Saturday, December 27, 2003

Elliptic Curves, Part I

The Directed Reading Program I took this quarter comes to an end with a presentation, Thursday of the first week of classes. So, over break, I'm preparing notes and slides for my presentation on elliptic curves. Unfortunately, to do so, I needed to find a graphing utility which can handle implicit functions, hopefully a free utility at that. Well, I can't recommend GrafEQ highly enough now. Not only does it handle elliptic curves like y^2 = x^3 - 5x +11, it can take on such bugaboos as y = x^x, as well as number-theoretical experiments like gcd(|10x|,|10y|) = 1.

As far as I can tell, GrafEq takes your relation between x and y and a 2-variable function in z, then analytically finds the zero set. It can take a while to work, if you give it a nasty function: for example, take a good implicit function 0 = f(x,y) and try 0 = cos(f(x,y)). But work it does, and I highly recommend it.

In a few days, I'll post a little explanation of elliptic curves and the group law (if I'm doing this right, it should be understandable to anyone who still remembers a good calculus course), along with a few pictures (if I can figure out how to upload to the U of C website).

And Christmas Day

Each year, the Christmas morning video shows the further development of a few cheesy traditions. We give our dog Nacci a new chew toy, and watch her rend it asunder in the space of a few minutes. Dad takes the bow from one of his gifts and affixes it to his head, acting as if nothing is out of the ordinary. Mom cheers over the bubble wrap in the box, ignoring the actual gift. I search my stocking for the button that plays Christmas music (a part of the stocking itself), and claim it as another present for me. My sister Katie gets increasingly embarrassed and begs us to stop.

This year added a few things. My grandma jitterbugged to her new Glenn Miller CD. Katie turned her herbal mask into a superhero's costume for the camera. Most dramatically, there was a loud crash in the living room as we sat in the family room; Dad had balanced our glass chess set atop the scanner, on the highest shelf, and eventually it slid off and shattered as we all jumped.

Later, we all headed to my aunt Jill's house for Christmas dinner (including multiple pies). For once, all of us sat at the adult table (only to discuss Hey Ya along with the Gnostic Gospels and college life). It was a nice ending to a most merry Christmas.

Tuesday, December 23, 2003

Sports, Desire and Life

Last night, I witnessed an incredible display on Monday Night Football. For those of you who don't typically follow football, here's a synopsis: Brett Favre lost his father to a heart attack the night before the game in Oakland. Brett decided to play after all, dedicating the game to his father, his first coach. Favre then performed unfathomably well, making perfect throws fifty yards out into double coverage, ignoring a fractured thumb on his throwing hand, amassing 300 yards and four touchdowns in the first half for a literally perfect rating and a Packers rout. It was incredible to watch, for just one half, this man play at a superhuman level, in the midst of all his pain.

It reminded me of a story I had seen some time ago about a high-school basketball player, whose grandfather passed away at the age of 62. The boy decided that as a tribute, he would go and score 62 points in the next game. Now, in high school, a team has to be good to score 60 points; while the player was talented, he had never before scored more than 25 in a game. But he would not be denied, making shot after shot, until he scored the last two on a layup while being fouled. With 62 points in hand, the game long won, but only a few points away from the state record, he purposefully flung his free throw twenty feet to the left, and then exited the game. He wasn't there for a record, but to honor his grandfather.

Nice stories, you may say, but so what? A sports contest doesn't really count for anything, after all. Well, the reason I found these important is because I think these are no flukes, but simply what a person can do when it's not just a game to him or her. Both athletes had something larger than their own egos and salaries to play for. It seems ridiculous to claim that most players don't want to win as much, but it's so true. Why do analysts talk so much about motivation, why can a team struggling make the playoffs destroy one who has already clinched, why do we see so many Cinderella stories of players who work their way past their more talented competitors? Because it's possible to desire victory more than your opponent does, and because that desire is as powerful as any raw talent.

This carries over into war as well as sports. A good commander makes his soldiers believe that they stand for something more important than their own lives; he makes them ready to die for the cause. If an army of 10,000 fought as hard as the 300 Spartans at Thermopylae, they would have been an irresistible force in ancient Greece.

We are capable of so much more than we actually achieve; we just do not will it, because our own happiness is not enough of a justification to dig deep. Reminds me of what I said before about (mythological or Hollywood) heroes versus saints. Heroes give us the comfortable fiction that we would act like them if endowed with their gifts. We could save the world like Superman/Wonder Woman, if we had those superpowers. Saints, however, hurt our egos by doing incredible things with no more raw intelligence, stamina, or charisma than you or I have, but with the desire to do what we are reticent to do. Saint Francis was not the smartest, strongest, most handsome, or the best leader. But he made sacrifices that I would be afraid to contemplate, and he changed the world.

Shoot, don't misinterpret me. Brett Favre is not a saint (as much as I, born in Green Bay, cheer for him), but last night he showed something that the saints illustrate in a different way: the unimaginable potential of a human being, fully alive.

P.S. Merry Christmas to you and your families. Or, if you prefer, Happy Decemberween. But in all seriousness, everyone I know is in my prayers tonight.

Friday, December 19, 2003

The New Political Science: Reinterpretation

As I haven't stimulated much controversy here yet, I thought I'd take the opportunity to restate something I've concluded over the last year. I'd said before that Bush is a political genius of sorts, but in reality all I can assert is that someone who writes his speeches is a political genius of the new type: he can get a politician to say X and have the nation hear Y. This skill evolved in the Reagan administration and was honed in the Clinton years.

In perhaps the most famous example, then-President Clinton faced down a news conference in early 1998 on the Lewinsky debacle, and solemnly told the nation X ("There is no sexual relationship"). While I'd say most Americans heard Y ("I did not cheat on my wife"), Clinton could later reinterpret his statement as Z ("The sexual relationship between Monica and I is over.") This was rather adept, since saying Y would have given political opponents a bald-faced lie they could resurrect later, and Z would have admitted the truth too early. As it happened, he was able to hold up that tension till August, when Middle America had ceased wanting to hear about it anymore.

George W. Bush- or rather, someone in the Bush Administration- has been even more clever and circumspect. Raise your hand if you think that George Bush stated before the invasion of Iraq that Saddam Hussein currently possessed weapons of mass destruction. Put your hands down, because he invariably left a little room to maneuver. I've looked through a few months of his releases, and found him saying that Iraq "desires weapons of mass destruction", and that there was a "grave and gathering danger" from Saddam. He said that Iraq "likely maintains stockpiles of chemical and biological agents", and is "willing to use weapons of mass destruction" (if he possesses them). In the closest quote I can find, at the time of his 48-hour ultimatum, he told the American people, "Intelligence gathered by this and other governments leaves no doubt that the Iraq regime continues to possess and conceal some of the most lethal weapons ever devised. This regime has already used weapons of mass destruction against Iraq's neighbors and against Iraq's people."

None of these are airtight or damning, however. After all, missiles are some of the most lethal weapons ever devised, and Saddam had a couple of those. Nothing I found would make a solid clip of "Bush lying" for, say, a Democratic ad. All the direct things we thought Bush said about Iraq and WMD were either set out by other administration officials, or conservative commentators, or were only what we thought the President meant. If you don't believe me, search through the transcripts yourself, looking for something that can't be reinterpreted away. (Although I thought it was a great touch that the official White House website gives these transcripts the header "Iraq: Denial and Deception"...)

Do I think that this counts as misleading? Yes. As lying? Maybe. It's saying something vague and letting the audience fill in the blanks with things you didn't say. But it's not just George W. Bush, and it's going to be the norm in government from now on. I think that nonverbal cues are beginning to lead us astray, so perhaps the best solution is to only read the transcripts of speeches and do our best to avoid leaps of implication. Welcome to post-Watergate politics: no more "plausible deniability", but it's all about "plausible reinterpretation".

Thursday, December 18, 2003

A Human-Based Animal Ethics

I'm bringing this topic up, not because I feel it's more relevant to my life than other ethical concepts, but because I had an idea about it. You see, most PETA-like statements of "animal rights" operate from the standpoint that animals are in some way morally equivalent to people, that their suffering is as terrible to them as our suffering, that they have souls in whatever sense we do, etc. I can't agree with this posture. I believe that people are of a different moral dignity, but I still (of course) think that inflicting unnecessary pain on animals is inexcusably immoral.

First off, I recognize that the pain animals feel is not quite the same as the suffering of human beings. An injured tiger may flee its attacker or respond with aggression, but neither are the human response of hatred; a bear may attack something that threatens her cubs, but will not later track down the interloper for revenge; a deer that breaks its leg does not curse its condition, nor ask why this happened. Of course, a human baby will not have adult reactions to pain either, but there are more vital reasons for protecting human children.

We recognize that animal pain is not on an equal standing with human pain when we witness, say, the widespread starvation of wild elk in Alaska. We may find that sad, but our concern doesn't extend to the level it would for an Inuit tribe starving to death in the same area. We would- at least we should- help the people in that situation, and we understand that the death of many elk is just part of the natural order in that case.

However, pain inflicted by humans on animals is a different case. Not because of some metaphysical value of animals, I think, but because we identify their agonies with ours. As much as we can reason about animal pain, the sight of a wounded animal can often move us as if it were a human who was suffering. To inflict needless pain on an animal, I think, has the psychological flavor of doing the same to a human. It's no coincidence that abuse of pets is a precursor of cruelty toward people; an abuser may "take out" his resentment of other people by attacking an animal, making it an effigy of sorts. It's as if he were hurting the person he hates.

To make my point, consider the Milgram experiment. Subjects believed that they were torturing someone else with electric shocks, although in reality the other person was an actor who felt no actual pain. A large number of subjects obeyed the researchers' commands to continue, even to the point that they were 'killing' the other person. Now, although nobody was in fact suffering, I believe that the subjects were acting unethically, in much the same way it's still wrong to shoot another person if the gun misfires. On some level, they were committing real cruelty, since they believed that they were doing exactly that.

What does this all mean? Society has been looking at animal cruelty the wrong way around; the real problem isn't that the animals feel unnecessary pain, but that people willingly inflict it, and that there is some moral equivalence on that end to cruelty towards other people. I don't have a problem with eating meat, as long as the companies are doing their best to be humane (this, of course, is a matter I haven't looked into closely yet; ah, how long should invincible ignorance last?). Am I being clear enough?

Of course, I don't think that most people would argue against my conclusion that animal cruelty is morally wrong. I suppose that makes this a philosophical irrelevancy. Oh well; I still need to practice my reasoning, even on easy targets.

P.S. The teaching in the Catechism, by the way (#339), also looks at it from the perspective of human actions, not of animal rights, but talks in terms of correctly using God's creation.

Wednesday, December 17, 2003

*It's been a few years since I would have identified myself as a "teen," but it still felt odd to actually turn twenty.

*My sister Katie was accepted to her dream school, NYU, for the theatre program. She hasn't calmed down entirely yet.

*My grandma has been living with us for the past month, since she's not ready yet for the stairs in her apartment. Thanks for all the prayers.

*I'm working on some of the stuff I promised, but since I have time now to edit, I don't want to throw out these ideas extemporaneously. I promise it'll get there...

P.S. I read my first Iraqi blog today. Absolutely spellbinding; check out the Dec. 12 and Dec. 9 entries. Thanks to Eve.

P.P.S. I saw "Return of the King" tonight. At the risk of sounding like every reviewer you've read, I'll tell you now: it works in just about every visceral and emotive way a movie can. Three and a half hours, and I was immersed in the world of Middle Earth for all but approximately two minutes of it. Words do not describe this experience, so I'll stop trying.

Thursday, December 11, 2003

Finals Over

So now I’ll have time to write some of the things I’ve been thinking. Expect over break several of the following:

Highlights and lowlights of this past quarter

Reflections on marriage in 2003

Proposal for an ethics of animal treatment

Ideas on prayer for the deceased and on time itself

Spirit-based and rule-based Catholic ethics

Elliptic Curves and Fermat’s Last Theorem

And more!

Hopefully I’ll be coherent... and interesting...

Tuesday, December 09, 2003

Disclaimer 2

I really shouldn't have to post this, but none of my religious reflections are intended as a proof of the Catholic faith or of God in general, or of my ethical and political views. Most of them are personal experience and ideas which you can peruse at your pleasure. I don't view this weblog as a vehicle to convert people (although if something you read does lead you to examine your thoughts on something, I'd thank God for it). So all comments, even "Pat, you don't make any sense here" or "I disagree with you completely", are OK with me (sans flaming and other irresponsible actions).

I Believe, Because It Is Difficult

Something recently occurred to me with regard to the Catholic Mass. I am at times capable of great focus. I can lose myself in a novel until the dawn is beginning to show. I can immerse myself in a math problem or an essay. I can focus during a completely dull lecture (modulo interrupting my notes with comical phrases once or twice). But I can't, for the life of me, keep my concentration during the Mass and particularly the Eucharistic Prayer.

What I've come to realize is that this means I must have some desire to avoid concentrating. When I can't will myself to do something I'm physically capable of doing, there is some emotional/psychological/spiritual factor at work. And the question is: why? If one suggests that somehow I know it's all empty, then how's that different from a novel I don't really enjoy? I stuck it out through Rabbit, Run, after all. That desire tells me that something is, in fact, happening in the Mass, and it's something that I don't fully like.

And humble submission to an infinitely superior God is certainly something I have desires against. I desire that submission, of course, but I also desire to be 'my own person', or more accurately 'my own God'. That's part and parcel of fallen human nature, as the Church puts it. So what exactly do I mean? I believe that the fact that I'm resisting something means that there is something- or Someone- there to be accepted or rejected. And I'm praying for a deeper acceptance of that One.

Human Dignity and Personhood Theory

A friend of mine believes that because of society's investment in education, people such as myself have a responsibility to survive that exceeds that of others. The hypothetical example that came up was that of sacrificing my life to save that of a child, which my friend thought would be morally indefensible: I'm just 'worth more', at least to society.

Somehow, I can't agree with that utilitarian line of reasoning, any more than I can stomach the 'personhood theory' asserted by people like Princeton's Peter Singer. According to it, people have no inherent worth except for their level of consciousness and productivity; Singer directly advocates infanticide, euthanization of the disabled, et cetera. Here's a link I chose because it lists his most startling conclusions without necessarily taking my side. I am trying to be balanced (if not fair). It just seems far too Brave New World to me, a new class system in the name of 'objectivity'.

But it's difficult without starting from religious principles to assemble a convincing argument in favor of human dignity. (Nota bene: last I checked, "difficult" is no synonym for "impossible".) And that's a problem. It's much, much easier to show that human life begins at conception than it is to convince someone else that we should respect that human life in and of itself. So what's a Catholic natural-law quasi-humanist to do in a post-Christian post-morality post-humanist post-post-modern world?

Finals Week

Monday was a brutal Representation Theory final. After writing out a lengthy pre-announced proof of the Frobenius Kernel Theorem, we had to choose five out of eight further problems. I couldn't understand three of the questions, and I blanked completely on two more. Thus I spent the last half-hour of the test staring at the exam booklet, wondering if everyone else had bombed the test as badly as I, and writing a haiku:

Thought I understood
Representation Theory;
Turns out I was wrong.

Fortunately, grad student Mike did the best thing any teaching assistant has ever done: he went over the problems with us immediately after the final. Turned out that everyone had, in fact, done about as badly as myself, that one of the questions I'd misunderstood was mostly right anyway, and that he'd lobby to weight the proof 80% and the rest 20%. I felt much better then.

European Civilization final was not stressful, and I think I'm prepared for Mathematical Logic. Sorry for the marathon post; good luck to all other Finals takers!

Saturday, December 06, 2003

Still Alive

Sorry for the lack of posting. I have much to say, but little time before Finals start. Of course, I just used 6 hours of that prep time on the William Lowell Putnam Exam, a completely voluntary national undergraduate math competition. What was I thinking, you ask?

Good question. I woke up at 7:30 today and headed to the math building. PS2 handed out the test envelopes. He warned us that the mean on this test is a couple points out of 120, and the median score is zero (in other words, at least half the students who take the Putnam get no points). We had 3 uninterrupted hours to work on the first 6 problems, then 3 more hours in the afternoon for 6 more problems.

Example: Let n an integer and take the sequence 1, 1/2, 1/3, ... , 1/n. Then average the adjacent terms, resulting in the sequence 3/4, 5/12, ... , (2n-1)/2(n)(n-1). Repeat this process of averaging neighbors until only one term remains; call this x. Show that x < 2/n.

I was rather proud of that problem, by the way. I know I didn’t make the papers, but I believe I scored in the low 30s, and that makes me feel good. Now if I could only understand Burnside’s Theorem in time for my Monday final...

Monday, December 01, 2003