itself, as physicists had expected based on the math of Einstein’s general theory of relativity.

Using math for understanding nature was unknown before Pythagoras. It was his idea. Previously math had been a tool for scribes or surveyors or cooks. “Pythagoras freed mathematics from these practical applications,” the Dutch mathematician B.L. van der Waerden wrote in his classic history of ancient math. “The Pythagoreans pursued mathematics as a kind of religious contemplation, as a way to approach the eternal Truth.”

As for the music of the spheres, one issue remained. If the heavens made harmonious sounds, why didn’t anybody hear them? Aristotle reported that the Pythagoreans “explain this by saying that the sound is in our ears from the very moment of birth and is thus indistinguishable from its contrary silence.”

Aristotle rejected that explanation, just as he rejected the idea of a “counter-Earth” as well as the whole notion that everything was made from numbers. And yet, the importance of numbers in science, first expressed by Pythagoras, ultimately proved to be much more resilient than most of Aristotle’s ideas. As experts on early Greek philosophy André Laks and Glenn Most have written, “Of all the early Greek philosophers,” Pythagoras “without a doubt exerted the longest-lasting influence until the beginning of modern times.”