Commentary

Pythagoras

Our intellectual forefathers in ancient Greece provided our scientific foundations, and many modern scientists, particularly theorists, provoke images of Plato and other philosophers of pure reason in their manner of conducting research. But while we can see the currents that flowed from the Greeks to modern times, their sources are alien to us. Reason and mysticism intertwine in the writings of Plato and other ancient philosophers. This is particularly true of the Pythagoreans, whose were among the first to apply mathematics to cosmology.

For the followers of Pythagoras, numbers held a special significance that guided their use in explaining the world. Pythagoras himself was as much a religious leader as a philosopher. Born in the middle of the 6th century B.C. on the Greek island of Samos, he traveled to Croton in southern Italy, where he founded his sect sometime after 530 B.C. To enter into his society, a novice had to place all of his property in common with the society, and he had to master five years of indoctrination. Pythagoras sat behind a screen when he taught the novices, and the novices observed strict silence in his presence. After five years, a novice was given a test; if he passed, we was permitted into the Pythagorean society, where he could gaze upon and speak to the great teacher, but if he failed, he was given his property back at 100% interest, and he was forbidden ever to approach the society again. I am tempted to claim this as the roots of the modern postgraduate education, but in truth, as difficult as graduate school is on both the intellectual and the psychological level, it is not an indoctrination into a mystical cult. And while modern astrophysics is kept from the public by its complexity, the beliefs of Pythagoras were kept from the public by a code of silence (Much of the information in this commentary is drawn from Kahn [2001]).¹

Members of the Pythagorean society were very powerful in the politics of southern Italy. During the era of Pythagoras, Croton was the dominant city of the region, and the influence of the Pythagorean society spread rapidly from Croton to other cities. Before long, the members of the Pythagorean society held powerful positions in every city in that region. And despite a revolt that removed the Pythagoreans from power and drove Pythagoras into exile, the Pythagorean society remained influential to the time of Plato in the early 4th century B.C.

The mathematical teachings within the Pythagorean tradition does not arise directly with Pythagoras, but with Philolaus, one of his intellectual offspring from the late 5th century B.C. Pythagorean thought split into two branches, the akousmatikoi, which focused on the religion (reincarnation), taboos, and rituals of Pythagoras, and the mathematikoi, which claimed to follow more closely the teachings of Pythagoras, and which were concerned with both religion and cosmology. The mathematical teaching of the Pythagoreans (not the Pythagorean theorem, which is wrongly attributed to Pythagoras) following the mathematikoi tradition. The teachings of Hippasus, and after him Philolaus, in this tradition center on numerology and the harmonics of music. They explained the cosmos in terms of the harmonic joining of discordant elements. From music, they took the octave (a ratio of 1:2), the fifth (the ratio of 3:2), and the fourth (the ratio of 4:3) as fundamental ratios. They constructed the perfect number 10 through the sum of the numbers in these harmonic ratios: 1 + 2 + 3 + 4 = 10. This is expressed in the Pythagorean symbol of the tetractus, which is an equilateral triangle made of four dots on a side and one in the middle.

This focus on harmonics and numerology drives the Pythagorean cosmology. The cosmology developed by Philolaus is of ten heavenly bodies, rather than the eight that the Greeks knew of. In addition to the Sun, Earth, Moon, and five known planets—Mercury, Venus, Mars, Jupiter, and Saturn&;, the Pythagorean add a counter-Earth and a central fire, both of which are invisible, to make the universe conform to their numerology. The universe is imagined as a sphere, with fire at the privileged places of the sphere—the center and the circumference—because fire is the most precious of the four elements; on the circumference are the stars, and at the center is the unseen central fire. Perhaps if the obsession over the perfect number 10 were absent, they would have correctly places the Sun at the center. From inner orbit to outer orbit, the order of the bodies is the counter-Earth, Earth, Moon, Sun, Venus, Mercury, Mars, Jupiter, and Saturn. The relative motions of the bodies in the solar system is thought of in terms of musical harmonics.

These ideas had a strong impact on the first modern astronomers. Copernicus hearkens back to the Pythagoreans when he states: “In the middle of all sits Sun enthroned. In this most beautiful temple could we place this luminary in any better position from which he can illuminate the whole at once?” And Kepler was driven by the Pythagorean desire to associate the motion of the planets with the Platonic solids. In this work Kepler did find a relationship, but not the one he expected. He derived a relationship, his third law of planetary motion, between a planet's orbital period and the average distance from the Sun; he found that the square of the period is proportional to the cube of the average distance. So from bad theory we get good science, which is not the last time this has happened.

The numerology we see in Pythagorean thought is now a distant memory. Modern theorists do at times fixate on certain characteristics within a mathematical theory, and they can overlook the right result by following a false principle, but no theorist today would assert a theory is correct based on the assertion of the perfection of a particular number. And the assertion of harmonics relationships, unless they arise naturally within a theory, would never be made a priori, because the development of the Fourier transform at the start of the 19th century removed their mystery; any motion can be expressed in terms of sines and cosines, and if the motion is periodic, then the motion is expressed as sines and cosines of the integer multiple of a fundamental frequency.

The patterns seen within a mathematical system are now what focus the theorist's mind. But like the modern chemist, who is descended from the alchemist, the modern theorist is descended from the Pythagoreans. Our modern theoretical objectivity evolved out of the mysticism of our forefathers.