Pythagoras: His Life and Teaching, a Compendium of Classical Sources (16 page)

Part Three

The Doctrine of Pythagoras

Section I. Mathematics
Section II. Philosophy
Section III. Symbols

Among the earliest coins of the Greek colonists in Southern Italy, this silver stater of Croton dates to the late 6th century
B.C.
It shows on its obverse the tripod of Apollo and the abbreviated name of the city, and on its reverse the incuse impression of that same design, less the inscription.
Photo courtesy of Numismatica Ars Classica

Section I. Mathematics

T
HE
M
ATHEMATICAL
S
CIENCES
P
REPARATIVE TO
P
HILOSOPHY

T
he mind being purified by discipline ought to be applied to things that are beneficial.
³¹⁹
These Pythagoras procured by some contrived ways, bringing it by degrees to the contemplation of eternal incorporeal things which are ever in the same state. He began this in an orderly manner from the most minute—lest by the suddenness of the change, it should be diverted and withdraw itself through its so great and long habit of perverse mental nutriment.

To this end, he first used the mathematical sciences, and those speculations which are intermediate between corporeals and incorporeals (for they have a threefold dimension like bodies, but they are impassible like Incorporeals) as degrees of preparation to the contemplation of the things that are. They divert, by an artificial reason, the eyes of the mind from corporeal things (which never are permanent in the same manner and estate), never so little to a desire of aliment. By means whereof, introducing the contemplation of things that are, he rendered men truly happy. This is the use he made of the mathematical sciences.

Hence it was that Justin Martyr, applying himself to a Pythagorean eminently learned, desirous to be his disciple, the teacher demanded whether he were versed in music, astronomy, and geometry. “Or do you think,” says he, “you may be able to understand anything that pertains to beatitude without having first learned these, which abstract the soul from sensibles, preparing and adapting her for her intelligibles? Can you without these contemplate what is honest and what is good?” Thus, after a long commendation of these sciences, he dismissed him, for that he had confessed himself ignorant of them.
³²⁰

Mathematics, Its Name and Parts

These sciences were first termed
[“mathematics”] by Pythagoras, upon consideration that all mental discipline is
Reminiscence. This comes not extrinsically to souls, as the phantasies, which are formed by sensible objects in the Imaginal world. Nor are they an advantageous accepted knowledge, like that which is placed in opinion. But it is excited from phenomena, and perfected intrinsically by the cogitation converted into itself.
³²¹

The whole science of mathematics, the Pythagoreans divided into four parts—attributing one to multitude, another to magnitude, and subdividing each of these into two.
³²²
For multitude either subsists by itself, or is considered with respect to another. Magnitude either stands still, or is moved. Arithmetic contemplates multitude in self; music with respect to another; geometry, unmovable magnitude; and trigonometry, moveable.

These sciences consider not multitude and magnitude simply,
³²³
but in each of these, that which is determinate. For sciences consider this abstracted from infinite; that they may not in vain attempt in each of these that which is infinite.
³²⁴
When therefore the wise persons say thus, we conceive it is not to be understood of that multitude which is in the sensible things themselves, nor of that magnitude which we perceive in bodies. For the contemplation of these, I think, pertains to Physic, not to mathematic. But the Maker of all things took union and division, identity and otherness, and station and motion, to complete the soul—and framed it of these kinds, as Timaeus teaches.

We must conceive therefore that the intellect—consisting according to the diversity thereof, and the division of proportions and multitude, and knowing itself to be both one and many—proposes number to itself; and produces them and the arithmetical knowledge of them. According to the union of multitude and communication with itself, and conjunction, it brings to itself music. For which reason, arithmetic excels music in antiquity, the soul itself being first divided by the Maker, then collected by proportions. And again, establishing the operation within itself according to its station, it produces geometry out of itself, and one figure, and the principles of all figures. But according to its motion, trigonometry: for she is moved by circles. It consists always in the same manner according to the causes of those circles, the straight and the circular. And for this
reason, likewise geometry is precedent to Trigonometry, as station is to motion.

But forasmuch as the soul produced these sciences—not looking on the excitation of Ideas, which is of infinite power, but upon the boundary of that which is limited in their several kinds
³²⁵
—therefore they say that they take infinite from multitude and magnitude, and are conversant only about finite. For the mind has placed in herself all principles both of multitude and magnitude; because being wholly of like parts within herself, and being one and indivisible, and again divisible and producing the world of ideas; it does share essential finiteness and infiniteness with the things which it does understand. But it understands according to that which is finite in them, and not according to the infiniteness of its life. This is the opinion of the Pythagoreans, and their division of the four sciences. Hitherto Proclus.

A
RITHMETIC

O
f these four methods, which is that which ought necessarily to be learned the first (viz. that which is by nature pre-existent to the rest and chief, being as it were principle and root, and mother of the rest)?
³²⁶
The answer is Arithemetic. Not only because it is pre-existent before the rest in the Intellect of the efficient God, as an ornative and exemplary reason—according to which the Maker of the Universe caused all things to be made out of matter to its proper end, as after a
[“a thing pricked,” i.e. “traced out beforehand”]† and archetypal pattern. But also because being naturally first generated,
³²⁷
it together takes away the rest with itself, but is not taken away with them. Thus animal is first in nature before man. For taking away animal, we take away man; but not in taking away man, do we take away animal. (Of this Nicomachus discourses more largely.)

As concerning Arithmetic, Timaeus affirms that Pythagoras addicted himself chiefly to it.
³²⁸
Stobaeus says that he esteemed it above all others, and brought it to light, reducing it from the use of trading.
³²⁹
Hence Isidore and others style him the inventor of arithmetic,
³³⁰
affirming he was the first who wrote upon this subject amongst the Greeks, which was afterwards more copiously composed by Nicomachus. He studied this science exceedingly, and so much did he prefer it above all the rest, that he conceived the ultimate good of man to consist in the most exact science of numbers.
³³¹

CHAPTER 1

N
UMBER
, I
TS
K
INDS
: T
HE
F
IRST
K
IND
, I
NTELLECTUAL IN THE
D
IVINE
M
IND

N
umber is of two kinds, the Intellectual (or immaterial) and the Sciential.
³³²
The intellectual is that eternal substance of number which Pythagoras, in his discourse concerning the gods, asserted to be the principle most providential of all Heaven and Earth and the nature that is between them.
³³³
Moreover, it is the root of divine beings, and of gods, and of daemons. This is that which he termed the principle, fountain, and root of all things. He defined it to be that which before all things exists in the Divine Mind;
³³⁴
from which and out of which all things are digested into order, and remain numbered by an indissoluble series.
³³⁵

For all things which are ordered in the world by nature—according to an artificial course in part and in whole—appear to be distinguished and adorned by Providence and the All-creating Mind according to number. The exemplar is established by applying (as the reason of the principle before the impression of things) the number pre-existent in the intellect of God, Maker of the world. This only in intellectual, and wholly immaterial, really a substance according to which as being the most exact artificial reason, all things are perfected—Time, Heaven, Motion, the Stars—and their various revolutions.

CHAPTER 2

T
HE
O
THER
K
IND OF
N
UMBER
: S
CIENTIAL, ITS
P
RINCIPLES

S
ciential number is that which Pythagoras defines as the extension and production into act of the seminal reasons which are in the Monad, or a heap of Monads, or a progression of multitude beginning from Monad, and a regression ending in Monad.
³³⁶

The Pythagoreans affirmed the expositive terms whereby even and odd numbers are understood to be the principles of Sciential numbers.
³³⁷
For example, of three insensible things, the Triad; of four Insensibles, the Tetrad; and so of other numbers.

They make a difference between the Monad and One, conceiving the Monad to be that which exists in intellectuals. One exists in numbers.
³³⁸
(Or as Moderatus expresses it, Monad amongst numbers, One amongst things numbered, one body being divisible into infinite. Thus numbers and things numbered differ, as incorporeals and bodies.
³³⁹
)

In like manner is two amongst numbers. The Duad is indeterminate. Monad is taken according to equality and measure, Duad according to excess and defect. Mean and measure cannot admit more and less, but excess and defect (seeing that they proceed to infinite) admit it. Therefore they call the Duad indeterminate, holding number to be infinite.
³⁴⁰
Not that number which is separate and incorporeal, but that which is not separate from sensible things.
³⁴¹

CHAPTER 3

T
HE
T
WO
K
INDS OF
S
CIENTIAL
N
UMBER
, O
DD AND
E
VEN

O
f sciential numbers, Pythagoras asserted two orders: one bounded, Odd; the other infinite, Even.
³⁴²
Even number (according to the Pythagorean definition) is that which at once admits division into the greatest and the least; into the greatest magnitudes (for halves are the greatest parts); the least in multitude (for two is the least number) according to the natural opposition of these two kinds.
³⁴³
Odd is that which cannot suffer this, but is cut into two unequals.

Herein the Pythagoreans differ from the Platonists, in that they hold not all number to be infinite, but only the Even. For even number is the cause of section into equal parts, which is infinite, and by its proper nature generates infinity in those things in which it exists.
³⁴⁴
But it is limited by the Odd; for that being applied to the Even, hinders its dissection into two equal parts.

Odd number is said to have been found by Pythagoras to be of Masculine virtue,
³⁴⁵
and proper to the Celestial Gods (to whom they sacrificed always of that number),
³⁴⁶
and to be full and perfect.
³⁴⁷
Even number is indigent, and imperfect, and Female, and proper to the subterranean deities, to whom they sacrificed Even things.
³⁴⁸

Moreover, whatever is generated of Odd number is male, whatsoever of Even is female; for Even number is subject to section and passion. Odd is void of both, and is efficacious. Wherefore they call one the Male, the other the Female.
³⁴⁹
A number, which arises out of the power and multiplication of Even and Odd, is called
Hermaphrodite.
³⁵⁰

This opinion Pythagoras seems to have derived from Zaratas, his Master. He called Duad the Mother of number, Monad the Father; and therefore they said that those numbers which resemble Monad (viz. the Odd), are the best.
³⁵¹

Odd Numbers they called Gnomons, because being added to Squares, they keep the same figures; so Gnomons do in Geometry.
³⁵²