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 - 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.

Video Series - Ancient Philosophy of Mathematics

I have started a series of videos on the Ancient Philosophy of Mathematics on my YouTube channel.  The series currently has 6 videos and I will be adding more in the future.  In this first set of videos, we discuss the Pythagorean and Platonic perspectives on mathematical philosophy.  The focus will be on metaphysics and ontology, symbolism and contemplation, and anagogue, or spiritual ascent.  We will be drawing material from the Introduction to Arithmetic by Nicomachus of Gerasa as well as Proclus' Commentary on Euclid's Elements.

This first video in the series will outline the distinction between Pythagorean and Platonic approaches to mathematical philosophy.  The Pythagorean approach focused on symbolism and instituted the "quadrivium", the 4-fold breakdown of all mathematics into arithmetic, harmonics, geometry, and astronomy.  The Platonic approach focused on the ontological status of mathematical objects, and grounding mathematics into a metaphysical hierarchy.

In this second video, we will outline the very basics of Proclus' philosophy of mathematics.  Proclus lived in the 5th century AD in Athens and wrote commentaries on Plato, but also on Euclid's Elements.  His commentary on Euclid is the only systematic philosophy of mathematics from antiquity, which is notable for classifying mathematics within a Platonic metaphysical hierarchy of being.  We will demonstrate Proclus' classification of mathematical objects onto the level of Understanding, which is below Intellect and above Opinion.

In part 3, will give the Pythagorean definition the Quadrivium following Nicomachus of Gerasa in his text Introduction to Arithmetic.  We begin with the division into multitide and magnitude, which is what we think of today as the discrete and continuous, or integers and real numbers.  Then multitude is split into arithmetic and harmonics, while magnitude is split into geometry and astronomy, thus establishing the Quadrivium as essentially a 4=2x2 system.

In this fourth video in the series, we will explain why arithmetic must go at the beginning of any study of the Quadrivium.  Again looking at the text of Nicomachus of Gerasa, the Introduction to Arithmetic, we will read his dialectic explaining how arithmetic naturally comes first.  If geometry were eliminated, we would have to eliminate arithmetic as well, for how could we define a triangle without the number 3?

In part five, we will explain the meanings of numbers as qualitative ideas, viewing numbers as symbols for archetypal ideas of an unfoldment process from the Monad (1) to the Decad (10).  Internalizing these meanings open us up to perceptions that lead to abilities like prophecy and divination.  We give keywords for each number, so that meditating on them unlocks the ineffable reality of numbers as Platonic ideas.

In part six, we will show how to construct meditation cards for doing the meditations on number symbolism using the meanings of numbers given in part five.  We explain the meditation process and give guidelines for how to achieve the best results, as well as indications of more advanced programming techniques that will come later.

I would appreciate any feedback you may have, please post your comments to the individual videos on the respective YouTube pages, or if you want to comment on the whole series, you can do that here.

Books mentioned in this post:

On The Pythagorean Definition of the Quadrivium

The philosophical tradition surrounding the name of Pythagoras derives its mistique from how little we know of its origins, while at the same time can claim the subsequent mathematical development and scientific advancements as verification of its basis in wisdom and truth.

Master Pythagoras

The Pythagoreans divided their teaching into a four-fold system called the Quadrivium, consisting of Numerics, Harmonics, Geometry, Cosmology. Nicomachus of Gerasa, well known as being the greatest Neo-Pythagorean of his time, can help us understand the meaning of this division of their teaching:

"Things, then, both those properly so called and those that simply have the name, are some of them unified and continuous, for example, an animal, the universe, a tree, and the like, which are properly and peculiarly called "magnitudes"; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called "multitudes", a flock, for instance, a people, a heap, a chorus, and the like.

"Wisdom, then, must be considered to be knowledge of these two forms.  Since, however, all multitude and magnitude are by their own nature of necessity infinite-- for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end, but proceeds therefore to infinity-- and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing either with magnitude, per se, or with multitude, per se, could never be formulated, for each of them is limitless in itself, multitude in the direction of the more, and magnitude in the direction of the less.  A science, however, would arise to deal with something separated from each of them, with quantity, set of from multitude, and size, set off from magnitude.

Again, to start afresh, since of quantity one kind is viewed by itself, having no relation to anything else, as "even", "odd", "perfect", and the like, and the other is relative to something else and is conceived of together with its relationship to another thing, like "double", "greater", "smaller", [etc], it is clear that two scientific methods will lay hold of and deal with the whole investigation of quantity; arithmetic, absolute quantity, and music, relative quantity.

"And once more, inasmuch as part of "size" is in a state of rest and stability, and another part in motion and revolution, two other sciences in the same way will accurately treat of "size", geometry the part that abides and is at rest, astronomy that which moves and revolves."    

 -- Nicomachus of Gerasa: Introduction to Arithmetic I - ch. 2-3

Another clear description of the way the Pythagoreans divided learning into the Quadrivium is given by the late Neo-Platonist Proclus:

"The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold.  A quantity can be considered in regard to its character by itself or in its relation to another quantity; magnitudes as either stationary or in motion.  Arithmetic, then, studies quantity as such; music the relations between quantities; geometry [studies] magnitude at rest, spherics [studies] magnitude inherently moving.  The Pythagoreans consider quantity and magnitude not in their generality, however, but only as finite in each case.  For they say that the sciences study the finite in abstraction from infinite quantities and magnitudes, since it is impossible to comprehend infinity in either of them.  Since this assertion is made by men who have reached the summit of wisdom, it is not for us to demand that we be taught about quantity in sense objects or magnitude that appears in bodies.  To examine these matters is, I think, the province of the science of nature, not that of mathematics itself."   

 -- Proclus: A Commentary on the First Book of Euclid's Elements - Prologue I ch. 7

 The study of arithmetic was given before anything else, so fundamental was the doctrine of Number for their philosophical school.  Nicomachus can help us see why arithmetic must be studied first:

"Which then of these four methods must we first learn?  Evidently, the one which naturally exists before them all, is superior and takes the place of origin and root and, as it were, of mother to the others.  And this is arithmetic, not solely because we said that it existed before all the others in the mind of the creating God like some universal and exemplary plan, relying upon which as a design and archetypal example the creator of the universe sets in order to their proper ends; but also because it is naturally prior in birth, inasmuch as it abolishes other sciences with itself, but is not abolished together with them.

"So it is with the foregoing sciences; if geometry exists, arithmetic must also needs be implied, for it is with the help of this latter that we can speak of triangle, quadrilateral, octahedron, icosahedron, double, eightfold, or one and one-half times, or anything else of the sort which is used as a term by geometry, and such things cannot be conceived of without the numbers that are implied with each one.  For how can "triple" exist, or be spoken of, unless, the number 3 exists beforehand, or "eightfold", without 8?  But on the contrary 3, 4, and the rest might be without the figures existing to which they give names.

"Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry.

"And once more is this true in the case of music; not only because the absolute is prior to the relative, as "great" to "greater" and "rich" to "richer" and "man" to "father", but also because the musical harmonies, diatessaron, diapente, and diapason, are named for numbers; similiarly all of their harmonic ratios are arithmetical ones, for the diatessaron is the ratio 4:3, the diapente that of 3:2, and the diapason the double ratio; and the most perfect, the didiapason, is the quadruple ratio.

"More evidently still astronomy attains through arithmetic the investigations that pertain to it, not alone because it is later than geometry in origin-- for motion naturally comes after rest-- nor because the motions of the stars have a perfectly melodious harmony, but also because risings, settings, progressions, retrogressions, increases, and all sorts of phases are governed by numerical cycles and quantites.

"So then we have rightly undertaken first the systematic treatment of this, as the science naturally prior, more honorable, and more venerable, and as it were, mother and nurse of the rest."    

 -- Nicomachaus of Gerasa: Introduction to Arithmetic I - ch. 4-5

It is recommended that a deeper understanding of the philosophy of the Quadrivium is gained by contemplation of these quotations from the mathematician-philosophers.

We shall be exploring the Quadrivium and related areas in future posts.

Proclus on the Mathematical Imagination

Proclus' Commentary on the First Book of Euclid's Elements offers penetrating insight on the nature of mathematical being, showing how the Platonic Ideas of the mathematical concepts are received by the mind (nous) and projected onto the imagination, which serves as a mirror for the reflections of the soul.  Geometrical figures are therefore rightly understood as doorways into the Ideal World.  Contemplation of the relation between the figures and the Ideas can lead us into the Via Mathesis, that is, the spiritual path of self-knowledge gained through mathematical insight.

Proclus (410 - 485 CE)

The following text acts like a seed, that when planted in consciousness and grown through meditative techniques yields a plentiful harvest of seed-bearing fruit (emphasis mine):

"Therefore just as nature stands creatively above the visible figures, so the soul, exercising her capacity to know, projects on the imagination, as on a mirror, the ideas of the figures; and the imagination, receiving in pictorial form these impressions of the ideas within the soul, by their means affords the soul an opportunity to turn inward from the pictures and attend to herself.

"It is as if a man looking at himself in a mirror and marveling at the power of nature and at his own appearance should wish to look upon himself directly and possess such a power as would enable him to become at the same time the seer and the seen.

"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."