Pythagoreanism may be summarized as a system of metaphysical speculation concerning the relationships between numbers and the causal agencies of existence. This school also first expounded the theory of celestial harmonics or "the music of the spheres".
John Reuchlin said of Pythagoras that he taught nothing to his disciples before the discipline of silence, silence being the first rudiment of contemplation.
Pythagoras taught that friendship was the truest and nearest perfect of all relationships. He declared that in Nature there was a friendship of all for all; of gods for men; of doctrines one for another; of the soul for the body; of the rational part for the irrational part; of philosophy for its theory; of men for one another; of countrymen for one another; that friendship also existed between strangers, between a man and his wife, his children, and his servants. All bonds without friendship were shackles, and there was no virtue in their maintenance. Pythagoras believed that relationships were essentially mental rather than physical, and that a stranger of sympathetic intellect was closer to him than a blood relation whose viewpoint was at variance with his own.
Pythagoras defined knowledge as the fruitage of mental accumulation. He believed that it would be obtained in many ways, but principally through observation. Wisdom was the understanding of the source or cause of all things, and this could be secured only by raising the intellect to a point where it intuitively cognized the invisible manifesting outwardly through the visible, and thus became capable of bringing itself en rapport with the spirit of things rather than with their forms. The ultimate source that wisdom could cognize was the Monad, the mysterious permanent atom of the Pythagoreans.
The God of Pythagoras was the Monad, or the One that is Everything. He described God as the Supreme Mind distributed throughout all parts of the universe — the Cause of all things, the Intelligence of all things, and the Power within all things. He further declared the motion of God to be circular, the body of God to be composed of the substance of light, and the nature of God to be composed of the substance of truth.
The sacredness of the triad and its symbol —the triangle— is derived from the fact that it is made up of the monad and the duad. The monad is the symbol of the Divine Father and the duad of the Great Mother. The triad being made of these two is therefore androgynous and is symbolic of the fact that God gave birth to His worlds out of Himself, who in His creative aspect is always symbolized by the triangle. The monad passing into the duad was thus capable of becoming the parent of progeny, for the duad was the womb of Meru, within which the world was incubated and within which it still exists in embryo.
To Pythagoras music was one of the dependencies of the divine science of mathematics, and its harmonies were inflexibly controlled by mathematical proportions. The Pythagoreans averred that mathematics demonstrated the exact method by which the good established and maintained its universe. Number therefore preceded harmony, since it was the immutable law that governs all harmonic proportions. After discovering these harmonic ratios, Pythagoras gradually initiated his disciples into this, the supreme arcanum of his Mysteries. He divided the multitudinous parts of creation into a vast number of planes or spheres, to each of which he assigned a tone, a harmonic interval, a number, a name, a color, and a form. He then proceeded to prove the accuracy of his deductions by demonstrating them upon the different planes of intelligence and substance ranging from the most abstract logical premise to the most concrete geometrical solid. From the common agreement of these diversified methods of proof he established the indisputable existence of certain natural laws.
The famous Pythagorean Υ signified the power of choice and was used in the Mysteries as emblematic of the Forking of the Ways. The central stem separated into two parts, one branching to the right and the other to the left. The branch to the right was called Divine Wisdom and the one to the left Earthly Wisdom. Youth, personified by the candidate, walking the Path of Life, symbolized by the central stem of the Υ, reaches the point where the Path divides. The neophyte must then choose whether he will take the left-hand path and, following the dictates of his lower nature, enter upon a span of folly and thoughtlessness which will inevitably result in his undoing, or whether he will take the right-hand road and through integrity, industry, and sincerity ultimately regain union with the immortals in the superior spheres.
Iamblichus gathered thirty-nine of the symbolic sayings of Pythagoras and interpreted them. These have been translated from the Greek by Thomas Taylor. Aphorismic statement was one of the favorite methods of instruction used in the Pythagorean university of Crotona. Ten of the most representative of these aphorisms are reproduced below with a brief elucidation of their concealed meanings:
The most famous of the Pythagorean fragments are the Golden Verses, ascribed to Pythagoras himself, but concerning whose authorship there is an element of doubt. The Golden Verses contain a brief summary of the entire system of philosophy forming the basis of the educational doctrines of Crotona, or, as it is more commonly known, the Italic School. These verses open by counseling the reader to love God, venerate the great heroes, and respect the dæmons and elemental inhabitants. They then urge man to think carefully and industriously concerning his daily life, and to prefer the treasures of the mind and soul to accumulations of earthly goods. The verses also promise man that if he will rise above his lower material nature and cultivate self-control, he will ultimately be acceptable in the sight of the gods, be reunited with them, and partake of their immortality. It is rather significant to note that Plato paid a great price for some of the manuscripts of Pythagoras which had been saved from the destruction of Crotona.