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30 April 2018

When Magic Gave Way to Numbers

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During their first 300,000 years of existence, human beings explained the phenomena that surrounded them (rains, death, harvest) by turning to the idea of magic and the influence of the gods. And so it was until around the sixth century B.C. when a revolution began in Ancient Greece in search of the basic principles to understand the observable world.

Rational thinking began to weaken the grip of mythological ideas; it was the awakening of science. Pythagoreanism was one of the pillars of that cultural eruption and also the first philosophical current in which women participated and stood out.

Pythagoras of Samos (569 BC – 475 B.C. approx.) and his followers sought to decipher the foundations of reality through numbers, and in doing so they created mathematical abstraction. Prior mathematics, developed empirically by the Egyptians and Mesopotamians, was a collection of rules for practical questions, such as dividing up a piece of land. It is possible that Pythagoras learned from them in his travels, but he took geometry and arithmetic much further. He was the first to observe that there is a set of axioms from which all other reasoning can be deduced—through demonstration, which the Pythagoreans established as a basic tool for constructing the framework of mathematics.

Pythagoras teaching a class of women. Source: Wikimedia

Mathematics stopped being a means, to become an end in itself. For the Pythagoreans, the search for knowledge was the way to fully realize this. Around this idea they created the Pythagorean community, which was also guided by ethical and moral principles, translated into a series of rules dictated by its leader. “The original Pythagoreans lived together in close-knit communities and abided by a strict discipline, which extended to dietary matters, wearing apparel, and the proper times to engage in sexual intercourse,” says the classical historian Sara Pomeroy in her book Pythagorean Women: Their History and Writing.

The first woman mathematician in history

In addition to the mysterious figure of Pythagoras, founder of the group, many other people laid the foundations of this current of thought. Among them were women like Theano, considered by some authors to be the first female mathematician in history. “Pythagoras was the first Greek philosopher to include women among his disciples,” notes Pomeroy. The Neo-Platonic philosopher Iamblichus (3rd century A.D.) lists 17 women among the 235 names of known Pythagoreans. “They were not “muted” like their respectable Athenian contemporaries in old Greece,” says Pomeroy. “Some of their witty and prudent remarks were quoted by later authors,” she adds. Of all of them, Theano was the most mentioned and influential Pythagorean woman.

Despite her indisputable notoriety, there is some confusion regarding her biography. Theano is usually considered to be the wife of Pythagoras and the daughter of Brontino (another member of the Pythagorean school); however, other sources identify her as a disciple of Pythagoras and the wife of Brontino. Some authors affirm that Theano was one of the survivors of the bloody attack on the sect in which Pythagoras died—according to different versions and legends—after which she dedicated herself to spreading the doctrine of the master.

Beyond this work, it is possible that Theano wrote some of her own texts that were not published. In addition to some moral reflections, she also researched cosmology, medicine and mathematics. She was particularly interested in the theory of the golden ratio and regular polyhedrons, also known as Platonic solids. One of them, the dodecahedron, had a prominent place in the arithmetical mysticism of the Pythagoreans.

There are only five regular polyhedra: the tetrahedron (composed of 4 equilateral triangles), the cube (6 squares), the octahedron (8 equilateral triangles), the dodecahedron (12 regular pentagons) and the icosahedron (20 equilateral triangles). Credit: Максим Пе

Devotion to numbers

For the Pythagoreans, the number was the essential material of all things. From their own experiments with the monochord they found that numbers described the theory of musical sounds and also explained the movement of the planets, as the Babylonian mathematicians already knew.

Guided by this devotion to numbers, they studied and classified them in many different ways: to start, they differentiated between odd and even numbers; then they classified them into prime numbers (those that can only be divided by one and themselves) and compound numbers (those that have more divisors); and, surprisingly, they managed to find perfect numbers and even pairs of friendly numbers. Perfect numbers are those that are equal to the sum of their divisors (e.g. 6 = 1 + 2 + 3), and one number is a friend of another if adding the divisors of one obtains the other, and vice versa. For example, 220 and 284:

– The divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, which add up to 284.

– The divisors of 284 are 1, 2, 4, 71 and 142, which add up to 220.

Pythagoreans found that numbers described the theory of musical sounds. Credit: Melchoir

In addition, the Pythagoreans attributed mystical characteristics to numbers. The number one represented the origin, odd numbers were masculine and even ones feminine. They believed that all nature and objects were characterized by numbers, which were their primary material and the cause of their modifications and permanent states.

It was precisely that vision, which turned from numbers back towards magic, that was the beginning of the end for the Pythagoreans. They discovered that reality could not be measured with whole numbers (or quotients of them), but that more complicated values ​​were needed: irrational numbers, with their infinite decimal figures that are not repeated periodically (like the number π). It was one of the most surprising discoveries made by the Pythagoreans, possibly by applying the Pythagorean theorem to something as simple as calculating the diagonal of a square of side 1. These incommensurable numbers destroyed their vision and showed that mathematics, and the world, were much more complex than they expected.

Ágata Timón

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