The Science of Discworld II

and this is quite a bit smaller … but still enormous. Mind you, most of those books wouldn’t make much sense either; they’d read something like ‘Cabbage patronymic forgotten prohibit hostile quintessence’ continuing at book length. 4 So maybe we ought to work with sentences … 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, of all the books that could have existed. In particular, we’re never going to run out of new books to write.
    Poincaré’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 millionnumbers. 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.
    Now, the income of a taxi-driver depends on circumstances. On good days, with lots of passengers around, he will do well; on bad days, he won’t. A rational taxi-driver will therefore work longer on good days and give up early on bad ones. However, a study of taxi-drivers in New York carried out by Colin Camerer and others shows the exact opposite. The taxi-drivers seem to set themselves a daily target, and stop working once they reach it. So they work shorter hours on good days, and longer hours on bad ones. They could increase their earnings by 8 per cent just by working the same number of hours every day, for the same total working time. If they worked longer on good days and shorter on bad ones, they could increase their earnings by 15 per cent. But they don’t have a good enough intuition for economic phase space to appreciate this. They are adopting a common human trait of placing too much value on what they have today, and too little on what they may gain tomorrow.
    Biology, too, has been invaded by phase spaces. The first of these to gain widespread currency was DNA-space. Associated with every living organism is its genome, a string of chemical molecules called DNA. The DNA molecule is a double helix, two spirals wrapped rounda common core. Each spiral is made up of a string of ‘bases’ or ‘nucleotides’, which come in four varieties: cytosine, guanine, adenine, thymine, normally abbreviated to their initials C, G, A, T. The sequences on the two strings are ‘complementary’: wherever C appears on one string, you get G on the other, and similarly for A and T. So the DNA contains two copies of the sequence, one positive and one negative, so to speak. In the abstract, then, the genome can be thought of as a single sequence of these four letters, something like AATG-GCCTCAG … going on for rather a long time. The human genome, for example,