The 9th Carnival of Mathematics is live at JD2718, in a cleverly alphabetical form.
The ABCs of Good Math
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FPP: No you cant haz yr LOLCODE: Sortle Instead
All week, I’ve been buried by a wave of requests to write about LOLCODE today. Normally, I do try to honor requests from readers, but from the time I started my friday pathological languages, I’ve always tried to stick to languages that actually had *something* interesting about their semantics. LOLCODE is funny because of its goofy grammar; but it’s really incredibly dull semantically. And while there are lots of programs written in it,
there’s no implementation (at least not yet).
Anyway, what I decided to do instead is a twistedly beautiful language called [Sortle][sortle]. Sortle is a very simple language which is based on insertion sort. The basic idea of how it works is: it takes the lines of the program, and sorts them. Then it executes them; when a line is executing, it can modify itself; when it gets modified, the interpreter re-sorts the line into its correct position in the sorted order according to its new value. So control flow is completely driven by the sort-order of the lines of the program.
Behe's Dreadful New Book: A Review of "The Edge of Evolution"
I’ve gotten my hands on a review copy of Michael Behe’s new book, “The Edge of Evolution”. The shortest version of a review is: Bad science, bad math, and bad theology, all wrapped up in a pretty little package.
As people who’ve followed his writings, lectures, and court appearances know, Behe is pretty much a perfect example of the ignoramus who makes a bad argument, and then puts his fingers in his ears and shouts “La la la, I can’t hear you” whenever anyone refutes it. He *still* harps on his “irreducible complexity” nonsense, despite the fact that pretty much *every aspect* of it has been thoroughly refuted. (The entire concept of IC is a pile of rubbish; the entire argument about IC is based on the idea that evolution is a strictly additive process, which is not true; there are numerous examples of observed evolution of IC systems, both in biology and in evolutionary algorithms. But none of these facts makes a bit of difference: like the energizer bunny of ignorance, he just keeps going, and going…)
Anyway, the new book is based on what comes down to a mathematical argument – a mathematical argument that I’ve specifically refuted on this blog numerous times. I’m not mentioning that because I expect Behe to read GM/BM and consider it as a serious source for his research; even if I were an expert in the subject (which I’m *not*), a blog is *not* a citable source for real research. But I mention it because the error is so simple, so fundamental, and so bleeding *obvious* that even a non-expert can explain what’s wrong with it in a spare five minutes – but Behe, who apparently spent several *years* writing this book still can’t see the problem. (In fact, one of the papers that he cites as *support* for this ridiculous theory contains the refutation!)
The Disgrace of Memorial Day
Today is Memorial Day, and I feel compelled to say something about it.
We’re in the middle of a horrible and pointless war. A war that we started, based on a bunch of lies. Since we did this, we have caused the deaths of hundreds of thousands of innocent Iraqis, and thousands of American soldiers. And we did it for no reason.
As the situation has grown progressively worse, and more and more people have been maimed and killed, we’ve heard an endless drumbeat from Bush supporters: me must support the troops!
I support American soldiers. As the son of a WW2 veteran, I grew up with a lot of respect for soldiers. People who join the military voluntarily give up many of the freedoms that we take for granted, and allow themselves to be put into situations that people like me can’t even really begin to imagine. We ask the members of our military to go and put themselves into a situation where they have to pick up a weapon, point it at someone, and kill them in cold blood – because they’re a member of the enemy’s army. We ask them to put themselves into a position where other people are going to try to kill them. These are horrible things, things that most of us can’t even really contemplate. Because my father put himself into that situation many years ago, he understood what it meant, in a way that I can only imagine.
When soldiers go to war, they’re taught to dehumanize the enemy. They don’t do that because they’re bad people. They do that because they have to: normal people, sane people, can’t pick up a gun, look through its sights at another human being that they don’t know, and pull the trigger and watch them die. People put into that situation hesitate – and hesitation in battle costs lives. As horrible as that sounds to us sitting in our comfortable homes, that’s what must happen to create effective combat soldiers.
When we ask people to do that on our behalf, we take on a great responsibility. We are asking them to do terrible things, things that will, under the best of circumstances, leave deep emotional scars. What happens when we put them into the hell of war is our responsibility. When we send them to war, we are obligated to respect the kind of sacrifices that we ask them to make; to make sure that we only ask them to do it an a cause that’s truly worthy of the price that they will pay; and to care for them and their families after the war is over.
In this war, that hasn’t happened. We’ve asked people to kill and die for no good reason; and while doing it, we have consistently neglected the soldiers we put in harms way, and the families that they left behind.
Our president criticizes people who want to get our soldiers out of harms way as “not supporting the troops” – while opposing funding for medical care for soldiers, housing wounded people in rat-infested hell-holes, denying financial assistance to them and their families. He talk about sending people back into combat three and four times as “support”. He sends them to die, and never attends a funeral, never watches the coffins coming home, never takes any responsibility for the horrors he’s inflicted on them and their families. He works hard to oppose things as simple as funding medical care for returning soldiers. But people who fight him on that, he tars as “not supporting the troops”.
He’s given the orders to teach them them to torture people, given the orders to tell them to torture people, and them blamed them for doing it. The horrors that he has chosen to inflict on them are, in his eyes and the eyes of his supporters, unimportant. He feels no responsibility for what he’s made them do. He’s sent near-children to the front lines, and watched as they’re punished for following his orders, while pardoning – or even rewarding the people who created the policies and gave the orders.
We should honor and respect the people who make sacrifices for our country. Instead, we spit on them, and call it support.
Today is memorial day, the day when we are supposed to remember the people who gave their lives for our country. And instead of honoring them, we’re sending more of them to die for no good reason, in a phony pointless war. Honoring them means making sure that we never ask them to sacrifice themselves unless there is a real need. We deserve to be deeply ashamed of what we’ve permitted in our names. On this day when we honor them, we should be begging them for forgiveness, for what we made them give, and what we made them do.
And meanwhile, our president’s idea of celebrating memorial day is giving a five minute speech, and then rushing off to his barbecue.
The Axiom of Choice
The Axiom of Choice
The axiom of choice is a fascinating bugger. It’s probably the most controversial statement in mathematics in the last century – which is pretty serious, considering the kinds of things that have gone on in math during the last century.
The axiom itself is quite simple, and reading an informal description of it, it’s difficult to understand how it managed to cause so much trouble. For example, wikipedia has a rather nice informal statement of it:
given a collection of bins each containing at least one object, exactly one object from each bin can be picked and gathered in another bin
The Axiom of Infinity
The axiom of infinity is a bundle of tricks. As I said originally, it does two things. First, it gives us our first infinite set; and second, it sets the stage for representing arithmetic in terms of sets. With the axiom of infinity, we get the natural numbers; with the natural numbers, we can get the integers; with the integers, we can get the rationals. Once we have the rationals, things get a bit harder – but we can get the reals via Dedekind cuts; and by transfinite induction, we can get the transfinite numbers. But before we can get to any of that, we need a sound representation of the naturals in terms of sets.
Popups
We’ve been having some problem with popups showing up on scienceblogs. It’s not deliberate: Seed does not accept popup ads. But it appears that one snuck in somehow, but no one is sure where it’s coming from.
If you see a popup on GM/BM, please do me a favor, and post a comment here telling me:
- Who the popup was for,
- What ads appeared on the page when the popup popped up.
The sooner we figure out where the popup is coming from, the sooner we can get rid of it.
The Axiom of Extensionality
Some of the basic axioms of ZFC set theory can seem a bit uninteresting on their own. But when you take them together, and reason your way around them, you can find some interesting things.
Let’s start by looking at the axiom of extensionality. Pretty simple, right? All it does is define what set equality means. It says that two sets are equal if, and only if, they have the same members: that is, a set is completely determined by its contents.
How much more trivial can a statement about sets get? It really doesn’t seem to say much. But what happens when we start thinking through what that means?
The way we normally think of sets, they’re collections of objects. So, imagine a set like {red, green, blue}, where the three values are atoms: that is, they’re single objects, not collections. What does the axiom of extensionality say about that? It says that red, green, and blue are not atoms?
Why? Well – let’s look at the axiom of extensionality again: (∀A,B: A=B ⇔
(∀C: C∈A ⇔ C∈B)). So – does red = blue? Well, if they’re atoms, then
yes, red=blue, because nothing is in red, and nothing is in blue. Since neither has any members, they’re equal.
In fact, if we follow that reasoning through, there’s only one possible atom: the only set with no members is the empty set. So anything else we want, we’re going to have to represent using some kind of collection.
As a result of that, along with the axiom of specification, we can show that the axiom
of the empty set is actually redundant. After all – the axiom of specification basically
says that if you can describe a collection of values by a predicate, that collection is a class. So take the predicate P(x)=false; that’s a set with no values. Also known as the empty set. So the empty set exists, and it’s the only set with no members – aka the only atom.
The Axiom of Pairing
The axiom of pairing is an interesting beast. It looks simple, and in fact, it
is simple. But it opens up a range of interesting things that we’d like to be able
to do. For example, without the axiom of pairing, we wouldn’t be able to formulate the
cartesian products of sets – and without cartesian product, huge ranges of interesting and
important areas of mathematics would be inaccessible to us. (Note that I’m saying that
pairing is necessary, not that it’s sufficient. You also need replacement
to get the projection functions that are part of the usual definition of the cartesian
product.)
The Man I called Fink
My father died on sunday.
To some degree, I’m still in shock. Even though we knew it was coming, when something like this happens, no amount of preparation really helps. He’d been sick with an antibiotic resistant infection since November, and on thursday, refused to let them give him a feeding tube. So we really knew, almost to the day, when he was going to die. And yet, when it finally happened, it was still a shock.
We buried him yesterday. I didn’t speak at the funeral, because I couldn’t. Every time I try to talk about him, my voice just shuts down. But my fingers don’t. So if you’ll bear with me, I want to say a little bit about my father.
If you enjoy reading this blog, you owe him. He’s the person who got me interested in math, and who taught me how to teach. To give you an example that I remember particularly vividly: when I was in fourth grade, he was doing work on semiconductor manufacturing. He
had brought home a ream of test data from a manufacturing run, and was sitting at our dining room table, paper spread out around him, doing an analysis of the data. I walked in, and asked what he was doing. He stopped working, and proceeded to explain to me what he was doing. That evening, he taught me about bell curves, linear regression, and standard deviation. He was able to make all of that understandable – both how to do it, and why you do it – to a fourth grader.
Until I went to college, I pretty much didn’t learn math in school. He taught me. I learned algebra, geometry, and calculus from my father, not from my math teachers. He bought me my first book on programming.
He was a soldier in World War two. He dropped out of high school, and lied about his age in order to be allowed to enlist. As far as I know, that was one of the only times in
his life that he lied about something important. But to him, doing his part to defend his country was more important than the rule about how old he had to be to enlist. He didn’t end up fighting; his unit didn’t get deployed to the front. He would up spending his time in the army in India.
After coming home, he went to college on the GI bill, and became a physicist. But calling him a physicist is a little deceptive: he was a very hands on person. He wound up
doing work that most people would call electrical engineering. He worked on semiconductors mainly for satellites and military applications. During his career, he worked on things ranging from communication satellites, to the Trident and MX missiles, to the power system for the Galileo space probe. I had my disagreements with him over working on the missiles; he believed very strongly in the whole idea of deterrence, that his work would not be used to harm people, but would prevent another war. And it does appear that he was right about that. But I was always prouder of his work on Galileo.
Music was an incredibly important thing to him. He made all three of his kids learn
music. Back when my brother and I were in high school, he used to spend something around 12
hours a week driving to music lessons or rehearsals. And he never missed a concert that one
of his kids played in – from the time we started playing instruments in elementary school,
all the way until he was hospitalized last november. My brother and sister both ended up going into music: my brother majored in music performance and composition in college; my sister in music education. Musically, I’m the black sheep of the family.
He was a survivor of cancer. 20 years ago, he developed an aggressive muscular cancer in his leg. Being incredibly lucky, even though he stalled for months after noticing a lump in his leg, it was operable, and between surgery and radiation therapy, he survived it. A year later, there appeared to be a recurrence; it turned out to just be scar tissue, but in the surgery where they discovered that, they needed to do an arterial graft, which caused intermittent trouble for the rest of his life.
Since high school, I called him “Fink”. I don’t even remember why. But he bore it with pride. When I had kids, he wanted them to call him grandfink.
He died of an antibiotic resistant infection. As long as I live, I’ll never be able to forgive the Doctors who took care of him. The illness that killed him started with an infection in his little toe. Due to a spectacularly stupid series of errors – where basically repeated infections with antibiotic resistant bacteria were not treated properly – he developed antibiotic resistant pneumonia, which is what ended up killing him.
He was 80 years old. He was an amazing person. And he will be missed.