[225] Plato, Republic, vii. pp. 523-524.
[226] Plato, Republic, vii. p. 524 B-C.
Perplexity arising from the One and Many, stimulates the mind to an intellectual effort for clearing it up.
Now arithmetic, besides its practical usefulness for arrangements of war, includes difficulties and furnishes a stimulus of this nature. We see the same thing both as One and as infinite in multitude: as definite and indefinite in number.[227] We can emerge from these difficulties only by intellectual and abstract reflection. It is for this purpose, and not for purposes of traffic, that our intended philosophers must learn Arithmetic. Their minds must be raised from the confusion of the sensible world to the clear daylight of the intelligible.[228] In teaching Arithmetic, the master sets before his pupils numbers in the concrete, that is, embodied in visible and tangible objects — so many balls or pebbles.[229] Each of these balls he enumerates as One, though they be unequal in magnitude, and whatever be the magnitude of each. If you remark that the balls are unequal — and that each of them is Many as well as One, being divisible into as many parts as you please — he will laugh at the objection as irrelevant. He will tell you that the units to which his numeration refers are each Unum per se, indivisible and without parts; and all equal among themselves without the least shade of difference. He will add that such units cannot be exhibited to the senses, but can only be conceived by the intellect: that the balls before you are not such units in reality, but serve to suggest and facilitate the effort of abstract conception.[230] In this manner arithmetical teaching conducts us to numbers in the abstract — to the real, intelligible, indivisible unit — the Unum per se.
[227] Plato, Republic, vii. p. 525 A. ἅμα γὰρ ταὐτὸν ὡς ἕν τε ὁρῶμεν καὶ ὡς ἄπειρα τὸ πλῆθος.
[228] Plato, Republic, vii. p. 525 B. διὰ τὸ τῆς οὐσίας ἁπτέον εἶναι γενέσεως ἐξαναδύντι, &c.
[229] Plato, Republic, vii. p. 525 D. ὁρατὰ ἢ ἁπτὰ σώματα ἔχοντας ἀριθμοὺς, &c.
[230] Plato, Republic, vii. p. 526 A. εἴ τις ἔροιτο αὐτούς, Ὦ θαυμάσιοι, περὶ ποίων ἀριθμῶν διαλέγεσθε, ἐν οἷς τὸ ἓν οἷον ὑμεῖς ἀξιοῦτέ ἐστιν, ἴσον τε ἕκαστον πᾶν παντὶ καὶ οὐδὲ σμικρὸν διαφέρον, μόριόν τε ἔχον ἐν ἑαυτῷ οὐδέν; τί ἂν οἴει αὐτοὺς ἀποκρίνασθαι; Τοῦτο ἔγωγε, ὅτι περὶ τούτων λέγουσιν ὧν διανοηθῆναι μόνον ἐγχωρεῖ, ἄλλως δ’ οὐδαμῶς μεταχειρίζεσθαι δυνατόν.
Geometry conducts the mind to wards Universal Ens.
Geometrical teaching conducts the mind to the same order of contemplations; leading it away from variable particulars to unchangeable universal Essence. Some persons extol Geometry chiefly on the ground of its usefulness in applications to practice. But this is a mistake: its real value is in conducing to knowledge, and to elevated contemplations of the mind. It does, however, like Arithmetic, yield useful results in practice: and both of them are farther valuable as auxiliaries to other studies.[231]