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.

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 Character and Virtue of Perfect Numbers

In keeping with our presentation of mathematical ideas from the qualitative symbolic perspective, we now turn to a consideration of the perfect numbers.  A perfect number is equal to the sum of its proper divisors.  This is in contrast to a deficient number, for which the sum of the divisors is less; and a super-abundant number, whose divisors yield a sum greater than the number itself.  This classification is enlightening when we consider numbers as having a personality, a quality of character, and an ethical behavior, rather than as pure quantity.

Examples of perfect numbers are 6, 28,  and 496.  Let's see why each of these fits the definition of a perfect number to illustrate the concept:

• The proper divisors of 6 are 1, 2 and 3.  When we take the sum of these, 1+2+3 = 6.
•  The proper divisors of 28 are 1, 2, 4, 7, and 14.  Their sum 1+2+4+7+14 = 28.
•  The proper divisors of 496 are 1, 2, 4, 8, 16, 31, 62, 124 and 248.  Their sum 1+2+4+8+16+31+124+248 = 496.

The perfect numbers are extremely rare among the integers.  There are currently only 47 known perfect numbers.  The discovery of perfect numbers is inherently connected to the discovery of Mersenne primes, and the there is a famous distributed-computing project devoted to their discovery, known as the Great Internet Mersenne Prime Search (GIMPS).

The deficient numbers and super-abundant numbers are in direct contrast to the perfect numbers.  Here are some examples: 21 is deficient since the sum of its divisors is 1+3+7 = 10 which is less than 21; while 12 is super-abundant as the sum of its divisors is 1+2+3+4+6 = 16 which is greater than 12.

We begin with a quantitative phenomenon and proceed to explore the qualitative symbolism.  If a deficient number had a personality, how would it behave?  It is best described as someone who is lazy, poor, unskilled, weak, or uneducated.  If a super-abundant number had a personality, how would it behave?  It could be described as greedy, prideful, self-aggrandizing, indulging or possessive.  What about perfect numbers?  Perhaps accommodating, comforting, helpful, caring, assisting, or welcoming.

The great English Platonist Thomas Taylor writes eloquently on this classification of numbers in his Theoretic Arithmetic of the Pythagoreans:

The super-abundant numbers are such as by an immoderate plenitude, exceed as it were, by the numerosity of their parts, the measure of their proper body.  On the contrary, the deficient numbers being as it were, oppressed by poverty, are less than the sum of their parts.

Such therefore are these numbers, the former of which in consequence of being surpassed by its parts, resembles one born with a multitude of hands in a manner different from the common order of nature, or one whose body is formed from the junction of three bodies, or any other production of nature which has been deemed monstrous by the multiplication of its parts.  But the latter of these numbers resembles one who is born with a deficiency of some necessary part, as the one-eyed Cyclops, or with the want of some other member.

Between these however, as between things equally immoderate, the number which is called perfect is alloted the temperament of a middle limit, and is in this respect the emulator of virtue; for it is neither extended by a superfluous progression, nor remitted by a contracted diminution; but obtaining the limit of a medium, and being equal to its parts, it is neither overflowing through abundance, nor deficient through poverty.

With that said, Taylor then goes on to describe the virtues of the perfect numbers:

Perfect numbers therefore, are beautiful images of the virtues which are certain media between excess and defect, and are not summits, as by some of the ancients they were supposed to be.

And evil indeed is opposed to evil, but both are opposed to one good. Good however, is never opposed to good, but to two evils at one and the same time.  Thus timidity is opposed to audacity, to both which the want of true courage is common; but both timidity and audacity are opposed to fortitude.

Craft also is opposed to fatuity, to both of which the want of intellect is common; and both these are opposed to liberality.  And in a similar manner in the other virtues; by all which it is evident that perfect numbers have a great similitude to the virtues.

But they also resemble virtues on another account, for they are rarely found, as being few, and they are generated in a very constant order.  On the contrary, an infinite multitude of superabundant and deficient numbers may be found, nor are they disposed in any orderly series, nor generated from any certain end; and hence they have a great similitude to the vices, which are numerous, inordinate, and indefinite.

Thomas Taylor, 1758-1835

What we see happening with the qualities of perfect numbers is a function of balance and centering.  And this is exactly how the quantitative aspect of the number is a balance between the too-little of the deficient number and the too-much of the super-abundant number, and holds the position of a center between these extremes.

Are we then surprised that the perfect numbers are so few and far between?  How many people in this world have a harmonious balance between giving and taking?  Indeed it is a rare quality.  The qualities of the perfect numbers are to be imitated by those who are cultivating virtues, which should be everyone!

The study of the perfect numbers was initiated by the Pythagoreans, and mathematicians are still exploring their mysteries to this day.  It is still an open problem to determine if there are infinitely many perfect numbers, and whether there exists any odd perfect numbers, all the known examples being even.

The Pythagoreans, a secret society of spiritual mathematicians

 Let's explore the examples of perfect numbers and see how they come up in geometry and spiritual symbolism, beginning with the number 6.

A deep mathematical mystery is that 6 circles will exactly fit around a central circle with the same radius and whose centers intersect on the circumference, as shown here:

Six around the One

This figure is called the Seed of Life and it is the primary archetype for all Sacred Geometry, resulting in a progression into the Flower of Life and the Tree of Life, as well as sacred figures.

The fact that the sixth circle completes the figure, resulting in this mysteriously symmetrical arrangement, could be why the Book of Genesis describes the creation of the world in six days, with the seventh day being a day of rest, which would be a return to the central circle.

Notice that this pattern contains the blueprint for the Cube, one of the five regular solids known colloquially as the Platonic solids, to which Plato assigned the element of Earth.

Blueprint of the Cube

Isometric view of the Cube

The Cube is the regular solid with 6 square faces.  The 6 faces correspond to the 6 directions of space: up, down, front, back, left, and right.  This mystery is described in the Kabbalistic text Sepher Yetzirah, where God creates the Cube of Space by infinite extent of His being through the 6 directions, and then sealing them with the 6 permutations of the three-letter Divine Name IHV (Yod-Heh-Vav).  6 is the 3rd Triangular Number with 1+2+3=6, while the 6th Triangular Number is 21, since 1+2+3+4+5+6=21 and 21 is the sum of the letters IHV (10+5+6=21). 

Among the other Platonic solids, the Octahedron is the dual solid to the Cube, and has 6 vertices.  The Tetrahedron has 6 edges.

The Octahedron has 6 vertices

The Tetrahedron has 6 edges

Now let's consider 28.  There are approximately 28 days in a lunar cycle.  It is the 7th Triangular Number with 1+2+3+4+5+6+7 = 28.  This is interesting in light of Genesis 1:1 "In the beginning, God created the Heaven and the Earth" which in Hebrew has a total of 7 words and 28 letters.

Genesis 1:1 - Seven words, Twenty-eight letters
28 is a perfect number

This definitely suggests that the plan that God had in mind when creating the Heavens and the Earth was based on perfection from the very beginning.

How about 496?  It is the 31st Triangular Number, and 31 is the Divine Name El (Aleph-Lamed) attributed to the 4th Sephiroth in Kabbalah.  The 4th Triangular Number is 10, which shows up as the Tetraktys.

The Tetraktys, representing the Decad
1+2+3+4 = 10

The Hebrew word for Kingdom is Malkut, which Kabbalah uses as a name for the 10th Sephiroth.  Malkut is the Kingdom and represents the Divine archetype of the physical world.  The letters which comprise the word Malkut add to 496!  This says that the physical world is a perfect Kingdom, but we need to see it as being filled with divinity in order to realize our citizenship in this Kingdom.  Now modern physics has even discovered the significance of the perfect number 496, since in order for superstring theory to make sense, the dimension of the gauge group must be 496:

The number 496 is a very important number in superstring theory. In 1984, Michael Green and John H. Schwarz realized that one of the necessary conditions for a superstring theory to make sense is that the dimension of the gauge group of type I string theory must be 496. The group is therefore SO(32). Their discovery started the first superstring revolution. It was realized in 1985 that the heterotic string can admit another possible gauge group, namely E8 x E8.
-- Source: http://en.wikipedia.org/wiki/Number_496

The purpose of this article was to weave through the interplay of quantitative and qualitative aspects of numbers, so we can see the difference of these perspectives and yet how the qualitative aspects are derived from the quantitative properties.  This perspective has us bouncing back and forth between the left and right hemispheres of the brain, and through the harmonization of these polar opposites we find fertile ground for the growth of new ideas and insights into how our minds work and who we are spiritually, keeping with the program of Via Mathesis.

Books mentioned in this article:

Rekindling the Flame of Platonic Virtures in Mathematics Education

I think that today's mathematicians have lost the understanding of the spiritual virtues of mathematical discipline.  In my previous post, Geometric Objects as Platonic Forms, I described the distinction between the shapes perceived through the senses vs. the ideal forms perceived through thought.  The education philosophy that we have today is quite different from that of Plato's Academy.  While many are aware that the inscription over the door of the Academy barred those "ignorant of geometry" from entering, it is not clear why, and there is a real mystery on how exactly they were using geometry once you got past the guardians of the knowledge.  In this post I will try to reconstruct the Platonic virtues of mathematical education, show how modern mathematics has forgotten about this program.

The School of Athens

While it is true that you can't help but be taken to the world of ideas when doing theoretical mathematics, for the most part this happens unconsciously, and when pressed to explain why they do what they do, we usually don't get a very articulate response.  "Because it is beautiful" certainly works at face value, but there must be something deeper pushing mathematicians to work like they do.  An example of this perspective is a familiar quote from the early 20th century Cambridge mathematician G. H. Hardy his famous Apology:

"The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics."

G. H. Hardy, 1877-1947

If you read Hardy carefully, he is definitely not a Platonist.  He seems to think that the "patterns" belong to the mathematician.  Plato would have said that the mathematical truths exist eternally in the Mind of God, and that we as mathematicians are like explorers and surveyors in this realm, and analogous to cartographers who report their findings back to the community so that others may be lead to the same reality. 

There is much subtlety in this distinction.  Consider another famous quotation from the Apology: "A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas."  While he is certainly aware of the indestructibility of mathematical objects and theorems, he is definitely saying that they do not have an existence before the mathematician writes down the proof. 

Hardy was in a state of depression when he wrote the Apology, as he was in the later stages of life and his mental acuity was waning, and could not work as efficiently as he could when younger.  He writes that "no mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game."  He thinks that doing mathematics is only about proving theorems, and this is the majority opinion in today's academia.  After all, if this is what you are getting paid to do, then its hard to blame him. 

But where is the spiritual virtue in this?  Doesn't mathematics enable us to know true freedom, to know the eternal vs. the perishable, to understand who we are and how we work on the inside?  In short, Hardy has no patience for a serious Via Mathesis, which I define as spiritual self-knowledge gained through mathematical discipline. 

For Hardy, discovering a theorem that has already been proved by someone else has no value.  This shows that the emphasis in modern mathematical academic circles is not in aquiring spiritual self-knowledge, but in getting new theorems published.  I'm not saying that there is no value in this, because of course it has tremendous value to add to the collection of human knowledge and is a testament to our superiority over nature, and puts us in a different class of being above the animal realm.  I'm saying that there is more to the story than this pursuit, and makes people think that unless they are proving new theorems, then they aren't real mathematicians.

If working through the struggles and travails of trying to prove a theorem that you know has already been proved leads to spiritual self-knowledge, then this is a virtuous task.  This is so much more important for a Platonist who treats mathematics more as a system of spiritual self-initiation into the world of ideas, rather than someone who is just trying to get new results published with the hopes of becoming "immortalized" in Hardy's narrow-minded sense:

"Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. 'Immortality' may be a silly word, but probably a mathematician has the best chance of whatever it may mean."

 The Platonists, following the Pythagoreans, affirmed the doctrine of the immortality of the soul and metempsychosis, or reincarnation, as described in the Phaedo and Phaedrus, and is alluded to in the last book of the Republic with the Myth of Er.  If the soul is immortal, then we have the best means of becoming conscious of our own immortality by learning what is immortal in the soul, and these are the mathematical ideas.  Then proving any theorem by yourself, regardless if it is new, and it could be something from classical geometry such as Euclid, will lead to the kind of spiritual self-knowledge I am referring to as the Platonic virtue of mathematical discipline.  For Hardy, however, 'immortality' is only granted to those whose names get published in academic journals and in books on the history of mathematics.  Rekindling the flame of Platonic virtues in today's mathematical education is what I am hoping to do with my work.

Geometric Objects as Platonic Forms

It is in fact true that no one has ever seen a true circle with their eyes.  When we explore the meaning of this seemingly preposterous proposition, we shall discover the right way to view mathematical objects, especially geometrical ones.  In the terminology of Platonic philosophy, there is a difference between the geometric objects perceived with the organs of physical sense-perception vs. the pure thought-picture which is apprehended by the mind alone.  This distinction is an exercise in discrimination between the real and the unreal, which is a main goal of Platonic education.

Plato, holding a manuscript of the Timaeus

In Plato's Timaeus, we find that sorting out the subtleties of this distinction is the necessary prerequisite for understanding the rest of Timaeus' exegesis describing the creation of the world.  In other words, unless we comprehend the difference between the material vs. the form/idea, then we are incapable of understanding anything else.

We must, then, in my judgment, first make this distinction: what is that which is always real and has no becoming, and what is that which is always becoming and never real?  That which is apprehensible by thought with a rational account is the thing that is always unchangeably real; whereas that which is the object of belief together with unreasoning sensation is the thing that becomes and passes away, but never has real being.
-- Plato, Timaeus (27D-28A)

The geometrical objects which we see with our eyes are not the real ones, since there is always some imperfection due to their being drawn with pencil or similar apparatus.  Euclid defines a line as "breadthless length".  However, upon closer scrutiny, a drawn line will be seen to have some measure of thickness, albeit on the scale of millimeters, but nonetheless it is not what Euclid meant by a line

A circle, but this not the Platonic form of the circle.

Therefore, a circle drawn on paper and seen with the eyes is not the real circle.  It is in the category of "the thing that becomes and passes away, but never has real being."  Eventually the paper on which the circle is drawn will decay and the drawn circle with it.  As an experiment, draw a nice circle on paper with a compass, then burn the paper.  You will then understand that, although the drawn circle has passed away, the idea of the circle remains.  This will help you perceive that the real circle is "that which is apprehensible by thought" and is "the thing which is always unchangeably real".

A similar thought is presented by the Neoplatonist Porphyry in his Life of Pythagoras:

As the geometricians cannot express incorporeal forms in words, and have recourse to drawings of figures, saying "This is a triangle", and yet do not mean that the actually seen lines are the triangle, but only what they represent, the knowledge in the mind, so the Pythagoreans used the same objective method in respect to first reasons and forms

Why then do mathematicians employ drawings in their work?  Because they are a powerful tool to lead the mind into the higher realm of the forms, like doorways that allow us to pass from the material world into the ideal world.  Practice with theoretical geometry is really an exercise in training the mind to gain freedom of mobility in this realm.  To this wrote Nicomachus of Gerasa in his Introduction to Arithmetic:

For it is clear that these studies are like ladders and bridges that carry our minds from things apprehended by sense and opinion to those comprehended by the mind and understanding, and from those material, physical things, our foster-brethren known to us from childhood, to the things with which we are unacquainted, foreign to our senses, but in their immateriality and eternity more akin to our souls, and above all to the reason which is in our souls.

With patience and persistence, anyone can begin a study of mathematics as a spiritual discipline and find themselves in the world of ideas.  You will find that a new power of freedom opens inside the soul, a freedom which leads to real knowledge of the self, and to a true happiness that lasts.  We will continue exploring the methods of how this is achieved on Via Mathesis.