Saturday, October 16, 2004

reductionism in geometry

Here is, some, background.

Analysis is an investigative technique where you break the subject down into parts. It is a very successful technique and leads to some important simplifications in our view of the universe. For example, we analyze human bodies into parts. If someone is sick, it is often because one of the parts is not functioning correctly. Identify the faulty part, and you are halfway to a solution.

Analysis within a single domain is an extremely useful, even vital form of reasoning. A domain (as I defined it in a previous post) is a class of things. Concepts are in a different domain from objects. Physical objects are in a different domain from thoughts. You can tell if two objects are from different domains because they have different properties; not just different values of the same properties, but entirely different properties.

For example, physical objects have a size, a mass, a temperature. Every physical object has those properties to a greater or lesser extent. Thoughts have none of those properties at all. Thoughts have subjects and emotional content and other non-physical properties. Physical objects have none of those properties. That is why thoughts are in a different domain from physical objects.

It makes sense to analyze an object into other objects that share the same domain. You analyze a physical object into other physical objects that have smaller size and mass. It makes a lot less sense to analyze an object into other objects from a different domain. When you do this, you have to show not only how the original object is made up of the other objects, but also how its properties are made up from aspects of the other domain. This sort of analysis is called reduction because you reduce one domain to another.

One success story for reductionism can be taken from geometry. Ancient geometry had two distinct concepts: points and lines. Some modern geometry reduces the domain of a lines to the domain of points. This mathematical form of reductionism offers some insights that can be used to talk about other kinds of reductionism.

As in any other instance of reduction, the point reductionist is faced with a serious problem from the very beginning: lines clearly aren't points. Lines have direction, points don't. Lines have length, points don't. Lines can intersect with other, different lines, points do not intersect with other, different points.

In geometry, we get around this problem by defining a line as a set of points. A set of points is not a point: for example, a set of points can intersect with another, different set of points, a single point can't do that. A set of points has a cardinality, a single point doesn't.

In mathematical logic we say that a set of points has a different type than a point. This difference in type is obviously similar to a difference in domains and the point reductionists exploit it to get their reduction. If a set of points were the same type as a point, we would have to explain how something that seems so different from a point (namely a line) can be reduced to something that is the same as a point. But now we can say even though lines are reduced ultimately to points, they are things of a different type and that is what gave us the illusion that they are of a different domain.

So, does this example from geometry prove that reduction is sometimes possible?

I'll give you a moment to think about it. You might want to compare and contrast the point/line dichotomy with the physical object/thought dichotomy.

Back? OK, here's my take: the problem with this example from geometry is that points and lines are only from partially different domains. Even though lines have some properties that points don't have, there are other kinds of properties that they both have in common. They both have positions, for example. In fact, it was always understood that lines contain points or even that points are "parts" of lines in some sense. The only thing the point reductionist really had to do was explain how lines could be dispensed with entirely in favor of points.

By contrast, it is highly controversial whether physical objects can be "parts" of thoughts in any sense or thoughts parts of physical objects. Yet there is some overlap of properties. Both a thought and physical object, for example, exist at some particular time.

It turns out that the concept of domain is not perfectly precise. Some kinds of properties are shared by many different kinds of things. We will have to examine each individual case of reductionism to see if it is plausible or not.

To begin, lets examine the reduction of line to point. Have modern mathematicians proven that a line is nothing more than a set of points? Many people think they have. But couldn't we just as well reduce points to sets of lines? Define a point as the set of all lines that pass though it.

So, if a line can be reduced to a set of points and a point can be reduced to a set of lines, what possible justification can you have for claiming that one of these reductions is somehow more fundamental? Isn't it more reasonable to just observe that point and lines are distinct concepts and that there are interesting homomorphisms between individuals of one and sets of the other?

It turns out that even this uncontroversial example of reduction isn't especially convincing. It's only a mathematical convenience. It turns out to be convenient to use sets of points rather than lines for some purposes. Our innate talent for physical analysis misleads us into thinking that we have found the parts of lines when all we have found is an interesting homomorphism.

I just realized that I never got around to explaining what a homomorphism is when I was talking about abstract algebras. I'll try to rectify that in the near future. For now, think of a homomorphism as a mapping from one domain to another in a special way. For example, you can map lines into sets of points in lots of different ways, but there is one "right" way to do it: map each line into the set of points that are in the line.

the death of civility?

Kerry has been heavily criticized for bring up Cheney's lesbian daughter in the debate. When he was citified for it, he went on the attack, partly through surrogates. A campaign spokeswoman said Cheney's daughter was "fair game" and Edwards's wife said the girl's mother was ashamed of her.

The most striking thing about this to me is the tone of it. Is it just my imagination or would this have been unthinkable just a few years ago? What would it cost Kerry to say "I didn't mean to give offense and I apologize for speaking carelessly?" In my view, something like that would completely negate what he said and even increase his stature.

But Kerry and his team don't agree. They seem to think that any civility at all toward the opponent would be harmful to them. Have things really gone that far? Have the Democrats gone so far over the edge that they wouldn't even let their candidate apologize for a ham-handed comment in a debate?

How much farther are things going to go?

the internet is sick today

I don't know how much of it is my own intranet and how much is external, but all kinds of things are going wrong lately. Right now, for example, I can get to Blogger to post a message but I can't access my blog to read it. It may just be a problem with our domain name servers, but I don't know who to call about it.

I'll probably bore you all with details when I find out what is going on.

Friday, October 15, 2004

stopping at almost nothing

Ann Coulter's latest column is not her best, but it ends with a great line:
As if it means something, Kerry keeps vowing: "I will never stop at anything to hunt down and kill the terrorists." But he will stop at the Iraqi border. Or if the French and Germans aren't on board. Or we don't have United Nations approval. Or it would require investigating a Muslim under the Patriot Act.

standing up for the equipment

Mostly Cajun almost got himself court-marshalled once. He seems to have been a bit overprotective of that helpless little tank of his...

the Guardian

Tim Blair reports that the Guardian, a British newspaper, has gotten a voter list from Clark County, Ohio and they are encouraging British citizens to write to those people about the election (link from Instapundit).

There is a lot of sport made about it (read the comments too, some are very funny), but there is also a bit of anger. I don't see why people should be angry about this. What's wrong with encouraging people to communicate? Tim Blair and his commenters are interpreting this an attempt to influence an American election, and they're right, but the point is that the attempt is by perfectly legitimate means.

Isn't this what blogging is all about? Wouldn't we all like it if some European leftists were influenced by some conservative American blogs? Influenced to the point of changing the way they vote? We American bloggers who write some political commentary, would any of us claim that we only want to influence other Americans? I doubt it.

We have opinions. We think our opinions are right. We think we have good reasons for our opinions and that if we could adequately express these reasons to the other side, then we might get some of them to change their minds. The Guardian editors are no different, they just have different opinions.

And who knows, maybe some Ohioans will write back and change the mind of some Guardian readers. Lot's of Guardian readers probably have no idea how little they know about the world. They think that they get lots of good international news from the Guardian and BBC and Reuters. They have no idea how distorted their news is. Maybe a letter from Ohio is just what they need to clue them in that they need to get on the Internet and find out what is really going on.

I not only approve the Guardian effort, I'd encourage other newspapers around the world to do the same thing. Let's have political pen pals from all over. Get the newspapers out of middle and let people talk.

stem cell research

Donald Crankshaw has some comments about fetal stem cell research. One thing he doesn't do is explain what is morally objectionable about it. I think it is important to say this almost every time you talk about the subject because a lot of people still don't understand what is going on: fetal stem cell researchers are trying to find ways to use body parts from dead babies in medicine.

This is a gruesome and horrific practice. It's the stuff of horror movies. You expect to read about these things in WWII medical experiments in Nazi Germany and Japan, not in modern-day America.

I was disappointed that Bush didn't point this out in the debates. I believe if more people understood this, there would be a lot less support for the practice.

Thursday, October 14, 2004

sorry about the delays

My site is loading slowly. It's not the new image though, it's Haloscan, which seems to be having some problems with domain name servers. I'm not getting any comments or trackback, but there is a delay while the browser tries to load them.

It this localized or is everyone seeing it?

Wednesday, October 13, 2004

new logo, new links

That new logo at the top is pretty cool, huh? It was done by a friend of mine, John Vu. And that's pretty much what I look like in my metahero costume when I'm going around the land fighting for justice and human dignity and the right for men to enjoy wet T-shirt contests even though we respect women and would never view them as mere sex objects (I mean under normal circumstances, such as when they aren't competing in a wet T-shirt contest).

John doesn't have a web site but if you like the drawing and want to see about getting one for your site, email me and I'll pass it on.

And, as long as I'm not on the subject: I've added some new blogs over at the right.

Beyond Salvage hosted the current Storyblogging Carnival and does some good photoblogging too.

Blaster's Blog is one I found through Tom Harrison's page. He's got some interesting political stories.

political violence

Michelle Malkin brings together a bunch of the stories of Democrat violence against Republicans. I predict that it's going to get worse as it becomes more and more evident that Kerry is going to lose.

Left-wing rabble rousers have been doing everything they can to build up the hatred and anger of their side. All that rage has to go somewhere, and I predict that there will be some serious felonies committed before it is all over.

I also predict that the rabble rousers will deny responsibility while defending the felons. They will tell us they were provoked, that they were afraid their freedoms were being taken away. That the war in Iraq is mass murder so they were trying to stop mass murder. They will, in effect, use the arguments that the KKK and the abortion bombers have used to defend their violence. The very arguments that the left was so primly outraged about.

Tuesday, October 12, 2004

croc-wrastlin' granny

A woman in Australia jumped on a crocodile that was dragging her son-in-law toward the water. Not exactly what you'd expect until you get to the kicker: it was dark and she thought it had her grandchild. Don't get between granny and the baby, guys.

Grandmother and son-in-law are both recovering. The baby is fine too.

There may be a movie about it.

Christopher Reeves, rest in peace

Christopher Reeves played Superman. After he made four movies as the man of steel, he was thrown from a horse and paralyzed from the neck down. The accident that paralyzed him was one of the most powerful ironies of our time.

Reeves represented Superman: faster than a speeding bullet, more powerful than a locomotive, able to leap tall buildings with a single bound and invulnerable to almost everything in the universe that can hurt mere mortals. Superman was the fantasy of godhood brought to man. Through the fantasy, we also became all-powerful and invulnerable. It could be a heady experience. He was, in a way, an alternative to Christ.

Christ offered hope in suffering; Superman offered a life without suffering. Christ taught us that if you had the right spirit then you could have joy no matter what your outward circumstances; Superman taught us that if you were strong enough you could make your circumstances be whatever you want.

When Reeves was paralyzed, we saw the man who represented power reduced to powerlessness. Superman was faster than a speeding bullet; Reeves could not even walk. Superman was more powerful than a locomotive; Reeves was not able to lift his own arm. Superman was able to leap tall buildings; Reeves needed someone to feed him. Superman was invulnerable; Reeves died from a bed sore.

In 1978 when Reeves made the first Superman movie, I was seventeen years old. I was young and strong. I thought I was bulletproof and I would take risks just for the sake of taking risks. Over the years, that changed. I started to feel the effects of age. I had aches and pains. I didn't heal as fast as I used to. When I decided to get back in shape, I would hurt myself just by exercising.

It was a depressing experience, and a little bit frightening. And I was still adjusting to it when Superman was brought low. Reeve's accident wouldn't have effected me so much if it had happened a bit earlier before I started the long decline or a little later after I had become inured to it. But because of what was going on in my life at the time, I found it very troubling. If that could happen to Superman, it could happen to anyone.

Still, life goes on. And Reeves proved that you can continue to be a part of life even if you have no physical power left. The man that once offered us an alternative to Christ's solution, proved the power of that solution.

I don't mean to imply that Reeves was a Christian. I don't know if he was. But I do know that he carried on with a full life even though his range of activity was greatly truncated. He proved that you don't need to be physically powerful to be a powerful influence. The lesson of the latter part of his life was far greater than the lesson of the earlier part.


Oops. I wasn't able to post when the announcement for the Storyblogging Carnival came out and then when I could post I forgot. I hope I remember to post an announcement today...

Nader and Perot

It just occurred to me: Bush didn't insist on a three-way debate with Ralph Nader.

He could have. The Bush team probably figured that Kerry had the advantage in a debate so they were better off not having one at all. If they had insisted that Ralph Nader be allowed in like the Democrats insisted for Ross Perot, there would have been no downside. If Kerry refused, Bush could get out of the debate without refusing to debate. If Kerry accepted, Ralph probably would have gotten a surge out of it.

Bush showed a lot of class by not taking advantage of this situation the way the Democrats took advantage of the situation with his father. I'm sure the leftist blogs are grudgingly congratulating him for his classiness too. Anyone have any examples for me to link to?

OK, I was kidding about that last bit.

Monday, October 11, 2004

I love blues

I figured out my posting problem. I'd share it with you but it's kind of embarrassing so I thought I'd distract you with this: Hey! Radio DGCI is having a blues night! Check it out.

this is a test

I can't seem to publish a post. This is a test to see if I can publish this one.

Sunday, October 10, 2004

I hate RealMedia

First the rational comments and then the rant:

RealMedia created a proprietary multimedia format with the express purpose of making everyone's life more difficult and leveraging a profit out of the difficulties they created. It's sort of like a company going into the car-making and road-building business at the same time, and deliberately building roads that only their cars can drive on. It's unethical, and I think if people had understood what they were doing, they never would have been successful. Microsoft and Apple have done the same thing, and I think everyone involved in all these efforts ought to be lined up so everyone who wants to can go down the line and slap all of them.

Please don't use proprietary media formats. MPEG and MP3 are perfectly adequate for all multimedia applications and all cars can drive on them.

begin rant:
I was forced to use RealPlayer to listen to DGCI's radio station. The player froze my computer. I pressed the reset button and instead of seeing a reboot, I saw a dialog from RealPlayer asking if I'm really sure I want to reboot because I'm in the middle of listening to something.

How freaking unbelievably arrogant do you have to be to think your application is so freaking important that it should delay a reboot because the user is listening to a freaking song! It's a freaking song! I can buy the freaking CD and listen to it any time I want!

Those dialogs are supposed to prevent you from losing work. When you spend an hour creating a file and then reboot without saving, it costs you time. What the hell does it cost you if a freaking song is interrupted?

And the freaking song is playing on the freaking computer! If I'm listening to it and I'm rebooting the computer then --listen up you RealMedia morons-- I freaking know the song is going to stop! It's playing on the freaking computer! I'm rebooting the freaking computer! The freaking computer can't play a song while it freaking reboots! Everyone freaking knows that! Shut up and let it reboot you freaking morons!

Thanks. I feel better now.