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?

Muddled metaphors.

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

Alice’s number.

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.