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.