Friday, 24 September 2010

The wonder of cardinality

Cardinality: one leg per horseman, but we don't need to know how many
The cardinality of which I speak has nothing to do with the Pope's visit to the UK, or even Cardinal Fang from the Spanish Inquisition sketch on Monty Python. No, it's maths, but a truly wondrous aspect of maths.

Cardinality is part of set theory. It defines the size of a set. But the clever thing about cardinality is that you don't have to know how big a set is to see if it's the same size as another set. All you have to do is pair off the items in one set with the items in the other. If they have one-to-one correspondence, the sets have the same cardinality. They're the same size.

This may sound trivial but it's really quite profound. I first came across this when writing my book on Infinity. It's very useful to be able to do this with an infinite set, because it's a touch tedious counting the contents - but provided you can set up a way to pair off the items in the sets working all the way through in a repeatable pattern, you can establish the two infinite sets have the same cardinality. (Things get particularly exciting when you discover there are some infinite sets that are so big you can't pair them off this way - but that's a different story.)

That sounds a bit abstract, but in fact this approach to cardinality has been used since time immemorial in real life, practical applications. Imagine you are shepherd with what we now know is a flock of 100 sheep. But you can only count up to 3. How are you going to see if you've got all your sheep back in the fold? Easy. As they leave the fold, make a mark on a piece of wood or a slate with a line for each sheep. Then on the way back in, run your finger along the tally as each sheep comes back. You can check you've got all your sheep without ever knowing how many there are.

This technique, where you score a mark on a piece of wood (or whatever) is where we get the idea of keeping score or scoring from. And these score marks were often grouped into 20s which - you guessed it - makes 20 a score. As in three score years and ten.

Nice one, cardinality.

Okay, I can't resist:

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