Syrianus on Theorems, Proofs, and Imagination in Geometry

The Neoplatonist Syrianus was well-known as the teacher of Proclus, and while the latter is perhaps more famous for having produced a greater literary output, he is consistent in his writings for awarding due credit to his great teacher.  We have very little of Syrianus' original writings that have come down to us-- basically all that remains is two commentaries on Aristotle's Metaphysics, one is on books 3-4 and the other is on books 13-14.  It is in books 13-14 that Aristotle takes a strong position against the Pythagorean and Platonic theories of mathematical Forms, and Syrianus finds opportunity to set the record straight about these doctrines.

I shall not try to summarize these commentaries in this post, merely to mention that they are valuable and worth studying in the context of mathesis, and to post insightful quotations to generate interest in these texts.  The two extant commentaries have recently been translated by Dillon and O'Meara and are available through Cornell University Press.

There is a very insightful passage from the commentary on books 13-14, discussing the significance of the use of diagrams in geometrical proofs.  It is a good example of the Platonist doctrine that mathematical theorems reside in the soul, but that the soul develops these reason-principles (logoi) through discursive thinking (dianoia) and projects them onto the screen of imagination.  If drawn diagrams are used, it is only to assist the soul in grasping the primary Forms.

Geometry aims to contemplate the actual partless reason-principles of the soul, but, being too feeble to employ intellections free of images (aphantastoi), it extends its powers to imagined and extended shapes and magnitudes, and thus contemplates in them these former entities.  Just as, when even the imagination does not suffice for it, it resorts to the reckoning-board (abakion), and there makes a drawing of a theorem, and in that situation its primary object is certainly not to grasp the sensible and external diagram, but rather the internal, imagined one, of which the external one is a soulless imitation; so also when it directs itself to the object of imagination, it is not concerned with it in a primary way, but it is only because through weakness of intellection it is unable to grasp the Form which transcends imagination that it studies at this imaginative level.  And the most powerful indication of this is that, whereas the proof is of the universal, every object of imagination is particular (merikon); therefore the primary concern was never with the object of imagination, but rather with the universal and absolutely immaterial.

There is much that is worthy of contemplation in this thought, especially regarding the meaning of mathesis.  The goal of mathesis is to be able to perceive and work with the reason-principles of the soul, and thereby gain a measure of self-knowledge that could not be attained otherwise.  When study geometry in the Platonic fashion, we are not primarily concerned with producing a body of theorems in the way modern mathematical research proceeds, but we care much more about being able to look into the depths of our own souls and find out more of who we are on the inside.  Syrianus' student Proclus wrote that the imagination was like a mirror into which we can perceive the contents of the soul, and this is done through geometrical study in the fashion described here by Syrianus and elsewhere by Proclus, especially in his Commentary on the First Book of Euclid's Elements, where he writes:

In the same way, when the soul is looking outside herself at the imagination, seeing the figures depicted there and being struck by their beauty and orderedness, she is admiring her own ideas from which they are derived; and though she adores their beauty, she dismisses it as something reflected and seeks her own beauty.  She wants to penetrate within herself to see the circle and the triangle there, all things without parts and all in one another, to become one with what she sees and enfold their plurality, to behold the secret and ineffable figures in the inaccessible places and shrines of the gods, to uncover the unadorned divine beauty and see the circle more partless than any center, the triangle without extension, and every other object of knowledge that has regained unity.

Books mentioned in this article: