Emil Artin on the Balancing of Visual Wholism and Logical Order

"We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt he must always fail. Mathematics is logical to be sure; each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that its perception should be instantaneous. We all have experienced on some rare occasions the feeling of elation in realizing that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications. How can this be achieved? Clinging stubbornly to the logical sequence inhibits visualization of the whole, and yet this logical structure must predominate or chaos would result."

-- Emil Artin, Bulletin of the American Mathematical Society, 1953

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

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:

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.