With the further development of the number-theory Pythagoreanism becomes entirely arbitrary and without principle. We hear, for example, that 1 is the point, 2 is the line, 3 is the plane, 4 is the solid, 5 physical qualities, 6 animation, 7 intelligence, health, love, wisdom. There is no principle in all this. Identification of the different numbers with different things can only be left to the whim and fancy of the individual. The Pythagoreans disagreed among themselves as to what number is to be assigned to what thing. For example, justice, they said, is that which returns equal for equal. If I do a man an injury, justice ordains that injury should be done to me, thus giving equal for equal. Justice must, therefore, be a number which returns equal for equal. Now the only numbers which do this are square numbers. Four equals two into two, and so returns equal for equal. Four, then, must be justice. But nine is equally the square of three. Hence other Pythagoreans identified justice with nine.
According to Philolaus, one of the most prominent Pythagoreans, the quality of matter depends upon the number of sides of its smallest particles. Of the five regular solids, three were known to the Pythagoreans. That matter whose smallest particles are regular tetrahedra, said Philolaus, is fire. Similarly earth is composed [{38}] of cubes, and the universe is identified with the dodecahedron. This idea was developed further by Plato in the "Timaeus," where we find all the five regular solids brought into the theory.
The central fire, already mentioned as identified with the unit, is a characteristic doctrine of the Pythagoreans. Up to this time it had been believed that the earth is the centre of the universe, and that everything revolves round it. But with the Pythagoreans the earth revolves round the central fire. One feels inclined at once to identify this with the sun. But this is not correct. The sun, like the earth, revolves round the central fire. We do not see the central fire because that side of the earth on which we live is perpetually turned away from it. This involves the theory that the earth revolves round the central fire in the same period that it takes to rotate upon its axis. The Pythagoreans were the first to see that the earth is itself one of the planets, and to shake themselves free from the geocentric hypothesis. Round the central fire, sometimes mystically called "the Hearth of the Universe," revolve ten bodies. First is the "counter-earth," a non-existent body invented by the Pythagoreans, next comes the earth, then the sun, the moon, the five planets, and lastly the heaven of the fixed stars. This curious system might have borne fruit in astronomy. That it did not do so was largely due to the influence of Aristotle, who discountenanced the theory, and insisted that the earth is the centre of the universe. But in the end the Pythagorean view won the day. We know that Copernicus derived the suggestion of his heliocentric hypothesis from the Pythagoreans.
The Pythagoreans also taught "The Great Year," probably a period of 10,000 years, in which the world comes into being and passes away, going in each such period through the same evolution down to the smallest details.
There is little to be said by way of criticism of the Pythagorean system. It is entirely crude philosophy. The application of the number theory issues in a barren and futile arithmetical mysticism. Hegel's words in this connection are instructive:--
"We may certainly," he says, "feel ourselves prompted to associate the most general characteristics of thought with the first numbers: saying one is the simple and immediate, two is difference and mediation, and three the unity of both these. Such associations however are purely external; there is nothing in the mere numbers to make them express these definite thoughts. With every step in this method, the more arbitrary grows the association of definite numbers with definite thoughts ... To attach, as do some secret societies of modern times, importance to all sorts of numbers and figures is, to some extent an innocent amusement, but it is also a sign of deficiency of intellectual resource. These numbers, it is said, conceal a profound meaning, and suggest a deal to think about. But the point in philosophy is not what you may think but what you do think; and the genuine air of thought is to be sought in thought itself and not in arbitrarily selected symbols." [Footnote 3]
[Footnote 3: Hegel's Smaller Logic, translated
by Wallace, second edition, page 198.]