Evolving Thoughts

Er, count the mistakes? (And I'm not even a logician.)

This is not xkcd at its best.

Posted by: Jason | August 27, 2008 4:53 AM

And what's a logician if not a pedantic specialist?

My real issues with the cartoon are:

A: It's easy to make a cartoon which illustrates Russell's Paradox better than this one.

B: IMO, many OTHER xkcd cartoons are pure genius. Like the previous one about quantum teleportation, for example. :-)

OK, the pedantry starts here. Non-logicians stop reading now. Go do something useful.

1. Speling of "Russell".

2. This is the big one: how have Russell and Whitehead failed to make their chart (as depicted in the cartoon)? They haven't, as far as we can see. Gödel's turned on by anything not on the list. If it's a complete list of fetishes (ex hypothesi) then he can't be turned on by anything not on it, on pain of contradiction. Let's suppose, for simplicity, that contradictions are bad. Even so, as long as Gödel's not turned on by anything, there's absolutely no failure.

This is completely different from Russell's Paradox, in which there is no get out of jail free card for the empty set.

3. Even if Gödel is turned on by something, Russell and Whitehead (sic -- he probably means Whitehead and Russell, since that's how they were known at the time they worked together) are still OK provided Gödel is WRONG to say that what he's turned on by isn't on their list. Which is very plausible since he isn't actually looking at it.

4. Even if Gödel is turned on by something and he's RIGHT to say that it's not on their list, Russell and Whitehead STILL may not have failed, depending on what logic we use. By this point I'm on thin ice, but we should at least say that the logic is up for grabs, since all three characters are logicians.

At this point I've got bored of pedantry, but I bet there are more issues to be found.

And yes I know my umlauts aren't showing properly.

I fixed 'em: JSW

Posted by: Jason | August 27, 2008 6:38 AM

I admit it, Jason, I'm turned on by your post.

Posted by: csrster | August 27, 2008 6:58 AM

csrster: made me laugh and nearly snort muesli out of my nose. (You probably didn't want to know that, but at least it's back on topic for a biology blog.)

Posted by: jason | August 27, 2008 7:04 AM

I've realised that since Thony is a (or an) historian, he might be interested in this: according to Nicholas Griffin, people changed from saying "Whitehead and Russell" to "Russell and Whitehead" when Russell got his Nobel Prize. This is anecdotal, of course. But that was DECADES after they'd done their work together. Isn't it weird how a Nobel Prize can rewrite history (albeit in a fairly minor way in this case)? And of course Russell's Nobel Prize was in literature, not maths, so it's not like it marked a claim that he'd done more work than Whitehead or anything like that.

Posted by: Jason | August 27, 2008 7:24 AM

Jason said: "how have Russell and Whitehead failed to make their chart (as depicted in the cartoon)? They haven't, as far as we can see. G�del's turned on by anything not on the list. If it's a complete list of fetishes (ex hypothesi) then he can't be turned on by anything not on it, on pain of contradiction. Let's suppose, for simplicity, that contradictions are bad. Even so, as long as G�del's not turned on by anything, there's absolutely no failure."

Sorry, Jason - you just missed it. The point actually goes through (if you interpret the comic with some charity). Point is that G�del is turned on by ANYTHING not on the list, INSOFAR as it is not on the list. The point, then, is NOT to illustrate Russell's paradox at all (if it were, the comic would presumably feature Frege asking Russell), the point is that the list is necessarily incomplete - i.e. to make a joke related to G�del's first incompleteness theorem (although this particular one would have worked just as well, maybe even better, with Cantor). It's not sufficient, then, for Russell and Whitehead to construct a list of every known fetish; they would have to compile a list of everything whatsoever. First problem, of course, is that they'd probably include a lot of non-fetishes on such a list, contradicting the criteria for inclusion on the list (i.e. probably a lot of them would be be (G�del's) fetishes only insofar as they're NOT on the list, and nobody's fetishes otherwise); second, to compile a list of absolutely everything is impossible anyway (I mean, e.g. how many subsets of the set of all natural numbers do G�del have a crush on?).

Posted by: G.D. | August 27, 2008 9:16 AM

(and note that R&W could NOT solve this by including "Anything not on our list" on their list. THAT would lead to paradox: If they included it, then it WOULD be on the list, but in that case it wouldn't be Gödel's fetish anymore and hence not a member of the set of fetishes. So their list is incomplete if it's consistent ...)

Posted by: G.D. | August 27, 2008 9:36 AM

At xkcd's site (http://xkcd.com/), if you hover your mouse over the cartoon, you'll shortly get a little yellow box with a comment in it. This is a standard feature of xkcd's work. The comment for this cartoon reads:

"They eventually resolved this self-reference, but Cantor's everything-in-the-fetish-book-twice" parties finally sunk the idea."

I think G. D.'s interpretation (#11) is correct and his point about Cantor is also made in the original source, rather nicely I thought.

Posted by: Elihu M. Gerson | August 27, 2008 10:07 AM

Jason: My one mistake is the spelling of Bertrand Arthur William 3rd. Earl Russell's name. My second mistake for pedants (and yes I am one) is exactly the fact that it is 'Whitehead and Russell' and not vice versa.

As for the rest I think you have missed the point of the joke. Whitehead and Russell set out to show in PM that the whole of mathematics could be deduced from the axioms of formal logic. Gödel came along and showed, using a variant of the du Bois- Reymond diagonal process (which is normally incorrectly attributed to Cantor), that any list of true mathematical statements is incomplete because there are mathematically true statements that are not in the list. PM is of course an attempt to produce such a list (by deduction from the axioms of logic) and is mentioned explicitly in the title of Gödel's famous paper.

Posted by: Thony C. | August 27, 2008 12:08 PM

A quick but urgently necessary correction to my last post before the enraged meta-mathematicians descend on my head like a hoard of hungry locusts; L-S says that any first order axiom system that has a non-countable model also has a countable one! (Scrub the infinite and finite)The rest remains the same!

Memo to self: engage brain before posting on the inter-tubes!

Posted by: Thony C. | August 27, 2008 1:38 PM

People often argue with me when I quote Whitehead on his interesting v. true proposition. Comment sections are far more interesting than true, the truth of which adds much interest here.

Posted by: brewdog | August 28, 2008 9:25 AM