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"Let no one ignorant of geometry enter"
Tradition has it that this phrase (1) was engraved at the door of Plato's Academy, the school he had founded in Athens. But is this tradition trustworthy?
We should first notice that this tradition is known to us only through quite late sources, dating from more than 10 centuries after Plato: it is mentionned by Joannes Philoponus, a late neoplatonic Christian philosopher who lived in Alexandria in the VIth century A. D. of whom several commentaries on works by Aristotle are still extant, in his commentary on Aristotle's De Anima (in De An., Comm. in Arist. Graeca, XV, ed. M. Hayduck, Berlin 1897, p. 117, 29); by Elias, another late neoplatonic philospher from Alexandria of the VIth century A. D. who lived after Philoponus and was a Christian too, in his commentary of Aristotle's Analytics (in Cat., Comm. in Arist. Graeca, XVIII, pars 1, ed. A. Busse, Berlin 1900, p. 118, 18); and also by Joannes Tzetzes, byzantine author from the early XIIth century A. D., in his Chiliades (VIII, 973) (2), where it is quoted under the complete form shown in note 1.
The first two references come from commentaries on works by Aristotle, and it is a fact that the word ageômetrètos appears in his writings, for instance in the Posterior Analytics, I, xii, 77b8-34, where the word is used 5 times within a few lines, but he himself never mentions, at least in his extant works, this inscription at the doorstep of the Academy, where he learned, taught and lived for about 20 years.
If the late character of our sources may incite us to doubt the authenticity of this tradition, there remains that, in its spirit, it is in no way out of character, as can be seen by reading or rereading what Plato says about the sciences fit for the formation of philosophers in book VII of the Republic, and especially about geometry at Republic, VII, 526c8-527c11. We should only keep in mind that, for Plato, geometry, as well as all other mathematical sciences, is not an end in itself, but only a prerequisite meant to test and develop the power of abstraction in the student, that is, his ability to go beyond the level of sensible experience which keeps us within the "visible" realm, that of the material world, all the way to the pure intelligible. And geometry, as can be seen through the experiment with the slave boy in the Meno (Meno, 80d1-86d2), can also make us discover the existence of truths (that of a theorem of geometry such as, in the case of the Meno, the one about doubling a square) that may be said to be "transcendant" in that they don't depend upon what we may think about them, but have to be accepted by any reasonable being, which should lead us into wondering whether such transcendant truths might not exist as well in other areas, such as ethics and matters relating to men's ultimate happiness, whether we may be able to "demonstrate" them or not.
One last remark about the word ageômetrètos, the greek adjective used in the supposed inscription to qualify those forbidden from entering. This adjective is made up of the privative prefix a- combined with the verbal adjective geômetrètos derived from the verb geômetrein by addition of the -tos suffix. The original etymological meaning of geômetrein is "measure (metrein) the land (gè)", from which comes the meaning "practice geometry", a science whose birth was indeed linked to land surveying. Verbal adjectives formed with the suffix -tos originally indicate the possible (the equivalent of english adjective ending with -able or -ible), so that the original meaning of geômetrètos is "capable of practising geometry" or, in the passive sense, "capable of being an object of geometry", that is, "geometrical", in which case, it becomes synonymous with geômetrikos (of which "geometrical" is the english transposition). (3) In view of this, it might be better to translate the supposed inscription "let no one inapt to geometry come in" rather than "let no one ignorant of geometry enter". The warning doesn't so much concern those who are not yet confirmed geometers as it does those who don't have the right mind set to practise geometry, the ability to understand it.