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.
Continue reading Cantor Crankery and Worthless Wankery
It’s been quite a while since my last chaos theory post. I’ve
been caught up in other things, and I’ve needed to do some studying. Based
on a recommendation from a commenter, I’ve gotten another book on Chaos
theory, and it’s frankly vastly better than the two I was using before.
Anyway, I want to first return to dense periodic orbits in chaotic
systems, which is what I discussed in the previous chaos theory
post. There’s a glaring hole in that post. I didn’t so much get it
wrong as I did miss the fundamental point.
If you recall, the basic definition of a chaotic system is
a dynamic system with a specific set of properties:
- Sensitivity to initial conditions,
- Dense periodic orbits, and
- topological mixing
The property that we want to focus on right now is the
dense periodic orbits.
Continue reading More about Dense Periodic Orbits
In the Haskell stuff, I was planning on moving on to some monad-related
stuff. But I had a reader write in, and ask me to write another
post on data structures, focusing on a structured called a
A zipper is a remarkably clever idea. It’s not really a single data
structure, but rather a way of building data structures in functional
languages. The first mention of the structure seems to be a paper
by Gerard Huet in 1997, but as he says in the paper, it’s likely that this was
used before his paper in functional code — but no one thought to formalize it
and write it up. (In the original version of this post, I said the name of the guy who first wrote about zippers was “Carl Huet”. I have absolutely no idea where that came from – I literally had his paper on my lap as I wrote this post, and I still managed to screwed up his name. My apologies!)
It also happens that zippers are one of the rare cases of data structures
where I think it’s not necessarily clearer to show code. The concept of
a zipper is very simple and elegant – but when you see a zippered tree
written out as a sequence of type constructors, it’s confusing, rather
Continue reading Zippers: Making Functional "Updates" Efficient
A lot of people have been sending me links to a numerology article, in which yet another numerological idiot claims to have identified the date of the end of the world. This time, the idiot claims that it’s going to happen on May 21, 2011.
I’ve written a lot about numerology-related stuff before. What makes this example particularly egregious and worth writing about is that it’s not just an article on some bozo’s internet website: this is an article from the San Francisco Chronicle, which treats a pile of numerological bullshit as if it’s completely respectable and credible.
As I’ve said before: the thing about numerology is that there are so many ways of combining numbers together that if you’re willing to spend enough time searching, you can find some way of producing any result that you want. This is pretty much a classic example of that.
Continue reading The End Of The World is Coming in Just 501 Days!
As you’ve surely heard by now, on christmas day, some idiot attempted to
blow up an airplane by stuffing his underwear full of explosives and then
lighting his crotch on fire. There’s been a ton of coverage of this – most of
which takes the form of people running around wetting their pants in terror.
One thing which I’ve noticed, though, is that one aspect of this whole mess
ties in to one of my personal obsessions: scale. We humans are really,
really lousy at dealing with big numbers. We just absolutely
have a piss-poor ability to really comprehend numbers, or to take what we
know, and put it together in a quantitative way.
Continue reading Big Numbers and Air Travel