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.