Do math like a jellyfish

It occurred to me this afternoon as I sat down to blog that my blog can basically be distilled down to a few recurring themes:

1. Pro-life issues
2. Miscellaneous political issues
3. Science and religion
4. Nerdy math stuff

Aside from the random things I find on the Internet and just post to say, "That's cool!" I think this really sums up my blog. I didn't necessarily mean for this to happen (my blog is still what my title says it is) but I guess these are things I think about a lot.

Anyway this afternoon I was thinking about topic 4. Nerdy math stuff and I decided my blog could use a little diversion from the more serious (read: relevant) topics normally discussed.

So I started reading this book called Is God a Mathematician? which is about the question, "Does math describe something that actually exists, or is it all just made up in our minds?" In other words, do circles and lines and numbers actually exist in some Platonic universe, or are they just a convenient fiction?

Personally, my intuition says that yes, there is some Platonic universe in which all these concepts actually exist. I guess. I mean, there's nothing to force an intelligent creature to come up with the concepts of "1, 2, 3, 4, ..." and so on, but if they did, whatever theorems about those concepts they came up with, they would match up with ours. There's an actual logical structure to the universe that exists, whether we try to imagine our way around it or not.

But there was one motivating example that I found striking. It comes in the book as a quote from Sir Michael Atiyah, whom I had not heard of before. Here's his idea:

"[L]et us imagine that intelligence had resided, not in mankind, but in some vast solitary and isolated jelly-fish, buried deep in the depths of the Pacific Ocean. It would have no experience of individual objects, only with the surrounding water. Motion, temperature and pressure would provide its basic sensory data. In such a pure continuum the discrete would not arise and there would be nothing to count."

As a math grad student who is interested in analysis (for all of you non-math majors out there, think Calculus... on steroids) I am very much interested in the study of continuous things. So it strikes me as a good thought experiment to try to develop some sort of mathematics starting with the continuum, rather than with discrete objects.

This is hard. We human beings, perhaps thanks to God's blessing, started doing math by counting things. So all of our theorems from back in the day (before we invented weird things like Calculus) have to do with finite numbers, like 1, 2, 3, 4, and so on. What Atiyah is saying is, okay, now imagine there's nothing to count. You can't count one cow, two cows, three cows because there are no cows. It's really hard for us to imagine, because we're trained from the time we're little to count things.

So when we did finally start to study continuous things, we were able through a series of brilliant strokes to describe continuous things in terms of discrete things. (Or if you are currently unable to do this, that is why you never got Calculus. Sorry.) That is, if you want to understand a nice curve, just chop it up into little tiny pieces and "take the limit" as the number of pieces you chop goes to infinity. It's a brilliant strategy for moving out from the finite numbers you can count to the continuum.

I wonder if it's possible to work the other way around, or simply to bypass discrete math altogether, by starting with the continuum. What if we could do math like a jellyfish? What kind of mathematics would we come up with if all we had to experience was a set of continuous phenomena? Personally I can't quite imagine it at this point, but I'm going to try and think about it to see what I come up with.

This is what we mathematicians think about as we're walking home from lunch...

Yeah...

Pro-lifers gain a slight majority

Gallup has released a poll showing that a slight majority of Americans now self-identify as "pro-life." This has never happened since Gallup started polling this issue back in 1995.

I was curious about why this happened, and Gallup has some interesting theories. They point out that over the past year the number of Republicans self-identifying as "pro-life" has increased by 10%, whereas Democrats haven't shifted much on the issue. They offer this explanation:

"With the first pro-choice president in eight years already making changes to the nation's policies on funding abortion overseas, expressing his support for the Freedom of Choice Act, and moving toward rescinding federal job protections for medical workers who refuse to participate in abortion procedures, Americans -- and, in particular, Republicans -- seem to be taking a step back from the pro-choice position. However, the retreat is evident among political moderates as well as conservatives.

It is possible that, through his abortion policies, Obama has pushed the public's understanding of what it means to be "pro-choice" slightly to the left, politically. While Democrats may support that, as they generally support everything Obama is doing as president, it may be driving others in the opposite direction."

It's an interesting short-run analysis, but what I'm interested in is the long-run. The fact is, Gallup polling shows that there has been a roughly steady shift over the past 14 years toward the pro-life side. What does this mean?

One thing that makes these data complicated to analyze is that political orientation also changes over time, just as views on specific issues do. So it's not really sound to say, "Liberals didn't change their views on this." Maybe there are people out there who used to identify as liberal, but then changed their position on abortion, and so now they guess they're politically moderate, even though their basic outlook has remained the same in a lot of ways. You get the idea.

In any case, even if liberals really haven't budged on this issue (20% of them still identify as pro-life, by the way), moderates surely have (45% pro-life, up from 38% last year), and it really doesn't make sense to me to explain this using Barack Obama. For whatever reason (I can think of a couple), Obama is attractive to a lot of moderates (even Republicans), and I just don't see them becoming more pro-life because they're turned off by his political agenda.

What's most significant, in my opinion, is the shift in views on the actual question of when abortion should be legal. Some people have strange definitions of "pro-life" and "pro-choice," but there's no denying what these data mean:

• Last year, 28% of Americans said abortion should be legal in all circumstances. This year, only 22% said so.
• Last year, only 17% of Americans said abortion should be illegal in all circumstances. This year, 23% said so.

This represents a clear shift in actual opinion among Americans, not just a shift in self-labeling. In order to explain this, I just don't think you can deny the power of peaceful vigils like 40 Days for Life, the passionate presence of Students for Life of America, or the compassionate work of Feminists for Life.

I think the bottom line is, pro-lifers on the ground are getting it done, changing opinions with facts, with passion, and with compassion. Folks like Planned Parenthood are getting so desperate as to call for days of protest against charitable organizations that give women free resources for pregnancy and parenting. It's kind of sad.

Ideally, people would stop seeing this as a war between pro-life and pro-choice. But, as Barack Obama so tactfully pointed out at his Notre Dame graduation speech, "at some level, the views of the two camps are irreconcilable." Maybe pro-lifers really do just have to keep fighting to make themselves heard. The evidence shows it's working.

Torture and the Religious Right

I read the results of a recent survey on American opinion about torture. Apparently there has been lots of analysis of these results floating around the Internet. Here's the detail everyone's been making a big deal over:

• Out of the entire US population, about 49% believe that torture can sometimes (34%) or often (15%) be justified.
• Out of the regular church-attending US population, about 54% believe that torture can sometimes (38%) or often (16%) be justified.
• Contrast this with the people who rarely or never attend religious worship: only about 42% believe that torture is sometimes (30%) or often (12%) justified.

The conversation that's been going on around the Net is pretty fascinating. Naturally, secularists are inclined to believe that this indicates a moral superiority of secularism over religion. I imagine some religious folks are inclined to simply defend their views as actually more moral.

But two points are worth mentioning. For one thing, about the same percentage of all groups (25%) believe torture is never justified. If purists are inclined to believe the only moral position is no torture at all, then religious folks are on par with secularist--and vice versa.

My second point is that the religious people in the survey only included white evangelicals, white mainline Protestants, white non-Hispanic Catholics, and the religiously unaffiliated. I am profoundly curious to see what minority groups think of this question.

In any case, the question is extremely delicate. One of my favorite conservative writers, Thomas Sowell, has written an article defending the use of torture using some pretty common sense language:

"What if it was your mother or your child who was tied up somewhere beside a ticking time bomb and you had captured a terrorist who knew where that was? Face it: What you would do to that terrorist to make him talk would make water-boarding look like a picnic...

But if the United States behaves that way it is called "arrogance"-- even by American citizens. Indeed, even by the American president."

Probably most evangelical Protestants buy into this kind of reasoning. Also, there is in theory some theological backing for this position. If you believe that evil is a real force in the world, one that requires strong resistance, then it follows that you might have to get your hands dirty to do that. Christians may look to the stories of the Old Testament as inspiration.

But I do worry about this correlation between evangelicals and this very conservative opinion on torture. If we take the power of Christ seriously, shouldn't we remember that evil doesn't conquer evil? We should be careful not to become what we hate.

I don't really think it's about whether or not poor little terrorists deserve to be tortured. That is sometimes what liberals sound like when they chastise Republicans for their war-mongering--Oh, we just need to reason with them, and they will be impressed and have a change of heart. There's more than a hint of arrogance in this kind of reasoning, and it annoys me, too, just like I'm sure it annoys a lot of conservatives who take the evil of terrorism seriously.

My concern is that when we humans participate in something like torture, it hurts us, not just the person being tortured. Evil hurts the person doing evil, on a deep, spiritual level. That is my concern for evangelicals--we can't ignore the evil we are inflicting on ourselves by hardening our hearts so solidly against our enemies. Evil is found in every human heart, and it is dangerous to feed whatever lurks in our own hearts.

My answer to the survey probably would have been "rarely" justified. I think it's dangerous to permit much more than that, though I see how those who do have reasons.

But I also find it a little arrogant, honestly, to be a purist on the issue and answer "never." I am quite certain that the vast majority of people who would answer "never" have never had to defend their family (or country) from violent assault, never seen how genuinely evil people can be (not that I have). "Never" just seems like the answer of proud liberalism, born out of the ability to intellectually analyze the issue without ever really facing it.

Whatever people's thoughts on the issue are, I'm pretty happy to see how diverse American opinion is. It certainly shows there's a lot of room for conversation.

Chuck Norris can fight crime--when he's not even there

This is simply the best story you'll hear all year:

A Croatian bakery has a unique way of keeping away burglars - Chuck Norris.

The bakery suffered from regular break-ins until it put up a life-sized photo of the Hollywood tough-guy, well-known for B-movie action classics like Delta Force and Invasion USA, news.com.au reported.

The photo also has a sign, saying the shop is "under the protection of Chuck Norris".

Since putting up the photo one month ago, the bakery in the city of Split has had no burglaries.

A sales assistant at the bakery said the Chuck Norris photo started as a joke but "people seem to respect him", news.com.au reported.

"Thieves haven't been anywhere near us for ages."

Burglars aren't the only ones fooled either - several customers have asked whether they can get Norris' autograph, the assistant told media.

"They really believe he is sitting in our storeroom out the back ready to pounce on any burglars."

How many Chuck Norris facts can you think of? Here's one to add to your list--and it's a true story! (Note: I have not verified the credibility of this story.)

Also, since my blog is usually about ideas, I couldn't help but mention that these days Chuck Norris has gone from fighting bad guys in movies to fighting liberalism with opinion columns. That might be one reason for me to still vote Republican--I mean, if Chuck Norris says so, what you can say to that?

Something you definitely should read

http://www.foxnews.com/story/0,2933,519181,00.html

There's not much point in me giving you an analysis; I just think everyone should read the article. It's amazing.

A kink in the first amendment

Fox News reports on the case of a student from Capistrano Valley High School (California) suing a history teacher for making such comments as these:

"[W]hen you put on your Jesus glasses, you can't see the truth."

"Conservatives don't want women to avoid pregnancies — that's interfering with God's work."

"When you pray for divine intervention, you're hoping that the spaghetti monster will help you get what you want."

The comment that finally got the judge to rule in favor of the student was referring to creationism as "religious, superstitious nonsense." The court ruled that this, at last, was favoring irreligion over religion, which by recent judicial precedent is considered a violation of the establishment clause of the 1st amendment.

According to the article, the judge tried to "balance" between the rights of the teacher and the student given to them by the 1st amendment. Which brings up an important question: what is this whole debate really about? One thing it's not about in this case or perhaps in any case is whether science and religion are compatible. This case is about power--power over intellectual discussion and over how ideas are presented.

What I'm wondering about is how the first amendment can seriously be applied one way or another in these cases. Conservatives will surely applaud the outcome of this case, but the reasoning used by the courts in these cases is fundamentally inconsistent with the conservative notion of "strict constructionist" interpretation.

That is, a literal interpretation of the first amendment by no means implies that this teacher was in the wrong. One teacher acting out of order does not establish a state religion. Only by extending the establishment clause to provide more general protection could the judge make his decision.

It just seems to me that a "government of laws and not of men" is not going to hold up in these cases, because we're dealing with complicated power struggles. We've created a system in which teachers hired by the government have power over students who may or may not agree with their philosophy. Someone's freedom is going to have to give--there's no way the teacher's right to free speech can be absolutely sacrosanct when students are placed in submission under them.

This power struggle over issues like creation vs. evolution is complicated because the dynamics are different depending on where you are in the country. Many people would like to see this struggle in purely dualistic terms--"us" and "them." In reality, the lines are squiggly, blurry, and otherwise so ill-defined so as to be useless in particular cases such as the one in Capistrano Valley.

Wisdom is what's needed in these cases. I applaud the court's decision in this case. I don't think it was a conservative decision at all. I just think it was a wise decision. (Note that the student didn't demand monetary damages, just that teachers not be allowed to make such comments in the future.) In this case the power seemed too heavy in favor of this rather insensitive teacher, and it needed to be tilted back. In a different case that might not have been true.

As a Christian I think this country needs actions on the part of individuals that promote love. In particular, I think we need court decisions that promote free discussion and remove hostility from the classroom. Frankly, a line by line reading of the constitution will not bring this about. Only wisdom that is borne out through experience with real people will work.

An A's an A?

So I happened to get an e-mail a few days ago giving me a link to CampusBuddy.com's results on the grade distribution here at UVA. You can't find a similar one for my alma mater, so I didn't know what to expect, exactly.

Maybe I shoudn't be surprised, but it seems that 78% of grades given at UVA are at least a B, and 47% of grades given are at least an A-. In fact, all the grades less than a B-, i.e. C+ down through F, constitute a mere 14% of grades given.

There are almost as many A+'s (7% of all grades) as there are C's and C+'s combined (9%), which is far more than the total of D+'s down through F's (a mere 3%).

UVA students are, for the most part, really smart, so on the one hand, this amazing grade distribution makes some sort of sense. On the other hand, doesn't that mean an 'A' is worth a lot less? Almost half the grades given are at least an A-. Even the A+ just doesn't seem to mean a whole lot.

I know people who complain about grade inflation are usually seen as crabby students who came from private schools, but hear me out.

There are two issues at stake. One is that students aren't being challenged enough. If it's that common to get an A, perhaps it is because professors aren't willing to give assignments that require more creative or original thinking.

The other issue is that grade inflation seems to make the world more competitive in the long run. I see this in high school, anyway. Just because you're smart, you still can't differentiate yourself from your peers without participating in 1000 different activities to boost your resume. You have to stand out, and doing well in classes isn't nearly enough. I would have to guess this has some similar affect on college students, as well.

Additionally, you get the problem that if grades are not efficient means by which to differentiate yourself from your peers, then you probably aren't going to focus on your coursework as much as other things. Is this really what universities are intending? Some of them may be. I guess there's something to be said for the university to be a forum for student activism, etc. rather than be strictly a learning institution. I guess I'm a little old fashioned; I still like the idea of the university as simply a learning institution.

But I suppose there's no turning back from the American system as we know it. The demand these days is more for preparation for the "real world," and not so much for education in a more traditional sense. So I suppose that skills like convincing your professor to bump that B+ up to an A-, asking for extensions on homework and extra credit to make up failed assignments, and learning to differentiate between important classes and freebies are much more valuable these days than simply learning things like philosophy, literature, mathematics, history, and science. The latter honestly don't get you as far in the "real world."

I guess the only thing that gets me is how people are willing to spend more and more for an education that actually means less and less.