Language, Axioms and Bananas

Talking about Algebra with my grade 8 class today, I was struck again by how confusing and apparently random Maths can be. I remember being totally mystified by Maths at various points in my relationship with the subject. Occasionally I still am mystified.

But something that has helped me to deal with my mystification is to remember that Maths as we know it is not the be all and end all and only conceivable possible way. It could (theoretically anyway) have all happened completely differently.

How? I hear you cry.

Well, Maths is a language, just like other languages (despite having a few extra features and lower comprehensibility ratings than most). The fact that "banana" represents banana does not say something innate about the relationship between word and object (this is Wittgenstein and those guys, but watered down). It just happens to be the word that history and society and chance and whatever other factors have attached to our familiar yellow fruit.

Same with Maths. All of our Mathematical system is based on certain axioms (like the fact that a point has no dimensions). These axioms (stated most famously by Euclid) are assumed to be true in the proofs and ways of thinking about things which are so entrenched in us. It is very difficult to imagine them not being true. Yet in actual fact this is only the case because its convenient that it should be, or because we're used to it being so.


Plus, of course, in order to do anything interesting or useful we have to have a common frame of reference. So by all means lets argue about something more interesting than whether we should write 2x2 as 2^2 or 2_2; or whether 3ab really does mean 3 x a x b!

The other alternative is very much like saying that banana is a very bad word and difficult to pronounce, so from now on the yellow fruit shall be known only as Ba.

Proof and Imagination

A few years ago I saw a movie called Proof. It's about an elderly Maths professor and his daughter. He spends his whole life trying to prove a single theorem, and eventually gets Alzheimers or something like that. His daughter sacrifices her academic career to care for him... Anyway, I won't give the plot away, but the idea of the film centres around the great value of proof.

Now my idea of a mathematical proof tends to be a hand-wavy argument followed by the conclusion: "so it seems clear that..." But through the years I've changed my mind. The great beauty of a true proof (as recognised by Socrates, Aristotle, Bertrand Russell, and various others at different points in the history of thought) is that a proof is irrefutable. Every step follows inexorably on from the previous steps in logical and unarguable progression.

Stripped down to its bare bones, a proof is a statement of pure logic. It takes you from what you know (or what you assume, axiomatically or otherwise) to you guess but do not yet know. In this sense proof is another (more or less accessible, depending on your point of view) form of lucid imagination.

It's imagination that has a shape.