Monday, January 17, 2011

Welcome to Hilbert’s Hotel

[This is a continuation of a thread begun here.]

        To make any sense of the structure of the natural numbers and all they entail, we have argued (so far without proof), requires putting to work the whole apparatus of mathematics. Now, possibly anticipating such a move on our part, you (Ned the Nominalist), grudgingly grant Kronecker the strict construction of his text, and no more:  you draw the line at the positive integers as simply given, frozen, inert.  God, you concede, made the natural numbers (in a fit of prodigality which He no doubt later came to regret), “but He did not say you might actually add them, thus allowing them to become bigger than their britches, and to stand on one another’s heads; let alone to divide them (thus yielding the by-blow of the fractions), let alone to subtract them (thus yielding the aptly-named “negative” integers; as in, “Negative on that, Mac.”), let alone take their square roots (thereby whelping the well-dubbed “irrationals” – “wackos” would be more like it); let alone – O, let remotely alone – the square root of a negative integer, thus coughing forth the demon spawn that has no name in polite company.  You start messing with any of that, Mac, you’re ontologically on your own.”
            Fair enough.  Adding two to two is indeed to play with fire (the divine, the Heraclitean fire), and not much less so than calculating co-products on fibre bundles.  So let us indulge in no calculations with numbers.  Each is inviolable, isolated, vestal.  Let us simply put them into pots and take them out again, like Eeyore with his balloon.  How many go into a six-pack?  Six!  How many eggs in a dozen?  Twelve!   And how many integral guests in Hilbert’s Hotel, which was built to house all of them?  Why, all of them!
            And so we come (weary travelers) at last to Hilbert’s Hotel.  In austerity it somewhat resembles the castle of Kafka’s parable; but it is much more inviting than that somewhat dour institution, welcoming any new arrival if he can possibly be accommodated.  And on this dark evening, indeed, the hotel is is full, every one of its innumerable rooms filled to capacity, since each one can accommodate but a single integer, and innumerable integers have already taken up lodging for the night. And yet, lo, one more integer shows up, bindle over shoulder.  Whatever shall we do?
            The tale has been well told by Rudy Rucker in Infinity and the Mind, so I’ll do no more than to abbreviate the beginnings of that Thousand and One Nights account of all the goings-on at that inn, with more running about in the corridors  than in a farce by Feydeau, and refer the reader to that estimable work. [If you haven’t read it, do.  A fun read, with no pre-requisites, and it itself is a sort of pre-requisite to later parts of this ongoing essay.]  Put briefly: the concierge simply shifted the guest in room 1, to room 2; the guest in room 2, to room 3; and so on; so that the newcomer moved into room one and had a restful sleep. More awkward was the next night, when the still-full hotel saw the arrival this time of infinitely many guests.  But with a bit of baksheesh, the concierge found a way: guest 1 moves into room 2, guest 2 into 4, guest 3 into 6, and so on; while the grateful newcomers troop into the thus-vacated rooms 1, 3, 5, 7, 9, ….
            It gets hairier than that:  I refer you to Rucker, who doth a tale unfold, whose lightest word will make thy knotted and combinèd locks to part, and thine each particular hair to stand on end… culminating in the hilarious contretemps, when Groucho, arriving too late, discovers his stateroom to be already occupied by a decidedly fretful porpentine….

            So, welcome to the Hotel!  Enjoy your visit!

No comments:

Post a Comment