Math and Philosophy

Don't you love it when seemingly unrelated things come together?

A few years ago, I heard (or read, I don't remember which) Andrew Kern say that teachers ought to read The Meno, one of Plato's dialogues, regularly in order to become a better teacher.  I printed it out and tried reading it, but seriously.  It was heavy going and I didn't get very far.

When I noticed that Coursera was offering a class called Reason and Persuasion: Thinking Through Three Dialogues by Plato I immediately checked to see which three dialogues, and sure enough, The Meno was one of them, so I signed up.  The class started last month and it's been a huge help.  We spent two weeks on Euthyphro, and now I'm near the end of our second week of Meno.

Before starting this class, I had no idea that math would come up, but it plays an important role in The Meno, which you'll already know if you've ever read it.  The dialogue discusses the questions What is virtue? and Is virtue the sort of thing that can be taught?  But right smack in the middle of this dialogue is a geometry lesson.

While discussing this, the Coursera instructor mentions an essay by Bertrand Russell, "The Study of Mathematics," and quotes from it:

One of the chief ends served by mathematics, when rightly taught, is to awaken the learner's belief in reason, his confidence in the truth of what has been demonstrated, and in the value of demonstration. This purpose is not served by existing instruction; but it is easy to see ways in which it might be served. At present, in what concerns arithmetic, the boy or girl is given a set of rules, which present themselves as neither true nor false, but as merely the will of the teacher, the way in which, for some unfathomable reason, the teacher prefers to have the game played.

I don't know about you, but once I got to fourth grade math where they were actually teaching me new stuff (everything before that was pretty much just me getting to show off what I'd already figured out on my own in the normal course of childhood) this is exactly what math felt like to me -- arbitrary rules.

Russell goes to to critique my tenth-grade geometry class.  Do you remember geometry?  In that class they drive you insane by making your prove things that don't need to be proven -- you can tell just by looking at them.  Russell says this is a huge mistake:

[T]he learner should be allowed at first to assume the truth of everything obvious, and should be instructed in the demonstrations of theorems which are at once startling and easily verifiable by actual drawing . . . . In this way belief is generated; it is seen that reasoning may lead to startling conclusions, which nevertheless the facts will verify; and thus the instinctive distrust of whatever is abstract or rational is gradually overcome. *

This is what happens during the geometry lesson in The Meno.  Socrates calls over a slave boy, who has had no instruction in geometry, and draws a square that's two feet long on each side, and asks the boy what the square's area is.  The boy says that it is four square feet.

Next Socrates asks the boy what he would do in order to make a square whose area is twice this one.  The boy says that he would double the length of each side of the square.  They draw that out, but of course that turns out to make a square that's sixteen square feet, so the boy tries again -- three feet on a side, which still makes too big a square.

Of course, you probably know that the square root of 8 is not a rational number -- it's not something that can be expressed as a fraction.  It's less than 3, but it's more than 2 3/4.  In Socrates' time they didn't have a way of expressing that number since they weren't using the decimal system, so how in the world can this poor slave boy come up with the right answer?

Turns out it can be done and Socrates helps the boy figure out how to do it.  It's pretty clever, and I'll leave it to your imagination.**

The next passage quoted in my class was this one:

What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement. Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great work springs. Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world.

A lot of my reading the last couple of years about what math is has been hinting at this kind of thing.  Math is ennobling.  This was so exciting that I decided to find and read the entire essay.  Even if I hadn't had this introduction Russell would have won me over in his first paragraph:

In regard to every form of human activity it is necessary that the question should be asked from time to time, What is its purpose and ideal? In what way does it contribute to the beauty of human existence? As respects those pursuits which contribute only remotely, by providing the mechanism of life, it is well to be reminded that not the mere fact of living is to be desired, but the art of living in the contemplation of great things. [Emphasis added]

If you've read the John Gould Fletcher quote in my sidebar you'll know why that phrase resonates with me.

Our imaginations are meant to live in a kind of temple, and originally it was this idea that gave shape to education, or what we would call a Classical Education.  But in recent generations why we're teaching what we're teaching has been forgotten, or, as Russell puts it:

Dry pedants possess themselves of the privilege of instilling this knowledge: they forget that it is to serve but as a key to open the doors of the temple; though they spend their lives on the steps leading up to those sacred doors, they turn their backs upon the temple so resolutely that its very existence is forgotten, and the eager youth, who would press forward to be initiated to its domes and arches, is bidden to turn back and count the steps.

I haven't finished The Meno yet, so I've only got an inkling of what geometry has to do with virtue, but dear Lord, please save me from being a dry pedant!

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*Here, Russell is discussing teaching geometry with Euclid.  Euclid starts off with a long list of definitions and postulates, which is a fine way to organize the information, but a lousy way to teach it -- not that I know what the best way is.  I'm just reporting what I've learned so far.  :-p

**Hint:  It has to do with triangles.