§13.6ἐπεὶ δὲ διώρισται περὶ τούτων, καλῶς ἔχει πάλιν θεωρῆσαι τὰ περὶ τοὺς ἀριθμοὺς συμβαίνοντα τοῖς λέγουσιν οὐσίας αὐτοὺς εἶναι χωριστὰς καὶ τῶν ὄντων αἰτίας πρώτας.
Since these matters have been defined, it is well to consider again what happens in the case of numbers for those who say that they are separate substances and the primary causes of beings.
ἀνάγκη δʼ, εἴπερ ἐστὶν ὁ ἀριθμὸς φύσις τις καὶ μὴ ἄλλη τίς ἐστιν αὐτοῦ ἡ οὐσία ἀλλὰ τοῦτʼ αὐτό, ὥσπερ φασί τινες, ἤτοι εἶναι τὸ μὲν πρῶτόν τι αὐτοῦ τὸ δʼ ἐχόμενον, ἕτερον ὂν τῷ εἴδει ἕκαστον,—καὶ τοῦτο ἢ ἐπὶ τῶν μονάδων εὐθὺς ὑπάρχει καὶ ἔστιν ἀσύμβλητος ὁποιαοῦν μονὰς ὁποιᾳοῦν μονάδι, ἢ εὐθὺς ἐφεξῆς πᾶσαι καὶ συμβληταὶ ὁποιαιοῦν ὁποιαισοῦν, οἷον λέγουσιν εἶναι τὸν μαθηματικὸν ἀριθμόν (ἐν γὰρ τῷ μαθηματικῷ οὐδὲν διαφέρει οὐδεμία μονὰς ἑτέρα ἑτέρας)· ἢ τὰς μὲν συμβλητὰς τὰς δὲ μή (οἷον εἰ ἔστι μετὰ τὸ ἓν πρώτη ἡ δυάς, ἔπειτα ἡ τριὰς καὶ οὕτω δὴ ὁ ἄλλος ἀριθμός, εἰσὶ δὲ συμβληταὶ αἱ ἐν ἑκάστῳ ἀριθμῷ μονάδες, οἷον αἱ ἐν τῇ δυάδι τῇ πρώτῃ αὑταῖς, καὶ αἱ ἐν τῇ τριάδι τῇ πρώτῃ αὑταῖς, καὶ οὕτω δὴ ἐπὶ τῶν ἄλλων ἀριθμῶν· αἱ δʼ ἐν τῇ δυάδι αὐτῇ πρὸς τὰς ἐν τῇ τριάδι αὐτῇ ἀσύμβλητοι, ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων τῶν ἐφεξῆς ἀριθμῶν· διὸ καὶ ὁ μὲν μαθηματικὸς ἀριθμεῖται μετὰ τὸ ἓν δύο, πρὸς τῷ ἔμπροσθεν ἑνὶ ἄλλο ἕν, καὶ τὰ τρία πρὸς τοῖς δυσὶ τούτοις ἄλλο ἕν, καὶ ὁ λοιπὸς δὲ ὡσαύτως· οὗτος δὲ μετὰ τὸ ἓν δύο ἕτερα ἄνευ τοῦ ἑνὸς τοῦ πρώτου, καὶ ἡ τριὰς ἄνευ τῆς δυάδος, ὁμοίως δὲ καὶ ὁ ἄλλος ἀριθμός)· ἢ τὸν μὲν εἶναι τῶν ἀριθμῶν οἷος ὁ πρῶτος ἐλέχθη, τὸν δʼ οἷον οἱ μαθηματικοὶ λέγουσι, τρίτον δὲ τὸν ῥηθέντα τελευταῖον· ἔτι τούτους ἢ χωριστοὺς εἶναι τοὺς ἀριθμοὺς τῶν πραγμάτων, ἢ οὐ χωριστοὺς ἀλλʼ ἐν τοῖς αἰσθητοῖς (οὐχ οὕτως δʼ ὡς τὸ πρῶτον ἐπεσκοποῦμεν, ἀλλʼ ὡς ἐκ τῶν ἀριθμῶν ἐνυπαρχόντων ὄντα τὰ αἰσθητά) ἢ τὸν μὲν αὐτῶν εἶναι τὸν δὲ μή, ἢ πάντας εἶναι. —οἱ μὲν οὖν τρόποι καθʼ οὓς ἐνδέχεται αὐτοὺς εἶναι οὗτοί εἰσιν ἐξ ἀνάγκης μόνοι, σχεδὸν δὲ καὶ οἱ λέγοντες τὸ ἓν ἀρχὴν εἶναι καὶ οὐσίαν καὶ στοιχεῖον πάντων, καὶ ἐκ τούτου καὶ ἄλλου τινὸς εἶναι τὸν ἀριθμόν, ἕκαστος τούτων τινὰ τῶν τρόπων εἴρηκε, πλὴν τοῦ πάσας τὰς μονάδας εἶναι ἀσυμβλήτους.
And it is necessary, if indeed number is a certain nature and its substance is nothing other than this very thing, as some say, either that there is a first of it and a next, each being different in species,—and this either belongs directly to the units, and any unit is inassociable with any other unit, or they are all directly successive and any are associable with any, as they say mathematical number is (for in mathematical number no unit differs from any other); or some units are associable and others not (for example, if after the One the first is the Dyad, then the Triad, and so on with the other numbers, and the units in each number are associable, for example those in the first Dyad are associable with themselves, and those in the first Triad with themselves, and so on with the other numbers; but those in the Dyad-itself are inassociable with those in the Triad-itself, and likewise with the other successive numbers; and this is why mathematical number is counted, after the One, as two, i.e., another 'one' added to the previous 'one', and three as another 'one' added to these two, and the rest likewise; but this other number is counted, after the One, as another 'two' without the first 'one', and the Triad without the Dyad, and likewise with the other number); or one kind of numbers is such as the first mentioned, another is such as the mathematicians say, and a third is that mentioned last; further, these numbers must either be separate from things, or not separate but in sensible things (yet not in the way we first examined, but in the sense that sensible things are composed of numbers existing in them), or some of them are separate and others not, or all of them are.—These ways in which they can exist are of necessity the only ones, and almost all those who say that the One is the principle, substance, and element of all things, and that number is from this and something else, have stated some one of these ways, except that all units are inassociable.
καὶ τοῦτο συμβέβηκεν εὐλόγως· οὐ γὰρ ἐνδέχεται ἔτι ἄλλον τρόπον εἶναι παρὰ τοὺς εἰρημένους.
And this has happened reasonably; for it is not possible for there to be any other way besides those mentioned.
οἱ μὲν οὖν ἀμφοτέρους φασὶν εἶναι τοὺς ἀριθμούς, τὸν μὲν ἔχοντα τὸ πρότερον καὶ ὕστερον τὰς ἰδέας, τὸν δὲ μαθηματικὸν παρὰ τὰς ἰδέας καὶ τὰ αἰσθητά, καὶ χωριστοὺς ἀμφοτέρους τῶν αἰσθητῶν· οἱ δὲ τὸν μαθηματικὸν μόνον ἀριθμὸν εἶναι, τὸν πρῶτον τῶν ὄντων, κεχωρισμένον τῶν αἰσθητῶν.
Some, then, say that both kinds of numbers exist, that which has the prior and posterior being the Ideas, and the mathematical number being besides the Ideas and the sensible things, and both being separate from the sensible; others say that mathematical number alone exists, being the first of beings, separated from the sensible.
καὶ οἱ Πυθαγόρειοι δʼ ἕνα, τὸν μαθηματικόν, πλὴν οὐ κεχωρισμένον ἀλλʼ ἐκ τούτου τὰς αἰσθητὰς οὐσίας συνεστάναι φασίν· τὸν γὰρ ὅλον οὐρανὸν κατασκευάζουσιν ἐξ ἀριθμῶν, πλὴν οὐ μοναδικῶν, ἀλλὰ τὰς μονάδας ὑπολαμβάνουσιν ἔχειν μέγεθος· ὅπως δὲ τὸ πρῶτον ἓν συνέστη ἔχον μέγεθος, ἀπορεῖν ἐοίκασιν.
And the Pythagoreans also say there is one kind, the mathematical, except that it is not separated, but from it they say sensible substances are composed; for they construct the whole heaven out of numbers, though not of abstract units, but they assume the units to have magnitude; but how the first One was constructed having magnitude, they seem to be at a loss.
ἄλλος δέ τις τὸν πρῶτον ἀριθμὸν τὸν τῶν εἰδῶν ἕνα εἶναι, ἔνιοι δὲ καὶ τὸν μαθηματικὸν τὸν αὐτὸν τοῦτον εἶναι.
Another says that the first number, that of the Forms, is unique, and some say that the mathematical number is also this same one.
ὁμοίως δὲ καὶ περὶ τὰ μήκη καὶ περὶ τὰ ἐπίπεδα καὶ περὶ τὰ στερεά.
Likewise also concerning lengths, planes, and solids.
οἱ μὲν γὰρ ἕτερα τὰ μαθηματικὰ καὶ τὰ μετὰ τὰς ἰδέας· τῶν δὲ ἄλλως λεγόντων οἱ μὲν τὰ μαθηματικὰ καὶ μαθηματικῶς λέγουσιν, ὅσοι μὴ ποιοῦσι τὰς ἰδέας ἀριθμοὺς μηδὲ εἶναί φασιν ἰδέας, οἱ δὲ τὰ μαθηματικά, οὐ μαθηματικῶς δέ· οὐ γὰρ τέμνεσθαι οὔτε μέγεθος πᾶν εἰς μεγέθη, οὔθʼ ὁποιασοῦν μονάδας δυάδα εἶναι.
For some say that the mathematical objects and those after the Ideas are different; and of those who speak otherwise, some speak of mathematical objects and in a mathematical way, as many as do not make the Ideas numbers nor say that Ideas exist; others speak of mathematical objects, but not in a mathematical way; for they say neither that every magnitude is divided into magnitudes, nor that any units make a dyad.
μοναδικοὺς δὲ τοὺς ἀριθμοὺς εἶναι πάντες τιθέασι, πλὴν τῶν Πυθαγορείων, ὅσοι τὸ ἓν στοιχεῖον καὶ ἀρχήν φασιν εἶναι τῶν ὄντων· ἐκεῖνοι δʼ ἔχοντας μέγεθος, καθάπερ εἴηρται πρότερον.
But all who say that the One is the element and principle of beings assume numbers to consist of abstract units, except the Pythagoreans; and they assume them to have magnitude, as was said before.
ὁσαχῶς μὲν οὖν ἐνδέχεται λεχθῆναι περὶ αὐτῶν, καὶ ὅτι πάντες εἰσὶν εἰρημένοι οἱ τρόποι, φανερὸν ἐκ τούτων· ἔστι δὲ πάντα μὲν ἀδύνατα, μᾶλλον δʼ ἴσως θάτερα τῶν ἑτέρων.
In how many ways, therefore, it is possible to speak concerning them, and that all the ways have been mentioned, is clear from this; but all of them are impossible, though perhaps some more than others.