The esoteric teachings of Pythagoras in regard to number dealt principally with the significance of arithmetic and geometry, and emphasized the importance of the application of number to weights and measures. He was the first to explain the multiplication table to the Greeks. The leading idea of his system was that of the Unity in Multiplicity. Therefore the Pythagorean concept of harmony was based upon the relationship of the One and the Many, the idea of the One in Many and the Many in One—"as above, so below." By number Pythagoras meant not merely figures, but regulated motion or vibration, rhythm, law, and order; for he made number equivalent to intelligence. He said:
Number is that which brings what is obscure within the range of our knowledge, rules all true order in the universe and allows of no errors.
He assumed, as first principles, the numbers and the symmetries existing in them, which he called harmonies. He taught:
Virtue is a proportion or harmony. Happiness consists in the perfection of the virtues of the soul, the perfect science of numbers. Nature is an imitation of number.
Pythagorean arithmetic was concerned especially with the first ten digits, which were "hieroglyphic symbols, by means of which Pythagoras explained his ideas about the nature of things." He taught that unity, the monad or one, is no true numeral, for one multiplied any number of times by itself always equals one; that is, unity unlike the true numerals, has not an infinite series of varying powers, for its square, cube, and other powers, are one and all equal to one, the first term of the series. Another peculiarity, which proves unity not to be a true numeral, is its indivisibility into whole numbers.
The monad is God and the good, which is the origin of the one and is itself Intelligence. The monad is the beginning of everything. Unity is the principle of all things and from Unity went forth an infinite or indeterminate duality, the duad, which is subordinate to the monad as its cause.
Pythagoras taught that the duad, the first concept of addition, was the first true figure and regarded the one as a symbol for the Primitive Unity or the Deity, the Absolute, behind and above the indeterminate or infinite duad, which symbolized chaos or spirit-matter. The triad or the three, the monad plus the duad, symbolized the Divine, the Heavenly, as opposed to the Earthly.
The Pythagoreans say that the All and all things are defined by threes; for beginning, middle, and end constitute the number of all and also the number of the triad.
The tetrad or the four exists in two forms, its actual form the quaternary or the four, the symbol of Earth as opposed to Heaven, and its potential form, the tetraktys, which contains in germ the sum total of the universe, manifested and unmanifested, the Pythagorean dekad or ten, thus, 1 + 2 + 3 + 4 = 10. The tetraktys, therefore, was regarded as a very sacred symbol. The pentad or number five, symbolized man. The senary or number six, is, of course, composed of two threes, and was regarded as an abbreviation for the alpha and omega of evolutionary growth. The hebdomad or number seven, is the perfect number, par excellence, symbolizing both heaven and earth. In the words of H. P. Blavatsky