Pythagoreanism

While Pythagoras left no writings, the central focus of Pythagorean teachings remains clear. Number is a universal archetype, "the principle, source, and root of all things." More specifically, everything is composed of "the elements of number," which the Pythagoreans identified as the Limited and the Unlimited. The Pythagoreans engaged in the study of number theory or arithmetic (number in itself), geometry (number in space), harmonics or tuning theory (number in time), and astronomy (number in space and time). These descriptions correspond well with the modern definition of mathematics as "the study of patterns in space and time."

Since number and its qualities such as polarity, harmony, and proportion are archetypal principles that underlie physical manifestation, mathematics is discovered rather than invented, and possesses the power to reflect the essential nature of reality, rather than just modeling it or describing it. Moreover, in the Pythagorean view, because number is universal, it is also divine. While modern science is conceived in experimental terms, ancient Greek science was mathematical; and while mathematics has practical applications, Pythagoras's approach elevated mathematics to a study worth pursuing above any purely utilitarian ends for which it had previously been employed.

Pythagoras was the first person to call the universe a kosmos. The Greek term, which is the root of the word cosmetic, refers to an equal presence of order and beauty. The universe is a cosmos because the phenomena of nature embody geometrical form and proportion. These proportions allow things to unfold and function in the most elegant and efficient ways (which is a fact of nature), but also give rise to beauty (which is a value). In this way, the worlds of "fact" and "value" are not separate domains, but inherently related. In a larger sense, all things are related through whole–part and proportional relationships (analogia), as in an ecosystem. Because of this, the classical Pythagorean metaphor likens the universe to a living organism rather than an inanimate machine. Plato describes the Pythagorean view well when he describes the cosmos as "one Whole of wholes" and as "a single Living Creature which encompasses all of the living creatures that are within it" (Timaeus 33 A and 30 D).

The Pythagoreans (among whom Plato must be counted) perceived a fundamental relationship between proportion and the principle of justice, in which "each part of the whole receives its proper due." They believed that the essential nature of justice could be understood through the study of continuous geometrical proportion (analogia) and through the study of the mathematical ratios of the musical scale, in which the two extremes of the musical scale are bridged through various types of mathematical proportion. Central to the Greek concept of proportion is the idea of finding "means" or types of mediation between extremes. In tuning theory, the Pythagoreans identified the arithmetic, geometric, and harmonic means which underlie the musical scale, as well as the perfect consonances of music, which are mathematical ratios: the octave (1:2), the perfect fifth (2:3), and the perfect fourth (3:4). The Pythagoreans likened a just and well-ordered society to a well-tuned lyre. While each note retains its individuality, all are proportionally linked together in a larger whole to form a musical scale, and all are interdependent in terms of their reliance on one another. (See Plato, Republic 443 D–444). Justice is present in any well-functioning organism, society—and also the soul.

In Pythagorean thought, number gives rise to proportion, and proportion gives rise to harmony. The Greek word harmonia means "fitting together" or "joining together." Harmony and justice is the result of good proportion made manifest, and the kosmos itself is a harmony in which all of the parts are proportionally bound together. While every organism—including the cosmos—is a unity, it is harmony that allows the parts to function together as an integrated whole. Harmony, justice, and proportion relate to Greek medical theory, because healthy organisms possess a type of dynamic balance in which the various elements work together; when lack of harmony prevails, illness will result.

Understanding the cosmos is essential for self-understanding, because humanity is a microcosm, a reflection of the entire world-order in miniature. To know the powers present in the greater cosmos—including the divine powers of order, beauty, and reason—allows humans to become aware of the divine, universal principles reflected within our own being. Science was thus envisioned as a spiritual undertaking, because contemplation of the cosmic pattern aids in the assimilation of the soul to the divine. Once again Plato makes the point clearly when he writes that "a man should come to resemble that with which it delights him to associate.… Hence the philosopher through the association with what is divine and orderly (kosmios) becomes divine and orderly (kosmios) insofar as a man may" (Republic 500 C).

Pythagoreanism, with its emphasis on attuning the soul to a vision of the deepest level of reality through the study of mathematics and proportion, lies at the heart of Plato's educational program described in the Republic. Plato had personal contact with members of the Pythagorean school, including Archytas of Tarentum; the spirit of Pythagoreanism animated Plato's works and was transmitted through the later Platonic tradition. Where Plato parted company with Pythagoras was in his overwhelming emphasis on the transcendent. For Pythagoras, the divine order is immanent in the cosmos, where it may be encountered; but Plato emphasized a transcendent metaphysic of forms or examples, only accessible to the purified intellect, divorced from the phenomena of nature. In either case, both Pythagoreanism and Platonism stressed an epistemology in which human beings can know the deep structure of the world because of the mind's essential kinship with its archetypal structure.

Pythagoreanism remained active in medieval philosophical, scientific, and aesthetic thought, influencing the symbolism and architecture of the gothic cathedrals. It similarly influenced Islamic cosmological thought and architecture through the universal encyclopedia of the Brethren of Purity (Ikhwan al-Safa), compiled in the tenth century C.E. But it was with astronomer Johannes Kepler (1571–1630) that the Pythagorean tradition made its most spectacular triumph at the dawn of the modern era, in Kepler's discovery of the three laws of planetary motion.

Motivated by a belief that divine harmonies animate the celestial order, Kepler finally proved Plato's assumption that there was an elegant mathematical order underlying planetary motion despite the seeming, observed irregularities of retrograde motion. In this way, Kepler reconciled the mathematical science of the Greeks—and the Pythagorean spirit—with an absolutely rigorous empiricism that became the hallmark of modern physics.