At the same time, however, and even earlier the so-called Footnote 1Aristotle seems to have regarded Pythagoras as a legendary person. Pythagoreans applied themselves to mathematics, and were the first to develop this science Footnote 2Pythagoras himself (fl. B.C.) is said by Aristoxenus (ap. Stobaeus 1.20.1) to have been the first to make a theoretical study of arithmetic. ; and through studying it they came to believe that its principles are the principles of everything.
And since numbers are by nature first among these principles, and they fancied that they could detect in numbers, to a greater extent than in fire and earth and water, many analogues Footnote 3Cf. Aristot. Met. 14.6ff.. of what is and comes into being—such and such a property of number being justice , Footnote 4Apparently (cf. infra, Aristot. Met. 1.17) they identified these not only with properties of number but with numbers themselves. Thus justice (properly=squareness)=4, the first square number; soul or mind=1, opportunity=7 (Alexander). and such and such soul or mind , another opportunity , and similarly, more or less, with all the rest—and since they saw further that the properties and ratios of the musical scales are based on numbers, Footnote 5Pythagoras himself is credited with having discovered the ratios of the octave (2 : 1), the fifth (3 : 2) and the fourth (4 : 3). Burnet, E.G.P. 51. and since it seemed clear that all other things have their whole nature modelled upon numbers, and that numbers are the ultimate things in the whole physical universe, they assumed the elements of numbers to be the elements of everything, and the whole universe to be a proportion Footnote 6Or harmony.
Cf. Aristot. De Caelo 2.9, and E.G.P. 152. or number. Whatever analogues to the processes and parts of the heavens and to the whole order of the universe they could exhibit in numbers and proportions, these they collected and correlated;
and if there was any deficiency anywhere, they made haste to supply it, in order to make their system a connected whole. For example, since the decad is considered to be a complete thing and to comprise the whole essential nature of the numerical system, they assert that the bodies which revolve in the heavens are ten; and there being only nine Footnote 7Earth, sun, moon, five planets, and the sphere of the fixed stars. that are visible, they make the antichthon
Footnote 8i.e. counter-earth
; a planet revolving round the central fire
in such a way as to be always in opposition to the earth. the tenth.
We have treated this subject in greater detail elsewhere Footnote 9In the lost work On the Pythagoreans; but cf. Aristot. De Caelo 2.13. ; but the object of our present review is to discover from these thinkers too what causes they assume and how these coincide with our list of causes.
Well, it is obvious that these thinkers too consider number to be a first principle, both as the material Footnote 10See Burnet, E.G.P 143-146. of things and as constituting their properties and states. Footnote 11i.e., as a formal principle. Cf. Ross ad loc. The elements of number, according to them, are the Even and the Odd. Of these the former is limited and the latter unlimited; Unity consists of both (since it is both odd and even) Footnote 12Either because by addition it makes odd numbers even and even odd (Alexander, Theo Smyrnaeus) or because it was regarded as the principle of both odd and even numbers (Heath). ; number is derived from Unity; and numbers, as we have said, compose the whole sensible universe.
Others Footnote 13Zeller attributes the authorship of this theory to Philolaus. of this same school hold that there are ten principles, which they enunciate in a series of corresponding pairs: (1.) Limit and the Unlimited; (2.) Odd and Even; (3.) Unity and Plurality; (4.) Right and Left; (5.) Male and Female; (6.) Rest and Motion; (7.) Straight and Crooked; (8.) Light and Darkness; (9.) Good and Evil; (10.) Square and Oblong.
Apparently Alcmaeon of Croton speculated along the same lines, and either he derived the theory from them or they from him; for [Alcmaeon was contemporary with the old age of Pythagoras, and] Footnote 14This statement is probably true, but a later addition. his doctrines were very similar to theirs. Footnote 15He was generally regarded as a Pythagorean. He says that the majority of things in the world of men are in pairs; but the contraries which he mentions are not, as in the case of the Pythagoreans, carefully defined, but are taken at random, e.g. white and black, sweet and bitter, good and bad, great and small.
Thus Alcmaeon only threw out vague hints with regard to the other instances of contrariety, but the Pythagoreans pronounced how many and what the contraries are. Thus from both these authorities Footnote 16The section of Pythagoreans mentioned in 6, and Alcmaeon. we can gather thus much, that the contraries are first principles of things; and from the former, how many and what the contraries are.
How these can be referred to our list of causes is not definitely expressed by them, but they appear to reckon their elements as material; for they say that these are the original constituents of which Being is fashioned and composed.
From this survey we can sufficiently understand the meaning of those ancients who taught that the elements of the natural world are a plurality. Others, however, theorized about the universe as though it were a single entity; but their doctrines are not all alike either in point of soundness or in respect of conformity with the facts of nature.
For the purposes of our present inquiry an account of their teaching is quite irrelevant, since they do not, while assuming a unity, at the same time make out that Being is generated from the unity as from matter, as do some physicists, but give a different explanation; for the physicists assume motion also, at any rate when explaining the generation of the universe; but these thinkers hold that it is immovable. Nevertheless thus much is pertinent to our present inquiry.
It appears that Parmenides conceived of the Unity as one in definition, Footnote 17His argument was Everything that is is one, if
(πάντα ἕν, εἰ τὸ ὂν ἓν σημαίνει, Aristot. Phys. 187a 1); but he probably believed, no less than Melissus, in the material unity of reality. Cf. Melissus Fr. 8 (Diels). It has been suggested, however (by the Rev. C. F. Angus), that he was simply trying to convey in figurative language a conception of absolute existence. but Melissus Footnote 18Of Samos; defeated the Athenian fleet in B.C. as materially one. Hence the former says that it is finite, Footnote 19Melissus Fr. 8, ll. 32-3, 42-3. and the latter that it is infinite. Footnote 20Melissus Fr. 3. But Xenophanes, Footnote 21Of Colophon, b. 565 (?) B.C. Criticized and ridiculed most of the views of his day, especially the anthropomorphic conception of the gods. Burnet, E.G.P. 55 ff., esp. 61-62. Cf. Xenophanes Fr. 23 (Diels). the first exponent of the Unity (for Parmenides is said to have been his disciple), gave no definite teaching, nor does he seem to have grasped either of these conceptions of unity; but regarding the whole material universe he stated that the Unity is God.what is
has one meaning
This school then, as we have said, may be disregarded for the purposes of our present inquiry; two of them, Xenophanes and Melissus, may be completely ignored, as being somewhat too crude in their views. Parmenides, however, seems to speak with rather more insight. For holding as he does that Not-being, as contrasted with Being, is nothing, he necessarily supposes that Being is one and that there is nothing else (we have discussed this point in greater detail in the Physics Footnote 22Aristot. Phys. 1.3 ); but being compelled to accord with phenomena, and assuming that Being is one in definition but many in respect of sensation, he posits in his turn two causes, i.e. two first principles, Hot and Cold; or in other words, Fire and Earth. Of these he ranks Hot under Being and the other under Not-being. Footnote 23Cf. note on Aristot. Met. 3.13.
From the account just given, and from a consideration of those thinkers who have already debated this question, we have acquired the following information. From the earliest philosophers we have learned that the first principle is corporeal (since water and fire and the like are bodies); some of them assume one and others more than one corporeal principle, but both parties agree in making these principles material. Others assume in addition to this cause the source of motion, which some hold to be one and others two.
Thus down to and apart from the Italian Footnote 24The Pythagoreans; so called because Pythagoras founded his society at Croton. philosophers the other thinkers have expressed themselves vaguely on the subject, except that, as we have said, they actually employ two causes, and one of these—the source of motion —some regard as one and others as two. The Pythagoreans, while they likewise spoke of two principles, made this further addition, which is peculiar to them: they believed, not that the Limited and the Unlimited are separate entities, like fire or water or some other such thing, but that the Unlimited itself and the One itself are the essence of those things of which they are predicated, and hence that number is the essence of all things.
Such is the nature of their pronouncements on this subject. They also began to discuss and define the what
of things; but their procedure was far too simple. They defined superficially, and supposed that the essence of a thing is that to which the term under consideration first applies—e.g. as if it were to be thought that double
and 2
are the same, because 2 is the first number which is double another.
But presumably to be double a number
is not the same as to be the number 2.
Otherwise, one thing will be many—a consequence which actually followed in their system. Footnote 25i.e., the same number might be the first to which each of several definitions applied; then that number would be each of the concepts so defined. This much, then, can be learned from other and earlier schools of thought.