Monthly Archives: January 2010

Cantor Crankery and Worthless Wankery

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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More about Dense Periodic Orbits

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:

  1. Sensitivity to initial conditions,
  2. Dense periodic orbits, and
  3. topological mixing

The property that we want to focus on right now is the
dense periodic orbits.

In a dynamical system, an orbit isn’t what we typically think of as orbits. If you look at all of the paths through the phase space of a system, you can divide it into partitions. If the system enters a state in any partition, then every state that it ever goes through will be part of the same partition. Each of those partitions is called an orbit. What makes this so different from our intuitive notion of orbits is that the intuitive orbit repeats. In a dynamical system, an orbit is just a set of points, paths through the phase space of the system. It may never do anything remotely close to repeating – but it’s an orbit. For example, if I describe a system which is the state of an object floating down a river, the path that it takes is an orbit. But it obviously can’t repeat – the object isn’t going to go back up to the beginning of the river.

An orbit that repeats is called a periodic orbit. So our intuitive notion of orbits is really about periodic orbits.

Periodic orbits are tightly connected to chaotic systems. In a chaotic system, one of the basic properties is a particular kind of unpredictability. Sensitivity to initial conditions is what most people think of – but the orbital property is actually more interesting.

A chaotic system has dense periodic orbits. Now, what does that mean? I explained it once before, but I managed to miss one of the most interesting bits of it.

The points of a chaotic system are dense around the periodic orbits. In mathematical terms, that means that every point in the attractor for the chaotic system is arbitrarily close to some point on a periodic orbit. Pick a point in the chaotic attractor, and pick a distance greater than zero. No matter how small that distance is, there’s a periodic orbit within that distance of the point in the attractor.

The last property of the chaotic system – the one which makes the dense periodic orbits so interesting – is topological mixing. I’m not going to go into detail about it here – that’s for the next post. But what happens when you combine topological mixing with the density around the periodic orbits is that you get an amazing kind of unpredictability.

You can find stable states of the system, where everything just cycles through an orbit. And you can find an instance of the system that appears to be in that stable state. But in fact, virtually all of the time, you’ll be wrong. The most minuscule deviation, any unmeasurably small difference between the theoretical stable state and the actual state of the system – and at some point, your behavior will diverge. You could stay close to the stable state for a very long time – and then, whammo! the system will do something that appears to be completely insane.

What the density around periodic orbits means is that even though most of the points in the phase space aren’t part of periodic orbits, you can’t possibly distinguish them from the ones that are. A point that appears to be stable probably isn’t. And the difference between real stability and apparent stability is unmeasurably, indistinguishably small. It’s not just the initial conditions of the system that are sensitive. The entire system is sensitive. Even if you managed to get it into a stable state, the slightest perturbation, the tiniest change, could cause a drastic change at some unpredictable time in the future.

This is the real butterfly effect. A butterfly flaps its wings – and the tiny movement of air caused by that pushes the weather system that tiny bit off of a stable orbit, and winds up causing the diversion that leads to a hurricane. The tiniest change at any time can completely blow up.

It also gives us a handle on another property of chaotic systems as models of real phenomena: we can’t reverse them. Knowing the measured state of a chaotic system, we cannot tell how it got there. Even if it appears to be in a stable state, if it’s part of a chaotic system, it could have just “swung in” the chaotic state from something very different. Or it could have been in what appeared to be a stable state for a long time, and then suddenly diverge. Density effectively means that we can’t distinguish the stable case from either of the two chaotic cases.

Zippers: Making Functional "Updates" Efficient

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
than clarifying.

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The End Of The World is Coming in Just 501 Days!

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.

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Big Numbers and Air Travel

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.

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