1. Its Relative Position.—The development of the Ionic philosophy discloses the tendency to abstract matter from all else; though they directed this process solely to the determined quality of matter. It is this abstraction carried to a higher step, when we look away from the sensible concretions of matter, and no more regard its qualitative determinateness as water, air, &c., but only direct our attention to its quantitative determinateness,—to its space-filling property. But the determinateness of quantity is number, and this is the principle and standpoint of Pythagoreanism.

2. Historical and Chronological.—The Pythagorean doctrine of numbers is referred to Pythagoras of Samos, who is said to have flourished between 540 and 500 B. C. He dwelt in the latter part of his life at Crotonia, in Magna Grecia, where he founded a society, or, more properly, an order, for the moral and political regeneration of the lower Italian cities. Through this society, this new direction of philosophy seems to have been introduced,—though more as a mode of life than in the form of a scientific theory. What is related concerning the life of Pythagoras, his journeys, the new order which he founded, his political influence upon the lower Italian cities, &c., is so thoroughly interwoven with traditions, legends, and palpable fabrications, that we can be certain at no point that we stand upon a historical basis. Not only the old Pythagoreans, who have spoken of him, delighted in the mysterious and esoteric, but even his new-Platonistic biographers, Porphyry and Jamblichus, have treated his life as a historico-philosophical romance. We have the same uncertainty in reference to his doctrines, i. e. in reference to his share in the number-theory. Aristotle, e. g. does not ascribe this to Pythagoras himself, but only to the Pythagoreans generally, i. e. to their school. The accounts which are given respecting his school have no certainty till the time of Socrates, a hundred years after Pythagoras. Among the few sources of light which we have upon this subject, are the mention made in Plato’s PhÆdon of the Pythagorean Philolaus and his doctrines, and the writings of Archytas, a cotemporary of Plato. We possess in fact the Pythagorean doctrine only in the manner in which it was taken up by Philolaus, Eurytas and Archytas, since its earlier adherents left nothing in a written form.

3. The Pythagorean Principle.—The ancients are united in affirming that the principle of the Pythagorean philosophy was number. But in what sense was this their principle—in a material or a formal sense? Did they hold number as the material of things, i. e. did they believe that things had their origin in numbers, or did they regard it as the archetype of things, i. e. did they believe that things were made as the copy or the representation of numbers? From this very point the accounts given by the ancients diverge, and even the expressions of Aristotle seem to contradict each other. At one time he speaks of Pythagoreanism in the former, and at another in the latter sense. From this circumstance modern scholars have concluded that the Pythagorean doctrine of numbers had different forms of development; that some of the Pythagoreans regarded numbers as the substances and others as the archetypes of things. Aristotle, however, gives an intimation how the two statements may be reconciled with each other. Originally, without doubt, the Pythagoreans regarded number as the material, as the inherent essence of things, and therefore Aristotle places them together with the Hylics (the Ionic natural philosophers), and says of them that “they held things for numbers” (Metaph. I., 5, 6). But as the Hylics did not identify their matter, e. g. water, immediately with the sensuous thing, but only gave it out as the fundamental element, as the original form of the individual thing, so, on the other side, numbers also might be regarded as similar fundamental types, and therefore Aristotle might say of the Pythagoreans, that “they held numbers to be the corresponding original forms of being, as water, air, &c.” But if there still remains a degree of uncertainty in the expressions of Aristotle respecting the sense of the Pythagorean doctrine of numbers, it can only have its ground in the fact that the Pythagoreans did not make any distinction between a formal and material principle, but contented themselves with the undeveloped view, that, “number is the essence of things, every thing is number.”

4. The carrying-out of this Principle.—From the very nature of the “number-principle,” it follows that its complete application to the province of the real, can only lead to a fruitless and empty symbolism. If we take numbers as even and odd, and still farther as finite and infinite, and apply them as such to astronomy, music, psychology, ethics, &c., there arise combinations like the following, viz.: one is the point, two are the line, three are the superficies, four are the extension of a body, five are the condition (beschaffenheit), &c.—still farther, the soul is a musical harmony, as is also virtue, the soul of the world, &c. Not only the philosophical, but even the historical interest here ceases, since the ancients themselves—as was unavoidable from the arbitrary nature of such combinations—have given the most contradictory account, some affirming that the Pythagoreans reduced righteousness to the number three, others, that they reduced it to the number four, others again to five, and still others to nine. Naturally, from such a vague and arbitrary philosophizing, there would early arise, in this, more than in other schools, a great diversity of views, one ascribing this signification to a certain mathematical form, and another that. In this mysticism of numbers, that which alone has truth and value, is the thought, which lies at the ground of it all, that there prevails in the phenomena of nature a rational order, harmony and conformity to law, and that these laws of nature can be represented in measure and number. But this truth has the Pythagorean school hid under extravagant fancies, as vapid as they are unbridled.

The physics of the Pythagoreans possesses little scientific value, with the exception of the doctrine taught by Philolaus respecting the circular motion of the earth. Their ethics is also defective. What we have remaining of it relates more to the Pythagorean life, i. e. to the practice and discipline of their order than to their philosophy. The whole tendency of Pythagoreanism was in a practical respect ascetic, and directed to a strict culture of the character. As showing this, we need only to cite their doctrines concerning the transmigration of the soul, or, as it has been called, their “immortality doctrine,” their notion in respect of the lower world, their opposition to suicide, and their view of the body as the prison of the soul—all of which ideas are referred to in Plato’s PhÆdon, and the last two of which are indicated as belonging to Philolaus.