Pythagoras is famous for the Pythagorean theorem, the foundational geometric rule that describes the relationship between the sides of a right triangle. But the Greek philosopher who lived from roughly 570 to 490 BCE was far more than a mathematician. He founded a quasi-religious intellectual community, proposed that the soul is immortal and reincarnates, discovered the mathematical basis of musical harmony, and built an entire philosophy around the idea that numbers are the fundamental fabric of reality.
The Pythagorean Theorem
The theorem that bears his name is one of the most widely known results in all of mathematics: in any right triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides. Written as an equation, that’s a² + b² = c². If you know any two sides of a right triangle, you can calculate the third. The classic visual proof works by arranging four identical right triangles inside two squares of equal area, then showing that the leftover space in one arrangement forms two smaller squares (a² and b²) while the leftover space in the other forms one larger square (c²). Since the total areas are the same, a² + b² must equal c².
Whether Pythagoras himself proved the theorem is debated. Babylonian mathematicians used the relationship at least a thousand years earlier. But the Greeks, and the Pythagorean school in particular, are credited with providing a logical proof rather than simply observing that the numbers worked out in practice.
The Mathematics of Music
Pythagoras is also famous for discovering that musical harmony follows simple numerical ratios. The story usually involves a single-string instrument: shorten the string to half its length and the note jumps up exactly one octave, a ratio of 2:1. Shorten it to two-thirds its length and you get a perfect fifth, the interval between the first and fifth notes of a scale, at a ratio of 3:2. Combine those intervals mathematically and you arrive at a perfect fourth, with a ratio of 4:3.
This was a staggering insight at the time. It meant that something as subjective as “these two notes sound good together” could be reduced to precise fractions. For Pythagoras and his followers, it was proof that the universe itself was organized by number. They extended the idea into a concept sometimes called the “music of the spheres,” the belief that celestial bodies moving through space produce a kind of cosmic harmony governed by the same ratios.
“All Is Number”
The Pythagorean school’s motto, reportedly carved above the entrance, was “All is number.” They didn’t mean this as a loose metaphor. The inner circle of the school, known as the mathematikoi, believed the universe was literally built from whole numbers and their ratios. Every number from one to ten carried symbolic weight. Odd numbers were considered male, even numbers female, and the number one was neither, instead having the power to change each into the other. Four represented justice. Ten was considered the most perfect number because it is the sum of the first four integers: 1 + 2 + 3 + 4.
That sum had its own sacred symbol, the Tetractys, a triangular arrangement of ten points in four rows (one, two, three, and four points per row). The Pythagoreans packed layers of meaning into it. Each row corresponded to a dimension of space: one point is a position with no dimension, two points define a line, three define a plane, and four define a solid. The rows also mapped onto the four classical elements and the musical ratios that governed harmony. It was, for them, a compact diagram of reality itself.
The Crisis of Irrational Numbers
If everything is built from whole numbers and tidy ratios, then every length should be expressible as a fraction. This belief held until someone in the Pythagorean school, traditionally identified as Hippasus, examined something deceptively simple: the diagonal of a square. Take a square with sides of length 1. By the Pythagorean theorem, the diagonal’s length squared is 1² + 1² = 2, making the diagonal equal to the square root of 2. The problem is that no fraction of whole numbers produces this value. It cannot be written as a ratio, period.
This was devastating to a worldview that insisted all of reality could be described by whole-number relationships. The square root of 2 is what we now call an irrational number, meaning it can’t be represented as a ratio of integers. Legend holds that Hippasus was drowned at sea for revealing this secret to outsiders, though the story is likely myth. The mathematical crisis, however, was real, and the Pythagorean philosophy never fully recovered from it.
The Immortal, Reincarnating Soul
Pythagoras wasn’t only a mathematician. He was the leader of a philosophical and religious community, and his ideas about the soul were just as famous in the ancient world as his geometry. The core beliefs, recorded by the near-contemporary writer Dicaearchus of Messana, included four claims: the soul is immortal; after death it migrates into the bodies of other animals; events recur in fixed cycles so that nothing is truly new; and all living creatures belong to the same family.
This doctrine of transmigration, called metempsychosis, had practical consequences. One famous anecdote from the philosopher Xenophanes describes Pythagoras stopping a man from beating a dog, insisting he recognized the voice of a deceased friend in the animal’s cries. The Pythagoreans followed strict dietary and behavioral rules that likely stemmed from this belief that the boundaries between human and animal life were permeable.
The Pythagorean School at Croton
Pythagoras was born on the island of Samos, off the coast of modern Turkey. Around the age of forty, he emigrated to Croton in southern Italy, where he founded the school that would carry his ideas forward. It was part philosophical academy, part religious brotherhood. Members lived by a shared set of rules, studied mathematics and music, and were divided into an inner circle (the mathematikoi, who engaged directly with Pythagoras’s teachings) and an outer circle of listeners.
The school’s influence was enormous. Pythagorean ideas about number and harmony shaped Plato’s philosophy, fed into early astronomy, and established the principle that nature’s patterns can be described mathematically, an assumption that underpins all of modern science. Pythagoras left no writings of his own, and separating his personal discoveries from those of his students is nearly impossible. What survives is less a precise biography than a body of ideas so durable they’ve been taught continuously for over 2,500 years.