In the context of the ancient world, it makes perfect sense that Pythagoras would construct his school as a secret society. (Some Masonic lodges claim Pythagoras as an ancestor. Historians regard this as unlikely.) Because the Pythagoreans were so secretive, our knowledge of their teachings is limited. But we do know that they studied great deal more than just geometry. They were musicians who understood the relationships between the lengths of strings and the tones they made when plucked. Strings whose lengths had certain ratios were harmonious, while other ratios produced dissonance. To the Pythagoreans this meant that the ratios themselves were beneficial or harmful. This seems to have been the basis of an elaborate system of numerology. Most of the system is lost to us, but we do know (because Plato tells us) that they associated the concept of justice with the number four and the figure of the square. This association lives on in many of our modern expressions, such as a square deal or squaring a debt. It is no coincidence that the angles of a square are right angles. This association between abstract concepts and geometric figures or ratios meant that a geometric construction could also be a philosophical argument, in the same way that an allegorical story or song can relate some higher truth.
The Pythagoreans were the first people to call the universe a cosmos. The word cosmos, which comes from the same root as cosmetic, refers to the kind of beauty which arises from harmonious order. The word cosmos, then, does not just refer to the universe, it makes a statement about it – that the universe is harmonious and orderly and therefore beautiful.
The Pythagoreans are also believed to be the first people who knew about irrational numbers. This was a great mystery and consternation to them. Previously, the Greeks had believed that numbers, fractions, and lengths were all the same things. When it was discovered that the ratio between the diagonal of a square and its side was the square root of two, and that this could not be expressed as any fraction of integers, a philosophical crisis ensued. It was a great shock for people who based their worldview on numbers to realize that something as simple as the diagonal of a square could not be represented by anything they recognized as a number.
“The discovery of incommensurable ratios is attributed to Hippasus of Metapontum. The Pythagoreans were supposed to have been at sea at the time and to have thrown Hippasus overboard for having produced an element in the universe which denied the Pythagorean doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.” – Morris Kline, Mathematical Thought from Ancient to Modern Times
The mathematician Jonathan Holden tried to capture this jolt in his poem The Fall of Pythagoras.
Once all his calculations came out
like peas. You
could pop them
whole from the pod, perfect spheres.
And they
were sweet. The number Two
tasted the way deep fields taste
after
rain. You could take the trees
apart, find godlike shapes
if
you didn't look too hard,
which is why I feel sorry for
Pythagoras. He tried to take a pure
square apart, he was
tinkering
around too much. One hard look
at the diagonal, and
it was too
late. The quality of light
had already changed. It
lent the wind
forbidden possibilities, the clouds
this odd,
this brooding weight;
for the diagonal was the square root
of
Two. And what Two was made of –
this hard, sweet pea –
what all
these leaves, these animals and clouds,
the sun, even
these ugly dreams
he'd been having recently at night
were made
of, would not add up.
The harder you tried to add them
up, the
finer they became, the faster
sifted through your fingers
the
incommensurable parts of everything.
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