Saturday, September 17, 2016

Essential Thinkers #2 Pythagoras of Samos, Mystic and Mathematician

Probably born around the mid-sixth century BC no exact date is known as to when Pythagoras lived. Despite his name being familiar to every schoolchild for Pythagoras’ Theorem, which states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the remaining two sides, it is likely that this was known both to the Babylonians – where Pythagoras is thought to have travelled in his youth – and the Egyptians.

Pythagoras was a somewhat shadowy figure and like Socrates after him wrote nothing himself, preferring to leave his students to document his thoughts. Reputed to be a mystic as well as a thinker, the school he founded would nowadays be thought of as a religious cult that taught many unusual and strange doctrines including, notoriously, the veneration for – and abstinence from eating of – beans. Pythagoras also preached reincarnation and the transmigration of souls and is largely responsible for the modern belief in numerology, later popularised by Nostradamus.

According to Pythagoras, the ultimate nature of reality is number. This idea developed out of his theory of music, in which he proved that the intervals between musical tones could be expressed as ratios between the first four integers (the numbers one to four). Since part of Pythagoras’ religious teaching consisted in the claim that music has a special power over the soul, infused as it is into the very fabric of the universe, the belief that number is the ultimate nature of reality quickly followed.

The Pythagoreans went on to venerate certain numerical patterns, especially the so-called ‘tetractys of the decad.’ The tetractys is a diagram that represents the first four numbers in a triangle of ten dots:

Both the triangle and the number 10 – the decad – became objects of worship for the Pythagoreans. In Pythagorean thought, the number 10 is the perfect number because it is made of the sum of the first four integers, as shown in the tetractys. The integers themselves were thought to represent fundamental ideas – the number one representing the point, two the line, three the surface and four the solid. Further, it was thought that there were ten heavenly bodies – five planets, the sun, the moon, the earth and a mysterious and invisible ‘counter-earth’ (probably invented to make the celestial number up to ten) all revolving around a central fire.

After Pythagoras’ death, his school splintered into two camps. One maintained his religious and mystical teachings, while the other concentrated on his mathematical and scientific insights. The latter continued to believe the nature of the universe must be essentially arithmetical. Units of number, points, were somehow thought to possess spatial dimensions and be the ultimate constituents of objects. An idea later criticised by both Parmenides and Zeno. The Pythagorean cosmology also encountered grave problems due to one of Pythagoras’ own discoveries. For Pythagoras had shown how the ratio of the diagonal through a square to its sides could not be expressed as a whole number. The problem of ‘the incommensurability of the diagonal’ led to the discovery – or invention, depending on your philosophical point of view – of irrational numbers. Though a major problem for the Pythagorean cosmogony, irrational numbers have proven a major and lasting development in mathematical thinking.
[Summarized from Philosophy 100 Essential Thinkers by Philip Stokes, 2012. Also watch Brief History of the Pythagorean Theorem]

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