At any rate, even if we cut the numbers down in that manner, it turns out that the universe is not big enough to contain that many physical books. So it's a good job that L-space is available, and now we know why there's never enough shelf space. We like to think that our major libraries, such as the British Library or the Library of Congress, are pretty big. But, in fact, the space of those books that actually exist is a tiny, tiny fraction of L-space, all the books that could have existed. In particular, we're never going to run out of new books to write.

Poincare's phase space viewpoint has proved to be so useful that nowadays you'll find it in every area of science -and in areas that aren't science at all. A major consumer of phase spaces is economics. Suppose that a national economy involves a million different goods -cheese, bicycles, rats-on-a-stick, and so on. Associated with each good is a price, say .2.35 for a lump of cheese, .449.99 for a bicycle, .15.00 for a rat-on-a-stick. So the state of the economy is a list of one million numbers. The phase space consists of all possible lists of a million numbers, including many lists that make no economic sense at all, such as lists that include the .0.02 bicycle or the .999,999,999.95 rat. The economist's job is to discover the principles that select, from the space of all possible lists of numbers, the actual list that is observed.

The classic principle of this kind is the Law of Supply and Demand, which says that if goods are in short supply and you really, really want them, then the price goes up. It sometimes works, but it often doesn't. Finding such laws is something of a black art, and the results are not totally convincing, but that just tells us that economics is hard. Poor results notwithstanding, the economist's way of thinking is a phase space point of view.

Here's a little tale that shows just how far removed economic theory is from reality. The basis of conventional economics is the idea of a rational agent with perfect information, who maximises utility. According to these assumptions, a taxi-driver, for example, will arrange his activities to generate the most money for the least effort.