The Timpson Lyceum · History of Philosophy (2024)

Pythagoras of Samos: The Mystic Mathematician – Lecture 5 (History of Philosophy)

Overview

Pythagoras of Samos (~570–495 BC) is both a mystic and a mathematician, and the lecture keeps both halves in view. He founded a secretive religious-philosophical brotherhood at Croton in southern Italy, bound by strict communal rules and the doctrine of the transmigration of souls. Yet the same movement produced the deepest early insight into mathematics: that number and ratio are the essence of reality — captured in the slogan "all is number."

The Pythagoreans discovered that musical harmony is governed by simple whole-number ratios, which suggested the whole cosmos might be a "harmony" expressible in numbers. That triumphant vision then met its first great crisis: the discovery that some magnitudes (like the diagonal of a square) cannot be expressed as a ratio of whole numbers.

Key Concepts

• The Pythagorean brotherhood — A communal order at Croton with religious rules (the akousmata), shared property, secrecy, and famously peculiar prohibitions (e.g., against eating beans). Membership blended way-of-life with study.
• Transmigration of souls (metempsychosis) — The soul is immortal and is reborn into new bodies (including animals), which grounded their vegetarianism and their view of philosophy as purification of the soul.
• "All is number" — The central doctrine: things are what they are because of their numerical structure; numbers and their ratios are the fundamental principles (arche) of reality.
• Harmony of ratios (music) — Consonant musical intervals correspond to simple whole-number ratios of string length: the octave 2:1, the perfect fifth 3:2, the perfect fourth 4:3. Order in sound = order in number.
• Harmony of the spheres — Extending musical ratios to the heavens: the motions of celestial bodies were thought to embody numerical proportions, a cosmic music.
• Figurate numbers and the tetractys — Numbers arranged as shapes (triangular, square). The tetractys ($1+2+3+4=10$) was sacred and contained the musical ratios.
• The Pythagorean theorem — For a right triangle with legs $a,b$ and hypotenuse $c$: $a^2 + b^2 = c^2$. Long known empirically, it became emblematic of the school's marriage of number and space.
• The crisis of the incommensurable — The diagonal of a unit square has length $\sqrt{2}$, which is irrational — not expressible as a ratio of whole numbers. This directly threatened "all is number" (understood as whole-number ratios); legend says the Pythagoreans tried to suppress it (the story of Hippasus).

Worked Example

Proof that $\sqrt{2}$ is irrational — the incommensurable diagonal that shook the Pythagorean creed. A proof by contradiction.

1. Suppose, for contradiction, that $\sqrt{2}$ is rational: $\sqrt{2} = \dfrac{p}{q}$ with integers $p,q$ sharing no common factor (fraction in lowest terms).
2. Square both sides: $2 = \dfrac{p^2}{q^2}$, so $p^2 = 2q^2$.
3. Then $p^2$ is even, which forces $p$ to be even (the square of an odd number is odd). Write $p = 2k$.
4. Substitute: $(2k)^2 = 2q^2 \Rightarrow 4k^2 = 2q^2 \Rightarrow q^2 = 2k^2$.
5. So $q^2$ is even, hence $q$ is even.
6. But now both $p$ and $q$ are even — contradicting step 1, where the fraction was in lowest terms.
7. The assumption fails: $\sqrt{2}$ cannot be written as a ratio of whole numbers, so it is irrational. $\blacksquare$
8. Consequence: the diagonal and side of a square are incommensurable — a magnitude with no whole-number ratio — undermining the naive form of "all is number."

Summary

• Pythagoras fused a religious way of life (the brotherhood, transmigration of souls) with a revolutionary claim: reality is fundamentally mathematical ("all is number").
• The discovery that musical consonance follows simple ratios (2:1, 3:2, 4:3) suggested the whole cosmos is a numerical harmony.
• The school gave us figurate numbers, the sacred tetractys, and the theorem that bears its name, $a^2+b^2=c^2$.
• Its own mathematics then produced a crisis: $\sqrt{2}$ is irrational, so some magnitudes are incommensurable — a shock that reshaped Greek mathematics and metaphysics.
• Pythagoras thus stands at the root of the enduring idea that the book of nature is written in the language of number.