Science of Discworld III

though, what you reach is slightly less than two. But the amount by which it is less than two keeps shrinking. The sum sort of sneaks up on the correct answer, without actually getting there; but the amount by which it fails to get there can be made as small as you please, by adding up enough terms.
    Remind you of anything? It looks suspiciously similar to one of Zeno/Xeno’s paradoxes. This is how the arrow sneaks up on its victim, how Achilles sneaks up on the tortoise. It is how you can do infinitely many things in a finite time. Do the first thing; do the second thing one minute later; do the third thing half a minute after that; then the fourth thing a quarter of a minute after that … and so on. After two minutes, you’ve done infinitely many things.
    The realisation that infinite sums can have a sensible meaning is only the start. It doesn’t dispel all of the paradoxes. Mostly, it justsharpens them. Mathematicians worked out that some infinities are harmless, others are not.
    The only problem left after that brilliant insight was: how do you tell? The answer is that if your concept of infinity does not lead to logical contradictions, then it’s safe to use, but if it does, then it isn’t. Your task is to give a sensible meaning to whatever ‘infinity’ intrigues you. You can’t just assume that it automatically makes sense.
    Throughout the eighteenth and early nineteenth centuries, mathematics developed many notions of ‘infinity’, all of them potential. In projective geometry, the ‘point at infinity’ was where two parallel lines met: the trick was to draw them in perspective, like railway lines heading off towards the horizon, in which case they appear to meet on the horizon. But if the trains are running on a plane, the horizon is infinitely far away and it isn’t actually part of the plane at all – it’s an optical illusion. So the point ‘at’ infinity is determined by the process of travelling along the train tracks indefinitely. The train never actually gets there. In algebraic geometry a circle ended up being defined as ‘a conic section that passes through the two imaginary circular points at infinity’, which sure puts a pair of compasses in their place.
    There was an overall consensus among mathematicians, and it boiled down to this. Whenever you use the term ‘infinity’ you are really thinking about a process. If that process generates some well-determined result , by however convoluted an interpretation you wish, then that result gives meaning to your use of the word ‘infinity’, in that particular context.
    Infinity is a context-dependent process. It is potential.
    It couldn’t stay that way.
    David Hilbert was one of the top two mathematicians in the world at the end of the nineteenth century, and he was one of the great enthusiasts for a new approach to the infinite, in which – contraryto what we’ve just told you – infinity is treated as a thing, not as a process. The new approach was the brainchild of Georg Cantor, a German mathematician whose work led him into territory that was fraught with logical snares. The whole area was a confused mess for about a century (nothing new there, then). Eventually he decided to sort it out for good and all by burrowing downwards rather than building ever upwards, and putting in those previously non-existent foundations. He wasn’t the only person doing this, but he was among the more radical ones. He succeeded in sorting out the area that drove him to these lengths, but only at the expense of causing considerable trouble elsewhere.
    Many mathematicians detested Cantor’s ideas, but Hilbert loved them, and defended them vigorously. ‘No one,’ he declaimed, ‘shall expel us from the paradise that Cantor has created.’ It is, to be sure, as much paradox as paradise. Hilbert explained some of the paradoxical properties of infinity à la Cantor in terms of a fictitious hotel, now known as Hilbert’s Hotel.
    Hilbert’s Hotel has infinitely many rooms. They are numbered 1, 2, 3, 4 and so on indefinitely. It is an instance of actual infinity – every room exists now , they’re not still building room umpty-ump gazillion and one. And when you arrive there, on Sunday morning, every room is occupied.
    In a finite hotel, even with umpty-ump gazillion and one rooms, you’re in trouble. No amount of moving people around can create an extra room. (To keep it simple, assume no sharing: each room has exactly one occupant, and