Mathematics Magazine
Home Articles Math Book Applications Contact Us
Bookmark and Share Advertise Here. E-mail us your request for an advertising quote!

Pythagoras Philosophy

by Liliana Usvat

Like Thales, Pythagoras is rather known for mathematics than for philosophy. Anyone who can recall math classes will remember the first lessons of plane geometry that usually start with the Pythagorean theorem about right-angled triangles: a²+b²=c². In spite of its name, the Pythagorean theorem was not discovered by Pythagoras. The earliest known formulation of the theorem was written down by the Indian mathematician Baudhayana in 800BC. The principle was also known to the earlier Egyptian and the Babylonian master builders. However, Pythagoras may have proved the theorem and popularised it in the Greek world. With it, his name and his philosophy have survived the turbulences of history.

Pythagoras founded a society of disciples which has been very influential for some time. Men and women in the society were treated equally -an unusual thing at the time- and all property was held in common. Members of the society practised the master’s teachings, a religion the tenets of which included the transmigration of souls

It could be said that Pythagoras saw the study of mathematics as a purifier of the soul, just like he considered music as purifying. Pythagoras and his disciples connected music with mathematics and found that intervals between notes can be expressed in numerical terms. They discovered that the length of strings of a musical instrument correspond to these intervals and that they can be expressed in numbers. The ratio of the length of two strings with which two tones of an octave step are produced is 2:1.

Music was not the only field that Pythagoras considered worthy of study, in fact he saw numbers in everything. He was convinced that the divine principles of the universe, though imperceptible to the senses, can be expressed in terms of relationships of numbers. He therefore reasoned that the secrets of the cosmos are revealed by pure thought, through deduction and analytic reflection on the perceptible world.

This eventually led to the famous saying that “all things are numbers.” Pythagoras himself spoke of square numbers and cubic numbers, and we still use these terms, but he also spoke of oblong, triangular, and spherical numbers. He associated numbers with form, relating arithmetic to geometry. His greatest contribution, the proposition about right-angled triangles, sprang from this line of thought: Pythagoras Proposition“The Egyptians had known that a triangle whose sides are 3, 4, 5 has a right angle, but apparently the Greeks were the first to observe that 3²+4²=5², and, acting on this suggestion, to discover a proof of the general proposition. Unfortunately for Pythagoras this theorem led at once to the discovery of incommensurables, which appeared to disprove his whole philosophy. In a right-angled isosceles triangle, the square on the hypotenuse is double of the square on either side.

No texts by Pythagoras are known to have survived, although forgeries under his name — a few of which remain extant — did circulate in antiquity. Critical ancient sources like Aristotle and Aristoxenus cast doubt on these writings. Ancient Pythagoreans usually quoted their master's doctrines with the phrase autos ephe ("he himself said") — emphasizing the essentially oral nature of his teaching.

Pythagoras believed that all relations could be reduced to number relations. As Aristotle wrote:-

The Pythagorean ... having been brought up in the study of mathematics, thought that things are numbers ... and that the whole cosmos is a scale and a number.

Pythagoras studied properties of numbers which would be familiar to mathematicians today, such as even and odd numbers, triangular numbers, perfect numbers etc. However to Pythagoras numbers had personalities which we hardly recognise as mathematics today

Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. This feeling modern mathematics has deliberately eliminated, but we still find overtones of it in fiction and poetry. Ten was the very best number: it contained in itself the first four integers - one, two, three, and four [1 + 2 + 3 + 4 = 10] - and these written in dot notation formed a perfect triangle.

Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrational and the construction of the cosmic figures.

Theory: They taught that in Number only is absolute certainty to be found; that Number is the Essence of all things; that things are only a copy of Numbers; nay, that in some mysterious way, Numbers are things themselves. This Number theory was probably worked out from the fundamental conception, that, after destroying or disarranging every other attribute of matter, there still remains the attribute Number; we still can predicate that the thing is one. With this doctrine of Number was intimately connected that of the Finite and Infinite, corresponding respectively with the Odd and Even in Number; and from a combination of this Finite and Infinite it was taught that all things in the Universe result. The abstract principle of all perfection was One and the Finite; of imperfection, the Many and the Infinite

The Ethical teaching of the Pythagoreans was of the purest and most spiritual kind; Virtue was regarded as a harmony of the soul, a conformity with or approximation of the Deity; Self-restraint, Sincerity, and Purity of Heart were especially commended; and Conscientiousness and Uprightness in the affairs of life would seem to have been their distinguishing characteristics.

The Pythagorean system was carried on by a succession of disciples down to about 300 BC, when it seems to have gradually died out, being superseded by other systems of philosophy

The TEN PRINCIPLES of PYTHAGORAS Also known as the table of Opposites

  1. limit unlimited
  2. odd even
  3. one plurality
  4. right left
  5. male female
  6. at rest moving
  7. straight crooked
  8. light darkness
  9. good bad
  10. square oblong


  • Pythagoras calls drunkenness an expression identical with ruin.
  • The fundamental element of the world is the number 1.
  • Among what he calls his precepts were such as these:
  • Do not stir the fire with a sword.
  • Do not sit down on a bushel.
  • Do not devour the heart.
  • In the time of Pythagoras that proverbial phrase « Ipse dixit » was introduced into ordinary life.
  • Animals share with us the privilege of having a soul.

Pythagoras had also spoken about the five solid figures, called the mathematical solids. He said that…

  • The earth is made from a cube,
  • Fire from the pyramid,
  • Air from the octahedron,
  • Water from the eicosahedron,
  • And the sphere of the whole (the Aither) from the dodecahedron.

Pythagoreans held that the cosmos ‘Breathed in ‘from the Infinite Breath outside it: « aither ».

Pythagoras derived the world from the fire and the fifth element.

Philolaus the Pythagorean says that

  • FIRE is at the centre, calling it the heart of the universe;
  • second comes the COUNTER-EARTH, and
  • third the inhabited earth which in its revolution remains opposite the counter-earth,

wherefore the inhabitants of this earth do not see those of the other.

« They call the earth a star as being itself an instrument of time, for it is the cause of day and night.

Day it creates by being lit up on the side which turned toward the sun, and night through the cone of his shadow.

  • COUNTER EARTH was the name given by the Pythagoreans to the MOON and also HEAVENLY EARTH.»
  • Pythagoras maintained that the fire was at the centre of the cosmos. Many Pythagoreans followed his example and approved this ancient insight of HELIOCENTRICITY.

Pythagoras had also a certain opinion about the Nature of the Soul. He said one day: « In the case of the soul, the three parts that have to be brought into accord are of course reason, passion and desire. »

Pythagoras taught of the transmigration of the soul – man/plant/animal – and post-humous rewards for the good. It is recorded that he rememberd his past lives as Aethalides (son of Hermes, whence the gift of memory), Euphobus (from the Homer epics), Hermotimus and Pyrrus (a Delian fisherman) before his birth as Pythagoras. He used to say that he had received as a gift from Mercury the perpetual transmigration of his soul, so that it was constantly transmigrating and passing into all sorts of plants or animals.

"Chance favors the prepared mind." - Louis Pasteur