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.