One is the loneliest number according to Harry Nilsson, but to the Pythagoreans it was the most perfect. From the number one, every other number could be created, and from these numbers, points, lines, shapes, bodies, the very world. Pythagoreans believed that the world and matter were created from a ‘plurality of monads’, such as the individual stars of a constellation, or, indeed, as we now know, the atoms that make up a molecule. In this, and in so many other aspects, the Pythagoreans appear strikingly modern, and it is no wonder that philosophers of Ancient Greece are held up as a model of ‘rationality’ and detachment. But, alongside that ‘rationality’, they passed down something else stranger and more durable.

Pythagoras was one of the first Greek philosophers, operating in the sixth century BC, based in Croton in the South of Italy, then an area so thoroughly Greek that it was dubbed ‘Magna Graecia’. He is in the similar, regrettable position of Socrates in that he left no written work and his beliefs must be mediated through his students and their successors, the Pythagoreans. Their work has passed down through innumerable schools: from ancient academies to Islamic Madrasas, from Irish monasteries and Renaissance libraries, and, finally, to the smooth linoleum corridors of our modern laboratories and lecture halls.

Everyone knows Pythagoras’ theorem: the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of its other two sides. But this was only one of the many relationships between numbers that the Pythagoreans identified.

The world, for the Pythagoreans was literally composed of numbers, each with different identities. Odd numbers were thought of as limited (and therefore male), even numbers as ‘unlimited’ (and therefore female, obviously). The Pythagoreans were not the only ones to practice this ‘numerical mysticism’: you only have to look at a Celtic triskelion or Hindu trimurti to see the continent-spanning significance of three. Pythagorean numerology however, was one of the most intricate systems of numerical mysticism. Meaning was not created from just the numbers themselves, but deductively from their product. Consequentially, four (limited + limited) represented justice, five (two + three) represented marriage. Three was the first number to consist of beginning, middle and end, and represented totality.

The most important of these relationships was the ‘tetractys’, the first four numbers, representing reality. These numbers literally created the world, giving the four elements, even the four phases of the moon. One, two, three and four add up to ten: a perfect number.

The Pythagoreans are fairly sympathetic to modern sensibilities. They dressed in white, were vegetarian, vaguely deist and believed in the transmigration of souls. In this respect, they might not be so different from many New Age spiritual groups. Around ten per cent of Pythagoras’ followers were female and some, like Theano, could reach some prominence. This stood in marked contrast to the role (or lack thereof) of women in the broader Greek world, which Pericles sums up at the end of his stirring oration on Athenian democracy: ‘The great glory [for women] is being least talked about by the men.’

Alongside these points of contact with the modern world, there were several divergences as well. The Pythagoreans were perhaps the first to believe in a heliocentric orbit, but they also believed that the earth was accompanied by another, unseen earth orbiting on the other side of the sun. Handily, this brought the number of heavenly bodies to ten, the product of the tetractys. On a more disturbing note, initiation into the ranks of the Pythagoreans seems to have involved a five-year ‘silence’, and Pythagoras, at least in the later writings we have, seems to have occupied an exalted position not dissimilar to a cult leader. When one Pythagorean, Hippasus, revealed how to construct a dodecahedron, Pythagoras is alleged to have shoved him off a boat for revealing the secret. This may not, it has to be admitted, necessarily be a world away from the activities of some New Age spiritual groups.

The world of numbers was also a world of mysticism, and there was no firm distinction between mathematician and magician. Despite this, or rather because of this, numerical mysticism was not a trivial matter in the ancient world and Pythagoreans could go to great length to preserve their mystical mathematical knowledge: Timycha, a fourth-century Pythagorean, threatened with torture, bit her own tongue off rather than spill Pythagorean secrets. Even states could act in ways shaped by numerical mysticism. When the Romans were defeated at Lake Trasimene, their dictator consulted the sibylline books and arranged for 333 sestertia and 333.3 denarii to be spent on a festival in honour of the gods, seeing three as a most perfect number, ‘the first odd number… containing the first differences and the elements of every number’ and one suited to regain the god’s favour.

Of all of these, the most famous number from classical antiquity was 300. We know the number now for Leonidas’ brave contingent of Spartans at Thermopylae, but the ancient world is teeming with other 300s. Sparta’s original cavalry contingent was 300 strong, as was Athens’, as was Rome’s; the Roman senate was initially 300 strong; the Theban Sacred Band (composed of 150 pairs of male lovers) was 300. This, and all the many other instances suggests that something beyond just material demands drove the frequency of this charged number, composed of three and ten.

There is no firm distinction to be made between the ‘rational’ and the ‘irrational’ in the ancient world, and where we might expect now to find a binary opposition between the mathematical and spiritual, there was a merging. It is the relentlessly human search for patterns and meanings that led the Pythagoreans to ascribe the meaning of marriage to five, and it is that same search that spots the link between the hypotenuse, perpendicular and base. You cannot have one without the other.

It is tempting to imagine ourselves beyond the pull of mystical numbers and yet, we still knock three times on a door, count down from three to quell a noisy child, and nothing pleases us like a tricolon at the end of a speech: ‘veni, vidi, vici; blood, sweat, and tears’. The significance of numbers has changed, but their social pull is strong. What student hasn’t put 69 (the third most popular double number combination in passwords) at the end of their password, or 420 in their forum’s username? We cannot help ourselves. We are not purely rational, like the Greeks, but rather, irrational, like the Greeks.