Nikolaos Makris (nmakri@essex.ac.uk)
9 Nov 1995 20:18:36 GMT


A younger contemporary of Anaximenes, who spent his early life in the island of
Samos, developed the Ionian ideas in a new and most far-reaching way. Pythagoras,
experimenting with musical tones, discovered that the notes and harmonies of the scale
depended upon the numerical proportions borne to one another by the lengths of the
string of the monochord. This seems to have impressed upon Pythagoras the importance of
the numerical aspect of things, as opposed to what things were made of. Consequently,
he became the founder of theoretical mathematics, advancing beyond the empirical
discoveries of certain numerical relationships already made by the Egyptians and
We know very little about Pythagoras himself, but we do know that he was the head of
a great scientific and philosophical movement which spread all over Hellas (Greece) and
became a sort of religious sect, with "lodges" and schools in many places. We need not
go into the precise details of the various doctrines that the school developed. It will
be sufficient to list the more important ideas that emerged.
Pythagoras is credited with the dictum that "all things are numbers" and, as has been
said, the numerical aspect of things was the main interest of the school. But at that
early date there was no systematic numerical notation and numbers were represented by
dots or pebbles in the sand. The Pythagoreans became interested in the spatial
arrangemens of these dots and classified the numbers in accordance with them. Thus the
odd numbers were

square: ____________
* * * * |
_________ |
* * * |* |
_____ | |
* *| * |* |
__ | | |
*| *| * |* |

the even, oblong: ______________
* * * * *|
___________ |
* * * *| *|
________ | |
* * *| *| *|
_____ | | |
* *| *| *| *|

Triangular numbers *

had special properties, giving a sort of visual formula for obtaining the sum of
consecutive integers. Consequently, the numbers were called "schemata" (figures) or
"eidi" (forms). What is chiefly noticeable about these forms are the differences in
qualitative character that go along with the quantitative differences (squareness,
trianglarity, and the like); Pythagoras had found a concomitance between the
qualitative differences of sound and the quantitative differences of the sounding
objects. Next, the
geometrical shapes were seen to possess special geometrical properties (for example,
the special relations between the sides of a right angled triangle). Thus a solution of
the problem set by the Ionians was suggested: What accounts for the qualitative
differences of things? Not their matter (what they are made of), but the number or form
that makes them specifically what they are.
Another important point to notice is that the mathematical concepts represented by
the forms are unchanging and eternal. Squareness does not change into circularity or
circularity into triangularity, although material things may be changed from one of
these forms to another by modification of the quantities of their constituents. The
square is an eternal idea and so is the circle. Numbers also are eternal, each being
peculiarly and unchangeably itself, with its own special numerical properties. Likewise
qualitative forms are insusceptible of change: Middle C, redness, and the like. The
material things assume or display these qualities which are the same wherever they are
met, and the things change merely by changing from one to another in consequence of
alterations in the quantity (and ipso facto in the arrangement) of their matter.
With this theory of numbers the Pythagoreans combined various doctrines which they
had inherited from the Ionians. They made great play with the notion of opposites
derived from Anaximander, extending the number much beyond four to constitute a list of
ten pairs of categories ranged under one main opposition-that between the Unlimited (or
indefinite) and the Limit. Here, too, we see the influence of Anaximander's theory of
the Boundless, which was indefinite as to quality. For the Pythagoreans, definitive
qualities within the Unlimited were produced by the imposition upon it of number or
form,or precise limitation. In this list of opposites, the third opposition was that of
Many and One aligned under Unlimited and Limit. Thus the antitheses of matter and form
became associated with that between the Unlimited and the Limit as well as with the
Many and One, where the unlimited was thought of as indefinite and unintelligible and
the limit as that principle (or logos) that made things intelligible by delimitation.
The intelligible principle was the one, and the unintelligible, the unordered manifold.
These ideas were to bear fruit in later philosophy, especially in the work of Plato and
Pythagoreanism was not only a scientific school but also a religious sect, its
religious beliefs having much in common with the Orphic religion of Hellas (Greece).
The main religious tenet was belief in the divinity of the human soul. The Pythagoreans
believed that the soul was a divine and immortal being, imprisoned in the material body
by which it was polluted and from which it must be freed by purification. Purification
depended largely upon the individual's conduct and especially upon his observing a code
of prescribed rules. If the soul was not purified, it passed, on the death of the body,
into another body, possibly of some other species of animal, but if purification could
be achieved, the soul was released altogether from its connection with the body and
returned to the abode of the gods.
The Orphic version of this cult included elaborate purification rites involving the
use of music to induce an emotional state of ecstasy through which purification was
thought to be attained. The Pythagoreans seem to have refined the earlier practices.
Possibly they considered the music the most important factor and its mathematical
aspect as having magical properties. Eventuallly they came to think of the study of the
mathematical aspect as affording the proper process of purification. For it seemed to
the Pythagoreans that such intellectual activity was the truest and purest activity of
the soul, so that devotion to it would release and withdraw the soul from the body
altogether and thus enable it to attain its true immortality. Mathematics for the
Hellenes (Greeks) was synonymous with learning and the Pythagoreans included in it all
knowledge and wisdom. Thus the love of wisdom (philosophia) and the way of philosophy
came to be regarded as the true means of attaining purification of the soul and
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