Pythagoras & Euclid
The birth of proof — from mystical number philosophy to the axiomatic method
4.1 The Pythagorean Brotherhood
Pythagoras of Samos (c. 570–495 BCE) founded a philosophical and mathematical community in Croton, southern Italy. The Pythagoreans believed that “all is number” — that the fundamental nature of reality could be understood through mathematical relationships.
They discovered the mathematical basis of musical harmony (the ratios of string lengths that produce consonant intervals), studied the properties of whole numbers (perfect, amicable, and figurate numbers), and are traditionally credited with the first proof of the theorem that bears Pythagoras's name.
The Crisis of Irrationals
The Pythagorean worldview was shattered by the discovery that √2 is irrational — it cannot be expressed as a ratio of whole numbers. Legend holds that the discoverer, Hippasus, was drowned at sea for revealing this secret. Whatever the truth, the discovery forced Greek mathematics to develop a geometric, rather than purely arithmetic, foundation.
4.2 Euclid's Elements
Euclid (c. 325–265 BCE), working in Alexandria, compiled the Elements — thirteen books that organized virtually all known Greek mathematics into a single deductive system. Starting from five postulates and five common notions, Euclid proved 465 propositions covering plane geometry, number theory, and solid geometry.
The Elements is the most successful and influential textbook ever written. Over 1,000 editions have been published since the invention of printing. For more than two millennia, it was the gold standard of rigorous mathematical reasoning — Abraham Lincoln studied it to sharpen his logical thinking.
4.3 The Parallel Postulate
Euclid's fifth postulate — the parallel postulate — states that through a point not on a given line, there is exactly one line parallel to the given line. For centuries, mathematicians tried to prove this from the other four postulates, believing it was not truly independent.
These attempts failed — because the parallel postulate is independent. In the 19th century, Bolyai and Lobachevsky showed that denying it leads to perfectly consistent non-Euclidean geometries. This discovery revolutionized mathematics and ultimately provided the geometric framework for Einstein's general relativity.
4.4 The Legacy
The Pythagorean-Euclidean tradition established the paradigm that mathematics consists of theorems proved from axioms through logical deduction. This paradigm — extended and refined over the centuries — remains the foundation of all mathematical practice today.
Pythagoras
- • “All is number” philosophy
- • Mathematical basis of music
- • Number classification (perfect, amicable)
- • The Pythagorean theorem (as proof)
Euclid
- • Axiomatic-deductive method
- • 465 propositions in 13 books
- • Infinitude of primes (Book IX, Prop. 20)
- • Euclidean algorithm for GCD