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.

Are We Really Detached From the World of Mathematical Objects?

In reading this passage from Stewart Shapiro's book Philosophy of Mathematics: Structure and Ontology (Oxford University Press 1999), I came across this passage which sparked some thoughts for me:

From the realism in ontology, we have the existence of mathematical objects. It would appear that these objects are abstract, in the sense that they are causally inert, not located in space and time, and so on. Moreover, from the realism in truth-value, it would appear that assertions like the twin-prime conjecture and the continuum hypothesis are either true or false, independently of the mind, language, or convention of the mathematician. Thus, we are led to a view much like traditional Platonism, and the notorious epistemological problems that come with it. If mathematical objects exist outside the causal nexus, how can we know anything about them? How can we have any confidence in what the mathematicians say about mathematical objects? Under the suggested realism, this requires epistemic access to an acausal, eternal, and detached mathematical realm. This is the most serious problem for realism.

The main question I have, and that runs counter to the worldview of Via Mathesis, is this:  Are we really detached from the World of Mathematical Objects?  This big assumption is the real heart of why the "epistemological problems" that come with Platonism are seen as so intractable.  Who says we're detached from the mathematical realm, or from Plato's World of Ideas / Forms?  Why has this always been assumed without investigating the possibility that we might not be detached from this higher world?

The epistemic problem that philosophers face when considering the existence of mathematical objects could be greatly clarified by looking into the structure of the mind.  We begin by positing a three-component system to the overall mind:

1. Higher Mind - the 'conceiver'
2. Physical Mind - the 'receiver' 
3. Physical Brain - the 'perceiver'

The difficulty with epistemic access is due to thinking that the mind is only the physical mind.  The physical mind cannot apprehend mathematical objects, since it wasn't designed to do this.  Only the higher mind can do this.  Of course, every time philosophers create a division in the mind, they must provide an account of how the components interact.  Some would say this is already a known and difficult problem, and others have tried to provide an account and have gotten nowhere.  To this we must respond that philosophy is primarily a way of life.  When we intellectualize everything, we neglect the heart of philosophy.  Being the love of wisdom, or the love of learning, there must be great love at the heart of every philosopher.

We create philosophical accounts of reality as a way of clarifying our worldview, so we can orient our thoughts to a structure of beliefs.  For those who remain skeptical of a higher mind, they cannot perceive its functioning because their beliefs are such that it is not part of their worldview.  So we content ourselves with providing a limited account of the interaction between higher mind and physical mind, aware that this could all be doubted.  But knowing that it is more important to have our beliefs consistent with our worldview, we proceed with the premise that those who with an open heart will find all the evidence they need for the existence of a higher mind.

The higher mind is what apprehends mathematical objects, and comprehends them by unifying with them. Its comprehension of these objects gives rise to concepts which are projected into the physical mind, and then filter down into the structuring of the physical brain. So the brain must go through various stages of development to build a neurological network that can receive these types of thoughts from the higher mind.

Our beginnings of acquiring mathematical concepts arise from interacting with physical objects at a young age.  Playing with shapes and drawing pictures in kindergarten, we are reminded of the pure mathematical objects that exist above the physical world and prepared to receive concepts of them.  The higher mind can have encounters with pure mathematical objects whenever it wants to, but we are not able to receive thoughts from the higher mind about these objects until we have prepared the physical brain to receive such thoughts.

Every mathematician has had an experience of receiving an instantaneous 'download' of thought-information from the higher mind.  These are what are classically described as those 'Eureka!' moments when we see the solution to a problem in a flash of insight.  There are many stories of mathematicians who said they 'solved' a problem they'd been working on for years at very unexpected moments, such as in the shower, getting on a bus.  But it is misleading to think that they solved the problem 'in an instant' in these situations.  It is more like the work that was done leading up to that experience is what prepared the neurological circuitry of their brain to be able to receive a 'download' from the higher mind.

In our work as mathematicians, we do not work directly with the pure mathematical objects; instead we work with their projected concepts.  But the concepts are accurate enough so that mathematical statements expressed in the concept-language can be true or false statement about the real mathematical objects that exist in the higher world.  The concepts are accurate because the objective substance of mathematical  objects is thought-substance.

Via Mathesis is a path of spiritual self-knowledge, which is attained by way of mathematical knowledge, hence the name via mathesis.  These mathematical studies transform our physical mind to be able to communicate more easily with the higher mind.  The higher mind is to be valued much more than simply because it can apprehend pure mathematical objects, but because the higher mind guides our spiritual development and the evolution of our soul, leading us towards those things which we are meant to do.  Even if you're not excited by mathematical work, it is still a powerful means by which to build a relationship with the higher mind.  Of course, here on Via Mathesis we are exploring the full spectrum of spiritual mathematics, and will have much more interest in developing the connection between physical mind and higher mind.

Books mentioned in this article:

Proclus on the Range of Mathematical Thinking

"The range of mathematical thinking extends from on high all the way down to conclusions in the sense world, where it touches on nature and cooperates with natural science in establishing many of its propositions, just as it rises up from below and nearly joins intellect in apprehending primary principles.  In its lowest applications, therefore, it projects all of mechanics, as well as optics and many other sciences bound up with sensible things and operative in them, while as it moves upwards it attains unitary and immaterial insights that enable it to perfect its partial judgments and the knowledge gained through discursive thought, bringing its own genera and species into conformity with those higher realities and exhibiting in its own reasonings the truth about the gods and the science of being."

-- Proclus, Commentary on Euclid's Elements 19-20