The Pythagorean mathematical concepts, abstracted from sense impressions of nature, were... projected into nature and considered to be the structural elements of the universe. [Pythagoreans] attempted to construct the whole heaven out of numbers, the stars being... material points. ...they identified the regular geometric solids... with the different sorts of substances in nature. ...This confusion of the abstract and the concrete, of rational conception and empirical description, which was characteristic of the whole Pythagorean school and of much later thought, will be found to bear significantly on the development of the concepts of calculus. It has often been inexactly described as mysticism, but such stigmatization appears to be somewhat unfair. Pythagorean deduction a priori having met with remarkable success in its field, an attempt (unwarranted...) was made to apply it to the description of the world of events, in which the Ionian hylozoistic interpretations a posteriori had made very little headway. This attack on the problem was highly rational and not entirely unsuccessful, even though it was an inversion of the scientific procedure, in that it made induction secondary to deduction.

It has fallen to the lot of one people, the ancient Greeks, to endow human thought with two outlooks on the universe neither of which has blurred appreciably in more than two thousand years. ...The first was the explicit recognition that proof by deductive reasoning offers a foundation for the structure of number and form. The second was the daring conjecture that nature can be understood by human beings through mathematics, and that mathematics is the language most adequate for idealizing the complexity of nature into appreciable simplicity. Both are attributed by persistent Greek tradition to Pythagoras in the sixth century before Christ. ...there is an equally persistent tradition that it was Thales... who first proved a theorem in geometry. But there seems to be no claim that Thales... proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. ...in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world's earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of all by mutual consent to their master.

It seems to me that they do well to study mathematics, and it is not at all strange that they have correct knowledge about each thing, what it is. For if they knew rightly the nature of the whole, they were also likely to see well what is the nature of the parts. About geometry, indeed, and arithmetic and astronomy, they have handed us down a clear understanding, and not least also about music. For these seem to be sister sciences; for they deal with sister subjects, the first two forms of being.

If someone associates with a true Pythagorean, what will he will get from him, and in what quantity? I would say: statesmanship, geometry, astronomy, arithmetic, harmonics, music, medicine, complete and god-given prophecy, and also the higher rewards — greatness of mind, of soul, and of manner, steadiness, piety, knowledge of the gods and not just supposition, familiarity with blessed spirits and not just faith, friendship with both gods and spirits, self-sufficiency, persistence, frugality, reduction of essential needs, ease of perception, of movement, and of breath, good color, health, cheerfulness, and immortality.

The concept of a harmonious universe ordered according to "the Great Chain of Being" — a chain that connects the continuum of matter, body, mind, soul and spirit — stands as one of the most fundamental ideas of western thought. … It continues to be a profound influence upon the deepest strata of our thought. And yet a major rift has appeared in the consciousness of our time because the theme of harmonia has not been translated into the realm of human conduct. The challenge of our time may be to revive it, and make divine harmony "the great theme" of the next millennium. Any success we have in accomplishing this will be based, in large part, on the achievements of Pythagoras.

It is important to recognize that the materialist, scientific paradigm that dominates the late twentieth century world and provides the basis for its dominant institutions, has its basis in the life and work of Pythagoras, one of the most significant representatives of the perennial philosophy and a founder of the magical tradition. This spirit, which gave rise to our world view, is a spirit that must be recaptured if our civilization is to flourish. The choice is a clear one to many, and was summed up in a book title by the late Pythagorean and futurist Buckminster Fuller, Utopia or Oblivion.

At the dawn of our century, scientists were proclaiming that our understanding of the world was almost complete. Only one or two small problems in physics remained to be solved. One of these problems had to do with black body radiation and was solved by Max Planck. His solution, however, formed the foundation for quantum mechanics which was to sweep aside almost the whole edifice of fundamental assumptions in physics, and with it our understanding of the world. A hundred years later we are faced with a similar situation. The mechanistic viewpoint that began to dominate our world view in the seventeenth century has almost completed its hegemony. This paradigm, as historian Hugh Kearney points out, stems from only one of three main systems of thought that flowed from Greek thought into the modern world, each of which has dominated our world view at different points in our history. … In spite of the dominance of mechanistic thought in the contemporary world, a perplexing residue of the magical tradition still survives in the form of several issues, solutions to which do not appear possible within the context of a purely mechanical view of the world.

Pythagoras' teachings have enormous relevance in understanding both the sources of our culture and, perhaps more importantly, where it may be heading or may need to head. But to appreciate this we have to understand him in modern terms.

Pythagoras stands at the fountainhead of our culture. The ideas he set in motion were, according to Daniel Boorstin, "among the most potent in modern history," resulting directly in many of the pillars upon which the modern world is built. In particular, the very existence of science becomes possible only when it is realized that inner, purely subjective, mathematical forms have a resonance with the form and behavior of the external world — a Pythagorean perception. And a world at peace — that is to say, in a nuclear age, the survival of our planet — is predicated upon ideas of universal brotherhood to which Pythagoras, while not the sole author, made an enormous contribution. Even the seeming remoteness of Pythagorean teaching helps one to realize that the current world view, while it seems destined to dominate the planet, is fleeting and temporary and, like others before it, will pass.

Scholastic skeptics, as well as ignorant materialists, have greatly amused themselves for the last two centuries over the absurdities attributed to Pythagoras by his biographer, Iamblichus. The Samian philosopher is said to have persuaded a she-bear to give up eating human flesh; to have forced a white eagle to descend to him from the clouds, and to have subdued him by stroking him gently with the hand, and by talking to him. On another occasion, Pythagoras actually persuaded an ox to renounce eating beans, by merely whispering in the animal's ear! (Iamblichus: "De Vita Pythag.") Oh, ignorance and superstition of our forefathers, how ridiculous they appear in the eyes of our enlightened generations! Let us, however, analyze this absurdity. Every day we see unlettered men, proprietors of strolling menageries, taming and completely subduing the most ferocious animals, merely by the power of their irresistible will... Every one has either witnessed or heard of the seemingly magical power of some mesmerizers and psychologists. They are able to subjugate their patients for any length of time. Regazzoni, the mesmerist who excited such wonder in France and London, has achieved far more extraordinary feats than what is above attributed to Pythagoras. Why, then, accuse the ancient biographers of such men as Pythagoras and Apollonius of Tyana of either wilful misrepresentation or absurd superstition?