Pythagoras: His Life and Teaching, a Compendium of Classical Sources

by Wasserman, James

  For it is not lawful to bestow on everyone that which was acquired with so much labor, nor to reveal the mysteries of the Eleusian goddesses to profane persons. For they who do both these are alike unjust and irreligious. It is good to consider without ourselves, how much time was employed in taking away the spots that were in our beasts, that after five years we might be made capable of his discourses. For as dyers first wash and wring out the clothes they intend to dye—that they may take the dye so as that it can never be washed out or taken away—in like manner the divine prepared those who were inclined to philosophy, lest he might be deceived by those of whom he hoped that they would prove good and honest.

  For he used no adulterate learning, nor the nets wherewith many of the Sophists entangle the young men; but he was skillful in things divine and human. Whereas they, under the pretence of his doctrine, do many strange things, inveigling the young men unbeseemingly, and as they meet them, whereby they render their auditors rough and rash. For they infuse free theorems and discourse into manners that are not free but disordered. As if into a deep well full of dirt and mire we should put clear transparent water. It troubles the dirt and spoils the water. The same is it as to those who teach and are taught; for about the minds and hearts of such as are not initiated, there grow thick and tall coverts which darken all modesty and meekness and reason, hindering it from increasing there.

  Hence spring all kinds of ills, growing up and hindering the reason, and not suffering it to look out. I will first name their mothers, Intemperance and Avarice, both exceedingly fruitful. From intemperance spring up unlawful marriages, lust, drunkenness, perdition, unnatural pleasures, and certain vehement appetites leading to death and ruin. For some have been so violently carried away with pleasures that they have not refrained from their own mothers and daughters; but violating the commonwealth and the laws, tyrannically imprison men, and carrying about their Jalles317 (or stocks) violently, hurry them to destruction.

  From avarice proceed rapines, thefts, parricides, sacrileges, poisonings, and whatsoever is allied to these. It behooves therefore first to cut away the matter wherein these vices are bred with fire and sword, and all arts of discipline—purifying and freeing the reason from these evils—and then to plant something that is good in it.

  Thus Lysis. Neither is that expression, “If you are not changed, you are dead to me,” to be understood simply. For this, Hipparchus, because he communicated and publicly set forth by writing the Pythagorean doctrines, was expelled from the school, and a tomb was made for him as if he were dead (according to the custom formerly mentioned).318 So strict were the Pythagoreans in observance of this secrecy.

  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

  THE MATHEMATICAL SCIENCES PREPARATIVE TO PHILOSOPHY

  The 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 kinds325—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.

  ARITHMETIC

  Of th
ese 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

  NUMBER, ITS KINDS: THE FIRST KIND, INTELLECTUAL IN THE DIVINE MIND

  Number 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

  THE OTHER KIND OF NUMBER: SCIENTIAL, ITS PRINCIPLES

  Sciential 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

  THE TWO KINDS OF SCIENTIAL NUMBER, ODD AND EVEN

  Of 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

  CHAPTER 4

  SYMBOLIC NUMBERS

  The Pythagoreans, using the mathematical sciences as degrees of preparations to the contemplations of the things that are, were studiously addicted to the business of numbers for this reason. (So says Moderatus of Gades, who learnedly comprised their opinions in eleven books.353)

  Seeing they could not clearly explain the first forms and principles in discourse (those being the most difficult to understand and express), they had recourse to numbers for the better explication of their doctrine, imitating geometricians and such as teach to read. For as these, going about to explain letters and their powers, recur to marks—saying that these are, as it were, the first elements of learning—nevertheless, afterwards they tell us that they are not the elements, but that the true elements are known by them. And as the geometricians, not being able to express incorporeal forms in words, have recourse to the description of figures saying, “This is a triangle”; yet not meaning that this which falls under the sight is a triangle, but that which has the same figure and which is by the help thereof, and represents the knowledge of a triangle to the mind.

  The same did the Pythagoreans in the first reasons and forms. For seeing they could not in words express incorporeal forms and first principles, they had recourse to demonstration by numbers. And thus they called the reason of unity and identity, and equality, and the cause of amicable conspiration, and of sympathy, and of the conservation of the universe which continues according to the same, and in the same manner, ONE. For the One which is in particulars, is united to the parts and conspiring by participation of the first cause.

  But the two-fold reason of diversity and inequality, and of every thing that is divisible and in mutation, and exists sometimes one way, sometimes another, they called DUAD. For the nature of the Duad in particular things is such.

  These reasons are not only according to the Pythagoreans, and not acknowledged by others; but we see that other philosophers also have left certain unitive powers, which comprise all things in the Universe. And amongst them there are certain reasons for equality, dissimilitude, and diversity. Now these reasons—that the way of teaching might be more perspicuous—he called by the names of Monad and Duad. But it is all one amongst them if it be call
ed biform, or aequaliform, or diversiform.

  The same reason is in other numbers, for everyone is ranked according to some powers. In the nature of things exists something which has beginning, middle and end. To such a form and nature they attributed the number Three, saying that whatsoever has a middle is triform; so they called every perfect thing. And if anything be perfect, they affirm it makes use of this principle and is adorned according to it; which, since they could not name otherwise, they made use of the term TRIAD to express it. And when they endeavor to bring us to the knowledge thereof, they lead us to it by the form of this triad. The same in other numbers.

  These, therefore, are the reasons according to which the aforesaid numbers were placed. But these that follow are comprehended under one form and power, which they call Decad quasi Dechad.354 Wherefore they say that ten is a perfect number, even the most perfect of all numbers; comprehending in itself all difference of numbers, all reasons, species, and proportions. For if the nature of the universe be defined according to the reasons and proportions of numbers—and that which is produced, and increased, and perfected, proceeds according to the reasons of numbers—and the Decad comprehends every reason of number, and every proportion, and all species: Why should not nature itself be termed by the name of Ten, the most perfect number? Hitherto Moderatus.

  Thus, from the symbolical use of numbers proceeded a multiplex variety of names, attributed to them by Pythagoras and his followers. Of which we shall now speak more particularly, beginning with the Monad.