I've been seeing articles popping up all over the place about a recent
PLOS article called Order in Spontaneous Behavior. The majority of
the articles seem to have been following the lead of the Discovery Institute, which claims that the article demonstrates the existence of free will, which they argue is inconsistent with naturalism and darwinism.
Archive for: May, 2007
I've been seeing articles popping up all over the place about a recent
So, what's set theory really about?
We'll start off, for intuition's sake, by talking a little bit about what's now called naive set theory, before moving into the formality of axiomatic set theory. Most of this post might be a bit boring for a lot of you, but it's worth
being a bit on the pedantic side to make sure that we're starting from a clear basis.
A set is a collection of things. What it means to be a member of a set S is
that there's some predicate PS - that is, some way of describing things via logic - which is true only for members of S. To be a tad more formal, that means that for any possible object x, PS(x) is true if and only if
x is a member of S. In general, I'll just write S for both the set and the predicate that defines the set, unless there's some reason that that would be confusing and/or ambiguous. (Another way of saying that is that a set S is a collection of things that all share
some property, which is the defining property of the set. When you work through
the formality of what a property means, that's just another way of saying that there's a
Have you ever wondered about the real reason why math education in our schools is so awful? Why despite the best efforts of large numbers of parents, the schools seem to be incapable of figuring out why they're so dreadfully bad at recognizing the difference between a halfway decent math curriculum and a trendy piece of garbage?
Read below the fold for a perfect example of why. The short version: the people who are involved in running education in America consider it perfectly acceptable to be idiots when it comes to math.
While I've been writing about the Surreal numbers lately, it reminded me of some of the fun of Set theory. As a result, I've been going back to look at some old books. Since I've been enjoying it, I thought you folks would as well.
Set theory, along with its cousin, first order predicate logic, is pretty much the
foundation of nearly all modern math. You can construct math from a lot of
different foundations, but axiomatic set theory is currently pretty much the dominant approach. (Although Topoi seem to be making some headway...)
There's a reason for that. Set theory starts with some of the simplest ideas, and extends in a reasonably straightforward way to encompass the most astonishingly complicated one. It's truly remarkable in that - none of the competitors to set theory
as a foundation can approach the intuitive simplicity of set theory.
So I'm going to write a bit about set theory as I explore my old books. And I thought that a good place to start was Cantor's diagonalization. Cantor is the inventor of set theory, and the diagonalization is an example of one of the first major results that Cantor published. It's also a good excuse for talking a little bit about where set theory came from, which is not what most people expect. Set theory was originally created as a tool for talking about the relative sizes of different infinities.
I debated about whether or not I should write this post. But as you can see, in the
end, I overcame my better judgement, and so he we are.
Over the weekend, PZ wrote a Pharyngula post about the reaction people have had to Mitt
Romney's statement about evolution. He was pissed. And I agree with his initial reaction.
What we have is a politician basically saying "Yes, I agree with the facts". And somehow,
that's been taken by a seemingly huge number of people as something brave and bold,
something that should impress us. Nope, sorry folks: acknowledging that facts are facts is
not brave. I'm no more impressed with him for saying that evolution is true than
I'd be if he got up and said "I admit it: I believe that 2+2=4."
But typically, when discussing anything that involves religion, PZ went overboard. And
it ticked me off.
Sorry for the slow pace of things around here lately; life interferes sometimes. I've mentioned my fathers illness before; things took a drastic change for the worse a week ago, which made things more than a little bit crazy. Of course, life wouldn't be life in things happened one at a time; so naturally, my asthma picked now to act up as well, so I've been coughing my lungs out. It's been a fun week, really.
Anyway, I've been wanting to look a bit more at Pica. I've been thinking about how a
named process definition looks, and what it does. The type part of a definition is
relatively straightforward: a process is defined by a set of externally visible channels; each channel is defined by a type. In a named definition, there are two types of channels to think about. One is a set of channels supplied to the process when it's created; the other is a set of channels created by the process, but which are 'exported': that is, made visible to other processes. The type of a process is a tuple of all of the channels which
the process exports the outside world.
This isn't math, but I felt like commenting anyway. That shining example of
an Intelligent Design advocate, Dr. Michael Egnor, is back once again. And this time,
his point, such as it is, is to basically fling insults at PZ Myers. What did PZ do to bring on his ire?
Well, PZ was annoyed with Time magazine, because for their "Time 100" list, they
had Michael Behe write the entry about Richard Dawkins. The passage which Engor took such offense at was the following:
The incompetence is stunning. Richard Dawkins makes the Time 100 list, and who do they commission to write up his profile?
Michael F**king Behe.
That's not just stupid, it's a slap in the face. It would have been no problem to find a smart biologist, even one who might be critical of Dawkins' message, to write something that expressed some measure of respect from the editorial staff. But to dig up a pseudoscientific fraud whose sole claim to fame is that he has led the charge to corrupt American science education for over a decade is shameful.
Now, what's wrong with PZ's reaction, according to Egnor?
Yes. PZ used the word "Fucking" to refer to Behe.
I added the asterisks. Both Behe and Myers are college biology professors who teach young biologists and biochemists the methods of scientific inquiry and, by example, teach students the appropriate standards of scientific discourse.
Which professor is shamefully corrupting American science education?
Yup. That's it. It's the one word. In the mind of Michael Egnor, throwing off a nasty word in the heat of the moment is a grave offense, far worse than spending more than a decade as a professional liar, far worse an offense for a scientist than, say, pretending to know the content of several dozen papers that you've never read.
Now, I'm not exactly PZs biggest fan. I think he's abrasive and arrogant. But I also know, from the experience of reading his writings, that he's an intelligent, passionate advocate for science; he's a teacher who works hard at teaching his students real science, critical thinking, and the scientific method.
And I agree with him about the incredible stupidity of asking Behe to write the piece on Dawkins. Behe is not a legitimate scientist. Behe is a dreadful hack who's spent most of the last two decades hard at work on a program to deliberately and dishonestly misrepresent science as part of a political agenda. He's been caught lying repeatedly; his ideas, such as they are, have been discredited. He even made an ass of himself in the Kitmiller trial, by handwaving away a stack of papers that he'd never seen before, because even without reading them, he just knew that they contained
nothing of any relevance. Choosing him to write a profile of Richard Dawkins is just plain offensive. It's sort of like asking PZ to write a profile about the Pope's influence as a theologist.
So, Dr. Egnor. Which professor is shamefully corrupting American science education? The professor who is a passionate (if arrogant) teacher of real science? Or the dishonest (and arrogant) professor who is an advocate of introducing misrepresentations into the science curriculum in order to discredit scientific theories that disagree with his religious beliefs?
Let's try looking at one other little quote, which I think is pretty illustrative of the difference between the two as educators. PZ is a tireless advocate for the teaching of real science to all science students. Behe, on the other hand, at the end of his interview with the director of the movie "Flock of Dodos", said "Why should I care what gets taught in public school? My kids don't go to public schools."
Who's the teacher who really cares about science education? And who's the one corrupting it? Look at the facts, and it's pretty obvious.
Of course, Dr. Egnor won't do that. Because as he's demonstrated in the past, he
doesn't need to waste his time looking at petty little things like facts, because he already knows the truth. Just like Professor Behe.
- Baka Beyond, "Baka Play Baka": This is what happens when you take a bunch of great trad Irish musicians, and lock them into a room with a bunch of great African musicians from the Baka tribe in Cameroon. I don't know quite how to describe this. It really doesn't sound like anything else. You can tell that there's Irish roots, and you can hear some African things that sound a little bit like M'balah, but mostly, it's something different. Very cool stuff.
- Flook, "Beehive": Flook is, bar none, the greatest instrumental trad Irish band around. They've got the guy who I think is greatest tinwhistle player in the world, Brian Finnegan; Sarah Allen, who can somehow keep up with Brian while playing on a honking *huge* alto whistle while standing on one foot; John Joe Kelley, a man who somehow makes the Bodhran (a kind of drum which the scourge of sessions everywhere) into a delicate and expressive instrument (one of Flook's album liner notes quotes a review that says something like "Saying John-Joe plays the Bohran is like saying Everest is a bit of a climb"); and last but not least, Ed Boyd, a rhythm guitarist who demonstrates just why being a rhythm guitar player can be an artistic calling. If you've never heard Flook, go out any buy their albums. All of them. I've never met anyone who didn't like Flook.
- The Trey Gunn Band, "Gate of Dreams": a track from the band led by former King Crimson stick player Trey Gunn. This is probably my favorite track by the TGB, which unfortunately isn't saying that much. Trey is a brilliant player, but he's rather dull as a composer. His band's work tends to leave me very flat.
- The Flower Kings, "Days Gone By": This is very out of place in a shuffle. It's not really it's own song. It's the ending of a long piece told from the point of view of a self-hating vampire.
- Mouse on Mars, "Chartnok": Noisy electronica, recommended to me by someone who thought that if I liked postrock, I'd like this. They were wrong. Ick.
- Peter Hammill, "After the Show": live recording of a song by one of the founders of progressive rock. It's an incredibly sparse performance - Hammill on keyboards and vocals, plus an electric violin and bass. One of the most intense recordings I've ever heard. I get chills every time I hear this. I don't know that I'd call in beautiful music; but it's a brilliant piece of musical art which I love listening to.
- Godspeed you! Black Emperor, "Rockets Fall on Rocket Falls": Godspeed - the b est post-rock ensemble ever. Everything I've ever heard by them is amazing.
- Bela Fleck and the Flecktones, "Fugue" from Bach's "Prelude and Fugure No. 20": Ordinally, I love just about anything Bela Fleck does. Not this. There's nothing wrong, in principle, with playing with great classical music. Hell, I've heard ELP take on the Prelude and Fugure, and it was great. But this is a dreadful job of playing with it. Geez, Bela, what did Bach ever do to you to deserve this?
- Tony Trischka, "Armando's Children": Amazing coincidence that this came up now. Just what I needed after hearing that train-wreck of Bela's: Bela, along with his old Banjo teacher playing some brilliant newgrass. Now this is what I expect when I go listen to Bela - and it's even better when it's Bela playing along with one of the few people in the world who can keep up with - and even sometimes get a step or two ahead of him. Wow.
- Solas, "The Crested Hens": Solas is another dazzling traditional Irish band. Formed from a mixture of Irish and Irish-American musicians, led by the unbelievable Seamas Egan. This is a slow air featuring the wonderful violin playing of Winnifred Horan and low whistle by Seamus. Seeing them live back in March convinced me to go out and buy a low whistle. (I also have to say, after seeing them live, that I was very surprised by the violinist. On all of the photos on their album covers, she's always got this pissed-off look on her face, so I was expecting her to be a very grumpy performer. Turned out to be an incredibly silly, happy, funny person whose energy was dazzling. You could just see how the energy of a song would change when her violin part came in.)
Today's a mighty cool example of bizzare language design, called GammaPlex In terms of language
design, it's nothing particularly special: it's yet another stack language
with a befunge-like graphical syntax. What's unusual about GammaPlex is that it's strongly focused on graphics. It's got built in support for ascii graphics, OpenGL, and mouse input.
In the comments on my DMCA post, a reader asked me to comment on this piece of silliness. I try not to disappoint my readers, so here's my take. It's a pile of silliness with the distinct aroma of astrotur - silliness mixed with a bit of deliberate stupidity in order to obscure things.
The basic idea of it is: how dare we complain about the idea of copyrighting numbers! After all, everything you can do on a computer is ultimately stored in a form that can be interpreted as a great big number! So we're always copyrighting numbers: every book, every article, every poem, every story that's ever been copyrighted is really just a number. So why should we start complaining now unless we're just a bunch or dirty anticorporate hippies who are complaining because we want to stick it to the movie companies?