Read Online The Science of Discworld II by Terry Pratchett - Free Book Online

× 5 … × 5, with a hundred fives. This is 5 100 , which is rather big. To be precise, it is
78886090522101180541172856528278622
96732064351090230047702789306640625
    (we’ve broken the number in two so that it fits the page width) which has 70 digits. It took a computer algebra system about five seconds to work that out, by the way, and about 4.999 of those seconds were taken up with giving it the instructions. And most of the rest was used up printing the result to the screen. Anyway, you now see why combinatorics is the art of counting without actually counting ; if you listed all the possibilities and counted them ‘1, 2, 3, 4 …’ you’d never finish. So it’s a good job that the university administrator wasn’t in charge of car parking.
    How big is L-space? The Librarian said it is infinite, which is true if you used infinity to mean ‘a much larger number than I can envisage’ or if you don’t place an upper limit on how big a book can be, 3 or ifyou allow all possible alphabets, syllabaries, and pictograms. If we stick to ‘ordinary-sized’ English books, we can reduce the estimate.
    A typical book is 100,000 words long, or about 600,000 characters (letters and spaces, we’ll ignore punctuation marks). There are 26 letters in the English alphabet, plus a space, making 27 characters that can go into each of the 600,000 possible positions. The counting principle that we used to solve the car-parking problem now implies that the maximum number of books of this length is 27 600,000 , which is roughly 10 860,000 (that is, an 860,000-digit number). Of course, most of those ‘books’ make very little sense, because we’ve not yet insisted that the letters make sensible words. If we assume that the words are drawn from a list of 10,000 standard ones, and calculate the number of ways to arrange 100,000 words in order, then the figure changes to 10,000 100,000 , equal to 10 400,000 , 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