Science of Discworld III

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    The counting process only stops if you run out of breath; it does not stop because you’ve run out of numbers . Though a near-immortal might perhaps run out of universe in which to write the numbers down, or time in which to utter them.
    In short: there exist infinitely many numbers.
    The wonderful thing about that statement is that it does not imply that there is some number called ‘infinity’, which is bigger than any of the others. Quite the reverse: the whole point is that there isn’t a number that is bigger than any of the others. So although the process of counting can in principle go on for ever, the number you have reached at any particular stage is finite. ‘Finite’ means that you can count up to that number and then stop.
    As the philosophers would say: counting is an instance of potential infinity. It is a process that can go on for ever (or at least, so it seems to our naive pattern-recognising brains) but never gets to ‘for ever’.
    The development of new mathematical ideas tends to follow a pattern. If mathematicians were building a house, they would start with the downstairs walls, hovering unsupported a foot or so above the damp-proof course … or where the damp-proof course ought to be. There would be no doors or windows, just holes of the right shape. By the time the second floor was added, the quality of the brickwork would have improved dramatically, the interior walls would be plastered, the doors and windows would all be in place, and the floorwould be strong enough to walk on. The third floor would be vast, elaborate, fully carpeted, with pictures on the walls, huge quantities of furniture of impressive but inconsistent design, six types of wallpaper in every room … The attic, in contrast, would be sparse but elegant – minimalist design, nothing out of place, everything there for a reason. Then, and only then, would they go back to ground level, dig the foundations, fill them with concrete, stick in a damp-proof course, and extend the walls downwards until they met the foundations.
    At the end of it all you’d have a house that would stand up. Along the way, it would have spent a lot of its existence looking wildly improbable. But the builders, in their excitement to push the walls skywards and fill the rooms with interior decor, would have been too busy to notice until the building inspectors rubbed their noses in the structural faults.
    When new mathematical ideas first arise, no one understands them terribly well, which is only natural because they’re new . And no one is going to make a great deal of effort to sort out all the logical refinements and make sense of those ideas unless they’re convinced it’s all going to be worthwhile. So the main thrust of research goes into developing those ideas and seeing if they lead anywhere interesting. ‘Interesting’, to a mathematician, mostly means ‘can I see ways to push this stuff further?’, but the acid test is ‘what problems does it solve?’ Only after getting a satisfactory answer to these questions do a few hardy and pedantic souls descend into the basement and sort out decent foundations.
    So mathematicians were using infinity long before they had a clue what it was or how to handle it safely. In 500 BC Archimedes, the greatest of the Greek mathematicians and a serious contender for a place in the all-time top three, worked out the volume of a sphere by (conceptually) slicing it into infinitely many infinitely thin discs, like an ultra-thin sliced loaf, and hanging all the slices from a balance, to compare their total volume with that of a suitable shapewhose volume he already knew. Once he’d worked out the answer by this astonishing method, he started again and found a logically acceptable way to prove he was right. But without all that faffing around with infinity, he wouldn’t have known where to start and his logical proof wouldn’t have got off the ground.
    By the time of Leonhard Euler, an author so prolific that we might consider him to be the Terry Pratchett of eighteenth-century mathematics, many of the leading mathematicians were dabbling in ‘infinite series’ – the school child’s nightmare of a sum that never ends . Here’s one:
    1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …
    where the ‘…’ means ‘keep going’. Mathematicians have concluded that if this infinite sum adds up to anything sensible, then what it adds up to must be exactly two. 1 If you stop at any finite stage,