Instructions for Meditating on Geometric Objects - Circle

I would like to begin giving more detailed instructions on mathematical meditations.  It's easiest to begin with meditations on geometric objects, and with that we begin with the circle.

Draw a circle on paper using a compass. Then look at it for a minute.  Then close your eyes and see a memory image of the circle you were just looking at.  Do this for 1 minute, then let it fade.  Next, mentally visualize a circle drawn in thin white light on a black background.  Do this for 1 minute, then let it fade.  Next, mentally repeat the definition "a circle is a set of points with equal distance from a center" a total of 7 times.  Then sit in silence and feel yourself drawing closer to the pure idea of the circle. 

Understand that this idea is eternal, indestructible, and incorruptible.  You are now coming into contact with a pure mathematical object which subsists within the higher mind.  This is the real circle.  It is a perfect unity, a simultaneous whole and is without parts.  You are becoming more aware of your own eternal nature and your connection to the divine.

Once again, mentally repeat the definition "a circle is a set of points with equal distance from a center" a total of 7 times.  Then visualize a circle for 1 minute, and then draw one on paper using a compass.

When you are drawing the circle on paper, you are operating at the level of body.  When visualizing, you are on the level of soul.  When repeating the definition, you are at the level of intellect.  And when coming into direct union with the real circle you are at the level of The One.

Become aware of a qualitative change of consciousness as you move through the layers.  What do you experience as you move up to the next level?  Why is it important to come back down to the physical level?

Coming back down to the physical level is a powerful meditative movement.  It is a highly theurgical act.  Theurgy is the art of physical action which is in resonance with the downward movement of divine ideas into the physical world.  By engaging in this powerful, albeit simple theurgical ritual, we are resonating with the circular movement of the heavens and the earth, and thus becoming more aware of the emanation of providence into our world from the divine.

Proclus on the Meaning of Mathesis

I have been drawing great inspiration for my work from Proclus' Commentary on the First Book of Euclid's Elements.  The text is far more than a commentary on Euclid, since it includes two prefatory essays on the philosophy of general mathematics and the philosophy of geometry.  This is really the origin of what we today call the philosophy of mathematics, and Proclus, in his usual systematic fashion, herein establishes mathematics within the metaphysical hierarchy established in his other Neoplatonic works such as the Elements of Theology, which was by no coincidence written after the style of Euclid's Elements.  The commentary portion includes lengthy discussions of the metaphysical aspects to Euclid's definitions.  Learning all of this really brings the Euclid text to life, as we begin to see mathematical objects as real beings.  Geometric investigation then becomes an exploration of this ontological universe; the theorems established in geometrical discourse such as was stimulated by the Elements becomes a map of this higher world.

Proclus

 
At the end of the first essay on general mathematics, Proclus has a whole paragraph on the meaning of mathesis as I intend it to be used here.  The greek word is μαθησις and is translated by Morrow as "learning".  I would prefer it have been left untranslated since there is really no English equivalent and Proclus gives a thorough definition of what it is.  Here is what he says:

This, then, is what learning (mathesis) is, recollection of the eternal ideas of the soul; and this is why the study that especially brings us the recollection of these ideas is called the science concerned with learning (mathematike).  Its name thus makes clear what sort of function this science performs.  It arouses our innate knowledge, awakens our intellect, purges our understanding, brings to light the concepts that belong essentially to us, takes away the forgetfulness and ignorance that we have from birth, set us free from the bonds of unreason; and all this by the favor of the god who is truly the patron of this science, who brings our intellectual endowments to light, fills everything with divine reason, moves our souls towards Nous, awakens us as it were from our heavy slumber, through our searching turns us back upon ourselves, through our birthpangs perfects us, and through the discovery of pure Nous leads us to the blessed life.  And so, dedicating this composition to him, we proceed to delineate the theory of the science of mathematics.

Pay particular attention to this last line: "dedicating this composition to him".  Proclus is saying that his Commentary was intended as a hymn to the god of mathematics.   This god is Hermes.  This is consistent with Proclus' theurgy and shows how he sees the study of geometry as a theurgical and soteriological endeavor.  But we have lost this completely from the mathematics of today, and my purpose with all of my blogs and Youtube channel is to bring this back to life.

Books mentioned in this post:

Video - Geometric Theurgy 01 - Creation by Numbers

This video will begin the series on Geometric Theurgy by looking at a story of creation by numbers.  We will develop the symbolic meanings of numbers and explain why these meanings go with the numbers.  This is a profound exegesis on the Pythagorean tetraktys.  Learn how you are part of God's unfoldment of self-knowledge.  Once this is understood, you will be well prepared to begin practicing the highest form of theurgy-- creating your own universe.

Video - Ancient Philosophy of Mathematics 07 - The One, Limit, and Unlimited in Geometry

I have added another video to the video series Ancient Philosophy of Mathematics.  This is part seven, titled The One, Limit and Unlimited in Geometry.  We will explore the metaphysics of geometry through The One, Limit and Unlimited.  We will show how geometric constructions correspond to these first principles, with the point symbolizing The One, the circle symbolizing the Limit, and the line symbolizing the Unlimited. We give a basic description of the practice of geometric theurgy, a form of meditation that requires using body movement, thereby grounding the highest metaphysical principles down into the lowest level.  We read from Plato's Philebus and Proclus' Elements of Theology and Commentary on Euclid's Elements.  This video is the foundation for the next series of videos on geometric theurgy.