The problem of physical philosophy, as then conceived, was to find some primordial and fundamental nature, by and out of which the sensible universe was built up and produced; something which co-existed always underlying it, supplying fresh matter and force for generation of successive products. The hypotheses of Thales, Anaximander, and Anaximenes, to solve this problem, have been already noticed: Pythagoras solved it by saying, That the essence of things consisted in Number. By this he did not mean simply that all things were numerable, or that number belonged to them as a predicate. Numbers were not merely predicates inseparable from subjects, but subjects in themselves: substances or magnitudes, endowed with active force, and establishing the fundamental essences or types according to which things were constituted. About water,[25] air, or fire, Pythagoras said nothing.[26] He conceived that sensible phenomena had greater resemblance to numbers than to any one of these substrata assigned by the Ionic philosophers. Number was (in his doctrine) the self-existent reality — the fundamental material and in-dwelling force pervading the universe. Numbers were not separate from things[27] (like the Platonic Ideas), but fundamenta of things — their essences or determining principles: they were moreover conceived as having magnitude and active force.[28] In the movements of the celestial bodies, in works of human art, in musical harmony — measure and number are the producing and directing agencies. According to the Pythagorean Philolaus, “the Dekad, the full and perfect number, was of supreme and universal efficacy as the guide and principle of life, both to the Kosmos and to man. The nature of number was imperative and lawgiving, affording the only solution of all that was perplexing or unknown; without number all would be indeterminate and unknowable.”[29]
[25] Aristotel. Metaphys. A. 5, p. 985, b. 27. Ἐν δὲ τοῖς ἀριθμοῖς, ἐνδόκουν θεωρεῖν ὁμοιώματα πολλὰ τοῖς οὖσι καὶ γιγνομένοις, μᾶλλον ἢ ἐν πυρὶ καὶ γῇ καὶ ὕδατι, &c. Cf. N. 3, p. 1090, a. 21.
[26] Aristotel. Metaph. A. 9, p. 990, a. 16. Διὸ περὶ πυρὸς ἢ γῆς ἢ τῶν ἄλλων τῶν τοιούτων σωμάτων οὐδ’ ὁτιοῦν εἰρήκασιν, &c. (the Pythagoreans); also N. 3.
[27] Physic. iii. 4, p. 203, a. 6. Οὐ γὰρ χωριστὸν ποιοῦσι (the Pythagoreans) τὸν ἀριθμόν, &c. Metaphys. M. 6, p. 1080, b. 19: τὰς μονάδας ὑπολαμβάνουσιν ἔχειν μέγεθος. M. 8, p. 1083, b. 17: ἐκεῖνοι (the Pythagoreans) τὸν ἀριθμὸν τὰ ὄντα λέγουσιν· τὰ γοῦν θεωρήματα προσάπτουσι τοῖς σώμασιν ὡς ἐξ ἐκείνων ὄντων τῶν ἀριθμῶν.
[28] An analogous application of this principle (Number as the fundamental substance and universal primary agent) may be seen in an eminent physical philosopher of the nineteenth century, Oken’s Elements of Physio-Philosophy, translated by Tulk. Aphorism 57:—“While numbers in a mathematical sense are positions and negations of nothing, in the philosophical sense they are positions and negations of the Eternal. Every thing which is real, posited, finite, has become this, out of numbers; or more strictly speaking, every Real is absolutely nothing else than a number. This must be the sense entertained of numbers in the Pythagorean doctrine — namely, that every thing, or the whole universe, had arisen from numbers. This is not to be taken in a merely quantitative sense, as it has hitherto been erroneously; but in an intrinsic sense, as implying that all things are numbers themselves, or the acts of the Eternal. The essence in numbers is nought else than the Eternal. The Eternal only is or exists, and nothing else is when a number exists. There is therefore nothing real but the Eternal itself; for every Real, or every thing that is, is only a number and only exists by virtue of a number.”
Ibid., Aphorism 105-107:—“Arithmetic is the science of the second idea, or that of time or motion, or life. It is therefore the first science. Mathematics not only begin with it, but creation also, with the becoming of time and of life. Arithmetic is, accordingly, the truly absolute or divine science; and therefore every thing in it is also directly certain, because every thing in it resembles the Divine. Theology is arithmetic personified.” — “A natural thing is nothing but a self-moving number. An organic or living thing is a number moving itself out of itself or spontaneously: an inorganic thing, however, is a number moved by another thing: now as this other thing is also a real number, so then is every inorganic thing a number moved by another number, and so on ad infinitum. The movements in nature are only movements of numbers by numbers: even as arithmetical computation is none other than a movement of numbers by numbers; but with this difference — that in the latter, this operates in an ideal manner, in the former after a real.”
[29] Philolaus, ed. Boeckh, p. 139. seqq.
Θεωρεῖν δεῖ τὰ ἔργα καὶ τὰν ἐσσίαν (οὐσίαν) τῶ ἀριθμῶ καττὰν δύναμιν, ἅτις ἐντὶ ἐν τᾷ δεκάδι· μεγάλα γὰρ καὶ παντελὴς καὶ παντοεργὸς καὶ θείω καὶ οὐρανίω βίω καὶ ἀνθρωπίνω ἀρχὰ καὶ ἁγεμὼν ... ἄνευ δὲ ταύτας πάντα ἄπειρα καὶ ἄδηλα καὶ ἀφανῆ· νομικὰ γὰρ ἁ φύσις τῶ ἀριθμῶ καὶ ἁγεμονικὰ καὶ διδασκαλικὰ τῶ ἀπορουμένω παντὸς καὶ ἀγνοουμένω παντί. Compare the Fr. p. 58, of the same work.
According to Plato, as well as the Pythagoreans, number extended to ten, and not higher: all above ten were multiples and increments of ten. (Aristot. Physic. iii. 6, p. 203, b. 30).
The Monas — ἀρχή, or principle of Number — geometrical conception of number — symbolical attributes of the first ten numbers, especially of the Dekad.