The Mathematician in the Harbor
At the edge of the harbor in Pythagoreio, a modern bronze monument shows Pythagoras of Samos. Created by sculptor Nikolaos Ikaris in 1989, the statue shows the philosopher with one arm pointing to the sky and the other holding a right-angled triangle, a reference to his famous theorem. The town itself, a UNESCO World Heritage Site since 1992, is built over the ancient city of Samos, which reached its peak as a nautical and commercial power in the 6th century BCE, during Pythagoras's lifetime. This statue stands not far from where the philosopher himself was born around 570 BCE.
Pythagoras and his followers operated under the principle that the entire cosmos is ruled by numerical relationships. Their motto was "All is number." This idea emerged from tangible observations, most famously in music. The Pythagoreans discovered that consonant musical intervals correspond to simple integer ratios of string lengths. An octave has a ratio of 2:1, a perfect fifth a ratio of 3:2, and a perfect fourth a ratio of 4:3. This was one of the first successful descriptions of a physical fact in mathematical terms. This principle extended to the heavens in a concept called the "Harmony of the Spheres," which proposed that planets move according to mathematical equations, creating a form of celestial music.
A Crisis of Incommensurability
The Pythagorean belief system, based on the perfection of whole numbers and their ratios, faced a deep challenge from within its own geometry. The application of the Pythagorean theorem (a² + b² = c²) to a simple square with sides of length 1 results in a diagonal with a length equal to the square root of 2. This number, √2, cannot be expressed as a ratio of two whole numbers. It is what is now called an irrational number.
The existence of such "incommensurable" magnitudes was a philosophical shock. It suggested that the universe could not be fully explained by the neat ratios of integers that formed the foundation of the Pythagorean worldview. Ancient sources report that this discovery was initially kept a closely guarded secret. According to legend, the Pythagorean Hippasus of Metapontum was drowned at sea by his fellow philosophers for divulging the existence of irrational numbers to the outside world. This crisis shows how tightly their mathematical discoveries were bound to their philosophical and religious doctrines.