So, another bit of Cantor stuff. This time, it really isn’t Cantor
crankery, so much as it is just Cantor muddling. The href="http://rjlipton.wordpress.com/2010/06/11/does-cantors-diagonalization-proof-cheat/">post
that provoked this is not, I think, crankery of any kind – but it
demonstrates a common problem that drives me crazy; to steal a nifty phrase
from youaredumb.net, people who can’t count to meta-three really shouldn’t try
to use metaphors.
The problem is: You use a metaphor to describe some concept. The metaphor
isn’t the thing you describe – it’s just a tool that you use. But
someone takes the metaphor, and runs with it, making arguments that are built
entirely on metaphor, but which bear no relation to the real underlying
concept. And they believe that whatever conclusions they draw from the
metaphor must, therefore, apply to the original concept.
In the context of Cantor, I’ve seen this a lot of times. The post that
inspired me to write this isn’t, I think, really making this mistake. I think
that the author is actually trying to argue that this is a lousy metaphor to
use for Cantor, and proposing an alternative. But I’ve seen exactly this
reasoning used, many times, by Cantor cranks as a purported disproof. The
cranky claim is: Cantor’s proof is wrong, because it cheats.
Of course, if you look at Cantor’s proof as a mathematical construct, it’s
a perfectly valid, logical, and even beautiful proof by contradiction. There’s
no cheating. So where do the “cheat” claims come from?
A common way of describing Cantor’s proof is in terms of games. Suppose
I’ve got two players: Alice and Bob. Alice thinks of a number, and
Bob guesses. Bob wins if he guesses Alice’s number.
If Alice is restricted to a finite set of integers, then Bob will
win in a bounded set of guesses. For example, if Alice is only allowed
to pick numbers between 1 and 20, then Bob is going to win within 20 guesses.
If Alice is restricted to natural numbers, then Bob will win – but it
could take an arbitrarily long time. The number of steps until he wins is
finite, but unbounded. His strategy is simple: guess 0. If that’s not it, guess 1. If
that’s not it, guess 2. And so on. Eventually, he’ll win. And, in fact, after
each unsuccessful guess, Bob’s guess is closer to Alice’s number.
If Alice can use integers, then it gets harder for Bob – but it doesn’t
really change much. Still, in a finite but unbounded number of guesses, Bob
will get Alice’s number and win. Now, the “closer every guess” doesn’t really
apply any more – but something very close does: there are no steps where Bob
gets further away from the absolute value of Alice’s number; and
every two steps, he’s guaranteed to get closer to the absolute value of
We can make it harder for Bob – by saying that Alice can pick any
fraction. Now Bob’s strategy gets much harder. He needs to work out a system
to guess all the rationals. He can do that. But now the properties about
getting closer to Alice’s number no longer apply. He’s no longer doing things
in an order where his value is converging on Alice’s number. But still, after
a finite number of steps, he’ll get it.
Finally, we could let Alice pick any real number. And now,
the rules change: for any strategy that Bob picks for going through the
real numbers, Alice can find a number that Bob won’t even guess.
There’s a fundamental asymmetry there. In all of the other versions of the
game, Alice had to pick her number first, and then Bob would try to guess it. Now,
Alice doesn’t pick her number until after Bob starts guessing – and she
only picks her number after knowing Bob’s strategy. So Alice is cheating.
The game metaphor demonstrates the basic idea of Cantor’s theorem. The
naturals, integers, and rationals are all infinite sets, but they’re all
countable. In the game setting, even if Alice knows Bob’s strategy,
she can’t pick a number from any of those sets which Bob won’t guess
eventually. But with the real numbers, she can – because there’s something
fundamentally different about the real numbers.
Of course, if it’s a game, and the only way that Alice can win is
by knowing exactly what Bob is going to do – by knowing his complete
strategy from now to infinity – then the only way that Alice can win is
by cheating. In a game, if you get to know your opponent’s moves in advance,
and you get to plan your moves in perfect anticipation of every
move that they’re going to make — you get to change your move
in reaction to their move, but they don’t get to respond likewise
to your moves — that is, by definition, cheating. You’ve got an unfair
advantage. Bob has to pick his strategy in advance and tell it to Alice, and
then Alice can use that to pick her moves in a way that guarantees that
Bob will lose.
The problem with this metaphor is that Cantor’s proof isn’t a
game. There are no players. No one wins, and no one loses. The
whole concept of fairness makes no sense in the context of Cantor’s
proof. It makes sense in the metaphor used to explain Cantor’s
proof. But the metaphor isn’t the proof. A proof isn’t a competition.
It doesn’t have to be fair; it only has to be correct.
The fact that what Cantor’s proof does would be cheating if it were a game
is completely irrelevant.
This kind of nonsense doesn’t just happen in Cantor crankery. You see the
same problem constantly, in almost any kind of discussion which uses
metaphors. There are chemistry cranks who take the metaphor of an electron
orbiting an atomic nucleus like a planet orbits a sun, and use it to create
some of the most insane arguments. (The most extreme example of this in my
experience was a guy back on usenet, who called himself Ludwig von Ludvig,
then Ludwig Plutonium, and then href="http://www.iw.net/~a_plutonium/">Archimedes Plutonium. He went
beyond the simple orbit stuff, and looked at diagrams in physics books of
“electron clouds” around a nucleus. Since in the books, those clouds are made
of dots, he decided that the electrons were really made up of a cloud of dots
around the nucleus, and that our universe was actually a plutonium atom, where
the dots in the picture were actually galaxies.) There are physics bozos who
do things like worry about the semi-dead cats. There are politicians who worry
about new world orders, because of a stupid flowery metaphorical phrase that
someone used in a speech 20 years ago.
It’s amazing. But there’s really no limit to how incredibly, astonishingly
stupid people can be. And the idea of an imperfect metaphor is, apparently,
much too complicated for an awful lot of people.