A question was sent out to a Latin teacher’s list about Plato’s motto, which a sender’s colleague in math wanted to put above his door. Michael Hendry of Curculio remembered the first word of the Greek and replied with the quote and the citation as given in the LSJ.
I was intrigued and did a little reading via TLG, responding with what follows, though without the Greek, as e-mail still tends to revert to 7-bit ASCII, even in the 21st century. (The translations are mine):
The scholiast on Aelius Aristides 125.14 (Dindorf, Vol. 3) says the following:
ἐπεγέγραπτο ἔμπροσθεν τῆς διατριβῆς τοῦ Πλάτωνος ὅτι ἀγεωμέτρητος μηδεὶς εἰσίτω· ἀντὶ τοῦ ἄνισος καὶ ἄδικος. ἡ γὰρ γεωμετρία τὴν ἰσότητα καὶ τὴν δικαιοσύνην τηρεῖ.
‘In front of Plato’s school had been inscribed, “Let noone enter un-geometried” rather than “unequal” or “unjust,” for geometry maintains equality and justness.’
I assume geometry was among the lower mathematical pursuits required for the study of philosophy, and it seems plausible that Plato would have framed its usefullness in terms of qualities of the soul.
At any rate, Pseudo-Galen (post 2 A.D.?) quotes the phrase at the beginning of ‘On the divisions of philosophy,’ and makes geometry a preliminary to theology:
ὁ μὲν οὖν Πλάτων εἰς φυσιολογικὸν καὶ θεολογικὸν αὐτὸ διαιρεῖ· τὸ γὰρ μαθηματικὸν οὐκ ἠβούλετο εἶναι μέρος τῆς φιλοσοφίας, ἀλλὰ προγύμνασμά τι ὥσπερ ἡ γραμματικὴ καὶ ἡ ῥητορική· ὅθεν καὶ πρὸ τοῦ ἀκροατηρίου τοῦ οἰκείου ἐπέγραψεν ‘ἀγεωμέτρητος μηδεὶς εἰσίτω’. τοῦτο δὲ ὁ Πλάτων ἐπέγραφεν, ἐπειδὴ εἰς τὰ πολλὰ θεολογεῖ καὶ περὶ θεολογίαν καταγίνεται· συμβάλλεται δὲ εἰς εἴδησιν τῆς θεολογίας τὸ μαθηματικόν, οὗτινός ἐστιν ἡ γεωμετρία.
‘Plato divided it (theoretical philosophy) into physiology and theology. In fact, he did not want mathematics to be a part of philosophy, but a sort of progymnasma like grammar and rhetoric. That’s why, before his private lecture-room, he inscribed “Let no one enter un-geometried.” He inscribed this since he discoursed on theology in all matters and dwelt on theology, and included mathematics, of which geometry is a part, into theology’s forms of knowledge.’
I like the notion that it wasn’t the school but rather Plato’s personal lecture space, and that the teacher wants to hang it above the door to his own classroom.
As one of my old humanities professors used to say in his booming baritone, ‘mathematics is gymnastics for the mind!’