From the standpoint of the naïve logic of modern positivism, Pythagoras’ philosophy may not appear particularly profound, and his reflections may seem little more than childish games with numbers. Yet at the foundation of his thought lies a profound mystical insight: between entirely different objects—for example, six jugs of wine and six bulls—there is something fundamentally shared, as though the number six itself possessed a reality of its own. What fundamentally distinguishes six from seven? Why does 3 × 2 × 1 equal 3 + 2 + 1, while 7 × 1 does not equal 7 + 1? Where do prime numbers come from at all? Today, the power and universality of mathematics seem obvious to us, but at a time when mathematics was largely confined to elementary counting, insights of this kind represented a genuine breakthrough: they pushed apart the boundaries between the literal and the abstract, between everyday experience and universal order.
It is significant that counting with natural numbers is not a universal human ability: there are peoples who possess language but whose counting systems are limited to concepts such as “one,” “two,” and “many”—that is, to distinctions that remain within the limits of immediate perception. The greatness and influence of Pythagoras lie not so much in his mathematical discoveries—the relations between the sides of a right triangle were known before him—as in the force and radicalism of his thought. With Pythagoras begins the transition from mathematics as a technique of counting to the understanding of number as a principle and foundation of being.
Pythagoras and his school were the first to see in numbers not merely tools for solving problems, but an authentic structure organizing the cosmos—and this was their true discovery. According to legend, after proving the famous relation, Pythagoras sacrificed a hundred bulls. The significance of this story lies not so much in whether it is historically true as in what it expresses about the experience of that age: the proof of a mathematical relation could be experienced as an epiphany, as the manifestation of divine order within the world of things. It is here, at this threshold, that number first appears not as an instrument for practical purposes but as the manifestation of a hidden order—as the language of the gods, heard by humanity for the first time.
This step marked the turning point at which mathematics first acquired an ontological meaning, while the question of the reality of number became one of the central mysteries shaping the very fabric of philosophy. Pythagoras’ achievement was to establish radically new ontological coordinates by revealing the true place of mathematics—not as the business of accountants, but as the vocation of priests.
In essence, this was the principal achievement of the Pythagoreans: they transformed mathematics into a spiritual science, even if they did so through a gesture that may appear rather naïve from a modern perspective—the attribution of metaphysical status to numbers.
The Pythagoreans had a profound influence on Plato and his theory of Forms, since mathematics provides perhaps the clearest demonstration of the division of reality into two levels.
The productivity of the idea of sacralizing numbers was already questioned in antiquity. Plato himself attempted to overcome some of its difficulties by treating proportions and relations, rather than numbers themselves, as possessing a higher significance—an approach that was, incidentally, inherited in many respects by medieval Europe—while Aristotle was openly skeptical of Pythagorean number metaphysics. Nevertheless, Pythagoreanism accomplished two crucial things. First, it became one of the origins of the division of the world into two levels and, consequently, one of the beginnings of metaphysics itself. Second, it gave a powerful impulse to the understanding of mathematics as a form of priestly knowledge, legitimizing and motivating mathematical inquiry for generations of thinkers. The idea of a divided reality would subsequently run through the entire history of Western philosophy—from Plato to Descartes and beyond.
