The Philosophy of the Pythagoreans | Research & Encyclopedia Articles

The way of thinking about the world that came to be known as philosophy emerged in the sixth century B.C. among groups of Greek thinkers scattered around the Mediterranean region. The Pythagoreans were one of the most influential of these groups. During their approximately two hundred years as an organized community, the Pythagoreans spread ideas about numbers, nature, and man that were profoundly important to the subsequent study of mathematics, music, and astronomy.

As with most figures of the ancient world, factual evidence about Pythagoras and the community he founded is quite minimal. Historians agree, however, that Pythagoras of Samos (c. 560-c. 480 B.C.) founded a community of like-minded men among a Greek colony on the southern coast of Italy around 530 B.C. While it is tempting to describe this group in a modern way as a "school," in fact it began as a kind of religious community or cult. Their primary motivation for intellectual pursuits was as a means of achieving spiritual purification. By emphasizing contemplation and study, especially of numbers and numerical relationships, and practicing physical asceticism, the Pythagoreans hoped to bring their souls into harmony with the greater cosmos and therefore to escape the cycle of the transmigration of souls. This doctrine of reincarnation, or the "wheel of birth," held that souls were immortal but would be forced to pass through life again and again in various animal forms until sufficient purity was achieved. The Pythagoreans sought to achieve this purity directly through their philosophy and contemplative life.

Their beliefs in the transmigration and immortality of souls and in the purifying potential of philosophy were central to the religion of the Pythagoreans. They additionally believed in abstinence from various physical and dietary practices, in strict measures of loyalty and secrecy, and in the mystical significance of certain symbols. But the most influential of the Pythagorean beliefs was certainly their contention that all of reality was mathematical. Their elevation of numbers and numerical relationships to a position of such philosophical importance brought them to unique insights about subjects from acoustics to astronomy.

The precise history and activities of the Pythagoreans are not clearly established. It seems, however, that their society flourished into the fifth century B.C. throughout a portion of southern Italy near Crotona. Their power and influence had negative consequences, as in the middle of the fifth century they became entangled with political disputes in the region and ultimately were violently suppressed. The surviving members of the Pythagoreans then scattered throughout Greek-speaking regions outside of Italy. While their period as an influential religious brotherhood had come to an end, individuals and small groups continued to perpetuate the Pythagorean philosophy well into the following century.

While Pythagorean ideas about ethics and metaphysics appear in many important Greek treatises, including the works of Plato (c. 427-347B.C.) and Aristotle (384-322 B.C.), the mathematical ideas promulgated by the Pythagoreans had the greatest influence on the subsequent history of philosophy and science. Because the Pythagoreans maintained an oral rather than a written tradition, we have no direct evidence of their beliefs or discoveries and must rely on the accounts written by later philosophers who made use of or commented upon Pythagorean ideas. Aristotle, for example, contended that the Pythagoreans believed the entire universe to be essentially musical, and therefore mathematical. Whether the Pythagoreans believed this literally or merely as a model of some sort, we cannot know. But we can be certain that numbers and musical harmony were the most significant concepts in the Pythagorean understanding of the world. The most common characterization of the worldview of the Pythagoreans is the complete identification of reality with numbers.

The Pythagoreans studied the properties of numbers, the reflection of numerical properties in geometrical figures, and the existence of numerical relationships in the natural world. They were the first to study sums of series. By representing these series geometrically, they were able to classify different patterns resulting from summing different sets of numbers. Among students of elementary mathematics, the Pythagoreans are best known for one such geometrical relationship. They observed that in right-angled triangles, the square of the hypotenuse is equal to the sum of the squares of the other sides. This "Pythagorean theorem" remains one of the touchstones of elementary geometry, and is a clear example of the style of investigation and discovery typically attributed to the Pythagoreans. Another Pythagorean contribution important to elementary geometry was their recognition that the diagonal and the side of a square are incommensurable (that is, their ratio will be an irrational number).

In contrast to our modern understanding of these mathematical ideas, the Pythagoreans came to many of their discoveries by manipulating pebbles to represent numbers. These pebbles have their own legacy in modern mathematics; the Greek word for pebble was calculus and gave us the English term "calculate." The Pythagoreans represented numbers as triangular, square, or rectangular, depending on whether that number of pebbles could be arranged symmetrically in one shape or the other. For example, three is a triangular number, while four is a square number.

No set of ideas was more important to the worldview and philosophy of the Pythagoreansthan those relating numbers to musical harmony. The Pythagoreans identified the fundamental musical intervals of the octave, the fifth, and the fourth by using ratios, another conceptual discovery, and by demonstrating the production of harmonics on their stringed instruments. That is, the harmonic of the octave is produced by touching the string at one-half of its length, while the fifth is produced at two-thirds of the length. Building on this foundation, the Pythagoreans produced a system of musical scales and chords.

The Pythagoreans extended their understanding of musical intervals to their study of the heavenly bodies. An earlier philosopher, Anaximander (c. 610-c. 546 B.C.), suggested that the heavenly bodies consisted of three movable wheels. The Pythagoreans built upon this idea by identifying the intervals between the three wheels with the musical intervals of the octave, fifth, and fourth. They were the first to distinguish the diurnal revolution of the heavens from east to west from those of the Sun, Moon, and planets from west to east. Pythagoras is also credited with discovering the sphericity of Earth. Centuries later, Nicolaus Copernicus (1473-1543) credited the Pythagoreans for these astronomical concepts as important precursors to his own hypothesis that Earth and other planets rotate about the Sun, rather than the Sun rotating around Earth.

Tracing the impact of the Pythagoreans is somewhat frustrating. Later Greek philosophers including Plato, Aristotle, and Euclid (c. 330-c. 260 B.C.) explicitly and tacitly credited the Pythagoreans with many ideas, beliefs, and discoveries. But since these very sources are in many cases our only documentation of the work of the Pythagoreans, it is impossible to disentangle original ideas from their subsequent interpretations. If we are to give the Pythagoreans their broadest possible due, then they would get credit for the birth of the study of mathematics for its own sake rather than in support of commercial or other activities. The formal study of music and harmony, and the practice of music theory, are also achievements that can be confidently attributed to the Pythagoreans.

The boldest ideas of the Pythagoreans—their characteristic views that nature is made up of numerical relationships and the musical harmonies that correspond to those relationships—have reappeared in different forms throughout the history of science. The concept of the "harmony of the spheres" of the universe, for example, was a guiding principle for astronomer Johannes Kepler (1571-1630). Efforts to explain nature by mathematics have often yielded descriptions of nature as inherently mathematical, and these have at times shaded into purely mathematical accounts of natural phenomena. Most intriguing of all, however, may be the science of digitization, which electronically represents sound, images, and information of all kinds as numbers. The digital world can be seen as the ultimate realization of the Pythagorean ideal of nature as number. Although no causal connection could ever be drawn from Pythagoras in the fifth century B.C. to twenty-first century digital music, the latter no doubt owes some kind of conceptual debt to the imagination of the Pythagoreans.

Dreyer, J.L.E. A History of Astronomy from Thales to Kepler. New York: Dover Press, 1953.

Furley, David. The Greek Cosmologists. Cambridge: Cambridge University Press, 1987.

Lindberg, David. The Beginnings of Western Science. Chicago: University of Chicago Press, 1992.

Lloyd, G.E.R. Early Greek Science: Thales to Aristotle. Cambridge: Cambridge University Press, 1970.

Neugebauer, Otto. The Exact Sciences in Antiquity. Princeton: Princeton University Press, 1952.

O'Meara, Dominic. Pythagoras Revived: Mathematics and Philosophy in Late Antiquity. Oxford: Clarendon Press, 1989.

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