Saturday, August 21, 2004

any Kerry supporters out there?

Donald Sensing at One Hand Clapping has offered to publish an essay from a Kerry supporter who can argue why Kerry would make a good president. If you are a Kerry supporter, this is a great opportunity to present your case to thousands of people. Sensing's audience tends to be anti-Kerry but also serious and intellectual. They will give you a fair hearing if you make a good argument.

So, do you think Kerry is a genuinely good presidential candidate rather than just being not-Bush? Do you have a good argument for this position? Then I encourage you to enter Sensing's contest.

Friday, August 20, 2004

boobs for safety

Boob jobs have saved lives in real-life combat situations. This isn't theory people, it's actual experience with actual flying bullets. This news article tells of a woman who was hit by a bullet during a shootout between police and drug traffickers. Her life was very possibly saved by her breast implants.

So now, along comes Mary Carey, alleged porn star, involving herself in military affairs as though her "acting" experience gives her expertise in martial technology. What does a porn actress know about the combat effectiveness of head-light enhancements? She thinks just because she's been around a lot of after-market chest equipment she knows their ballistic properties? I don't think so. Even if she does girl-on-girl action (not having followed her career, I couldn't say) so that she's had a lot of hands-on experience with plastic mammaries, that still doesn't provide the appropriate expertise to weigh in on this issue.

Boob jobs save lives. That's the bottom line. And I salute the US military for recognizing this fact and making this life-saving technology available to service personnel and their spouses.

And if this activity produces more esthetically pleasing military bases, well, so much the better.

on sets and numbers

In the spirit of my ongoing campaign to alienate most of my readers, I've decided to add another post on mathematics (or more properly, on the foundations of mathematics which is a more interesting topic).

Probably the first symbols of numbers were collections of stones or shells or other small tokens. A herder could count his sheep (or his children) by keeping a small bag of stones and marking them off, each one to a single animal. If he had a special stone for each sheep, then this wasn't mathematics, it was naming. But if any stone could match any sheep, then the bag of stones represented a number and the herder was doing primitive mathematics.

So what is a number? Numbers don't seem to act like properties. You can ask what is the color of something, or what is its temperature and the answer is pretty much determined by the object. You can't ask what is the number of something. What is the number of a basketball team? Is it one team or twelve players? It depends on how you describe the team.

What is a set? People sometimes visualize sets as collections or groups of things, but this is only an analogy to what sets really are. A set is really the extension of a property. So once I explain what a property and an extension is, then you'll know what a set is.

A property is pretty much what you think it is. It's a description. For example "__ is red" is a property. Some things are red and some things are not. Here is an interesting pair of properties:

P1: __ is a prime number between 2 and 10
P2: __ is an odd number less than 8

What's interesting is that every number that satisfies P1 also satisfies p2 and the reverse. So are P1 and P2 the same property? Clearly not. P1 is about prime numbers and P2 is about odd numbers. They are entirely different concepts. But when two properties have this feature, we say that they are coextensive or that they have the same extension. The extension of a property is all and only those objects that satisfy the property. A set is the extension of a property viewed as an object in its own right. A collection or class. (The intension of a property is the underlying concept).

For many purposes, you only care about the extension of a property and not the intension. For example these two properties are coextensive:

P3: __ is an odd number
P4: __ is a number having a 1, 3, 5, 7, or 9 as the last decimal digit

We know that these two sets are coextensive, so we switch between them at will. There are 951 people who are going to vote yes or no. Can we be sure there will be no tie? We want to know if 951 is an odd number so we apply P4 to find out, rather than P3, which is the question we really want to answer. Since they are coextensive it doesn't matter. They describe the same set.

As sets are derived from properties, so operations and relations on sets are derived from operations and relations on properties. You can operate on properties using "and", "or", and "and not" (called logical connectives). P3 can be combined with

P5: __ is less than 8

to produce

P6: __ is an odd number and is less than 8

which is coextensive with P2. This corresponds to set intersection. If S3 is the extension of P3 and S5 is the extension of P5, then S6=S3*S5 is the extension of P6 (using * to represent set intersection). Similarly

P7: __ is an odd number or is less than 8

to produce the set union S7=S3+S5, which is the extension of P7. Finally,

P8: __ is an odd number and is not less than 8

has the set difference S8=S3-S5 as its extension.

Reducing a property to its extension is a process of abstraction. You throw away some details so that you can reason more generally. In this case, you throw away all details except the identities of the objects that satisfy the property. The underlying concept is entirely discarded.

You can think of this process of abstraction as being defined by the relation of coextension. If property p is coextensive with q and q is coextensive with r, then p is coextensive with r. We say that p, q, and r are all equivalent with respect to coextension or that they are all in the same equivalence class. Each set corresponds to an equivalence class of properties.

Of course coextension isn't the only relation we can use to abstract properties. Another one we can use is cocardinality (I made that word up to go with coextension, the usual term is "equinumerous"). Cocardinality throws out even more detail than coextension does; we don't even keep track of the identities of the elements. Two properties p and q are set to be cocardinal if there is any one-to-one mapping from the values that satisfy p to the values that satisfy q. Within the mapping, each value that satisfies p has exactly one corresponding value that satisfies q and each value that satisfies q has exactly one corresponding element that satisfies p. For example given

P9: __ is an odd number less than 6
P10: __ is an even number less than 7

the set of objects that satisfy P9 is S9={1,3,5} and the set that satisfies P10 is S10={2,4,6}. We can map S9 to S10 like this: 1<->2, 3<->4, 5<->6. Since we can create this mapping, P9 and P10 are cocardinal. Compare this to coextension, where we don't throw away the identities of the elements. Two properties are coextension only if there is a one-to-one mapping that specifically maps objects on one side to equal objects on the other. Cocardinality allows you to map objects on one side to any object on the other side.

Like coextension, cocardinality can be used to produce an abstraction. As the abstraction based on coextension produces sets, the abstraction based on cocardinality produces cardinal numbers. Every property is abstracted to the cardinal number that goes with its equivalence class.

The operations and relations on properties extend to cardinal numbers much as they do to sets. In order to avoid confusion, I'm going to switch from numbers to letters for my examples:

P11: __ is a vowel
P12: __ is a liquid consonant
P13: __ is a letter less than b.

The extension of P11 is {a,e,i,o,u} with cardinality N11=5, the extension of P12 is {l,r} with cardinality N12=2, and the extension of P13 is {a} with cardinality N13=1. Consider

P14: __ is a vowel or a liquid consonant
P15: __ is a vowel or a letter less than b

The extension of P14 is {a,e,i,o,u,l,r} with cardinality N14=7=N11+N12. So "or" corresponds to addition. But notice that the cardinality of P15 is not N11+N13. This is because addition only applies to disjoint properties. Properties that have no elements in common.

P16: __ is a vowel and not a letter less than b
P17: __ is a vowel and not a liquid

Here, "and not" corresponds to subtraction (N16=N11-N13) but only in a restricted case: where the second property is contained in the first.

Multiplication is a bit more complex and I'll handle it in a later post.

Tuesday, August 17, 2004

some people never learn

Joel Mowbray reports that the State Department is still approving 90% of all visa applications from Saudi Arabia. Nineteen of the 9/11 hijackers were from that Saudia Arabia. According to the 9/11 commission report, that's because it was too hard to get visas for people from other countries. Furthermore, all 19 of the visas should have been rejected according to US law.

So now it looks like al Qaeda has big pre-election plans and they have to get people into the country to carry them out. So I have a question: if another three thousand people are killed by Saudi terrorists that were let in on flaky visas, then can we fire some of the idiots at State and replace them with people who will do the job?

I blame George Bush for this. Yet another reason I'm going to have to hold my nose when I vote for him.

Oh, I also blame Colin Powell for this. Yet another reason to hope that he keeps his promise and resigns after the election.

Monday, August 16, 2004

algebras, theories, and calculuses

In the post on non-numeric algebras, it may have occurred to you that I was leaving out some kinds of rules. For example, how do I know that I can add a number to both sides of an equation and the equation remains true? Is this part of the algebra of arithmetic? The rule might look something like this:

substitution in +: if a=b then a+c=b+c

This rule says that if you know that a is the same as b, then you can infer that a+c is the same as b+c. This rule is not part of the algebra of arithmetic. Part of the reason is that the rule would be too limited in scope if it were part of arithmetic. The rule isn't just true for addition, but also for all other functions. A more complete rule would be

substitution: if a=b then f(a)=f(b)

where f is any function or operation at all. We can say even more than that. Let P be any property at all. We write P(x) to say that x has property P. Then

indistinguishability: if a=b then P(a) if and only if P(b)

This says that there is no property at all that lets you distinguish between two equal values. All of this is part of the theory of equality, not the algebra of arithmetic. Algebras just assume this theory of equality. A theory is like an algebra except that where algebras deal with operations, theories deal with relations (like equality). (Beware: "theory" is one of the most confusingly overloaded words in mathematics. In different contexts it means different things). Here are the other rules associated with the theory of equality:

reflexivity: a=a
symmetry: if a=b then b=a
transitivity: if a=b and b=c then a=c

Well, that's one answer to the question, "how do I know that I can add a number to both sides of an equation and the equation remains true?". The other answer is that we are dealing with an algebra rather than with a calculus. A calculus is a set of rules for transforming expressions or equations. It is different from an algebra or a theory in that it views the equations as objects in their own right rather than as merely symbols for communication. This is not an obvious point, so let me try to illustrate with an example. Consider a sentence:

s1: Joe was wearing a brown sock and a black sock.

There are two radically different kinds of questions I can ask about this sentence. The first group of questions is like this:

q1a: How many socks is Joe wearing?
q1b: What is wrong with the way Joe is dressed?
q1c: What can you guess about Joe's sock drawer or laundry basket?

These questions are all about Joe and his socks. They rely on your understanding of the meaning of the sentence. The other sort of questions are like this:

q2a: How many times does the word "sock" appear in the sentence?
q2b: How many words occur twice in the sentence?
q2c: What is the subject of the sentence?

These questions are about the sentence itself. They don't require any understanding of the sentence meaning. Even q2c only requires an understanding of the sentences structure, not its meaning. For example you can pick out the subject of

s2: The plu are delarkable this morning.

and you can do it without having any idea what the sentence means.

This is the difference between equations in an algebra and equations in a calculus. An algebra only uses equations as a way to express meaning. They deal with the meaning. You can use any knowledge you have, including a theory of equality because you are dealing with real mathematical objects. A calculus deals only with the equations themselves, ignoring the meaning. (What you learn in "calculus" textbooks is not really a calculus in this sense).

In a calculus, you need rules like substitution in +. You can't rely on outside knowledge because there isn't any outside knowledge other than the structure of the equation. If we were interested in a calculus of arithmetic, we would write transformation rules for formulas rather than rules expressing properties of numbers and operators. A transformation rule says, "if you have an equation that looks like this, you can change it to an equation that looks like that." Here is how transformation rules are typically written:

+ introduction: a=b

Given any equation that looks like the one on the top, you can transform it to the one on the bottom.

If you are dealing with a calculus, you aren't allowed to do anything that isn't strictly permitted by the transformation rules. By contrast, if you are dealing with an algebra or theory, you can use your outside knowledge to reason about the equation.

Sunday, August 15, 2004

Hey, what's this slime doing here?

Andrew Stuttaford reviews Alien Vs. Predator:
Paper-thin characterization, laughable dialog, wooden acting, lotsa slime, Spud from Trainspotting, hoky mythologizing, sub-Tomb Raider sets, a weird X-Files reference (oh yes), interesting details for Alien completists, Frank Black, and great, great creature punch-ups, this movie has it all, except, disappointingly, gratuitous nudity (that pesky PG-13, I suppose).
I just saw it today, and I have to agree 100%. The lack of gratituitous nudity was especially disappointing given how hot the female star, Sanaa Lathan is. She did a good job in the female-action-hero role, too. I'm not a big fan of the female-action-hero genre, but Lathan at least looks athletic and they kept the action believable.

And although there were some great creature punch-ups, there was a little too much of the fast camera motion, poor lighting and tight close-ups, all intended to give the impression of action without showing the shoddy details. I hate that. Show me the shoddy details!

Some features: a genius archeologist who can read ancient hieroglyphics with such subtlety as to know that they are talking about aliens from other planets (rather than, say, gods) and to figure out the entire plot from almost no information, a bunch of well-armed guys and automatic weapons on an archeological dig, a wise hero who tries to warn them of their foolish recklessness from the very beginning but ends up going anyway, just to try to save who she can, an ancient pyramid with traps and secrets, a great warrior teaming up with a plucky girl. It was basically Alien/Aliens/Predator/Stargate/Raiders of the Lost Ark/True Grit. One big happy family in one movie. Thankfully there was no hint of Tomb Raider.

I'll echo Stuttaford and give it two severed thumbs up.