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PRESOCRATIC ROOTS: PYTHAGORAS AND FOLLOWERS
Pythagorean esotericism
In the 6th Century BCE, Greater Greece stretched west all
they way to Sicily and southern Italy. At this time its western
colonies were suddenly reinvigorated by an influx of refugees chased
out of Ionia by the conquering Persians. Pythagoras, who was
born on the island of Samos some time before 550 BCE, at a time when
the whole of eastern Greece was becoming unsafe, settled at Croton in
Sicily, probably before 520 BCE. A great many legends soon sprang up
around this mysterious figure. By the end of the 5th
Century BCE there was no longer any reliable source of information on
the life and works of Pythagoras or his immediate successors.
What is clear is the reason behind this mystery: it lies in the
doctrine itself and the teaching methods of the man who was called the
Master. Nothing Pythagoras taught was to be written down or
divulged to the uninitiated, and even the disciples were divided into
two classes, the μαθηματικοί 
(mathematikoi), who were students privileged to know the
thoughts of the Master, and the ᾽ακουσματικοί
(akousmatikoi), the mere listeners, allowed to know a little of
his teaching but unworthy of the name of Pythagoreans. Furthermore, to
avoid the common illusion by which some people think they made a
discovery themselves, when in fact they merely happen to be the medium
through which that discovery is crystallised, all thoughts and new
ideas that came out of the reflections of the Pythagorean circle were
attributed to Pythagoras himself, even long after he had
died. So it is probable that the famous theorem that bears his name
was devised long after his time.
Secrecy must have been carefully guarded, and relatively few of the
Pythagoreans left any written records. It is even said that some were
punished by death (by Fate, or with the help of zealous members of the
brotherhood, as in the case of Hippasus) because they divulged
a little of the esoteric knowledge they had acquired. In its
cosmogony, the Pythogorean school draws partly on Milesian thought,
but most of the thinking associated with the school is probably of
eastern rather than Greek origin. Science and ethics were closely
mingled, brought together via an explicit dualist hypothesis into a
religious theory (whence the importance of the esoteric aspect). Body
and soul (here Ψυχή ,
but regarded as immaterial) were separate entities (1);
and the immortal soul could inhabit
different bodies as a result of metempsychosis (2),
sometimes the bodies of all kinds of animals. This property of the
soul, and the numerical constraints associated with all objects,
whether material or immaterial, involved a great many necessities and
incompatibilities. The Pythagoreans respected many taboos and followed
a very strict moral code, taught via a special system of education (3)
. All passion, all excess, was to be avoided,
so as to preserve the harmony of the soul (4).
The Pythagoreans also believed adamantly in true friendship, and each
member of the School would have done everything for a fellow member in
difficulties, ruling out all rivalry and jealousy. The fact that all
discoveries were attributed to Pythagoras himself helped to
avoid any tension caused by the vain desire for intellectual property.
According to Aristotle, to whom we owe most of what we know
about the Pythagoreans, they held that the primary nature of things
was Number. Some even regarded things as being made of
numbers. Eurytus, a pupil of the Pythagorean Philolaus,
demonstrated this in graphic fashion. He used small coloured stones,
which he stuck on a wall prepared with plaster, to show that the
number of mankind was 250, and that of plants was 360 (5).
Others gave a more subtle role to number by associating an explanatory
figure with each thing (for instance, a square had the number four),
or by postulating that relationships between numbers explained
objects, in the same way as the proportions in a recipe.
(Gnomon) had been invented by Anaximander. Two
arithmetico-geometric figures illustrate the issues that particularly
interested the Pythagoreans. The Τετρακτύς
(Tetractys), used by the Pythagoreans as a symbol of membership,
is a triangle formed from the first four whole numbers: •
• •
• • •
• • • •
τετρακτύς
which add up to 10, the sacred Decad, symbol of the pairs of opposites
that give birth to the Universe.
(Gnomon), a pair of recurring figures produced by moving
regularly spaced points, enables all the geometric figures to be
represented (6).
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τετράγωνον
. . . . . .
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ἑτερόμηκες
The odd Γνωμών
generates all the odd numbers from the Monad or unity. Because it is
symmetrical, it also generates the square and figures related to it.
The even Γνωμών ,
generates the even numbers and rectangular figures. The fundamental
role of geometrical construction was part of the more general emphasis
laid by Pythagoras and his disciples on the role of memory, or
rather the act of accessing memory , or recall (ἀνάμνησις
anamnesis), in the processes leading to perfect knowledge. It
was not enough to know that everyone had a soul which had had numerous
experiences during its former lives; it was necessary to be able to
revive the memory of the past at any time. The μαθηματικοί
had to train their memory every evening by remembering all the events
of the past day. By doing this they hoped to become able to remember
the history of their souls, and perhaps to avoid it having to pass
through an appropriate rite of purification during the cycle of
rebirth (the same preoccupation is found with Empedocles).
This is summed up in the admirable words of Alcmaeon of Croton:
τους ανθρώπους διά τούτο απόλλυσθαι, ότι ου δύνανται
την αρχήν τω τέλει προσάψαι
Human beings perish because they are not able
to join their beginning to their end.
Difficult as it is to know the exact origins of geometry (and we know
how interested Thales was in this), the early
Pythagoreans were probably the first to study polygons and the regular
polyhedra. They discovered all the regular convex polyhedra: the
tetrahedron, the cube, the octahedron, the dodecahedron and the
icosahedron, and used them to explain the geometric properties of all
the objects met with in the Universe (7). Hippasus
of Metapontium was expelled from the Pythagorean school because
he betrayed the convention by revealing how the dodecahedron is
constructed, and its relation with the regular pentagon (the faces of
the dodecahedron are pentagonal). It is even said that he was drowned.
Finally, it was during the early life of the Croton school that the
notion of incommensurability was discovered (through the study of
right-angled triangles whose sides are measured in whole numbers), and
that the problem of squaring the circle was first addressed.
Music was another fundamental question in the early days of
Pythagoreanism, and it is thought that Hippasus first discovered the
rules that govern the pitch of sounds, and especially the
interdependence between intervals in music and certain numerical
relationships.
Later, the rules of Pythagorean cosmogony were codified, and summed up
in the form of a set of ten Oppositions:
περας και απειρον Limited Unlimited
περιττον και αρτιον Odd Even
εν και πληθον One Many
δεξιον και αριστερον Right Left
αρρεν και θυλη Male Female
ηρεμουν και κινουμενουν Immobile Mobile
ευθη και καμπυλον Straight Curved
φως και σκοτον Light Dark
αγαθον και κακον Good Evil
τετραγωνον και ετερομηκες Square Rectangular
The most remarkable thing about this varied list (about which
Aristotle was fairly sarcastic) is the association between the
Unlimited, the Even, and Evil (Plato later developed this further). It
implied an association between infinity and imperfection,
disproportion, or ὕβρις
(hybris). In contrast, the finite (and usually every cosmos (κόσμος,
especially our own, was conceived as closed and limited), together
with regularity, symmetry (for instance, the symmetry of polygons or
regular polyhedra) were regarded as symbols of perfection. In
appropriate combinations, this set of ten pairs of opposites should be
enough to provide an accurate representation of the whole world.
The Monads were principles of Unity, unmixedness, measure, and purity.
They were seen as points in space, with a certain breadth. Organised
in space and set in motion, they could generate all possible shapes,
first the simple geometric shapes (such as those found from the Γνωμών)
and then the others.
The texts of Archytas (a contemporary of Plato and
pupil of Philolaus) are particularly revealing. He discussed
the properties of the Monad (which divides the world into two: odd and
even, through simple self-addition of the Monad, one), defined
arithmetical, geometrical and harmonic progression (8),
and used them as the basis for the rules of music. Above all he gave
geometrical proof for a number of theorems involving the generation of
three-dimensional figures from the regular movement of the simplest
figures: the cylinder is described by a straight line following a
parallel path around a circle; the torus by a circle whose centre
moves around a larger circle; and the cone by a triangle revolving
around one of its sides.
Archytas was also among
those who demonstrated that, despite what the apparent perfection of
the Universe might lead us to believe, in fact it is infinite. To do
this, he took the Milesians’ spherical conception of the Universe and
argued thus: if I had reached the outer limit, in other words the
sphere of the fixed stars, would I, or would I not, be able to stretch
out my hand, or my staff, beyond that? And the answer is yes, at any
point on this apparent frontier. This implies that the laws of physics
are the same at all points in the Universe, and that it would be
absurd to imagine that it has any limit.
But the most revealing insight into Pythagorean cosmology probably
comes from what is left of the writings of Philolaus of Tarentium.
In this cosmogony, the Earth was no longer the centre of the world.
For Philolaus, as for his predecessors, the Universe was made
of pairs of opposites, the Limited (the Monad) and the Unlimited (the
Decad) (9). The Decad
contained in itself the intrinsic nature of the point (unity), the
straight line (two), the surface (three, since a triangle is the
simplest surface) and space (four, represented by the tetrahedron, the
first regular polyhedron and the first three-dimensional figure) (10).
But it did not stop there: the number five
generated Quality and Colour; six, the impulse of life (ψύχωσις),
seven, mind (νοῦς) health and light; and eight was the number
of love (ἔρως), friendship (φιλία), cleverness (μῆτις)
and intention (ἐπίνοια). But it was not so much the numbers
themselves as the geometrical properties that went with them that were
the basis for understanding the world (11).In
his Timaeus, Plato later took up this Pythagorean hypothesis,
and associated four of the regular solids with the four elements: the
tetrahedron with Fire, the octahedron with Air, the icosahedron with
Water and the cube with Earth. He went on to postulate that the
dodecahedron was the (limited!) figure that symbolised the Universe.
The whole world was organised around a nucleus, the central Fire, with
the Ten Celestial Bodies arranged concentrically around it (12).
From the periphery inwards, there were the
stars, the five planets (Saturn, Jupiter, Mars, Venus and Mercury),
the Sun and the Moon, and finally, between the Moon and the central
Fire was ᾽Αντιχθον ,
the Counter-Earth. This last heavenly body, which hid the centre of
the Universe from our sight, and was itself invisible (since the face
of the Earth was always turned away from it, just as the dark side of
the Moon is always invisible from Earth), was inhabited by living
beings. Philolaus divided this Universe into three main
regions, the outer region or Όλυμπος
(Olympos), which contained pure elements; the intermediary
region or Κόσμος
(Cosmos) containing the five planets, the Sun and the Moon; and
the inner region, Οὐρανός
(Ouranos), home of Earth and Counter-Earth. The Earth and its
counterpart revolved together around the central fire which he called
the Hearth, in the same direction as the Sun and the Moon but in a
different orbital plane. The Sun was a transparent lens that
concentrated the light of the fire in the Olympian Ether. The Moon was
inhabited (13) and its
day lasted for fifteen earth days, making its inhabitants fifteen
times stronger and more beautiful than humans.
The Pythagorean tradition involved not only a cycle in the life of the
soul, but also a cycle in astronomical phenomena, which could be
detected using the right arithmetic. The Great Year (the period of
these phenomena) had been the subject of scholarly conjectures, and Œnopides
of Chios (who was probably not a Pythagorean, but was certainly
well informed about the work of the sect) had calculated the length of
the Great Year as 59 years of 365 days and 22/59ths (14).
Looking at his tables again, Philolaus realised that a small
modification would give a Great Year of 93 = 729 months,
one month less than Œnopides’ year, which he thought much more
satisfactory, giving 364 and a half days per year (15).
This made it possible to endow number with
enormous power over the laws that governed the Universe.
The Pythagoreans were not only interested in cosmology. Like their
predecessors and their contemporaries, they were fascinated by
biology, which they approached via the study of acoustics and the
physics of music (for which they are still known today), and also by
the interpretation of the intrinsic properties of living beings. In
music, besides Pythagoras himself the most prominent figures were Hippasus
and Archytas, who established the relationship between the
sound produced by metal discs according to their thickness, and by
wires according to their length, and calculated the lengths that
produced the main musical intervals (octave, fifth and fourth). Philolaus
specified the notes of the scale by defining the major fifth and the
major fourth, as well as the composition of the octave (five tones and
two semi-tones), thus laying the foundation for modern music.
In biology, the ideas of the Pythagoreans were strongly marked by
their dualist thinking. Through metempsychosis, a soul could inhabit
any body, implying that there are souls in residence everywhere in the
world. Aristotle reports that the disciples of Pythagoras
thought the rays of the sun were alive and that one could actually see
the movement of souls in the shining spots that dart here and there in
sunlight, even when there is no wind. The air was full of souls,
influencing our dreams and our actions, and warning of the future
through premonitions. The Pythagoreans were also concerned with a
number of medical questions, and long before the school of Hippocrates
of Cos, it was a man from western Greece, Alcmaeon of Croton,
who created the first real medical school in Greece, around 500 BCE.
He studied the foundations of sensory perception and made numerous
anatomical observations. He is even said to have been the first to
have dared carry out a radical excision of the eyeball. But his most
fundamental contribution was to recognise the brain as the seat of all
sensations. It received signals from the ears (whose hollow structure
enabled sounds to be concentrated), from the eyes (which transmitted
light via filaments that conducted Fire to the centre of the brain),
from the nose, tongue and all the rest of the body. These various
signals were conducted along appropriate pathways and arranged by the
brain into a harmonious structure, and it was this ability to bring
sensations together to form a whole that made the brain the seat of
thought. Memory, and faith (not based on reason), were made up of
stored perceptions, and when memories were stabilised they formed
knowledge (16). Alcmaeon
also distinguished intelligence as the ability to organise sensations,
and attributed it to humans alone, while other living beings had only
sensory perception and were not capable of understanding.
The fundamental role he gave to the brain led him to study
embryogenesis in respect of the formation of the head, which he
thought was formed first. His experimental material was birds’ eggs,
and he believed in the female origin of all offspring, in contrast to
the generally accepted opinion which attributed it to the male semen.
For Alcmaeon (as
reported in a fragment of the doxographical tradition), all
tissue was made up of opposite qualities (more of them than in the
most orthodox Pythagorean tradition):
Alcmaeon said that the equality (ἰσονομία
(wet, dry, cold, hot, bitter, sweet, etc.)
maintains health but that monarchy among them produces disease. [...]
Alcmaeon thought that disease arose because of an excess of heat or cold, which in turn arose because of an excess
or deficiency in nutrition. Disease is said to arise in the blood, the marrow or the brain.
It can also be caused by external factors such as the water,
the locality, toil, or violence.
Many other Pythagoreans were interested in biology, but only a few
slim fragments of their thought have survived. Menestor used
the theory of opposites to develop a study of botany, including
germination, fructification and the role of the plant’s environment.
And just as Empedocles did for the animal kingdom, he used the
idea of a process of dissolution (σῆψις ,
sepsis) to explain the taste of vegetables. Hippo tried
to explain biological cycles in terms of the number 7, and Philolaus,
unlike Alcmaeon, explained disease as resulting from an excess
of certain fluids found throughout the body, such as bile. Finally, Archytas
attempted to explain biological form in terms of intrinsic
mathematical properties (17).Why
were the parts of vegetables and animals usually rounded, rather than
triangular or polygonal? Archytas explained this through
natural movement, in other words movement that respects all
constraints equally, as was observed, for instance, in the uniform
rotation of a triangle around one of its sides, describing a cone.
Following the constraints of symmetry generated rounded surfaces. This
idea may have been suggested by the way trees grow in concentric
rings. In the same way, the human trunk was thought to be made up of
concentric layers. We can note here an analogy between this imagery
and the structure of the Pythagorean Universe.
The disciples of Pythagoras spoke in the name of a truth
expressed through Number. Other contemporaries of theirs claimed to
represent the truth in other ways, and we shall look at them next.
1: Matter and soul are absolute opposites,
as with the dualistic Asian gods. (back to text)
2: This was proven by the memory of past
events that some people have, even of events before their birth. Like
all properties of the Universe, this was subject to numerical or
arithmetical laws. Some even made the calculation: the transmigration
of souls took six years. (back to text)
3: In this ideal upbringing, all the phases
of human development from conception to adulthood are considered to be
crucially important. While stockbreeders pay enormous attention to the
way their animals are reared, people seem to give little importance to
such problems in the case of their own offspring. Whereas in fact they
should pay constant attention to the way their children are brought up
. For instance, sexual relations should begin late (after the age of
20 for young men) and should go together with the harmonious
development of the body and intellectual knowledge. Boys and girls
should learn self-control through a life of hard work, avoiding all
excess. And since all passion is incompatible with acquiring supreme
knowledge, it goes without saying that the sole aim of sex should be
procreation, and should be restricted to those who are capable of
bringing up their children, for aimless procreation without a proper
upbringing for the resulting children is the origin of Evil. Children
should be taught the rules of best behaviour very early, together with
the first rudiments of knowledge (reading and writing); young men
should learn the laws of the State; adults should undertake active
work and public service; and the old, who have theoretical knowledge
and good judgement, are able to give good advice. However the time at
which these various dispositions appear should not be considered as
fixed for all, but should in reality be correlated with the harmonious
development of the personality. Adolescence is the most difficult
period, and requires a great deal of attention, to correct the
mistakes of childhood (which are not faults!) and those of maturity,
especially passions that are too strong. Violence must never be used,
and even verbal violence must be avoided. In any case, good behaviour
is the ideal of love, beauty and knowledge – the only possessions
worth having, despite what the thoughtless believe. (back
to text)
4:...είναι διο πολλοί φασι των σοφών οι τας
αληθινές φύσεις εν τοίς ρυθμούς και τους μέλεσιν οργής μεν αρμονίαν
είναι την ψυχήν, οι δ' έχειν αρμονίαν
Some believe soul and harmony to be the same thing, for others harmony
(among other things) is a property of the soul. (back to
text)
5: This may seem very primitive to us, but
is it really so far distant from the recent enthusiasm for crude
analogies that mask reality and try to explain everything in terms of
"catastrophes", or even more crudely, "dissipative structures"? (back
to text)
6: The main characteristic of Γνωμών
(the Gnomon) seems to be the presence of a right angle (the same issue
that led to Pythagoras’ theorem), together with the recurrence
that enables the same form to be preserved through successive
homothetic transformations, adding a "set square" shape to each new
figure. Given these factors, Γνωμών was probably an
instrument, perhaps a set square, used for plotting land and in
architecture. (back to text)
7: This explanatory analogy was taken up at
length by Plato in Timaeus. It has also been used as a theme
in modern analogies such as those developed by René Thom in his
catastrophe theory. (back to text)
8: The first two are well known. A harmonic
progression would be 1, 1/3 , 1/5 , 1/7 , and so on. The harmonic mean
takes its name from the fact that it produces the three main musical
intervals formed by notes produced by strings whose lengths are in a
ratio of 6:3 = 2:1 (octave); 6:4 = 3:2 (fifth); 4:3 (fourth). (back
to text)
9: Philolaus chose not the Dyad but
the Decad as the principle that produced the Unlimited. This choice
reflected a very long eastern tradition that had led the sect to use
the Tetractys as its symbol, and also to conceive the world in terms
of ten pairs of opposites. Philolaus justified this choice
through an anthropological argument, as reported in fragment DK A13,
commenting that not only the Greeks but all nations gave a special
role to the figure 10 in their counting systems. (back
to text)
10: Further evidence of the major
importance of the Decad: a tetrahedron has four faces and six sides,
adding up to ten. (back to text)
11: This makes geometry the queen of
sciences:
γεωμετρια ἀρχη και μητροπολις ... των ἀλλων [μαθηματων]. (back
to text)
12: Note the intense interest in the
possible discovery of a tenth planet . (back to text)
13: Only 125 years ago, in his Popular
Astronomy (Astronomie Populaire) Camille Flammarion
imagined beings living on the moon. And if Orson Welles is to be
believed, Martians landed on Earth in 1938. (back to
text)
14: A solar year is now given as 365 days
5 hours 48 minutes and 46 seconds. (back to text)
15: This kind of ‘improvement’, which
consists in distorting reality to make it fit a theoretical model,
which is seen as "intuitive" or "revealed", is still a powerful
driving force. We will see endless examples of it. (back
to text)
16: “Then I will tell you, said Socrates.
When I was young, Cebes, I had a prodigious desire to know that
department of philosophy which is called Natural Science; this
appeared to me to have lofty aims, as being the science which has to
do with the causes of things, and which teaches why a thing is, and is
created and destroyed; and I was always agitating myself with the
consideration of such questions as these: Is the growth of animals the
result of some decay which the hot and cold principle contracts, as
some have said? Is the blood the element with which we think, or the
air, or the fire? or perhaps nothing of this sort – but the brain (ἐγκεφαλον)
may be the originating power of the perceptions of hearing and sight
and smell, and memory (μνήμη) and opinion (δόξα) may
come from them, and science may be based on memory and opinion when no
longer in motion, but at rest.” Plato, Phaedo 96 a-b; translated by
Benjamin Jowett (1871) (back to text)
17: Here we come back to a question that
has always haunted the best mathematicians – and Archytas was
certainly a great mathematician – as can be seen today in the work of
René Thom or his imitators. Diels, H. and W. Kranz, 1952, Die
Fragmente der Vorsokratiker (in three volumes), 6th edition,
Dublin and Zürich: Weidmann. ((back to text)
