Doesn’t this kind of logic remind me of something?

From The Hitchhiker’s Guide to the Galaxy, we learn this about the universe:

Population: None.

It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the universe can be said to be zero. From this is follows that the population of the whole Universe is also zero, and that any people you may meet from time to time are merely the products of a deranged imagination.

Oh yeah, now I know: Political debates!

Douglas Adams is hilarious, and if you haven’t already read the book I’m quoting, well you should. I have not finished it yet, but it’s only a matter of time, and I enjoy every minute of it. Let me have a little go at the piece above (unfortunately, I have had to make a more extensive argument than what I first had in mind – what I put down as notes in the book – because WordPress doesn’t seem to like mathematical symbols):

a) That the size of the universe is infinite does not mean that the ‘number of worlds’ is also infinite, just because ‘there’s room for them’. It simply does not follow. There’s room for a lot more than 9 planets in our solar system…

b) Even if we accept the premise that there’s an infinite number of planets, the fact that only a subset of those planets are inhabitable does not mean that there can be only a finite number of inhabitable planets. There might be, but then again there might not be. One can draw a finite subset from an infinite set, one always can. But one could also consider the marginal planet a potentially habitable planet – if you take that view, the number of inhabitable planets goes to infinity too (a constant less than one but greater than zero times infinity is still infinity). By taking this view, the next inference, that the average population of all the planets in the universe could be said to be zero, also breaks down. If both the total number of planets and the number of inhabitable planets go to infinity, you can’t say very much about the average – you end up dividing infinity with infinity, which is undefined. The only thing we know for sure in that case about the total average is that the average population of all planets is smaller than the average population of the inhabited planets alone (because the total number of planets approaches infinity faster than the number of inhabitable planets do); but we already knew that.

c) The best part is the last: From this is follows that the population of the whole Universe is also zero… I can’t even spot a faulty argument here, there is no argument, he just jumps from average population: 0 to total population: 0. The structure of this argument is quite close to the standard political argument as politicians present them: “Here we have a problem – the government must intervene”. The two arguments look equally valid to me.