Pythagoras

Pythagoras (Πυθαγόρας, Pythagóras; Pythagoras; /pɪˈθæɡərəs/) was a Greek philosopher and mathematician, born on the island of Samos around 570 BCE and believed to have died around 495 BCE in Metapontum (southern Italy). He is best known for founding the Pythagoreanism movement, which combined religious rites, ascetic practices, and philosophical inquiry; his teachings and beliefs have had a profound influence on mathematics, philosophy, and mysticism. Pythagoras traveled widely in his youth, including visits to Egypt and perhaps Babylon, where he was exposed to a wide range of mathematical and philosophical ideas.

He emigrated to southern Italy about 532 BCE, apparently to escape tyrannical rule, and eventually settled in Croton, Magna Graecia, where he established a school that was part philosophical community and part religious sect. Pythagoras is most famous for the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Although this principle was likely known before Pythagoras, he is credited with its first proof.

It is difficult to distinguish Pythagoras' teachings from those of his disciples; he wrote no known books, and Pythagoreans invariably supported their doctrines by indiscriminately citing their master's authority. Pythagoras is generally credited with the theory of the functional significance of numbers in the objective world, and in music; other discoveries often attributed to him (the incommensurability of the side and diagonal of a square, for example, and the Pythagorean theorem for right triangles) were probably developed only later by the Pythagorean school. More probably, the bulk of the intellectual tradition originating with Pythagoras himself belongs to mystical wisdom rather than to scientific scholarship.

Pythagoras and his followers believed in the profound spiritual significance of numbers and mathematical relations, contributing to the development of number theory and the idea that mathematical principles underpin the structure of the cosmos. Pythagoras taught the doctrine of metempsychosis, the belief in the transmigration of souls, or reincarnation, asserting that souls are immortal and move among living beings in a cycle of rebirths until they achieve purification. The Pythagorean community lived a communal, ascetic lifestyle, adhering to strict dietary rules, engaging in rituals, and following ethical teachings that emphasized harmony, self-discipline, and the pursuit of philosophical wisdom. The community's teachings influenced both Plato and Aristotle and laid the groundwork for many later developments in Western philosophy and science. Pythagoras' later years were marked by political turmoil in Croton, leading to conflict with the ruling aristocracy; his school was eventually attacked, and Pythagoras was forced to flee. He died in Metapontum.

Much of what is known about Pythagoras comes from later sources, as he left no written records. Early accounts by philosophers like Heraclitus and later works by Plato, Aristotle, and the biographer Diogenes Laërtius, among others, provide the basis for our understanding of his life and teachings. These sources often mix historical facts with legend, making it difficult to separate Pythagoras' actual teachings from those of his later followers. The first fragmentary accounts of his life came in the fourth century BCE, about 150 years after his death.

The confusing amalgam of ideas that represent Pythagoreanism 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.
5. the Pythagorean theorem.
6. the demand that members of the order shall observe a strict loyalty and secrecy.

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 centering on the Ionian coast of Asia Minor, which was preoccupied with determining what the basic cosmic substance is. 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.

The Pythagoreans also displayed an interest in metaphysics, though they claimed to find its key in mathematical form rather than in any homogenous basal material. They accepted the essentially Ionian doctrines that the world is composed of opposites (wet-dry, hot-cold, and so on) 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 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 religion more inseparably than does any other movement in ancient Greek thought.

Pythagoras appears to have taught by pregnant, cryptic akousmata (Greek: literally, 'something heard') or symbola ('symbols'). 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. He seems to have claimed a semidivine status in close association with Apollo; he believed that he was able to remember his earlier incarnations and, hence, to know more than others knew. Research in the 20th century CE emphasized shamanistic traits deriving from the ecstatic cult practices of Thracian medicine men in the early Pythagorean outlook (and Pythagoras may have kept a Thracian slave).

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, et cetera. He seems also to have taught purification of the soul by means of music and mental activity 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.

The sacred decad (the sum of the first four numbers) 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.' Speculation on number and proportion led to an intuitive feeling of the harmonia ('fitting together') of the kosmos ('order of things'); and the application of the tetraktys to the theory of 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 constituted a decisive step toward a metaphysic in which form is the basic reality.

From the Ionians, the Pythagoreans adopted the idea of cosmic opposites, which they 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, et cetera), is an active force effecting order, harmony; and 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 person 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.

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 to the total universe, conceived as the One. There was also a cosmogonical theory (a theory of the origins of the cosmos) 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.

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. In their speculation on odd and even numbers, the early Pythagoreans used gnōmones ('carpenter's squares').

Gnomon numbers, represented by dots or pebbles, were arranged in the manner shown in the figure. If a series of odd numbers are 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.

The square numbers of the gnomons were early associated with the Pythagorean theorem (likely to have been used in practice in Greece 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, 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.

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 (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. In the 4th century, Pythagorizing mathematicians made a significant advance in the theory of irrational numbers, such as the-square-root-of-n (√n), n being any rational number, when they developed a method for finding progressive approximations to √2 by forming sets of so-called diagonal numbers.

The tetrakys was thought to correspond to the point, line, triangle, and tetrahedron, since they are determined by one, two, three, and four points respectively. The Pythagorans 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. The five regular solids were:

• Tetrahedron - four triangular faces
• Cube - six square faces
• Octahedron - eight triangular faces
• Dodecahedron - 12 pentagonal faces
• Icosahedron - 20 triangular faces

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 (see golden ratio). Some advance in geometry was made at a later date, by 4th-century Pythagoreans; 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; on the whole the numerous achievements of non-Pythagorean mathematicians were in fact more conspicuous than those of the Pythagoreans.

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 ÷ 4/3, and the semitone by 256:243 (fourth minus two tones); i.e., 4/3 ÷ (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).

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.

The school apparently founded by Pythagoras at Croton in southern Italy seems to have been primarily a religious brotherhood centered 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 BCE a coup by a rival party caused Pythagoras to take refuge in Metapontum, where he died.

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, who went to Greece and formed small Pythagorean circles in Thebes and Phlious.

Little is known about Pythagorean activity during the latter part of the 5th century BCE. The differentiation of the school into two main sects, later called akousmatikoi (from akousma, 'esoteric') and mathēmatikoi (from mathēmatikos, '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.