MATHEMAGICAL

Dick Pountain/13 February 2006/16:19/Idealog 139

I've just finished John D. Barrow's 'The Infinite Book', an entertaining review of the various kinds of infinity, and one story in it in particular set me to to thinking about maths, reality and computers. This story, Hotel infinity, is often attributed to the famous German mathematician David Hilbert. The hotel in question has an infinite number of rooms. You turn up at reception one night but the clerk tells you he's sorry, they're fully booked. However the manager, overhearing him, tells you not to worry, he'll find you a room. He then tells the clerk to move the occupant of room 1 into room 2, the occupant of room 2 into room 3, and so on, forever, which frees up room 1 for you immediately, while no-one is left without a room.

 The story is meant to point up the odd properties of countable infinities (for example, even if you brought an infinite number of friends with you, the manager could still devise a slightly more involved procedure that will accommodate you all) but it does more. It's a pure maths story, shorn of physical considerations about whether the universe is big enough to contain an infinite hotel, or if it could, who'd pay the infinite cost of the infinite number of bricks and infinite weeks of labour ('sorry mate, we 'ad another job on this morning'.)

 Let's for the moment grant the existence of Hotel Infinity. The manager's solution of moving his guests serially will take an infinite time to complete, which isn't entirely a bad thing as it means most of them will sleep the whole night and have checked out before the signal even reaches them: it's only a few hundred who get moved in the middle of the night who'd be upset. This could be avoided by moving the guests in parallel. If the clerk had an infinite switchboard furnished with a broadcast button, he could telephone all the guests simultaneously and ask them 'could you please move into the room with the next higher number'. This should complete in just the time it takes to change rooms, so all the guests get a good night's sleep.

 However it creates a serious headache for the hotel's comms engineer. The infinite number of phone cables leading to the  rooms need to be routed through physical space and, whatever topology you dream up, this is tricky. The cables will be pretty long and the trunking pretty thick, infinitely so in fact, but this scarcely matters since the messages can't travel faster than light anyway. Wi-Fi and Bluetooth aren't much help.

 Mathematics resembles magic in the sense that it chooses to ignore some or all of the constraints imposed by existing in the physical world. Its abstract symbols (for example infinite hotels) are manipulated without any expenditure of energy and often without duration either. Barrow's book contains another amusing example that makes the point. Imagine you have an infinitely long beam of 2x2" timber. Saw it up into 2" lengths and each will make a 2" cubic block. Now arrange these blocks into a sequence of nested shells: start with one; surround it with 26 more to make a cube of side 3 blocks; surround that with 98 more to make a cube of side 5; and so ad infinitum. This single thin length of wood, which stretched from horizon to horizon and out of sight, can thus be made to fill the whole universe. Applying real world physical contstraints, you would of course wear out a infinite number of saw blades, consume an infinite amount of energy, and generate an infinite volume of sawdust in the process, and you'd also have to walk an infinite distance along the wood (pulling the wood toward you isn't on since it has infinite inertia).

 Computers occupy a realm halfway between the timeless, no-energy world of mathematical abstraction and the slow, heavy world of matter. The theory underlying computers may be mathematical but they're products of engineering. They can represent real objects - for example in pictures or sounds - by sampling them digitally and storing the list of numbers generated, but this process doesn't come quite free. It costs energy (just listen to those fans) and it takes time to sample the bits, to move and store them and to process them. Computer storage space is never infinite nor are the numbers they work on: computers can't handle the countably-infinite natural numbers but only finite subsets of them (modulo 2-to-the-power of the word-width), and ditto with floating-point approximations of the real numbers.

It's this ambiguous position in the middle ground between the worlds of maths and physics that makes computers so useful. Their digital representations of real objects may be neither energy-free nor timeless, but they need so much less energy and time than the objects they represent that they allow us to play 'what if' games with the world. An AutoCAD drawing takes much less energy to make than a physical model, so you can modify it more quickly and more often. A computer can approximate atmospheric movements faster than real-time, which is what enables weather forecasts rather than mere weather descriptions. Computer games work because virtual people and cars cost almost nothing to reproduce and don't complain, whereas real gladiatiors and chariots were strictly for Roman emperors...