The world of Pythagoras' youth was a rapidly changing place, and, in his time, change was generally for the better. Yet he was unhappy.
Society was progressing out of a broad dark ages, and it found its material conditions improving. But spiritually and intellectually people found a growing alienation. Political and social ties were changing, and so too were previously stable religious forms and cultural modes.
A little background might be in order. The high civilizations of the eastern mediterranean, the "Bronze Age" Mycenaean, Egyptian, Anatolian, and Levant cultures, had developed from far more ancient Mesopotamian and Egyptian roots---and had followed the same arc into destruction. They had crested perhaps a millennium prior, and by Pythagoras' time (the 500's BC) these civilizations and those before were long, long gone. They were as much a memory to the bustling sea-side port town of Pythagoras' boyhood as Rome was to the artists of renaissance Florence.
In both cases, the long centuries of intervening time were troubled. The older cultures (extinguished by 1200 BC) had crumbled from over-extended militaries, collapsing agricultural productivity, declining trade, persistent warfare, and finally political revolutions and a destructive migratory period: Germans and the Huns in the Roman case, the Sea Peoples, Dorians, Indo-Europeans, and others in the Mediterranean case. What followed was recognizably similar: centuries of dark ages where economies were localized, cities virtually non-existent, populations in decline, social organization built around honor and kinship hierarchies, art reverting to simpler forms, written literature all but gone, and religious observance, though fragmented, absolute.
By about 800 BC, the alphabet (from obscure origins) was introduced into the eastern mediterranean cultural basin. This lead first to more effective trade, then to revitalized and more democratized governments (the common people could read and write), and then to new literary forms. Persistent warfare was brought under control, possibly through sea-faring innovations and new technologies in iron-based weaponry, which allowed strengthened municipalities to exert control and offer protection in larger geographical areas and in sea lanes.
These trends were a few centuries old when Pythagoras was born---roughly as old as colonial times are to us. By then the motley, atomized world of the past had coalesced into more uniform associations of varying sizes and strengths (the city-states and their dependents) that were connected by a variety of cultural and economic ties, and were sometimes in conflict. Broadening trade brought wide-reaching cultural exchange: the Ionians (Athens and most of the greek colonies) wrestled with incorporating the new ideas; the Dorians (Sparta and laconia) generally retrenched and built internal responses to re-inforce the old mores.
To some, this was confusion. To others, enlightenment. Pythagoras was born in Samos, a port city of the Anatolian littoral and one of the wealthiest independent cities in the Ionian cultural sphere----and at the crossroads of the growing trade networks of the mediterranean. Exactly what influenced Pythagoras' upbringing is difficult to determine, but ancient writers agree he had traveled widely in youth, learning everything he could about mind, spirituality, science, politics, and ethics. He was certainly in Egypt for a time, likely in the levant, possibly in mesopotamia, and could have been as far away as India.
As a result he was among the very most learned of his day. And what did he do with his knowledge? What anyone would do! He started a revolutionary humanistic movement.
I say humanistic, because he was among the first of his age to believe in the supremacy of the human mind: he did not deny the supernatural, but he completed the break, begun two generations earlier with Thales, from his era's confining religious heritage. To him, knowledge and society's norms were not of the gods, but could be questioned, and, maybe, re-formed. They were subjects of human understanding; humans were not subject to them.
Hubris? In the eyes of his contemporaries, yes, very much so. In fact he was considered dangerous: Pythagoras' political movement was violently suppressed, and at age 75 he was forced to flee his adopted hometown of Croton (which is still a town in southern Italy), and likely died stateless. Apparently nobody else wanted to admit an incendiary like this into their protection.
Of course Pythagoras' ethical, political, and spiritual ideas have little resonance today. But his mathematical achievement outlasted not just his political organization, not just the rise and fall of Greek civilization, but survives intact to the present and has become part of our common world culture. Specifically, the achievement begun by him and his contemporaries, and completed by the time of Euclid 200 years later, was this: formal proof.
What could Pythagoras do with proof? Well for one he could show that his rulers were idiots. They believed they had Knowledge (big knowledge with a big K) and so should govern the cities. But Pythagoras knew they could not answer the simplest of queries.
For instance, what is the ratio of the hypotenuse of an isosceles right triangle to its side length?
Pythagoras could show that there was no such ratio! This kind of pure thought, of knowledge-as-such, made up the cornerstone of Pythagoras' thinking. The direct human contact with truths, whether simple or profound, mathematical or political, was Pythagoras' revolution.
Returning to math, in class we gave two proofs that such a ratio does not exist: Pythagoras' original proof using the method of infinite descent, and a simpler proof using Euclid's theory of factorization applied to fractions. Here we will give just the simpler proof.
Proof that the square root of 2 cannot be expressed as a fraction.
For a proof by contradiction, assume that there is a ratio that squares to 2. Assume m/n is that ratio, meaning m and n are natural numbers and m2/n2=2. By using Euclid's theorem of unique factorization and then canceling common terms, we know we can write such an m and n with no common factors.
Rearranging, m2=2n2. Therefore m2 is even, so m is even, so m=2q for some natural number q.
The equation is then 4q2=2n2, so 2q2=n2. This means n2 is even, so therefore n is even.
And we have derived a contradiction: although m and n had no common factors, we proved that m and n have a common factor of 2. This contradiction implies that the assumption, that the ratio m/n squares to 2, must be false.