{"id":861,"date":"2010-06-17T13:45:15","date_gmt":"2010-06-17T13:45:15","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2010\/06\/17\/metaphorical-crankery-a-bad-metaphor-is-like-a-steaming-pile-of\/"},"modified":"2010-06-17T13:45:15","modified_gmt":"2010-06-17T13:45:15","slug":"metaphorical-crankery-a-bad-metaphor-is-like-a-steaming-pile-of","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2010\/06\/17\/metaphorical-crankery-a-bad-metaphor-is-like-a-steaming-pile-of\/","title":{"rendered":"Metaphorical Crankery: a bad metaphor is like a steaming pile of …"},"content":{"rendered":"

So, another bit of Cantor stuff. This time, it really isn’t Cantor
\ncrankery, so much as it is just Cantor muddling. The post
\nthat provoked this<\/a> is not, I think, crankery of any kind – but it
\ndemonstrates a common problem that drives me crazy; to steal a nifty phrase
\nfrom youaredumb.net, people who can’t count to meta-three really shouldn’t try
\nto use metaphors.<\/p>\n

The problem is: You use a metaphor to describe some concept. The metaphor
\nisn’t<\/em> the thing you describe – it’s just a tool that you use. But
\nsomeone takes the metaphor, and runs with it, making arguments that are built
\nentirely on metaphor, but which bear no relation to the real underlying
\nconcept. And they believe that whatever conclusions they draw from the
\nmetaphor must, therefore, apply to the original concept.<\/p>\n

In the context of Cantor, I’ve seen this a lot of times. The post that
\ninspired me to write this isn’t, I think, really making this mistake. I think
\nthat the author is actually trying to argue that this is a lousy metaphor to
\nuse for Cantor, and proposing an alternative. But I’ve seen exactly this
\nreasoning used, many times, by Cantor cranks as a purported disproof. The
\ncranky claim is: Cantor’s proof is wrong, because it cheats<\/em>. <\/p>\n

Of course, if you look at Cantor’s proof as a mathematical construct, it’s
\na perfectly valid, logical, and even beautiful proof by contradiction. There’s
\nno cheating. So where do the “cheat” claims come from?<\/p>\n

<\/p>\n

Muddled metaphors.<\/p>\n

A common way of describing Cantor’s proof is in terms of games. Suppose
\nI’ve got two players: Alice and Bob. Alice thinks of a number, and
\nBob guesses. Bob wins if he guesses Alice’s number.<\/p>\n

If Alice is restricted to a finite set of integers, then Bob will
\nwin in a bounded set of guesses. For example, if Alice is only allowed
\nto pick numbers between 1 and 20, then Bob is going to win within 20 guesses.<\/p>\n

If Alice is restricted to natural numbers, then Bob will win – but it
\ncould take an arbitrarily long time. The number of steps until he wins is
\nfinite, but unbounded. His strategy is simple: guess 0. If that’s not it, guess 1. If
\nthat’s not it, guess 2. And so on. Eventually, he’ll win. And, in fact, after
\neach unsuccessful guess, Bob’s guess is closer<\/em> to Alice’s number.<\/p>\n

If Alice can use integers, then it gets harder for Bob – but it doesn’t
\nreally change much. Still, in a finite but unbounded number of guesses, Bob
\nwill get Alice’s number and win. Now, the “closer every guess” doesn’t really
\napply any more – but something very close does: there are no steps where Bob
\ngets further away<\/em> from the absolute value of Alice’s number; and
\nevery two steps, he’s guaranteed to get closer to the absolute value of
\nAlice’s number.<\/p>\n

We can make it harder for Bob – by saying that Alice can pick any
\nfraction. Now Bob’s strategy gets much harder. He needs to work out a system
\nto guess all the rationals. He can do that. But now the properties about
\ngetting closer to Alice’s number no longer apply. He’s no longer doing things
\nin an order where his value is converging on Alice’s number. But still, after
\na finite number of steps, he’ll get it.<\/p>\n

Finally, we could let Alice pick any real<\/em> number. And now,
\nthe rules change: for any strategy that Bob picks for going through the
\nreal numbers, Alice can find a number that Bob won’t even guess.<\/p>\n

There’s a fundamental asymmetry there. In all of the other versions of the
\ngame, Alice had to pick her number first, and then Bob would try to guess it. Now,
\nAlice doesn’t pick her number until after<\/em> Bob starts guessing – and she
\nonly picks her number after knowing Bob’s strategy. So Alice is cheating.<\/p>\n

The game metaphor demonstrates the basic idea of Cantor’s theorem. The
\nnaturals, integers, and rationals are all infinite sets, but they’re all
\ncountable. In the game setting, even if<\/em> Alice knows Bob’s strategy,
\nshe can’t<\/em> pick a number from any of those sets which Bob won’t guess
\neventually. But with the real numbers, she can – because there’s something
\nfundamentally different about the real numbers. <\/p>\n

Of course, if it’s a game, and the only way that Alice can win is
\nby knowing exactly what Bob is going to do – by knowing his complete
\nstrategy from now to infinity – then the only way that Alice can win is
\nby cheating. In a game, if you get to know your opponent’s moves in advance,
\nand you get to plan your moves in perfect anticipation<\/em> of every
\nmove that they’re going to make — you get to change<\/em> your move
\nin reaction to<\/em> their move, but they don’t get to respond likewise
\nto your moves — that is, by definition, cheating. You’ve got an unfair
\nadvantage. Bob has to pick his strategy in advance and tell it to Alice, and
\nthen Alice can use that to pick her moves in a way that guarantees that
\nBob will lose.<\/p>\n

The problem with this metaphor is that Cantor’s proof isn’t a
\ngame<\/em>. There are no players. No one wins, and no one loses. The
\nwhole concept of fairness makes no sense<\/em> in the context of Cantor’s
\nproof. It makes sense in the metaphor<\/em> used to explain Cantor’s
\nproof. But the metaphor isn’t the proof. A proof isn’t a competition.
\nIt doesn’t have to be fair<\/em>; it only has to be correct<\/em>.
\nThe fact that what Cantor’s proof does would be cheating if it were a game
\nis completely irrelevant.<\/p>\n

This kind of nonsense doesn’t just happen in Cantor crankery. You see the
\nsame problem constantly<\/em>, in almost any kind of discussion which uses
\nmetaphors. There are chemistry cranks who take the metaphor of an electron
\norbiting an atomic nucleus like a planet orbits a sun, and use it to create
\nsome of the most insane arguments. (The most extreme example of this in my
\nexperience was a guy back on usenet, who called himself Ludwig von Ludvig,
\nthen Ludwig Plutonium, and then Archimedes Plutonium<\/a>. He went
\nbeyond the simple orbit stuff, and looked at diagrams in physics books of
\n“electron clouds” around a nucleus. Since in the books, those clouds are made
\nof dots, he decided that the electrons were really made up of a cloud of dots
\naround the nucleus, and that our universe was actually a plutonium atom, where
\nthe dots in the picture were actually galaxies.) There are physics bozos who
\ndo things like worry about the semi-dead cats. There are politicians who worry
\nabout new world orders, because of a stupid flowery metaphorical phrase that
\nsomeone used in a speech 20 years ago.<\/p>\n

It’s amazing. But there’s really no limit to how incredibly, astonishingly
\nstupid people can be. And the idea of an imperfect metaphor is, apparently,
\nmuch too complicated for an awful lot of people.<\/p>\n","protected":false},"excerpt":{"rendered":"

So, another bit of Cantor stuff. This time, it really isn’t Cantor crankery, so much as it is just Cantor muddling. The 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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[11],"tags":[],"class_list":["post-861","post","type-post","status-publish","format-standard","hentry","category-cantor-crankery"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-dT","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/861","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=861"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/861\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=861"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=861"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=861"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}