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:

Comments

Why I hate opera

If I'm honest, the title of this post is an exaggeration to make a point. I don't really hate opera. There are a couple of operas - notably Monteverdi's Incoranazione di Poppea and Purcell's Dido & Aeneas - that I quite like. But what I do find truly sickening is the reverence with which opera is treated, as if it were some particularly great art form. Nowhere was this more obvious than in ITV's 2010 gut-wrenchingly awful series Pop Star to Opera Star , where the likes of Alan Tichmarsh treated the real opera singers as if they were fragile pieces on Antiques Roadshow, and the music as if it were a gift of the gods. In my opinion - and I know not everyone agrees - opera is: Mediocre music Melodramatic plots Amateurishly hammy acting A forced and unpleasant singing style Ridiculously over-supported by public funds I won't even bother to go into any detail on the plots and the acting - this is just self-evident. But the other aspects need some exp

Is 5x3 the same as 3x5?

The Internet has gone mildly bonkers over a child in America who was marked down in a test because when asked to work out 5x3 by repeated addition he/she used 5+5+5 instead of 3+3+3+3+3. Those who support the teacher say that 5x3 means 'five lots of 3' where the complainants say that 'times' is commutative (reversible) so the distinction is meaningless as 5x3 and 3x5 are indistinguishable. It's certainly true that not all mathematical operations are commutative. I think we are all comfortable that 5-3 is not the same as 3-5. However. This not true of multiplication (of numbers). And so if there is to be any distinction, it has to be in the use of English to interpret the 'x' sign. Unfortunately, even here there is no logical way of coming up with a definitive answer. I suspect most primary school teachers would expands 'times' as 'lots of' as mentioned above. So we get 5 x 3 as '5 lots of 3'. Unfortunately that only wor

Why backgammon is a better game than chess

I freely admit that chess, for those who enjoy it, is a wonderful game, but I honestly believe that as a game , backgammon is better (and this isn't just because I'm a lot better at playing backgammon than chess). Having relatively recently written a book on game theory, I have given quite a lot of thought to the nature of games, and from that I'd say that chess has two significant weaknesses compared with backgammon. One is the lack of randomness. Because backgammon includes the roll of the dice, it introduces a random factor into the play. Of course, a game that is totally random provides very little enjoyment. Tossing a coin isn't at all entertaining. But the clever thing about backgammon is that the randomness is contributory without dominating - there is still plenty of room for skill (apart from very flukey dice throws, I can always be beaten by a really good backgammon player), but the introduction of a random factor makes it more life-like, with more of a sense

Search This Blog