§11.4ἐπεὶ δὲ καὶ ὁ μαθηματικὸς χρῆται τοῖς κοινοῖς ἰδίως, καὶ τὰς τούτων ἀρχὰς ἂν εἴη θεωρῆσαι τῆς πρώτης φιλοσοφίας.
But since the mathematician also uses common principles in a specific way, to investigate the principles of these would also belong to first philosophy.
ὅτι γὰρ ἀπὸ τῶν ἴσων ἴσων ἀφαιρεθέντων ἴσα τὰ λειπόμενα, κοινὸν μέν ἐστιν ἐπὶ πάντων τῶν ποσῶν, ἡ μαθηματικὴ δʼ ἀπολαβοῦσα περί τι μέρος τῆς οἰκείας ὕλης ποιεῖται τὴν θεωρίαν, οἷον περὶ γραμμὰς ἢ γωνίας ἢ ἀριθμοὺς ἢ τῶν λοιπῶν τι ποσῶν, οὐχ ᾗ δʼ ὄντα ἀλλʼ ᾗ συνεχὲς αὐτῶν ἕκαστον ἐφʼ ἓν ἢ δύο ἢ τρία·
For that when equals are subtracted from equals, the remainders are equal, is common to all quantities, but mathematics, having selected a certain part of its proper matter, makes its investigation about this, for instance about lines or angles or numbers or some other of the remaining quantities, not, however, qua beings but insofar as each of them is continuous in one, two, or three dimensions.
ἡ δὲ φιλοσοφία περὶ τῶν ἐν μέρει μέν, ᾗ τούτων ἑκάστῳ τι συμβέβηκεν, οὐ σκοπεῖ, περὶ τὸ ὂν δέ, ᾗ ὂν τῶν τοιούτων ἕκαστον, θεωρεῖ.
Philosophy, on the other hand, does not inquire about particular things insofar as some accident belongs to each of these, but investigates being, insofar as each of such things is a being.
τὸν αὐτὸν δʼ ἔχει τρόπον καὶ περὶ τὴν φυσικὴν ἐπιστήμην τῇ μαθηματικῇ· τὰ συμβεβηκότα γὰρ ἡ φυσικὴ καὶ τὰς ἀρχὰς θεωρεῖ τὰς τῶν ὄντων ᾗ κινούμενα καὶ οὐχ ᾗ ὄντα (τὴν δὲ πρώτην εἰρήκαμεν ἐπιστήμην τούτων εἶναι καθʼ ὅσον ὄντα τὰ ὑποκείμενά ἐστιν, ἀλλʼ οὐχ ᾗ ἕτερόν τι)· διὸ καὶ ταύτην καὶ τὴν μαθηματικὴν ἐπιστήμην μέρη τῆς σοφίας εἶναι θετέον.
And it is in the same relation with respect to physical science as to mathematics; for physics investigates the accidents and principles of beings insofar as they are in motion and not qua beings (whereas we have said that the first science is about these things insofar as the subjects are beings, and not insofar as they are anything else); therefore both this science and mathematical science must be posited as parts of wisdom.
§11.5ἔστι δέ τις ἐν τοῖς οὖσιν ἀρχὴ περὶ ἣν οὐκ ἔστι διεψεῦσθαι, τοὐναντίον δὲ ἀναγκαῖον ἀεὶ ποιεῖν, λέγω δὲ ἀληθεύειν, οἷον ὅτι οὐκ ἐνδέχεται τὸ αὐτὸ καθʼ ἕνα καὶ τὸν αὐτὸν χρόνον εἶναι καὶ μὴ εἶναι, καὶ τἆλλα τὰ τοῦτον αὑτοῖς ἀντικείμενα τὸν τρόπον.
But there is in beings a certain principle about which it is not possible to be mistaken, but on the contrary, it is necessary always to do the opposite, I mean to speak the truth, for instance, that it is not possible for the same thing at one and the same time to be and not to be, and other such principles that are opposed to one another in this manner.
καὶ περὶ τῶν τοιούτων ἁπλῶς μὲν οὐκ ἔστιν ἀπόδειξις, πρὸς τόνδε δὲ ἔστιν· οὐ γὰρ ἔστιν ἐκ πιστοτέρας ἀρχῆς αὐτοῦ τούτου ποιήσασθαι συλλογισμόν, δεῖ δέ γʼ εἴπερ ἔσται τὸ ἁπλῶς ἀποδεδεῖχθαι.
And concerning such things there is no demonstration absolutely, but there is against a particular person; for it is not possible to make a syllogism from a principle more trustworthy than this itself, yet it is necessary if there is to be absolute demonstration.
πρὸς δὲ τὸν λέγοντα τὰς ἀντικειμένας φάσεις τῷ δεικνύντι διότι ψεῦδος ληπτέον τι τοιοῦτον ὃ ταὐτὸ μὲν ἔσται τῷ μὴ ἐνδέχεσθαι ταὐτὸ εἶναι καὶ μὴ εἶναι καθʼ ἕνα καὶ τὸν αὐτὸν χρόνον, μὴ δόξει δʼ εἶναι ταὐτόν· οὕτω γὰρ μόνως ἂν ἀποδειχθείη πρὸς τὸν φάσκοντα ἐνδέχεσθαι τὰς ἀντικειμένας φάσεις ἀληθεύεσθαι κατὰ τοῦ αὐτοῦ.
But against one who says that opposite assertions are true, he who shows that it is false must assume some such thing which will be the same as the impossibility of the same thing being and not being at one and the same time, but will not seem to be the same; for only in this way could a demonstration be made against one who asserts that opposite statements can be true of the same thing.
τοὺς δὴ μέλλοντας ἀλλήλοις λόγου κοινωνήσειν δεῖ τι συνιέναι αὑτῶν· μὴ γιγνομένου γὰρ τούτου πῶς ἔσται κοινωνία τούτοις πρὸς ἀλλήλους λόγου;
Now those who are to join in discussion with one another must understand something of themselves; for if this does not happen, how will there be any sharing of discussion for them with one another?
δεῖ τοίνυν τῶν ὀνομάτων ἕκαστον εἶναι γνώριμον καὶ δηλοῦν τι, καὶ μὴ πολλά, μόνον δὲ ἕν· ἂν δὲ πλείονα σημαίνῃ, φανερὸν ποιεῖν ἐφʼ ὃ φέρει τοὔνομα τούτων.
Each of the names must therefore be known and signify something, and not many things, but only one; and if it signifies more than one, it must be made clear to which of these the name refers.
ὁ δὴ λέγων εἶναι τοῦτο καὶ μὴ εἶναι, τοῦτο ὅ φησιν οὔ φησιν, ὥσθʼ ὃ σημαίνει τοὔνομα τοῦτʼ οὔ φησι σημαίνειν· τοῦτο δʼ ἀδύνατον.
He, then, who says that this is and is not, denies what he says, so that he denies that the name signifies what it signifies; but this is impossible.
ὥστʼ εἴπερ σημαίνει τι τὸ εἶναι τόδε, τὴν ἀντίφασιν ἀδύνατον ἀληθεύειν.
So that if "to be this" signifies something, it is impossible for the contradiction to be true.
ἔτι δʼ εἴ τι σημαίνει τοὔνομα καὶ τοῦτʼ ἀληθεύεται, δεῖ τοῦτʼ ἐξ ἀνάγκης εἶναι· τὸ δʼ ἐξ ἀνάγκης ὂν οὐκ ἐνδέχεταί ποτε μὴ εἶναι· τὰς ἀντικειμένας ἄρα οὐκ ἐνδέχεται φάσεις καὶ ἀποφάσεις ἀληθεύειν κατὰ τοῦ αὐτοῦ.
Furthermore, if the name signifies something and this is true, this must of necessity be; and that which is of necessity cannot ever not be; therefore, opposite assertions and negations cannot be true of the same thing.
ἔτι δʼ εἰ μηθὲν μᾶλλον ἡ φάσις ἢ ἡ ἀπόφασις ἀληθεύεται, ὁ λέγων ἄνθρωπον ἢ οὐκ ἄνθρωπον οὐθὲν μᾶλλον ἀληθεύσει· δόξειε δὲ κἂν οὐχ ἵππον εἶναι φάσκων τὸν ἄνθρωπον ἢ μᾶλλον ἢ οὐχ ἧττον ἀληθεύειν ἢ οὐκ ἄνθρωπον, ὥστε καὶ ἵππον φάσκων εἶναι τὸν αὐτὸν ἀληθεύσει (τὰς γὰρ ἀντικειμένας ὁμοίως ἦν ἀληθεύειν)· συμβαίνει τοίνυν τὸν αὐτὸν ἄνθρωπον εἶναι καὶ ἵππον ἢ τῶν ἄλλων τι ζῴων. —ἀπόδειξις μὲν οὖν οὐδεμία τούτων ἐστὶν ἁπλῶς, πρὸς μέντοι τὸν ταῦτα τιθέμενον ἀπόδειξις.
Furthermore, if the assertion is no more true than the negation, he who says "man" or "not-man" will be no more speaking the truth; but it would seem that he who says that a man is not a horse is either more or no less speaking the truth than he who says "not-man", so that he will also be speaking the truth in saying that the same is a horse (for it was assumed that opposite statements are equally true); it follows, therefore, that the same is a man and a horse or some other of the animals. —There is, then, no demonstration of these things absolutely, but there is a demonstration against one who posits this.
ταχέως δʼ ἄν τις καὶ αὐτὸν τὸν Ἡράκλειτον τοῦτον ἐρωτῶν τὸν τρόπον ἠνάγκασεν ὁμολογεῖν μηδέποτε τὰς ἀντικειμένας φάσεις δυνατὸν εἶναι κατὰ τῶν αὐτῶν ἀληθεύεσθαι· νῦν δʼ οὐ συνιεὶς ἑαυτοῦ τί ποτε λέγει, ταύτην ἔλαβε τὴν δόξαν.
And quickly would one, by questioning Heraclitus himself in this manner, have forced him to admit that opposite assertions can never be true of the same things; but as it is, not understanding himself what he actually says, he adopted this opinion.
ὅλως δʼ εἰ τὸ λεγόμενον ὑπʼ αὐτοῦ ἐστὶν ἀληθές, οὐδʼ ἂν αὐτὸ τοῦτο εἴη ἀληθές, λέγω δὲ τὸ ἐνδέχεσθαι τὸ αὐτὸ καθʼ ἕνα καὶ τὸν αὐτὸν χρόνον εἶναί τε καὶ μὴ εἶναι· καθάπερ γὰρ καὶ διῃρημένων αὐτῶν οὐδὲν μᾶλλον ἡ κατάφασις ἢ ἡ ἀπόφασις ἀληθεύεται, τὸν αὐτὸν τρόπον καὶ τοῦ συναμφοτέρου καὶ τοῦ συμπεπλεγμένου καθάπερ μιᾶς τινὸς καταφάσεως οὔσης οὐθὲν μᾶλλον ἢ ἡ ἀπόφασις τὸ ὅλον ὡς ἐν καταφάσει τιθέμενον ἀληθεύσεται.
And generally, if what is said by him is true, not even this very thing would be true, I mean that it is possible for the same thing at one and the same time to be and not to be; for just as when they are separated the assertion is no more true than the negation, in the same manner also in the case of the combined and compounded statement, as though it were a single assertion, the whole posited as in assertion will be no more true than the negation.
ἔτι δʼ εἰ μηθὲν ἔστιν ἀληθῶς καταφῆσαι, κἂν αὐτὸ τοῦτο ψεῦδος εἴη τὸ φάναι μηδεμίαν ἀληθῆ κατάφασιν ὑπάρχειν.
Furthermore, if it is not possible to assert anything truly, even this very statement would be false, namely, to say that no true assertion exists.
εἰ δʼ ἔστι τι, λύοιτʼ ἂν τὸ λεγόμενον ὑπὸ τῶν τὰ τοιαῦτα ἐνισταμένων καὶ παντελῶς ἀναιρούντων τὸ διαλέγεσθαι.
But if there is something, the statement of those who raise such objections and completely destroy discussion would be refuted.