Colorado Trip

(In which I talk about my personal life; if you don’t know me, you probably won’t be interested.)

I just got back from a trip to Colorado. J’s friend M got married on Friday, and we were invited to Silverthorne, Colorado for the ceremony.

We were (rightly, I think) concerned about the flooding in and around Denver, which probably explains how the car rental guy successfully upsold me a Ford Explorer. Pro: decent pickup, plenty of room, more features than I could count. Con: it’s the size of a battleship. I deliberately avoided visiting downtown Denver because I was afraid I’d have to maneuver it through an underground parking lot and scrape the roof off.

The trip out to Silverthorne went well. Bad traffic and rain, but no worse than some of the stuff I’ve seen on the Beltway.

Just for the record, Colorado is incredibly scenic. Even J, who normally doesn’t care for mountains, was enchanted, even if she had trouble breathing at ten thousand feet. I saw a moose outside our room, quietly ignoring us.

One highlight of the wedding was a replica TARDIS, built by the groom’s brother, and augmented with software written by the groom, that served as a photo booth. There was a table with silly hats, mustaches on sticks, and similar accessories. Guests could accouter themselves, then have their photo taken in the TARDIS, and these went into the wedding photo album.

The next day, we decided to avoid the floods in Denver by heading south into (more of) the mountains, via smaller-than-interstate roads, to Leadville and Colorado Springs.

It’s well known that Colorado Springs is home to the Air Force academy and NORAD, the Santa-tracking arm of the US military. But slightly less well known is the fact that Leadville is home to the Mining Hall of Fame and Museum.

The Hall of Fame part is eminently skippable, unless you’re in the mining industry and want to see plaques immortalizing so-and-so digging the first Germanium mine in Indiana or inventing a new drill or whatever, but the rest of the museum is more interesting than you’d expect, which is how it often goes with these things. It had some walk-through replicas of mines, to show you what conditions people worked in when the Colorado mines were first dug, as well as a section on the history of mining, which fits in nicely with the sort of westward-expansion history that I like.

There were also exhibits aimed at giving the visitor a proper appreciation for the role of mining in the world we inhabit. But I confess I only gave a cursory glance at the World of Molybdenum exhibit.

Most immature-giggle-producing item in the gift shop: the book Beyond the Glory Hole: A Memoir of A Climax Miner by James Ludwig. It probably helps to know that Climax, Colorado is a mining town, and that a glory hole is an opening at the top of a mine. But feel free to tee-hee anyway.

Having spent the night in Colorado Springs, we went to visit the Garden of the Gods park, but just as we were getting our bearings in the visitors’ center, they announced that the river through the park was rising due to the rain (see flooding, above) and that the park was being closed as a precautionary measure. We stuck around a bit to see if the sun would come out and make the powers that be change their mind, but it didn’t and they didn’t. So we moved on.

We did stop by Focus on Your Own Damn the Family headquarters. Apparently there are all sorts of activities that you can bring your Jesus-approved, one-man-one-woman family to, but on this Sunday they were observing the fourth commandment (or third, for you Catholic and Lutheran heathens) by being closed, so all we saw was a bunch of buildings in an office park.

Oh, except for one thing: a statue of three children, the eldest playing with and protecting the younger two, one of whom is reading a book that says:

Jesus is the Good Shepherd.
A good shepherd always takes care of his sheep.
Sheep say
Baa!
Baa!
Baa!.

And with that, onward and northward. It turns out that Denver is far flatter than we had expected, what with it being the mile-high city and all. But it’s right at the junction between the boring-as-Kansas plains, and the totally-not-flat Rockies. So you get to pick the kind of slope you want to live on.

We found the Soundwalk, an art installation in Denver consisting of a series of grates that play water sounds on a loop. Or more precisely, J found it by being startled by what sounded like a toilet flushing under her feet. Either way, it’s cool.

The following day, we visited the Denver Museum of Nature & Science. I confess that the Smithsonian has spoiled me, and I was a bit taken aback at the thought of paying admission to a museum, but it was worth it. Especially if you have (or are) kids with a modicum of curiosity.

At the airport, I got a TSA Junior Officer from the first TSA inspector.

Me: Look what I got!
J: How’d you get that?
Me: Um, I asked for it.

The tram station at Denver airport has a series of posters of local celebrities. On one, I first saw the Hugo. Then I looked up and saw the face of Connie Willis. Then a pile of her books. And only then did I notice the text that said who the face belonged to. Yes, I successfully made my fanboy roll.

All in all, lots of fun. We need to go back, some time when it’s less floody. If things go as well as this first time, I might even get J to hate snow a little less.

I Am Chase and/or Sanborn

Or, How I Learned to Stop Worrying and Love the Intuition Pump

So the other day, I found myself inside a philosophical intuition pump. But first, a bit of background:

Qualia (singular quale), in philosophy, are basically the sense of perceiving something. If you’ve ever had that discussion about “how do I know that what you see as red is what I see as red?” or “what’s it like to be someone who actually enjoys Brussels sprouts?”, then you’ve thought about qualia.

Daniel Dennett wrote Quining Qualia, a paper that argues against qualia, or at least against the notion that they’re a useful concept. In it, he introduces several intuition pumps, analogies intended to help us wrap our minds around the problem. One of them is:

Intuition pump #7: Chase and Sanborn. Once upon a time there were two coffee tasters, Mr. Chase and Mr. Sanborn, who worked for Maxwell House […] Along with half a dozen other coffee tasters, their job was to ensure that the taste of Maxwell House stayed constant, year after year. One day, about six years after Mr. Chase had come to work for Maxwell House, he confessed to Mr. Sanborn:

I hate to admit it, but I’m not enjoying this work anymore. When I came to Maxwell House six years ago, I thought Maxwell House coffee was the best-tasting coffee in the world. I was proud to have a share in the responsibility for preserving that flavor over the years. And we’ve done our job well; the coffee tastes just the same today as it tasted when I arrived. But, you know, I no longer like it! My tastes have changed. I’ve become a more sophisticated coffee drinker. I no longer like that taste at all.

Sanborn greeted this revelation with considerable interest. “It’s funny you should mention it, ” he replied, “for something rather similar has happened to me.” He went on:

When I arrived here, shortly before you did, I, like you, thought Maxwell House coffee was tops in flavor. And now I, like you, really don’t care for the coffee we’re making. But my tastes haven’t changed; my . . .tasters have changed. That is, I think something has gone wrong with my taste buds or some other part of my taste-analyzing perceptual machinery. Maxwell House coffee doesn’t taste to me the way it used to taste; if only it did, I’d still love it, for I still think that taste is the best taste in coffee. Now I’m not saying we haven’t done our job well. You other tasters all agree that the taste is the same, and I must admit that on a day-to-day basis I can detect no change either. So it must be my problem alone. I guess I’m no longer cut out for this work.

Chase and Sanborn are alike in one way at least: they both used to like Maxwell House coffee, and now neither likes it. But they claim to be different in another way. Maxwell House tastes to Chase just the way it always did, but not so for Sanborn. But can we take their protestations at face value? Must we? Might one or both of them simply be wrong? Might their predicaments be importantly the same and their apparent disagreement more a difference in manner of expression than in experiential or psy chological state? Since both of them make claims that depend on the reliability of their memories, is there any way to check on this reliability?

So the other evening, I opened a bottle of riesling and poured myself a glass. It was quite good, [insert a bunch of pretentious oenological terms like “fruity” and “bouquet”]. I recapped the bottle and put it back in the fridge.

The next evening, I opened the bottle and poured myself a second glass. But this time, it tasted distinctly more sour than I remembered it.

My first thought was “Well, crap. This means that I have to either drink a whole bottle in one sitting, or pay $12 for one glass of wine.” But I asked frequent Epsilon Clue commenter Fez, who knows more about wine than I do, and he said that my story didn’t match his experience; that he’ll often recork an opened bottle and drink it the next day.

So now I’m not sure what’s going on. It’s possible that this particular bottle went sour overnight. Maybe reds last longer than whites. Maybe whatever I had for dinner those two nights affected my taste buds. Maybe something else.

But what’s interesting to me is that what was originally intended as a hypothetical example to make a philosophical point has become more concrete and personal for me, with literally tens of dollars at stake.

I’d Rather Have a Long List of Scary Warnings than Nothing at All

I recently participated in a coversation—or maybe I’m conflating two or more conversations, but no matter—in which my interlocutor said that she prefers alt-med natural remedies because mainstream drugs all have a long list of scary potential side effects.

But when I asked whether alt-med drugs actually lower cholesterol or help prevent heart attacks or whatever they claim to do, she said that people who sell alternative medicines tend to avoid making medical claims. They’ll say the product “enhances well-being” or some such, but not “this product helps regulate LDL”.

Because what happens is this: if you make a specific claim about physiological effects or the like, that’s a medical claim, and the FDA expects you to back it up. So Pfizer comes along and says, “this new drug, XYZ, improves blood-clotting.” The FDA says, “Oh, yeah? Show me.” And so Pfizer performs studies, or cites independent studies, that show that yes, as a matter of fact, patients who receive XYZ tend to clot better than patients who don’t, even after taking into account other possible explanations, like luck or the placebo effect. And the FDA says “All right, you’ve made your case. You can claim that XYZ improves blood-clotting in your advertisements.” At least, that’s how we want it to go; how we hope that it goes.

Unfortunately, the world is complicated, and it’s never as simple as “take this drug and you’ll get better.” Different people have different bodies and react to things differently—for instance, I have a friend who doesn’t drink caffeine because it puts him to sleep. So at best you’ll have “take this drug, and it’ll most likely help, but it might not do anything.” More often, you get a drug that does what it’s intended to do in the majority of cases, but also has a list of possible, hopefully rare, side effects. But the more participants in the study (which is good), the greater the chance that one of them will have a heart attack or something that can be plausibly be attributed to the drug being studied. So the Scary List O’ Adverse Effects grows.

So yeah, traditional herbal remedies that don’t have words like “vomiting” or “stroke” on the label look appealing by comparison. But that’s only because the people selling the herbs aren’t required to test them, or to publish the negative results. If someone out there did make a specific claim, like “echinacea helps relieve flu symptoms”, and the FDA said “Oh, yeah? Show me”, and they showed ’em, and ran tests and studies and such, there would almost certainly be some adverse side effects to report. If you’re not seeing any, then either someone’s hiding them, or else no one’s looked for them.

In the real world, everything has problems. Saying you prefer alternative remedies to conventional medicine because it doesn’t have a scary list of adverse effects is like getting your financial advice from a psychic instead of an investment banker because instead of scary disclaimers about lawsuits and patents and the possibility of losing all your money, she just has the friendly statement “For entertainment purposes only.”

In Defense of Gussie Fink-Nottle

For those who may have forgotten, Gussie Fink-Nottle is a character in the Jeeves stories by P.G. Wodehouse. He is the series’s stereotypical nerd: socially inept, a teetotaler, and physically unimpressive. His most memorable trait, however, is his fascination with newts.

Clearly Wodehouse tried to find the least interesting subject he could think of, to allow his character to easily bore all the other characters to tears by going on at length about his pathetic pet subject.

I don’t remember his early life ever being discussed in any detail, but I imagine that, as a weakling, he was never any good at sports and thus never developed an interest in them. Unable to hold his liquor, he never got into the habit of meeting with the chaps over drinks and experiencing the sorts of things that only seem to happen during alcohol-fueled debaucheries. His social ineptitude meant that he never became a lothario. Eventually, he was forced to become interested in that most uninteresting of subjects, newts.

But I would look at it from another angle: there is an infinite number of subjects in the world. What is it about newts that’s so interesting that Gussie would choose to devote his life to them?

Stephen Jay Gould, as I recall, did his graduate research on snails. Carl Sagan was interested in points of light in the sky. Bertrand Russell worked on breaking down existing mathematical proofs into longer chains of simpler steps. In each case, they found something interesting in what might appear to be an unintersting subject.

Likewise, I sometimes wonder what makes people want to go into professions like accounting or proctology. It can’t just be the money, can it? Presumably there’s something there that I don’t see, some hidden bit of beauty that I haven’t seen or had explained to me.

I don’t want to think, “Wow, what a loser, for being interested in something as boring as newts.” Rather, I want to ask, “What is it about newts that’s so interesting?”

XKCD: Beauty

Learning to Learn

The Aug. 29, 2011 episode of 60-Second Science talks about a finding that drawing helps scientists develop their ideas.

I can’t say I’m terribly surprised at this. Drawing seems to me to be more concrete than speech (or raw thought). Just as a simple example, I can say “two circles”, or I can draw two circles. If I draw two circles, rather than just talking about them, I must necessarily place them next to each other, or one above the other; close together or far apart; of equal or different sizes; and so on. Depending what the circles represent, these small choices might matter, and force me to think about some aspect of the problem.

Chabris and Simon’s The Invisible Gorilla describes something similar: pick some object that you know well — the example they use is that of a bicycle — and draw a diagram of it. No need for artistic verisimilitude, just try to get all the important parts and how they relate to each other. Now, compare your drawing to the real thing. Are the pedals attached to the frame? Do the pedals go through the chain? Is the chain attached to both wheels, by any chance? According to the authors, a lot of people make glaring mistakes. I think it’s because, while people know how to use a bicycle (or a stove, or a TV set), we rarely if ever need to think about the way the parts have to fit together to actually work.

Which brings me to my own field:

It has often been said that a person does not really understand something until after teaching it to someone else. Actually a person does not really understand something until after teaching it to a computer, i.e., expressing it as an algorithm.

— Donald E. Knuth, in American Scientist:61(6), 1973, quoted here

What I mean is that if you really want to understand something, the best way is to try and explain it to someone else. That forces you to sort it out in your mind. And the more slow and dim-witted your pupil, the more you have to break things down into more and more simple ideas. And that’s really the essence of programming. By the time you’ve sorted out a complicated idea into little steps that even a stupid machine can deal with, you’ve learned something about it yourself.

— Douglas Adams, Dirk Gently’s Holistic Detective Agency

Computers have a nasty habit of doing exactly what you tell them, and only what you tell them (or at least they did back when I learned programming; since then, they’ve occasionally attempted to be helpful, which usually means they’re not even doing what you tell them). This means that to write any kind of program, you have to think about absolutely every step, and make decisions about everything. And the machine isn’t at all shy about letting you know that YOU GOT IT WRONG HAHAHAHA LOSER!, although it usually lets you know through a cryptic error message like segmentation fault (core dumped) or dropping your Venus probe into the Atlantic.

But in most disciplines, we are not so lucky to have such stupid students, or to receive the kind of feedback that programmers do, so we need to resort to other methods.

Explaining things to someone else helps, probably because it forces you to explicitly state a lot of the things that you can just gloss over when you’re thinking about it. John Cleese has talked about the importance of test audiences in improving movies: they’ll tell you about all sorts of problems with the film that you never would have noticed otherwise. One of the cornerstones of science is peer-review, which basically means that you throw your ideas out there and let your colleagues and rivals take pot-shots at them. And the study I mentioned at the the top of this post says that it helps to draw pictures of what you’re thinking about.

It seems to me that the common element is looking at every aspect of a design, the better to try and make its flaws evident. The human brain is a remarkable organ, but it’s also very good at rationalizing, at overlooking details, at making connections that aren’t there, and the like.

But the good news is that we do have techniques like doodling, explaining, soliciting feedback, and so on. And that suggests that we can learn to think better. Genius may not be something innate, something bestowed by whichever Fates decided your genetic makeup, but rather something that you can learn over time and improve through practice, like playing piano or baking a soufflé.

I hope this is the case. It would mean that our children can be better than we are, and there’s something we can do about it. Heck, it would mean that we can improve ourselves.

Faith and Confidence

The other week, J. and I were talking about Harold Camping and his amusing predictions of imminent doom, and she said that while he is, of course, a loon, she admires his faith and how it allows him to persist in the face of adversity and ridicule. This isn’t the first time I’ve run across this sentiment, so I thought it might be worth addressing.

While it’s nice to be able to overcome obstacles, that’s only half the story. If I may use an extended (and therefore ultimately friable) analogy, as is my wont, confidence is like the engine in a car: the more powerful it is, the faster you’ll get where you’re going, and the fewer potholes will stop you.

But that’s no good unless you’re going someplace worth going. A powerful engine can get you to Fort Lauderdale for spring break, but it can also send you flying through the front window of a Wendy’s. That is to say, if you have full faith and confidence in a wrong idea, it can allow you to ignore obstacles like valid criticism and do something damnfoolish or worse (9/11 springs to mind).

To a great extent, confidence is a good thing. But this has to be tempered with reason. Sometimes, when people tell you you’re an idiot for pursuing your dream, it’s because they’re jealous or don’t realize the brilliance of your idea. But a lot of times, it’s because you’re an idiot. Remember that dotcom you worked for in the nineties, when you though that if you could just convince enough people to subscribe to your kielbasa-review site, that network effects would kick in and you’d make enough money to topple Wal-Mart? Yeah, that’s what I’m talking about. You were an idiot.

Now personally, I find that what helps my confidence is knowing what the hell I’m doing. Granted, not knowing what you’re doing can mean that you don’t know the conventional wisdom that says that what you’re doing is impossible, and can therefore surprise people when you do it and demonstrate that the conventional wisdom was wrong. But usually, the conventional wisdom is right.

But the better you know your subject, and the better your critical thinking skills, the better you’re able to judge the merits of the conventional wisdom. You’re in a better position to figure out whether your idea is feasible or just a pipe dream. To figure out whether the obstacles and arguments against you are real or just illusory.

And, of course, there’s a lot to be said for plain old getting pumped up about something. But I’m the wrong person to ask about that. You may want to take a look at how it’s done at churches, concerts, and so on.

Subjective/Objective

This was originally posted at Secular Perspectives.

Here’s something that occurred to me recently. It’s nearly-trivial, but I found it interesting.

The reason a subjective statement, like “Beethoven’s ninth is his best symphony” is subjective is that a) it refers mental state, and b) that mental state can vary from person to person.

But it can be turned into an objective statement by simply saying whose mind it refers to: “Smith thinks that Beethoven’s ninth symphony is his best”. This is an objective statement, and its truth or falsehood can be ascertained simply by asking Smith. In a few years, maybe we’ll even have scanners that can read the answer in Smith’s brain.

Or instead of specifying a particular subject to whom the statement applies, we can specify a class of people, e.g., “Most music critics think that Beethoven’s ninth is his best”, or “Nobody likes being humiliated” (vs. “humiliation is bad”).

One consequence of this is that it helps put morality on a reality-based footing: a question like “should the US intervene in the Ivory Coast?” seems hopelessly subjective, but we can at least ask questions like, “how many Americans think the US should intervene?” and “how many Ivorians want the US to intervene?”. These questions, and their answers, are called polls, and they’re used all the time. (I’m not saying that complex moral questions should be decided by polling. But polls can provide an objective underpinning to moral arguments. For instance, if 98% of Ivorians hated Americans and wanted the US to stay the hell away, that would undercut arguments like “we should move in: we’ll be greeted as liberators”.)

Secular morality is often attacked for being too subjective. I hope the above helps correct that perception. The whole point of having a system of morality is, presumably, to improve the universe in some way, and hopefully allow us to be happier and get along with each other in the process. What “better” means, above, is subjective, but at the very least we can see what people think, and what most of us can agree on.

Me? On the Radio?

Yeah. The UMD Society of Inquiry has a radio show on WMUC (5 watts, broadcasting to the greater South Campus Commons Area. But it’s not owned by ClearChannel, which counts for something). One of the guys who was supposed to do the show today had to bail, so I got pulled in as an emergency backup replacement.

On the off chance that you’re interested, the show can be downloaded here (and the last 30 seconds here) until next Monday.

Creative Ignorance

The other day, I was in a discussion about whether computers can be creative. Personally, I thought the answer was a big “duh, yes”, if only because programs (often even my own) often do things that surprise me, but at least I managed to shift the conversation toward the question of “what is creativity? How will you recognize it when a computer achieves it?” And along the way, I noticed a couple of things about creativity.

For one thing, the perception of creativity can depend on the audience’s ignorance. Years ago, I wrote a custom email filter for my boss, because none of the commonly-used ones could easily do what he wanted (like filter on the number of people in the “To:” and “Cc:” lines). When I showed it to him, he thought it was the most amazing thing ever, that we should write a paper about it, and send it in to a research journal. I told him that this was too trivial, and that I couldn’t in good conscience call it groundbreaking or innovative, and that I’d be embarrassed to send it to a research journal.

In short, my boss thought my code was innovative because he knew far less than I did about the state of mail filters. And to this day, whenever I see a statue or painting or something and think, “Oh, that’s cleverly cool! I never would’ve thought of that”, I immediately have second thoughts along the lines of “Yes, but that’s because you don’t hang out with artists and go to galleries and such. The person who did this probably just took five or six ideas that were floating around the technisphere and tweaked them.”

A lot of the proposed definitions of “creativity” circled around the general idea of “using a tool in a new or unexpected way”. And it occurred to me that you don’t need intelligence to be creative in this way. If you don’t know what a tool is for, you won’t be burdened with preconceived ideas of how you ought to use it. In fact, that’s how natural selection works: it has no intelligence whatsoever, and doesn’t know that wings are “for” protecting eggs, and doesn’t punish those individuals that manage to use them for gliding or flying.

Of course, if you’re an adult human, then you’re intelligent (at least compared to natural selection or a bacterium), so this type of creativity is harder. But you can use first sight.

In Terry Pratchett’s A Hat Full of Sky, Tiffany Aching is said to have “first sight and second thoughts”. First sight is the ability to see what’s actually in front of you, rather than what you think is there.

There’s an old story about a student who was asked on a test to measure the height of a building with a barometer that I’m sure you’re all familiar with. Because the problem specified the use of a barometer, clearly the instructor expected students to use the barometer for the thing that barometers are supposed to be used for, namely measure air pressure.

The student’s smartass answers seem creative (oh, come on, admit it: you thought it was cool, the first time you heard the story) is that he ignores the fact that barometers are for measuring air pressure, and sees its other properties: it has mass, so it can be swung like a pendulum; it has length, so it can be used to count off units of height; it has value, so it can be offered as a bribe.

Outside of the world of contrived puzzles, first sight can also be useful, because it lets you stop asking “what is this for?” and start asking “what can I do with this?”. That last question, in turn, breaks down into sub-questions like “what tools do I have?”, “what properties do they have?”, and “how does this property help me solve my problem?”

For instance, spreadsheets are nominally for tabulating data, aggregating sums and averages of interesting numbers, and like that. But people have noticed that hey, Excel does arithmetic, so why not use it as a calculator? I’ve also worked with people who noticed that hey, it lays things out in neat columns, so why not use it as a to-do list?

When technology advances, old tools sometimes become cheap enough to do simple tasks. Car phones have existed for a long time, but if you grew up in the 1960s, you probably decided that they were just fancy toys that rich people used to flaunt their wealth. But in the 1990s, they became cheap enough that anyone could have one. So if you were running a business in the 90s and were expecting people to use pay phones to stay in touch with the office while they were traveling, you were going to have your lunch eaten by the people who had looked at the field the way it was, not the way you imagined it, and realized that they could just give all their salespeople and field techs cell phones.

On a grander scale, the Internet was originally set up for government researchers to share data, and as a nuclear-war-resistant means of communication for the military. It certainly wan’t built to help you find friends from High School or coordinate popular uprisings in the Middle East. That part came from people looking at the thing for what it was, and ignoring &mdash: or often ignorant of — what it was supposed to be for.

What’s interesting about this, I think, is that you don’t need to be a genius to be creative. In fact, you don’t even need intelligence at all. A lot of creationists look at the complexity of biological systems and can think only in terms of a superior intellect putting the pieces together to achieve a goal.

But if I’m right, then it’s possible to be creative simply by being to stupid to know what’s impossible. Creativity can be what Dennett called a crane, rather than a skyhook.

What Good Is Math?

I think most people, if you asked them, would say that teaching math in school is a good thing. If you ask why, the usual answer is something like, it allows you to figure out how much carpet and wallpaper you need to buy to redecorate your living room, to determine whether the 12-can pack of ravioli at Costco is cheaper than what you can find at Safeway, and so forth.

But that’s all elementary and High School level stuff: arithmetic, geometry, a dash of algebra. I’ve rarely used trigonometry after college, and I know one person who used calculus in quilting. I think it’s safe to say that most people never use calculus, differential equations, prepositional logic, etc. after college. But I still say there’s value in studying math beyond the practical.

Let me ask a contrasting question: why should kids play sports in school? Only a tiny minority of them will make a living playing or coaching sports, and less than half will even play on the office softball team or the like.

The answer I get most often is that sports teach teamwork: how to subsume your immediate desires for the greater good, make sure you do your part and trust your teammates to do theirs, and communicate effectively to make sure you’re not working at cross-purposes. You learn to win graciously and learn from your failures.

In other words, it’s not so much that playing sports develops muscle tone and general fitness. But rather, it’s an indirect way of teaching other skills like cooperation. They may not be taught explicitly, and there are other ways of teaching them, but this works.

The same thing, I think, applies to math. A few years after learning it, you probably won’t be able to prove that there is an infinite number of primes, or how to integrate a function. But that’s okay, because math teaches skills other than the purely pragmatic.

Proving theorems, for instance, forces you to distinguish between what you think is true, and what you can demonstrate; what looks right, and what is right. Geometry teachers always admonish students not to reason from the diagram because concrete examples are often misleading: just because line AB is perpendicular to line CD in this drawing doesn’t mean that that’s always the case. The point is to figure out what’s universally true, not just what’s true about this particular instance.

These are skills that apply to non-mathematical professions: judges and lawyers often deal with people who have clearly done something wrong, and need to distinguish between “I know that ain’t right” and whether the action in question is legal or not. Police officers likewise need to know the difference between “I know Jimmy’s been selling crack to students” and “I can prove to a jury that Jimmy’s been selling crack”. Programmers will find that they write code that works in situations that they didn’t imagine, because it’s a truism that end users will do weird things with your software that you never dreamed of. Financiers should be able to put together a portfolio that can survive unexpected catastrophic changes in the market. (And as much as it pains me to defend the Bush administration, Donald Rumsfeld was quite right in distinguishing between “known unknowns” and “unknown unknowns”, and trying to plan for both.)

Or take the discussion about defining information in the “I Get Email” thread; specifically, the exchange between Tom and Troublesome Frog on whether all living beings contain information. It seems that Tom doesn’t quite get the difference between ∀ xy p (no matter how you define information, all living beings contain information) and ∃ xy p (we can come up with a definition of “information” such that all living beings contain information).

Math is very big on abstraction. This starts in algebra, which is all about figuring things out about things that you know you don’t know much about. And it just gets more abstract from there. The first time you demonstrate that there is no solution for some equation, or better yet, that there can be no proof of a given proposition, can be quite a thrill.

And abstract thought is one of those things we humans are good at. It’s what allows us to formulate moral rules that apply to everyone, and see things like “the market” instead of a bunch of people trading stuff. It seems that we ought to learn to do it well.

One of my favorite types of SAT question is the one that presents a math problem, and one of the possible answers is “not enough information to solve the problem”. The lesson here isn’t just “know your limitations”. It also shows that just because something looks solvable doesn’t mean that it is; that just because something is printed in an official-looking book doesn’t make it kosher. And also, the sooner you figure out that a problem has no solution, the less time you’ll waste looking for one.

Finally, one thing that everyone should get out of any math class is that you can figure stuff out on your own, without looking the answers up in the back of the book. Yes, the same is true of science and other classes, but math is one of those branches where you don’t need fancy equipment to work on a problem and figure out the solution.

One problem I see is that a lot of people seem to think that knowledge is something that is handed down from on high, rather than something that can be created. This seems to be at the root of the claim that evolution is just another religion: “I have my priests who tell me that God created humans, and you have your priests who tell you humans evolved. The only difference between creation and evolution is which team you’re on.” There’s no arguing with someone who simply repeats what they were taught; but if you’re in an argument with someone who thinks that mere mortals can work out answers on their own, you might get somewhere. And that’s something we could use more of.