Pythagoreans

1. The Mysterious Pythagoreans

2. Pythagoras of Samos
   • background: Orphic Mysteries???
   • founds colony in Croton (extreme southern Italy)
   • his death in Metapontum

3. elements of the philosophy:
   • secrecy
   • purity
   • men & women
   • music
     • HARMONIA
     • intervals on a plucked string
       • octave (2:1)
       • fifth (3:2)
       • fourth (4:3)
       NB: these intervals are independent of the length diameter or material of the string
       A case can be made that this is the first completely general mathematical law of nature discovered in the Western tradition. We are now so familiar with such formulations that we tend to discount the genius of the discovery while emphasizing the recklessness of the generalization. Yet against a backdrop of mythology like that codified by Hesiod and a cosmology based in (physical) elements and paired oppositions such as we find in the Milesians the significance of this discovery is difficult to overstate. All discoveries of natural physical laws expressible in simple mathematical relations can in some important sense all be said to be ":Pythagorean." When Kepler generalized from his observations to the hypothesis that planetary orbits must be elliptical, he was thinking like a Pythagorean. His fervent hunch took him to the simplest possible interpretation: objects should orbit in circles. When that assumption could not predict interplanetary positions with anything like accuracy, he moved to the next simplest explanation. If they do not revolve around a single focus, perhaps there are two foci: eureka, ellipse! Likewise in the case of Newton. Building upon Galileo Galilie's discovery of the constant rate of acceleration of free-falling bodies, Newton was led to his discovery of the universal law of gravitation. Not the simplest law one could imagine, but still, given the variety, complexity, and palpable differences among the things of this world, remarkably simple: directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The assumptions at work include the following: other things being equal, realtionships will be fixed, arithimetically or algebraically expressible, and as simple as can be made consistent with the facts. By implication what things look like, feel like, taste like-- their textures, shapes, odors, opposites, composition-are all esentially irrelevant. The Pythagoreans devised or discovered (whichever you prefer) the reducibility of physical phenomena to mathematical relationships. This acheivement (for better and for worse) is one of the definitive steps in the development of Western thought.
   • mathematics
     • Pythagorean Theorem
     • our mathematical & musical vocabulary
       • rational and irrational numbers
       • chords
       • dissonance
         The specific discovery that led to this generalization involved the harmonics of plucked strings. Someone early on in the history of Pythagoreanism (whether Pythagoras himself or not) discovered or learned from some other source that the tones generated by vibrating chords of different lengths stand in determinate realtionships independent of the length, thickness, or composition of the taut string (provided only that sufficient forced exists to pluck it.) Any such chord, precisely halved in length, willl sound at what we recognize as one octave higher than the full length chord. Other so-called "consonant" intervals are similarly expressible as ratios of whole numbers. This appears to be the key to Pythagorean philosophy. (Its way of life may well derive from other sources.)
         Algebraists that we are, we tend to characterize such discoveries in numerical terms. The octives are in a ratio of 2:1; fifths are 3:2; fourths 4:3, etc. But it is important to bear in mind that the ancient Greeks did not have Arabic numerals. Thus they lacked the concepts of place-value and, as a result, decimal numbers. This point is essential to understanding the turmoil that must have accompanied the discovery of the square root of 2. Geometrically this value is easily expressed. Given an isoceles right triangle with legs set at 1 unit, the length of the hypotenuse will be the square root of 2. Although this can be expressed simply in geormetrical terms, arithmetically it cannot be expressed as the ratio of two whole numbers. From this stems our entire vocabulary of so-called "irrational" numbers.
   • metempsychosis
     • reincarnation
     • immortality
     • the body as tomb
       Early on the Pythagoreans probably adapted their discovery of musical harmonics to the explanation of life. How does some of the organized stuff of the world come to life? This question implicit in Milesian Philsophy receives a unique new interpretation in Pythagorean thought. Life arises, according to the Pythagorean view, from the balance or HARMONIA among the parts of a body or organism. Where elements venture too far out of balance, life gives out. This viewpoint builds on the assumptions about oppositions and order that we see in Anaximander and resonates through the ages in concepts of health (from Galen to the present day) based on a proper balance of humours or chemicals. In the case of the Pythagoresans, these thoughts were combined with the assumption of reincarnation (transmigration) or multiple lives.

4. form (ratio) as the ARCHE

5. Later Pythagoreanism
   • two types
     • those who learn (MATHEMATIKOI)
     • those who listen (AKOUSMATIKOI)
   • Philolaus of Croton or (???) Tarentum

6. influence on Plato

7. mysterious disappearance of the Pythagoreans (by 400 B.C.E.)

8. the legacy of Pythagoreanism
   • mystics & mathematicians
   • alchemy
   • the "music of the spheres"

revised September 14, 1996

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