READ PARTS I AND II HERE
Of all the thinkers, mathematicians, geometers and physicians among the Ancient Greek line-up of astonishing mortal men, among the most famous is Pythagoras. As a cultural export, his mathematical theories have outlived him by well over two thousand years. They have become part of the national curriculum of schools the world over, to the extent that when we think of Pythagoras we think of squaring the triangle. But that was not the whole story; Pythagoras had his own school, only the students who attended it may have found his curriculum a little bit surprising.
In the 6th century BC, when Pythagoras was alive, he established a school that was more like a sect. It had a set of rules, odd ones. As a sect it was not dangerous, but it was radical; Pythagoras told his students how to live, not just how to work. His line-up of regulations was obscure, more obscure than mathematics to a school child. They included requirements such as do not pick up what has fallen, or do not look in a mirror beside a light. Understandably, this been something of a puzzle to most people. Was there some sort of philosophical reasoning behind these regulations, and if so, what?
Reading down the list in Russell’s History of Western Philosophy, the regulations become even more unfathomable. Do not let swallows share your roof; do not stir the fire with an iron (although that could be common sense), and oddest of all is his requirement to abstain from eating beans. It could be that Pythagoras was just a man with a developed sense of decorum, but in my view, Pythagoras was suffering from a obsessive-compulsive disorder. Do not walk on highways, he adds. Do not touch a white cock. It is true that most Greeks were superstitious; the crossroads of a highway was considered dangerous to Thracians. Evil lurked at the crossings of the ways and it was safer to avoid them. Likewise, the Pythagorean requirement to avoid touching a white cock must have had its roots in the contemporary symbolism of the day. Cocks crowing in the morning meant that the time had come to cast off idleness and get to work, and the colour white is associated with the sacred. Clearly Pythagoras did not want a school of idle students. But what came out of this bizarre school, at the end of the day?
As we already know, Pythagoras was credited with squaring the triangle, or in other words, with the discovery that the square of the hypotenuse is equal to the sum of the squares of the other two sides. But as is often the case with discoveries, even as this theory was established, something came along to upset the apple cart of mathematical perfection. At some point during all of this, Pythagoras and his students stumbled upon an awkward numerical reality: irrational numbers.
An irrational number neither terminates nor repeats. As such, it defies any form of measurement. The discovery of the existence of irrationals threw Pythagoras into a bit of a dilemma. How should such things be dealt with, and what did they mean? In fact, Pythagoras could not answer his questions; it would be another 250 years before someone else did. When a man called Euclid stepped on to the scene of Greek mathematics, he resolved the problem of ‘unmeasurables’ or incommensurables, broadly speaking by separating geometry from arithmetic. The upshot of all this was very profound: you start with something that is self-evident, such as geometry, and end up with something that is anything but self-evident: something that defies rationality. This led to the following conclusion: there was more to mathematics than clear-cut solutions; there was a concept — the concept of eternity.
This new way of thinking soon became the keystone not only of scientific method but also of theology. Pythagoras’ irrational numbers had lit the way down a new path, one that did not terminate. Considering his obsessive tendencies I wonder if he would have really wanted to venture down it. But perhaps he did, because for all his obsessive regulations, Pythagoras was convinced that life too, like an irrational number, did not terminate. He believed in the reincarnation of the soul and the power of contemplative solitude. In a sense, Pythagoras’ school incarnated what he was destined to discover: mysticism.
Read Part IV next week…