How Pythagoras Turned Math Into a Tool For Understanding Reality


When you hear “the music of the spheres,” mathematics might not be your immediate association. However, the historical roots of this phrase are deeply intertwined with math. This concept highlights a pivotal moment in the intertwining of mathematics and science.

Historically, in places like ancient Mesopotamia and Egypt, math was primarily utilized as a functional instrument to simplify human activities.

Calculations were essential for determining the size of agricultural fields, monitoring workers’ compensation, or measuring ingredients for bread or beer. The use of mathematics as a means to probe the essence of physical existence wasn’t common.

This shifted when philosophers from ancient Greece began searching for scientific explanations for natural occurrences, sidestepping mythological explanations. Pythagoras of Samos, an enigmatic spiritual leader, was the pioneer in harnessing mathematics to delve into these mysteries.

Around 575 B.C., Pythagoras was born on Samos, an island near modern-day Turkey in the Eastern Mediterranean. He journeyed extensively, visiting places like Babylonia, Egypt, and possibly Persia, absorbing mathematical practices from these ancient civilizations.

It’s believed he might have even studied under the wise Thales in Miletus, close to his homeland, who initiated the Greek rational approach to understanding the world.

When Pythagoras reached about 40, he relocated to Croton, a Greek outpost in southern Italy. In this locale, he embarked on a novel chapter in ancient scientific exploration, blending faith, mathematics, and music in a sect committed to living in sync with the environment.

As described by the Greek philosophy historian, W.K.C. Guthrie, Pythagoras’s approach to philosophy transitioned from primarily explicating nature to “seeking a lifestyle that aligned the philosopher harmoniously with the cosmos.”

Understanding the cosmos is paramount if your aim is cosmological harmony. So, although Pythagoras essentially founded a spiritual movement, he and his disciples continued the Greek mission of unraveling the universe’s mysteries. This quest was where mathematics became invaluable.

For Pythagoras, numbers formed the essence of existence. While it might seem peculiar to those of us taught about molecules and atoms, the nature of reality was largely speculative in antiquity. Philosophers of the era had personal theories about the foundational elements of reality.

Take Thales, for instance, who believed everything originated from water. Anaximander, his student, proposed that existence fundamentally comprised an endless, indistinct substance named the Apeiron.

Anaximenes, succeeding Anaximander, favored air as the source, asserting everything could be rationalized by air either becoming dense or expanding. Heraclitus, another philosopher, believed fire was the ultimate source.

Numbers were Pythagoras’s pick. He conveyed to his disciples that numbers were the essence of all existence. Every Pythagorean was obligated to swear an oath acknowledging his discovery of numbers as the wellspring of the ever-changing natural world.

Specifically, Pythagoras attributed existence’s essence to the Tetractys: the first four numbers – 1, 2, 3, 4. Their sum, 10, was perceived by him as the ideal number, the numeral revealing nature’s secrets.

And the significance of 1, 2, 3, and 4? These integers were integral to generating melodious tunes. Consider strumming a stretched, constant-length string to yield a musical tone. Strumming a half-length string produces a tone an octave apart from the initial note.

If played together, these tones are consonant. Essentially, a string length proportion of 2:1 creates a melodious tone. Similarly, other euphonious musical intervals known as the fourth and fifth are characterized by string length proportions of 4:3 and 3:2, respectively.

Recognizing the significance of these ratios, Pythagoras deduced that their total, 10, was essential for a universal theory.

While Pythagoras might not have fully elaborated this theory, his successors envisioned the cosmos with celestial entities circling a “central flame” – distinct from the sun, another celestial entity.

The sun’s luminosity emanated from reflecting the central flame’s light onto Earth (with its inhabited regions perpetually turned away from this flame).

Pythagoreans theorized that the trajectories of these celestial entities produced delightful melodies. As Aristotle later elucidated, swift movements of these bodies would inevitably generate sounds, akin to swiftly moving objects on Earth (like arrows darting through the atmosphere).

The precise proportions of the planetary velocities (reliant on their proximity to the central flame) ensured these sounds were consonant. Thus, the moving planets yielded the “cosmic harmony.”

Subsequent Greek scholars hypothesized that each planet’s orbit was maintained by a spinning sphere, leading to the term “the music of the spheres.”

In essence, the Pythagoreans envisioned the cosmos as a colossal musical apparatus. As articulated by Aristotle, the Pythagoreans perceived “the entire cosmos as an interplay of musical harmony and numerals.”

However, this concept encountered a dilemma. The Pythagoreans recognized only eight celestial entities: Earth, moon, sun, Mercury, Venus, Mars, Jupiter, and Saturn, with a ninth sphere carrying the static stars.

For the universe to attain perfection, a 10th celestial body was necessary. The Pythagoreans introduced the idea of a “counter-Earth” – a planet orbiting the central flame closer than Earth.

This planet remained invisible, always obscured by the central flame. Instead of confining their cosmic understanding to observable elements, they invoked mathematical theory to suggest an imperceptible reality.

While this notion was inaccurate, it presaged contemporary mathematical applications in anticipating unobservable phenomena. In the 1800s, for instance, James Clerk Maxwell employed mathematical formulas to foretell the presence of radio waves.

Come the 1900s, Paul Dirac harnessed math to anticipate antimatter’s existence. Then, in the 2000s, astronomers identified gravitational waves—ripples in spacetime—as anticipated from Einstein’s general theory of relativity’s mathematical principles.

Prior to Pythagoras, the idea of harnessing mathematics to decipher the natural world was uncharted territory. Until then, mathematics served as an instrument for record-keepers, land measurers, or culinary experts.

Regarding the celestial harmonies, one question lingered: If the skies resonated with melodies, why were they inaudible to humans? Aristotle dismissed this theory, along with the “counter-Earth” concept and the overarching idea of numerals as the foundation of all things.

However, Pythagoras’s emphasis on the significance of numbers in scientific inquiry endured, outlasting many of Aristotle’s assertions.