Thursday, March 22, 2007

Penny-wise - Pound-foolish

Imagine a cube - 100 pence long, 100 pence wide and 100 pence high. This cube contains one million pennies, and so is worth £10,000.

Of course, since 100p = £1, we could equally well take this cube to be one pound coin long by one pound coin wide by one pound coin high. In other words, a cube containing just one £1 coin.

This proves the old saying of the title.

Another example of the hallucinatory effects of long periods in departure lounges ...

Postscript: 25/2/07.

You may think the above is quite frivolous. Yes, but also interesting as to what's wrong with the argument. The correct answer has to disentangle the following.

1. The relationship between, centimetres and metres, and cubic centimetres and cubic metres.

2. The relationship between pennies and pounds.

3. Would a correct analogy to (1) work if we had a concept of cubic pennies and cubic pounds?

4. The confusion between the purely logical relationship of (2) and the fact that both pennies and pounds, as coins, are three dimensional physical objects which can be placed in lines, and in the cells of a three dimensional cube: a process which conflates the distinctions of (1-3).

To properly make everything explicit you end up with a nicely elaborated commutative diagram (exercise for the reader).