||c. 580 BC-572 BC|
||c. 500 BC-490 BC|
||Metaphysics, Music, Mathematics, Ethics, Politics|
||Musica universalis, Golden ratio, Pythagorean tuning, Pythagorean theorem|
||Thales, Anaximander, Pherecydes|
||Philolaus, Alcmaeon, Parmenides, Plato, Euclid, Empedocles, Hippasus, Kepler|
Pythagoras of Samos (Greek: Πυθαγόρας; between
580 and 572 BC–between 500 BC and 490 BC) was an Ionian (Greek) philosopher and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician and scientist; however, careful scholarship in the past
three decades has found no evidence of his contributions to mathematics or natural philosophy. His name led him to be associated with Pythian Apollo; Aristippus explained his name by saying, "He spoke (agor-) the truth no less than did the Pythian (Pyth-),"
and Iamblichus tells the story that the Pythia prophesied that his pregnant mother would give birth to a man supremely
beautiful, wise, and of benefit to humankind.
Pythagorean thought was dominated by mathematics, but it was also profoundly mystical. In the
area of cosmology there is less agreement about what Pythagoras himself actually taught, but most scholars believe that the Pythagorean idea of the transmigration of the soul is too central to have been added by a later follower of Pythagoras. The Pythagorean conception
of substance, on the other hand, is of unknown origin, partly because various accounts of his teachings are conflicting. The
Pythagorean account actually begins with Anaximander's teaching that the ultimate substance of things is "the boundless," or what Anaximander called
the "apeiron." But the Pythagorean account says that it is only through the notion of the "limit" that the
"boundless" takes form.
Pythagoras wrote nothing down, and relying on the writings of Parmenides, Empedocles, Philolaus and Plato (people either considered Pythagoreans, or whose works are thought deeply indebted to Pythagoreanism)
results in a very diverse picture in which it is difficult to ascertain what the common unifying Pythagorean themes were.
Relying on Philolaus, whom most scholars agree is highly representative of the Pythagorean school, one has a very intricate
picture. Aristotle explains how the Pythagoreans (by which he meant the circle around Philolaus) developed Anaximander's ideas about the apeiron and the peiron, the unlimited and limited, by writing that:
for they [the Pythagoreans] plainly say that when the one had been constructed, whether out of planes or of surface or of
seed or of elements which they cannot express, immediately the nearest part of the unlimited began to be drawn in and limited
by the limit.
Continuing with the Pythagoreans:
Pythagoreans, too, held that void exists, and that it enters the heaven from the unlimited breath – it, so to speak,
breathes in void. The void distinguishes the natures of things, since it is the thing that separates and distinguishes the
successive terms in a series. This happens in the first case of numbers; for the void distinguishes their nature.
When the apeiron is inhaled by the perion it causes separation, which also apparently means that it "separates
and distinguishes the successive terms in a series." Instead of an undifferentiated whole we have a living whole of inter-connected
parts separated by "void" between them. This inhalation of the apeiron is also what makes the world mathematical, not just
possible to describe using math, but truly mathematical since it shows numbers and reality to be upheld by the same principle:
both the continuum of numbers (that is yet a series of successive terms, separated by void) and the field of reality, the
cosmos - both are a play of emptiness and form, apeiron and peiron. What really sets this apart from Anaximander's original
ideas is that this play of apeiron and peiron must take place according to harmonia (harmony), about which Stobaeus commentated:
- "About nature and harmony this is the position. The being of the objects, being eternal, and
nature itself admit of divine, not human, knowledge – except that it was not possible for any of the things that exist
and are known by us to have come into being, without there existing the being of those things from which the universe was
composed, the limited and the unlimited. And since these principles existed being neither alike nor of the same kind, it would
have been impossible for them to be ordered into a universe if harmony had not supervened – in whatever manner this
came into being. Things that were alike and of the same kind had no need of harmony, but those that were unlike and not of
the same kind and of unequal order – it was necessary for such things to have been locked together by harmony, if they
are to be held together in an ordered universe."
A musical scale presupposes an unlimited continuum of pitches, which must be limited in some
way in order for a scale to arise. The crucial point is that not just any set of limiters will do. One may not simply choose
pitches at random along the continuum and produce a scale that will be musically pleasing. The diatonic scale, also known
as "Pythagorean," is such that the ratio of the highest to the lowest pitch is 2:1, which produces the interval of an octave.
That octave is in turn divided into a fifth and a fourth, which have the ratios of 3:2 and 4:3 respectively and which, when
added, make an octave. If we go up a fifth from the lowest note in the octave and then up a fourth from there, we will reach
the upper note of the octave. Finally the fifth can be divided into three whole tones, each corresponding to the ratio of
9:8 and a remainder with a ratio of 256:243 and the fourth into two whole tones with the same remainder. This is a good example
of a concrete applied use of Philolaus’ reasoning. In Philolaus' terms the fitting together of limiters and unlimiteds
involves their combination in accordance with ratios of numbers (harmony). Similarly the cosmos and the individual things
in the cosmos do not arise by a chance combination of limiters and unlimiteds; the limiters and unlimiteds must be fitted
together in a "pleasing" (harmonic) way in accordance with number for an order to arise.
This teaching was recorded by Philolaus' pupil Archytas in a lost work entitled On Harmonics
or On Mathematics, and this is the influence that can be traced in Plato. Plato's pupil Aristotle made a distinction in his
Metaphysics between Pythagoreans and "so-called" Pythagoreans. He also recorded the Table of Opposites, and commented
that it might be due to Alcmaeon of the medical school at Croton, who defined health as a harmony of the elements in the body.
After attacks on the Pythagorean meeting-places at Croton, the movement dispersed, but regrouped
in Tarentum, also in Southern Italy. A collection of Pythagorean writings on ethics collected by Taylor show
a creative response to the troubles.
The legacy of Pythagoras, Socrates and Plato was claimed by the wisdom tradition of the Hellenized
Jews of Alexandria, on the ground that their teachings derived from those of Moses. Through Philo of Alexandria this tradition
passed into the Medieval culture, with the idea that groups of things of the same number are related or in sympathy. This
idea evidently influenced Hegel in his concept of internal relations.
The ancient Pythagorean pentagram was drawn with two points up and represented the doctrine
of Pentemychos. Pentemychos means "five recesses" or "five chambers", also known as the pentagonas — the five-angle, and
was the title of a work written by Pythagoras' teacher and friend Pherecydes of Syros.
The Monad was a symbol referred by the Greek philosophers as "The First", "The Seed", "The Essence", "The
Builder", and "The Foundation"
The Pythagoreans are known for their theory of the transmigration of souls, and also for their theory that numbers constitute the true nature of things. They performed
purification rites and followed and developed various rules of living which they believed would enable their soul to achieve
a higher rank among the gods. Much of their mysticism concerning the soul seem inseparable from the Orphic tradition. The Orphics included various purifactory rites and practices as well as incubatory
rites of descent into the underworld. Apart from being linked with this, Pythagoras is also closely linked with Pherecydes of Syros, the man ancient commentators tend to credit as the first Greek to teach a transmigration of souls.
Ancient commentators agree that Pherekydes was Pythagoras's most intimate teacher. Pherekydes expounded his teaching on the
soul in terms of a pentemychos ("five-nooks," or "five hidden cavities") — the most likely origin of the Pythagorean
use of the pentagram, used by them as a symbol of recognition among members and as a symbol of inner health (eugieia).
Neo-Pythagoreanism was a revival in the 2nd century BC, 1st century BC, and the next two centuries of various ideas traditionally associated with the followers of Pythagoras,
Notable Neo-Pythagoreans include first or second century B.C.E. writers who went by the names
Ocellus Lucanus, Timeus Locrus, and Archytas. First century C.E. Apollonius of Tyana is also considered a Neo-Pythagorean. Middle and Neo-Platonists such as Numenius and Plotinus also exhibited some Neo-Pythagorean influence.
In 1915 a subterranean basilica was discovered near Porta Maggiore on Via Praenestina, Rome where Neo-Pythagoreans held their meetings in the 1st century. The groundplan shows a basilica
with three naves and an apse similarly to early Christian basilicas that appeared only much later, in the 4th century. The
vaults are decorated with white stuccoes symbolizing Neo-Pythagorean beliefs but its exact meaning remains a subject of debate
- The Pythagorean idea that whole numbers and harmonic (pleasing) sounds are intimately connected in
music, must have been well known to lute-player and maker Vincenzo Galilei, father of Galileo Galilei. While possibly following Pythagorean modes of thinking, Vincenzo is known to have discovered a new mathematical
relationship between string tension and pitch, thus suggesting a generalization of the idea that music and musical instruments
can be mathematically quantitated and described. This may have paved to way to his son's crucial insight that all physical
phenomena may be described quantitatively in mathematical language (as physical "laws"), thus beginning and defining the era
of modern physics.
- Pythagoreanism has had a clear and obvious influence on the texts found in the hermetica corpus and thus flows over into hermeticism, gnosticism and alchemy.
- The Pythagorean cosmology also inspired the Arabic gnostic Monoimus to combine this system with monism and other things to form his own cosmology.
- The pentagram (five-pointed star) was an important religious symbol used by the Pythagoreans, which is often seen as
being related to the elements theorized by Empedocles to comprise all matter.
- The Pythagoreans were advised to "speak the truth in all situations," which Pythagoras said he learned from the Magi of Babylon.