§13.8#1πάντων δὲ πρῶτον καλῶς ἔχει διορίσασθαι τίς ἀριθμοῦ διαφορά, καὶ μονάδος, εἰ ἔστιν.
First of all, it is well to define what difference there is in number, and in a unit, if there is any.
ἀνάγκη δʼ ἢ κατὰ τὸ ποσὸν ἢ κατὰ τὸ ποιὸν διαφέρειν· τούτων δʼ οὐδέτερον φαίνεται ἐνδέχεσθαι ὑπάρχειν.
And it is necessary that they differ either in quantity or in quality; but neither of these seems to be possible to belong to them.
ἀλλʼ ᾗ ἀριθμός, κατὰ τὸ ποσόν.
But in so far as it is number, it differs in quantity.
εἰ δὲ δὴ καὶ αἱ μονάδες τῷ ποσῷ διέφερον, κἂν ἀριθμὸς ἀριθμοῦ διέφερεν ὁ ἴσος τῷ πλήθει τῶν μονάδων.
And if indeed the units also differed in quantity, then even a number equal in the multitude of its units would differ from another number.
ἔτι πότερον αἱ πρῶται μείζους ἢ ἐλάττους, καὶ αἱ ὕστερον ἐπιδιδόασιν ἢ τοὐναντίον;
Moreover, are the first units larger or smaller, and do those which are subsequent increase, or the contrary?
πάντα γὰρ ταῦτα ἄλογα.
For all these things are irrational.
ἀλλὰ μὴν οὐδὲ κατὰ τὸ ποιὸν διαφέρειν ἐνδέχεται.
But indeed, neither is it possible for them to differ in quality.
οὐθὲν γὰρ αὐταῖς οἷόν τε ὑπάρχειν πάθος· ὕστερον γὰρ καὶ τοῖς ἀριθμοῖς φασὶν ὑπάρχειν τὸ ποιὸν τοῦ ποσοῦ.
For no affection can possibly belong to them; for they say that quality belongs to numbers later than quantity.
ἔτι οὔτʼ ἂν ἀπὸ τοῦ ἑνὸς τοῦτʼ αὐταῖς γένοιτο οὔτʼ ἂν ἀπὸ τῆς δυάδος· τὸ μὲν γὰρ οὐ ποιὸν ἡ δὲ ποσοποιόν· τοῦ γὰρ πολλὰ τὰ ὄντα εἶναι αἰτία αὕτη ἡ φύσις.
Moreover, this could accrue to them neither from the One nor from the Dyad; for the former is not qualitative, and the latter is quantitative; for this nature is the cause of there being many existents.
εἰ δʼ ἄρα ἔχει πως ἄλλως, λεκτέον ἐν ἀρχῇ μάλιστα τοῦτο καὶ διοριστέον περὶ μονάδος διαφορᾶς, μάλιστα μὲν καὶ διότι ἀνάγκη ὑπάρχειν· εἰ δὲ μή, τίνα λέγουσιν; —ὅτι μὲν οὖν, εἴπερ εἰσὶν ἀριθμοὶ αἱ ἰδέαι, οὔτε συμβλητὰς τὰς μονάδας ἁπάσας ἐνδέχεται εἶναι, φανερόν, οὔτε ἀσυμβλήτους ἀλλήλαις οὐδέτερον τῶν τρόπων·
But if, after all, it is in some other way, this must be stated at the very beginning and defined concerning the difference of a unit, especially why it is necessary that it should exist; and if not, what difference do they mean? —It is clear, then, that if the Ideas are numbers, it is neither possible for all the units to be associable, nor for them to be inassociable with one another in either of the two ways.
ἀλλὰ μὴν οὐδʼ ὡς ἕτεροί τινες λέγουσι περὶ τῶν ἀριθμῶν λέγεται καλῶς.
But indeed, neither is the account well given which certain other people give of numbers.
εἰσὶ δʼ οὗτοι ὅσοι ἰδέας μὲν οὐκ οἴονται εἶναι οὔτε ἁπλῶς οὔτε ὡς ἀριθμούς τινας οὔσας, τὰ δὲ μαθηματικὰ εἶναι καὶ τοὺς ἀριθμοὺς πρώτους τῶν ὄντων, καὶ ἀρχὴν αὐτῶν εἶναι αὐτὸ τὸ ἕν.
And these are those who do not think that Ideas exist either absolutely or as being certain numbers, but that mathematical objects exist and numbers are the first of existents, and their principle is the One itself.
ἄτοπον γὰρ τὸ ἓν μὲν εἶναί τι πρῶτον τῶν ἑνῶν, ὥσπερ ἐκεῖνοί φασι, δυάδα δὲ τῶν δυάδων μή, μηδὲ τριάδα τῶν τριάδων· τοῦ γὰρ αὐτοῦ λόγου πάντα ἐστίν.
For it is absurd that there should be a certain first 'one' among the 'ones', as they say, but not a first dyad among dyads, nor a first triad among triads; for they are all of the same principle.
εἰ μὲν οὖν οὕτως ἔχει τὰ περὶ τὸν ἀριθμὸν καὶ θήσει τις εἶναι τὸν μαθηματικὸν μόνον, οὐκ ἔστι τὸ ἓν ἀρχή (ἀνάγκη γὰρ διαφέρειν τὸ ἓν τὸ τοιοῦτο τῶν ἄλλων μονάδων· εἰ δὲ τοῦτο, καὶ δυάδα τινὰ πρώτην τῶν δυάδων, ὁμοίως δὲ καὶ τοὺς ἄλλους ἀριθμοὺς τοὺς ἐφεξῆς)·
If, then, the case of number is thus, and one posits that mathematical number alone exists, the One is not the principle (for it is necessary that such a 'one' should differ from the other units; and if this is so, there must also be a certain first dyad among dyads, and likewise also with the other subsequent numbers).
εἰ δέ ἐστι τὸ ἓν ἀρχή, ἀνάγκη μᾶλλον ὥσπερ Πλάτων ἔλεγεν ἔχειν τὰ περὶ τοὺς ἀριθμούς, καὶ εἶναι δυάδα πρώτην καὶ τριάδα, καὶ οὐ συμβλητοὺς εἶναι τοὺς ἀριθμοὺς πρὸς ἀλλήλους.
But if the One is the principle, the case of numbers must rather be as Plato used to say, and there must be a first dyad and triad, and the numbers must not be associable with one another.
ἂν δʼ αὖ πάλιν τις τιθῇ ταῦτα, εἴρηται ὅτι ἀδύνατα πολλὰ συμβαίνει.
But if, on the other hand, one posits these things again, it has been said that many impossible consequences result.
ἀλλὰ μὴν ἀνάγκη γε ἢ οὕτως ἢ ἐκείνως ἔχειν, ὥστʼ εἰ μηδετέρως, οὐκ ἂν ἐνδέχοιτο εἶναι τὸν ἀριθμὸν χωριστόν.
But indeed, it must be either in this way or in that way, so that if in neither way, it would not be possible for number to be separable.
—φανερὸν δʼ ἐκ τούτων καὶ ὅτι χείριστα λέγεται ὁ τρίτος τρόπος, τὸ εἶναι τὸν αὐτὸν ἀριθμὸν τὸν τῶν εἰδῶν καὶ τὸν μαθηματικόν.
—And it is clear from these things also that the third way is the worst stated, namely, that the number of the forms and mathematical number are the same.
ἀνάγκη γὰρ εἰς μίαν δόξαν συμβαίνειν δύο ἁμαρτίας· οὔτε γὰρ μαθηματικὸν ἀριθμὸν ἐνδέχεται τοῦτον εἶναι τὸν τρόπον, ἀλλʼ ἰδίας ὑποθέσεις ὑποθέμενον ἀνάγκη μηκύνειν, ὅσα τε τοῖς ὡς εἴδη τὸν ἀριθμὸν λέγουσι συμβαίνει, καὶ ταῦτα ἀναγκαῖον λέγειν. —ὁ δὲ τῶν Πυθαγορείων τρόπος τῇ μὲν ἐλάττους ἔχει δυσχερείας τῶν πρότερον εἰρημένων, τῇ δὲ ἰδίας ἑτέρας.
For it is necessary that two errors combine in a single opinion; for neither is it possible for mathematical number to exist in this way, but he who posits it must spin it out by making peculiar hypotheses, and also he must state all the consequences that result for those who speak of number as forms. —But the way of the Pythagoreans in one respect has fewer difficulties than those previously mentioned, but in another respect has other peculiar difficulties.
τὸ μὲν γὰρ μὴ χωριστὸν ποιεῖν τὸν ἀριθμὸν ἀφαιρεῖται πολλὰ τῶν ἀδυνάτων· τὸ δὲ τὰ σώματα ἐξ ἀριθμῶν εἶναι συγκείμενα, καὶ τὸν ἀριθμὸν τοῦτον εἶναι μαθηματικόν, ἀδύνατόν ἐστιν.
For not making number separable removes many of the impossible consequences; but that bodies are composed of numbers, and that this number is mathematical, is impossible.
οὔτε γὰρ ἄτομα μεγέθη λέγειν ἀληθές, εἴ θʼ ὅτι μάλιστα τοῦτον ἔχει τὸν τρόπον, οὐχ αἵ γε μονάδες μέγεθος ἔχουσιν· μέγεθος δὲ ἐξ ἀδιαιρέτων συγκεῖσθαι πῶς δυνατόν;
For neither is it true to speak of indivisible magnitudes, and even if this were so in the highest degree, at least units do not have magnitude; but how is it possible for a magnitude to be composed of indivisibles?
ἀλλὰ μὴν ὅ γʼ ἀριθμητικὸς ἀριθμὸς μοναδικός ἐστιν.
But indeed, arithmetical number at least consists of units.