Pythagoras · Mathematics
How the diagonal of a simple square produced a number that no ratio of whole numbers could express - shattering the Pythagorean faith that all is whole-number ratio, and forcing mathematics to confront the irrational.
The deepest article of Pythagorean faith was that all is number - and by ‘number’ they meant the whole numbers (1, 2, 3, …) and their ratios. They believed that any two lengths, any two magnitudes whatever, are commensurable: that there always exists some common unit of measure, however small, that divides both of them a whole number of times. Take any two line segments; the Pythagoreans were certain you could find a tiny unit such that the first segment is, say, exactly 17 of those units and the second exactly 23 - so that their ratio is the whole-number ratio 17 to 23. This was not a casual assumption but the foundation of their entire worldview: it meant that all of reality, including all of geometry, could be expressed in terms of whole numbers and their ratios. The harmony of music was whole-number ratios; the structure of space was whole-number ratios; everything reduced, in the end, to the integers and their proportions. The whole numbers were the alphabet of the cosmos, and everything could be spelled with them. And then, with bitter irony, the Pythagoreans’ own most famous theorem - the relationship of the right triangle - destroyed this faith from within.
How can one be certain that no fraction whatsoever has a square equal to 2? Not by trying fractions one by one - there are infinitely many, and you could never finish. The Greeks proved it by one of the most elegant arguments in mathematics, a proof by contradiction. Suppose, for the sake of argument, that the square root of 2 could be written as a fraction, and write that fraction in lowest terms (so the numerator and denominator share no common factor - every fraction can be reduced this way). Call it a/b. Then a²/b² = 2, so a² = 2b². This means a² is even (it is two times something), and a number is even only if its own square is even - so a must be even. Write a = 2c. Then (2c)² = 2b², that is 4c² = 2b², so b² = 2c² - which means b² is even, and therefore b is even too. But now we have shown that both a and b are even - both divisible by 2. Yet we assumed the fraction a/b was in lowest terms, with no common factor! This is a flat contradiction: a and b cannot be both share no common factor and both be even. The assumption that led here - that the square root of 2 can be written as a fraction - must therefore be false. No such fraction exists. The square root of 2 is irrational. The proof is complete, certain, and devastating: it does not merely fail to find a fraction; it proves that none can possibly exist.
The discovery of irrational magnitudes was one of the most important turning points in the history of mathematics, and its significance far outlasted the crisis it caused. It revealed that the continuous world of geometric magnitude - the lengths, areas, and volumes of figures - is fundamentally richer than the discrete world of whole numbers and their ratios. The number line is not merely dotted with fractions; between and beyond the fractions lie infinitely many irrational numbers - the square root of 2, the square root of 3, and countless others, including (as was proved much later) numbers like π and e - that no ratio of integers can capture. In fact, in a precise sense established in the nineteenth century, the irrationals are vastly more numerous than the rationals: almost every point on the number line is irrational. The whole numbers and their ratios, which the Pythagoreans thought were everything, turned out to be a sparse scaffolding within a far larger continuum. This forced mathematics to confront the nature of the continuous, of magnitude, and ultimately of the real numbers themselves - a confrontation that drove some of the deepest developments in the subject, from the Greek theory of proportion to the nineteenth-century construction of the real number system. The crisis of the irrational, born from the Pythagoreans’ own theorem, did not destroy Greek mathematics; it deepened it, forcing it to a new level of rigour and revealing that the realm of number and magnitude was vastly stranger and grander than anyone had imagined.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned how the discovery of irrational numbers (like the square root of 2) shattered the Pythagorean belief that all magnitudes are ratios of whole numbers. Explain the discovery and why it was a crisis.
Leads to Aristotle.
Begin this lesson →epoché — a humanities education that remembers you.