Archive for the 'Cantor Crankery' category

Cantor Crankery and Worthless Wankery

Jan 29 2010 Published by under Cantor Crankery

Poor Georg Cantor.

During his life, he suffered from dreadful depression. He was mocked by
his mathematical colleagues, who didn't understand his work. And after his
death, he's become the number one target of mathematical crackpots.

As I've mentioned before, I get a lot of messages either from or
about Cantor cranks. I could easily fill this blog with nothing but
Cantor-crankery. (In fact, I just created a new category for Cantor-crankery.) I generally try to ignore it, except for that rare once-in-a-while that there's something novel.

A few days ago, via Twitter, a reader sent me a link to a new monstrosity
that was posted to arxiv, called Cantor vs Cantor. It's novel and amusing. Still wrong,
of course, but wrong in an amusingly silly way. This one, at least, doesn't quite
fall into the usual trap of ignoring Cantor while supposedly refuting him.

You see, 99 times out of 100, Cantor cranks claim to have
some construction that generates a perfect one-to-one mapping between the
natural numbers and the reals, and that therefore, Cantor must have been wrong.
But they never address Cantors proof. Cantors proof shows how, given any
purported mapping from the natural numbers to the real, you can construct at example
of a real number which isn't in the map. By ignoring that, the cranks' arguments
fail: Cantor's method still generates a counterexample to their mappings. You
can't defeat Cantor's proof without actually addressing it.

Of course, note that I said that he didn't quite fall for the
usual trap. Once you decompose his argument, it does end up with the same problem. But he at least tries to address it.

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Another Cantor Crank: Representation vs. Enumeration

Dec 09 2009 Published by under Cantor Crankery

I've been getting lots of mail from readers about a new article on Google's Knol about Cantor's diagonalization. I actually wrote about the authors argument once before about a year ago.

But the Knol article gives it a sort of new prominence, and since we've recently had one long argument about Cantor cranks, I think it's worth another glance.

It's pretty much another one of those cranky arguments where they say "Look! I found a 1:1 mapping between the natural and the reals! Cantor was a fool!"

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The Hallmarks of Crackpottery, Part 1: Two Comments

Oct 28 2009 Published by under Bad Logic, Cantor Crankery

Another chaos theory post is in progress. But while I was working on it, a couple of
comments arrived on some old posts. In general, I'd reply on those posts if I thought
it was worth it. But the two comments are interesting not because they actually lend
anything to the discussion to which they are attached, but because they are perfect
demonstrations of two of the most common forms of crackpottery - what I call the
"Education? I don't need no stinkin' education" school, and the "I'm so smart that I don't
even need to read your arguments" school.

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The Continuum Hypothesis Solved: All Infinities are the Same? Nope.

Jan 28 2009 Published by under Cantor Crankery

Of all of the work in the history of mathematics, nothing seems to attract so much controversy, or even outright hatred as Cantor's diagonalization. The idea of comparing the sizes of different infinities - and worse, of actually concluding that there are different infinities, where some infinities are larger than others - drives some people absolutely crazy. As a result, countless people bothered by this have tried to come up with all sorts of arguments about why Cantor was wrong, and there's only one infinity.

Today's post is another example of that. This one is sort of special. Unless I'm very much mistaken, the author of this sent me his argument by email last year, and I actually exchanged several messages with him, before he concluded, roughly "We'll just have to agree to disagree." (I didn't keep the email, so I'm not certain, but it's exactly the same argument, and the authors name is vaguely familiar. If I'm wrong, I apologize.)

Anyway, this author actually went ahead and wrote the argument up as a full technical paper, and submitted it to arXiv, where you can download it in all it's glory. I'll be honest, and admit that I'm a little bit impressed by this. The proof is still completely wrong, and the arguments that surround it range from wrong to, well, not even wrong. But at least the author has the Chutzpah to treat his work seriously, and submit it to a place where it can actually be reviewed, instead of ranting about conspiracies.

For those who aren't familiar with the work of Cantor, you can read my article on it here. A short summary is that Cantor invented set theory, and then used it to study the construction of finite and infinite sets, and their relationships with numbers. One of the very surprising conclusions was that you can compare the size of infinite sets: two sets have the same size if there's a way to create a one-to-one mapping between their members. An infinite set A is larger than another infinite set B if every possible mapping from members of B to members of A will exclude at least one member of A. Using that idea, Cantor showed that if you try to create a mapping from the integers to the real numbers, for any possible mapping, you can generate a real number that isn't included in that mapping - and therefore, the set of reals is larger than the set of integers, even though both are infinite.

This really bothers people, including our intrepid author. In his introduction, he gives his motivation:

Cantor's theory mentioned in fact that there were several dimensions for infinity. This, however, is questionable. Infinity can be thought as an absolute concept and there should not exist several dimensions for the infinite.

Philosophically, the idea of multiple infinities is uncomfortable. Our intuitive notion of infinity is of an absolute, transcendent concept, and the idea of being able to differentiate - or worse, to be able to compare the sizes of different infinities seems wrong.

Of course, what seems wrong isn't necessarily wrong. It seems wrong that the mass of something can change depending on how fast it's moving. It seems even more wrong that looked at from different viewpoints, the same object can have different masses. But that doesn't change the fact that it's true. Reality - and even worse, abstract mathematics - isn't constrained by what makes us comfortable.

Back to the paper. In the very next sentence, he goes completely off the rails.

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Revisiting Old Friends, the Finale

Nov 05 2007 Published by under Cantor Crankery

Now, it's time for the final chapter in my "visits with old friends" series, which brings us
back to the Good Math/Bad Math all-time reader favorite crackpot: Mr. George Shollenberger.

Last time I mentioned George, a number of readers commented on the fact that it's cruel to pick on poor George, because the guy is clearly not all there: he's suffered from a number of medical problems which can cause impaired reasoning, etc. I don't like to be pointlessly cruel, and in general, I think it's inappropriate to be harsh with someone who is suffering from medical problems - particularly medical problems that affect the functioning of the mind.

But I don't cut George any slack. None at all. Because much of what spews from his mouth isn't the
result of an impaired mind: it's the product of an arrogant, vile, awful person. Since our last contact
with George, aside from the humorous idiocy, he's also taken it upon himself to explain how we'll never
have a peaceful society in America until we get rid of all of those damned foreigners
, who have
"unamerican mindsets". That post was where I really started to despise George. He's not just a senile
old fool - he's a disgusting, horrible person, just another of the evil ghouls who used a horrible
event, committed by a severely ill individual, as a cudgel to promote a deeply racist agenda.

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