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2 ANCIENT AND MEDIEVAL SCHOOLS |
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The philosophical school and religious
brotherhood known as Pythagoreanism is believed to have been founded by
Pythagoras of Samos, who settled in Croton in southern Italy about 525 BC. |
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The character of the original
Pythagoreanism is controversial, and the conglomeration of disparate features
that it displayed is intrinsically confusing. Its fame rests, however, on some
very influential ideas, not always correctly understood, that have been ascribed
to it since antiquity. These ideas include those of (1) the metaphysic
of number and the conception that reality, including music and astronomy, is, at
its deepest level, mathematical in nature; (2) the use of philosophy as a means
of spiritual purification; (3) the heavenly destiny of the soul and the
possibility of its rising to union with the divine; (4) the appeal to certain
symbols, sometimes mystical, such as the tetraktys,
the golden section, and the harmony of the spheres (to be discussed below);
(5) the Pythagorean theorem; and (6) the demand that members of the order shall
observe a strict loyalty and secrecy. (see also number system) |
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By laying stress on certain inner
experiences and intuitive truths revealed only to the initiated, Pythagoreanism
seems to have represented a soul-directed subjectivism alien to the mainstream
of Pre-Socratic Greek thought centring on the Ionian coast of Asia Minor
(Thales, Anaximander, Anaxagoras, and others), which was preoccupied with
determining what the basic cosmic substance is. (see also Ionian
school) |
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In contrast with such Ionian naturalism,
Pythagoreanism was akin to trends seen in mystery
religions and emotional movements, such as Orphism, which often claimed
to achieve through intoxication a spiritual insight into the divine origin and
nature of the soul. Yet there are also aspects of it that appear to have owed
much to the more sober, "Homeric" philosophy of the Ionians. The
Pythagoreans, for example, displayed an interest in metaphysics (the nature of
Being), as did their naturalistic predecessors, though they claimed to find its
key in mathematical form rather than in any substance. They accepted the
essentially Ionian doctrines that the world is composed of opposites (wet-dry,
hot-cold, etc.) and generated from something unlimited; but they added the idea
of the imposition of limit upon the unlimited and the sense of a musical harmony
in the universe. Again, like the Ionians, they devoted themselves to
astronomical and geometrical speculation. Combining, as it does, a rationalistic
theory of number with a mystic numerology and a speculative cosmology with a
theory of the deeper, more enigmatic reaches of the soul, Pythagoreanism
interweaves Rationalism and irrationalism
more inseparably than does any other movement in ancient Greek thought. (see
also opposites, table of) |
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The problem of describing Pythagoreanism
is complicated by the fact that the surviving picture is far from complete,
being based chiefly on a small number of fragments from the time before Plato
and on various discussions in authors who wrote much later--most of whom were
either Aristotelians or Neoplatonists (see below History
of Pythagoreanism ). In spite of the
historical uncertainties, however, that have plagued searching scholars, the
contribution of Pythagoreanism to Western culture has been significant and
therefore justifies the effort, however inadequate, to depict what its teachings
may have been. Moreover, the heterogeneousness of Pythagorean doctrines has been
well documented ever since Heracleitus, a
classic early 5th-century Greek philosopher who, scoffing at Pythagoras'
wide-ranging knowledge, said that it "does not teach one to have
intelligence." There probably never existed a strictly uniform system of
Pythagorean philosophy and religious beliefs, even if the school did have a
certain internal organization. Pythagoras appears to have taught by pregnant,
cryptic akousmata ("something heard") or symbola. His pupils handed these on, formed them partly into Hieroi
Logoi ("Sacred Discourses"), of which different versions were
current from the 4th century on, and interpreted them according to their
convictions. |
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Western Philosophical
Schools and Doctrines |
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The belief in the transmigration of
souls provided a basis for the Pythagorean way of life. Some Pythagoreans
deduced from this belief the principle of "the kinship of all beings,"
the ethical implications of which were later stressed in 4th-century
speculation. Pythagoras himself seems to have claimed a semidivine status in
close association with the superior god Apollo; he believed that he was able to
remember his earlier incarnations and, hence, to know more than others knew.
Recent research has emphasized shamanistic
traits deriving from the ecstatic cult practices of Thracian medicine men in the
early Pythagorean outlook. The rules for the religious life that Pythagoras
taught were largely ritualistic: refrain from speaking about the holy, wear
white clothes, observe sexual purity, do not touch beans, and so forth. He seems
also to have taught purification of the soul by means of music and mental
activity (later called philosophy) in order to reach higher incarnations.
"To be like your Master" and so "to come nearer to the gods"
was the challenge that he imposed on his pupils. Salvation, and perhaps ultimate
union with the divine cosmos through the study of the cosmic order, became one
of the leading ideas in his school. (see also reincarnation) |
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The advanced ethics and political
theories sometimes ascribed to Pythagoreanism may to some extent reflect ideas
later developed in the circle of Archytas, the
leading 4th-century Pythagorean. But a picture current among the Peripatetics
(the school founded by Aristotle) of Pythagoras as the educator of the Greeks,
who publicly preached a gospel of humanity, is clearly anachronistic. Several of
the Peripatetic writers, Aristoxenus, Dicaearchus, and Timaeus, seem to have
interpreted some principles--properly laid down only for esoteric use in the
brotherhood--as though these applied to all mankind: the internal loyalty,
modesty, self-discipline, piety, and abstinence required by the secret doctrinal
system; the higher view of womanhood reflected in the admission of women to the
school; a certain community of property; and perhaps the drawing of a
parallelism between the macrocosm (the universe) and the microcosm (man), in
which (for instance) the Pythagorean idea that the cosmos is an organism was
applied to the state, which should thus mix monarchy, oligarchy, and democracy
into a harmonic whole--these were all universalized. |
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According to Aristotle, number
speculation is the most characteristic feature of Pythagoreanism. Things
"are" number, or "resemble" number. To many Pythagoreans
this concept meant that things are measurable and commensurable or proportional
in terms of number--an idea of considerable significance for Western
civilization. But there were also attempts to arrange a certain minimum number
of pebbles so as to represent the shape of a thing--as, for instance, stars in a
constellation that seem to represent an animal. For the Pythagoreans even
abstracted things "have" their number: "justice" is
associated with the number four and with a square, "marriage" with the
number five, and so on. The psychological associations at work here have not
been clarified. |
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Figure 1: The Tetraktys (see text). |
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The sacred decad in particular has a
cosmic significance in Pythagoreanism: its mystical name, tetraktys (meaning approximately "fourness"), implies 1 +
2 + 3 + 4 = 10; but it can also be thought of as a "perfect triangle,"
as in Figure 1. (see also cosmology
) |
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Speculation on number and proportion led
to an intuitive feeling of the harmonia ("fitting
together") of the kosmos ("the
beautiful order of things"); and the application of the tetraktys
to the theory of music (see below Music
) revealed a hidden order in the range of sound. Pythagoras may have
referred, vaguely, to the "music of the heavens," which he alone
seemed able to hear; and later Pythagoreans seem to have assumed that the
distances of the heavenly bodies from the Earth somehow correspond to musical
intervals--a theory that, under the influence of Platonic
conceptions, resulted in the famous idea of the "harmony
of the spheres." Though number to the early Pythagoreans was still a kind
of cosmic matter, like the water or air proposed by the Ionians, their stress
upon numerical proportions, harmony, and order comprised a decisive step toward
a metaphysic in which form is the basic reality. (see also music
theory) |
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From the Ionians, the Pythagoreans
adopted the idea of cosmic opposites, which they--perhaps secondarily--applied
to their number speculation. The principal pair of opposites is the limit and
the unlimited; the limit (or limiting), represented by the odd (3,5,7, . . .),
is an active force effecting order, harmony, "cosmos," in the
unlimited, represented by the even. All kinds of opposites somehow "fit
together" within the cosmos, as they do, microcosmically, in an individual
man and in the Pythagorean society. There was also a Pythagorean "table of
ten opposites," to which Aristotle has referred--limit-unlimited, odd-even,
one-many, right-left, male-female, rest-motion, straight-curved, light-darkness,
good-evil, and square-oblong. The arrangement of this table reflects a dualistic
conception, which was apparently not original with the school, however, or
accepted by all of its members. |
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The Pythagorean number metaphysic was
also reflected in its cosmology. The unit (1), being the starting point of the
number series and its principle of construction, is not itself strictly a
number; for, to be a number is to be even or odd, whereas, in the Pythagorean
view, "one" is seen as both even
and odd. This ambivalence applies, similarly, to the total universe, conceived
as the One. There was also a cosmogonical theory (of cosmic origins) that
explained the generation of numbers and number-things from the limiting-odd and
the unlimited-even--a theory that, by stages unknown to scholars, was ultimately
incorporated into Plato's philosophy in his doctrine of the derivation of sensed
realities from mathematical principles. |
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Pythagorean thought was scientific as
well as metaphysical and included specific developments in arithmetic
and geometry, in the science of musical tones and harmonies, and in astronomy. |
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Early Pythagorean achievements in
mathematics are unclear and largely disputable, and the following is, therefore,
a compromise between the widely divergent views of modern scholars. |
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Figure 2: Gnomons of Pythagorean number theory (see text). |
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In the speculation on odd and even
numbers, the early Pythagoreans used so-called gnomones
(Greek: "carpenter's squares"). Judging from Aristotle's account,
gnomon numbers, represented by dots or pebbles, were arranged in the manner
shown in Figure 2. If a series of odd numbers is
put around the unit as gnomons, they always produce squares; thus, the members
of the series 4, 9, 16, 25, . . . are "square"
numbers. If even numbers are depicted in a similar way, the resulting
figures (which offer infinite variations) represent "oblong" numbers,
such as those of the series 2, 6, 12, 20 . . . . On the other hand, a triangle
represented by three dots (as in the upper part of the tetraktys) can be extended by a series of natural numbers to form
the "triangular" numbers 6, 10 (the tetraktys), 15, 21. . . . This procedure, which was, so far,
Pythagorean, led later, perhaps in the Platonic Academy, to a speculation on
"polygonal" numbers. |
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Figure 3: Gnomon for Pythagorean theorem. The marked off "carpenter's
square"--comprising 3. . . |
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Probably the square numbers of the
gnomons were early associated with the Pythagorean theorem (likely to have been
used in practice in Greece, however, before Pythagoras), which holds that for a
right triangle a square drawn on the hypotenuse is equal in area to the sum of
the squares drawn on its sides; in the gnomons it can easily be seen, in the
case of a 3,4,5-triangle for example (see Figure 3),
that the addition of a square gnomon number to a square makes a new square: 32
+ 42 = 52, and this gives a method for finding two square
numbers the sum of which is also a square. |
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Some 5th-century Pythagoreans seem to
have been puzzled by apparent arithmetical anomalies: the mutual relationships
of triangular and square numbers; the anomalous properties of the regular
pentagon; the fact that the length of the diagonal of a square is
incommensurable with its sides--i.e., that
no fraction composed of integers can express this ratio exactly (the resulting
decimal is thus defined as irrational); and the irrationality
of the mathematical proportions in musical scales. The discovery of such
irrationality was disquieting because it had fatal consequences for the naive
view that the universe is expressible in whole numbers; the Pythagorean Hippasus
is said to have been expelled from the brotherhood, according to some sources
even drowned, because he made a point of the irrationality. |
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In the 4th century, Pythagorizing
mathematicians made a significant advance in the theory of irrational numbers,
such as the-square-root-of-n ({sqroot n}), n being any rational number, when they developed a method for
finding progressive approximations to {sqroot 2} by forming sets of so-called
diagonal numbers. |
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In geometry,
the Pythagoreans cannot be credited with any proofs in the Euclidean sense. They
were evidently concerned, however, with some speculation on geometrical figures,
as in the case of the Pythagorean theorem, and the concept that the point, line,
triangle, and tetrahedron correspond to the elements of the tetraktys,
since they are determined by one, two, three, and four points, respectively.
They possibly knew practical methods of constructing the five regular solids,
but the theoretical basis for such constructions was given by non-Pythagoreans
in the 4th century. |
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It is notable that the properties of the
circle seem not to have interested the early Pythagoreans. But perhaps the
tradition that Pythagoras himself discovered that the sum of the three angles of
any triangle is equal to two right angles may be trusted. The idea of geometric proportions
is probably Pythagorean in origin; but the so-called golden
section--which divides a line at a point such that the smaller part is to
the greater as the greater is to the whole--is hardly an early Pythagorean
contribution. Some advance in geometry was made at a later date, by 4th-century
Pythagoreans; e.g., Archytas offered
an interesting solution to the problem of the duplication of the cube--in which
a cube twice the volume of a given cube is constructed--by an essentially
geometrical construction in three dimensions; and the conception of geometry as
a "flow" of points into lines, of lines into surfaces, and so on, may
have been contributed by Archytas; but on the whole the numerous achievements of
non-Pythagorean mathematicians were in fact more conspicuous than those of the
Pythagoreans. (see also Pythagorean
theorem) |
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The achievements of the early
Pythagoreans in musical theory are somewhat less controversial. The scientific
approach to music, in which musical intervals
are expressed as numerical proportions, originated with them, as did also the
more specific idea of harmonic "means." At an early date they
discovered empirically that the basic intervals of Greek music include the
elements of the tetraktys,
since they have the proportions 1:2 (octave), 3:2 (fifth), and 4:3 (fourth).
The discovery could have been made, for instance, in pipes or flutes or stringed
instruments: the tone of a plucked string held at its middle is an octave higher
than that of the whole string; the tone of a string held at the 2/3 point is a
fifth higher; and that of one held at the 3/4 point is a fourth higher.
Moreover, they noticed that the subtraction of intervals is accomplished by
dividing these ratios by one another. In the course of the 5th century they
calculated the intervals for the usual diatonic scale, the tone being
represented by 9:8 (fifth minus fourth); i.e.,
3/2 {division} 4/3, and the semitone by 256:243 (fourth minus two tones); i.e.,
4/3 {division} (9/8 × 9/8). Archytas made some modification to this
doctrine and also worked out the relationships of the notes in the chromatic
(12-tone) scale and the enharmonic scale (involving such minute differences as
that between A flat and G sharp, which on a piano are played by the same key). |
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In their cosmological views the earliest
Pythagoreans probably differed little from their Ionian predecessors. They made
a point of studying the stellar heavens; but--with the possible exception of the
theory of musical intervals in the cosmos--no new contributions to astronomy
can be ascribed to them with any degree of probability. Late in the 5th century,
or possibly in the 4th century, a Pythagorean boldly abandoned the geocentric
view and posited a cosmological model in which the Earth, Sun, and stars circle
about an (unseen) central fire--a view traditionally attributed to the
5th-century Pythagorean Philolaus of Croton. |
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The life of Pythagoras and the origins
of Pythagoreanism appear only dimly through a thick veil of legend and
semihistorical tradition. The literary sources for the teachings of the
Pythagoreans present extremely complicated problems. Special difficulties arise
from the oral and esoteric transmission of the early doctrines, the profuse
accumulation of tendentious legends, and the considerable amount of confusion
that was caused by the split in the school in the 5th century BC. In the 4th
century, Plato's inclination toward
Pythagoreanism created a tendency--manifest already in the middle of the century
in the works of his pupils--to interpret Platonic concepts as originally
Pythagorean. But the radical skepticism as to the reliability of the sources
shown by some modern scholars has on the whole been abandoned in recent
research. It now seems possible to extract bits of reliable evidence from a wide
range of ancient authors, such as Porphyry and Iamblichus (see below Neo-Pythagoreanism
). |
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Most of these literary sources hark back
ultimately to the environment of Plato and Aristotle;
and here the importance of one of Aristotle's students has become obvious, viz.,
the musicologist and philosopher Aristoxenus,
who in spite of his bias possessed firsthand information independent of the
point of view of Plato's Academy. The role played by Dicaearchus,
another of Aristotle's pupils, and by the Sicilian historian Timaeus,
of the early 3rd century BC, is less clear. Recently, the reliability of
Aristotle's account of Pythagoreanism has also been emphasized against the
doubts that had been expressed by some modern scholars; but Aristotle's sources,
in turn, hardly lead farther back than to the late 5th century (perhaps to
Philolaus; see below Two Pythagorean sects
). In addition, there are scattered hints in various early authors and in
some not very substantial remains of 4th-century Pythagorean literature. The
mosaic of reconstruction thus has to be to some extent subjective. |
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Within the ancient Pythagorean movement
four chief periods can be distinguished: early Pythagoreanism, dating from the
late 6th century BC and extending to about 400 BC; 4th-century Pythagoreanism;
the Hellenistic trends; and Neo-Pythagoreanism, a revival that occurred in the
mid-1st century AD and lasted for two and a half centuries. |
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The background of Pythagoreanism is
complex, but two main groups of sources can be distinguished. The Ionian
philosophers (Thales, Anaximander, Anaximenes, and others) provided Pythagoras
with the problem of a single cosmic principle, the doctrine of opposites, and
whatever reflections of Oriental mathematics there are in Pythagoreanism; and
from the technicians of his birthplace, the Isle of Samos, he learned to
understand the importance of number, measurements, and proportions. Popular
cults and beliefs current in the 6th century and reflected in the tenets of
Orphism introduced him to the notions of occultism and ritualism and to the
doctrine of individual immortality. In view of the shamanistic traits of
Pythagoreanism, reminiscent of Thracian cults, it is interesting to note that
Pythagoras seems to have had a Thracian slave. (see also opposites,
table of, number system) |
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The school apparently founded by
Pythagoras at Croton in southern Italy seems to have been primarily a religious
brotherhood centred around Pythagoras and the cults of Apollo and of the Muses,
ancient patron goddesses of poetry and culture. It became perhaps successively
institutionalized and received different classes of esoteric members and
exoteric sympathizers. The rigorism of the ritual and ethical observances
demanded of the members is unparalleled in early Greece; in addition to the
rules of life mentioned above, it is fairly well attested that secrecy and a
long silence during the novitiate were required. The exoteric associates,
however, were politically active and established a Crotonian hegemony in
southern Italy. About 500 BC a coup by a rival party caused Pythagoras to take
refuge in Metapontum, where he died. |
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During the early 5th century,
Pythagorean communities, inspired by the original school at Croton, existed in
many southern Italian cities, a fact that led to some doctrinal differentiation
and diffusion. In the course of time the politics of the Pythagorean parties
became decidedly antidemocratic. About the middle of the century a violent
democratic revolution swept over southern Italy; in its wake, many Pythagoreans
were killed, and only a few escaped, among them Lysis
of Tarentum and Philolaus of Croton, who
went to Greece and formed small Pythagorean circles in Thebes and Phlious. |
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Little is known about Pythagorean
activity during the latter part of the 5th century. The differentiation of the
school into two main sects, later called akousmatikoi
(Greek: akousma, "something
heard," viz., the esoteric teachings) and mathematikoi (Greek: mathematikos,
"scientific"), may have occurred at that time. The acousmatics
devoted themselves to the observance of rituals and rules and to the
interpretation of the sayings of the master; the "mathematics" were
concerned with the scientific aspects of Pythagoreanism. Philolaus, who was
rather a mathematic, probably published a summary of Pythagorean philosophy and
science in the late 5th century. |
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In the first half of the 4th century,
Tarentum, in southern Italy, rose into considerable significance. Under the
political and spiritual leadership of the mathematic Archytas, a friend of
Plato, Tarentum became a new centre of Pythagoreanism, from which
acousmatics--so-called Pythagorists who did not sympathize with Archytas--went
out travelling as mendicant ascetics all around the Greek-speaking world. The
acousmatics seem to have preserved some early Pythagorean Hieroi Logoi and ritual practices. Archytas himself, on the other
hand, concentrated on scientific problems, and the organization of his
Pythagorean brotherhood was evidently less rigorous than that of the early
school. After the 380s there was a give-and-take between the school of Archytas
and the Academy of Plato, a relationship that makes it almost impossible to
disentangle the original achievements of Archytas from joint involvements (but
see above, Geometry
and Music
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Whereas the school of Archytas
apparently sank into inactivity after the death of its founder (probably after
350 BC), the Academics of the next generation continued
"Pythagorizing" Platonic doctrines, such as that of the supreme One,
the indefinite dyad (a metaphysical principle), and the tripartite soul. At the
same time, various Peripatetics of the school of Aristotle, including
Aristoxenus, collected Pythagorean legends and applied contemporary ethical
notions to them. In the Hellenistic Age, the Academic and Peripatetic views gave
rise to a rather fanciful antiquarian literature on Pythagoreanism. There also
appeared a large and yet more heterogeneous mass of apocryphal writings falsely
attributed to different Pythagoreans, as if attempts were being made to revive
the school. The texts fathered on Archytas display Academic and Peripatetic
philosophies mixed with some notions that were originally Pythagorean. Other
texts were fathered on Pythagoras himself or on his immediate pupils, imagined
or real. Some show, for instance, that Pythagoreanism had become confused with
Orphism; others suggest that Pythagoras was considered a magician and an
astrologist; there are also indications of Pythagoras "the athlete"
and "the Dorian nationalist." But the anonymous authors of this
pseudo-Pythagorean literature did not succeed in reestablishing the school, and
the "Pythagorean" congregations formed in early imperial Rome seem to
have had little in common with the original school of Pythagoreanism established
in the late 6th century BC; they were ritualistic sects that adopted,
eclectically, various occult practices. |
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With the ascetic sage Apollonius
of Tyana, about the middle of the 1st century AD, a distinct
Neo-Pythagorean trend appeared. Apollonius studied the Pythagorean legends of
the previous centuries, created and propagated the ideal of a Pythagorean
life--of occult wisdom, purity, universal tolerance, and approximation to the
divine--and felt himself to be a reincarnation of Pythagoras. Through the
activities of Neo-Pythagorean Platonists, such as Moderatus of Gades, a pagan
trinitarian, and the arithmetician Nicomachus of Gerasa,
both of the 1st century AD, and, in the 2nd or 3rd century, Numenius
of Apamea, forerunner of Plotinus (an epoch-making elaborator of
Platonism), Neo-Pythagoreanism gradually became a part of the expression of
Platonism known as Neoplatonism; and it did so
without having achieved a scholastic system of its own. The founder of a Syrian
school of Neoplatonism, Iamblichus, a pupil of
Porphyry (who in turn had been a pupil of Plotinus), thought of himself as a
Pythagorean sage and about AD 300 wrote the last great synthesis of
Pythagoreanism, in which most of the disparate post-classical traditions are
reflected. It is characteristic of the Neo-Pythagoreans that they were chiefly
interested in the Pythagorean way of life and in the pseudoscience of number
mysticism. On a more popular level, Pythagoras and Archytas were remembered as
magicians. Moreover, it has been suggested that Pythagorean legends were also
influential in guiding the Christian monastic tradition. |
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In the Middle Ages the popular
conception of Pythagoras the magician was combined with that of Pythagoras
"the father of the quadrivium"; i.e.,
of the more specialized liberal arts of the curriculum. From the Italian
Renaissance onward, some "Pythagorean" ideas, such as the tetrad, the
golden section, and harmonic proportions, became applied to aesthetics. To many
Humanists, moreover, Pythagoras was the father of the exact sciences. In the
early 16th century, Nicolaus Copernicus, who
developed the view that the Earth revolves around the Sun, considered his system
to be essentially Pythagorean or "Philolaic," and Galileo
was called a Pythagorean. The 17th-century Rationalist G.W.
Leibniz appears to have been the last great philosopher and scientist who
felt himself to be in the Pythagorean tradition. (see also humanism) |
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It is doubtful whether advanced modern
philosophy has ever drawn from sources thought to be distinctly Pythagorean. Yet
Platonic-Neoplatonic notions, such as the mathematical conception of reality or
the philosopher's union with the universe and various mystical beliefs are still
likely to be stamped as Pythagorean in origin. Even today a relatively
uncritical admiration of Pythagoreanism is common. (see also Platonism) |
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The history of the projection of
Pythagoreanism into subsequent thought indicates how fertile some of its core
concepts were. Plato is here the great catalyst; but it is possible to perceive
behind him, however dimly, a series of Pythagorean ideas of paramount potential
significance: the combination of religious esoterism (or exclusivism) with the
germs of a new philosophy of mind, present in the belief in the progress of the
soul toward the actualization of its divine nature and toward knowledge; stress
upon harmony and order, and upon limit as the good; the primacy of form,
proportion, and numerical expression; and in ethics, and emphasis upon such
virtues as friendship and modesty. The fact that Pythagoras, to later ages, also
became alternatively conceived of as a Dorian nationalist, a sportsman, an
educator of the people, or a great magician is a more curious consequence of the
productivity of his teaching. |
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(H.T.) |
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