[Most Recent Entries]
Below are the 14 most recent journal entries recorded in
|Friday, June 29th, 2007|
|Thursday, March 22nd, 2007|
For a few years now I've been training in various martial arts, and for most of those years I've been perfectly aware that training has its risks. When I got kicked in the eye, I didn't blame the person who kicked me, or the drill in which it happened, but I did take time (about the month I was putting atropine in my eye) to think about why I was training and whether the risks were worth it. In the end, I didn't go back to that club, but I have tried others since, with variable success. When, a few years later, I broke a friend's nose, he thanked me, a month or two later, for forcing him to think about what he was doing and what it meant to him. So I'm trying to take this broken rib as a blessing.
Martial arts are, like many people's hobbies, far from the state of the art. If you want to navigate a violent situation, you should probably be using an assault rifle and the support of your platoon. I think, for me at least, the purpose of a martial arts class is not to prepare one for any situation one is ever likely to encounter. At least, not to directly prepare one. The balance and coordination (I hesitate to say grace) that you learn, and the fitness that you build, do in fact prepare you well for real situations - climbing stairs, walking on ice, moving house... Of course, one could design an exercise program whose explicit purpose was to teach those things and thereby avoid the macho nonsense that too-often accompanies the martial arts. I suppose that's what yoga is (as I understand it).
For myself, though, I find that I need to be learning something. Aerobics are a great way to get in shape, but you really don't learn much doing them. And I don't think I have the patience for that.
Worse, in a way, I've spent many years learning various martial arts, but there are none I'm really good at, certainly none I could teach. Most martial arts are so complex, and most of us learn these physical skills so slowly, that one really needs to spend ten or more years training very seriously to be able to put the various skills together in a way that makes sense. So in a way, I feel that by quitting, or even by switching to a different school or art, I would be throwing away all that training. That's not quite true - one of the major skills I've learned is learning physical skills. It's something I noticed while leading beginner classes - some beginners are strong and fast, but you can tell the ones that have done serious physical training before because you only have to show them something once, then they can go off and practice it.
I have also learned enough, I think, to be able to express a real preference for one art over another. I have some idea what styles of movement and what practice formats suit me. Putting this into practice is not so easy though.
Martial arts, like most serious, competitive physical activites, carries its risks. There's almost always a danger of injury, from getting hit, from falling, from muscle pulls or joint sprains. Worse, training hard enough to reach your limits puts a great deal of wear on your joints; almost anyone who's been training for years has some lasting joint damage; many soft tissue injuries never heal. We all get old, though, sooner or later, and saving your health for your old age doesn't work and is a poor path to happiness. So, risks. We all have to decide how many and which ones. That martial arts are dangerous is no more a reason not to do them than the risk of a plane crash is a reason not to fly, if you decide it's worth it.
"Is it worth it" is a tricky question. You have to ask what it does for you, and you have to ask what the risks are. Training in a needlessly dangerous way, doing something you don't enjoy... well, that's a bad idea. Taking a few risks to do something I enjoy and learn a difficult skill is worth it to me. Which category is the training I'm doing in? I don't know.
Well. I have six weeks to think about it.
|Saturday, October 28th, 2006|
|Grading is always depressing
I just graded (two questions of) a stack of 170 calculus midterms. It's a slow, boring job at the best of times, but it always brings into sharp focus what a poor job we're doing of actually teaching calculus.
It's tempting to blame the students for being stupid, and I suppose a few of them must be, but there's no way that's the reason they do poorly. And doing poorly on an exam is one thing - it can happen to any of us - but many of them write down things that are plainly nonsense.
For example, one of the problems I was grading was to compute the sum of the series whose terms are n 2^n / 3^n. They were given the hint that it could be done using differentiation of known power series; in fact the way you do it is you start from the fact that the sum of x^n over all (nonnegative) n is 1/(1-x) and you take the derivative on both sides. I can understand (though it's a bit depressing) getting the derivative wrong. But one student tried to "take the derivative of the original series with respect to 2/3". Others simply treated the series as geometric and wrote the answer as n/(1-2/3). This shows not just a lack of understanding of calculus, but a lack of understanding of the basic language of mathematics.
This lack of understanding of the language of mathematics is a serious problem, and it's not really their fault. In our courses, we spend a few minutes, if that, talking about what our notation means and what the basic objects we're working with are, and then we spend hours talking about how to do calculations. The assignments are all about calculations, often very messy calculations. So what do the students think of as the important part of calculus?
What alternative do we have? How can we emphasize understanding what's going on? For mathematics majors, there's no problem: proofs do require exactly that understanding, and once people have done proofs for a while, there's no point giving them a question that is functionally equivalent to one they've done before because they can see the common structure behind them.
Am I really recommending we make the chemistry (say) students do proofs? I suppose I am. Not necessarily sophisticated or difficult proofs, but enough to force them to think about what objects they're using and how those objects fit together. It will, I think, if done right, make them much more effective at solving real problems.
Good luck convincing the administration of that, though.
|Saturday, October 21st, 2006|
|Rayleigh-Taylor instability and kids' toys
Wikipedia is cool.
I'll admit that it is occasionally unreliable (for example, it claims that the speed of light
is affected by gravity) and it's sometimes a mess but it's full of fascinating information. For example, Rayleigh-Taylor instability.
Rayleigh-Taylor instabilities occur when a less-dense fluid accelerates a more-dense fluid. This sounds fairly unexciting, I know. I had a toy in my room when I was a kid that demonstrates what happens: little variations in the interface amplify into interpenetrating "fingers". It made for a cool-looking little toy, but it never really seemed that interesting either.
But then I go reading about it on Wikipedia.
Turns out that when a supernova goes off, a spherical shell of matter is flung off. That shell of relatively dense matter crashes into the not so dense interstellar medium, and the interface breaks up into "fingers", giving nebulas their characteristic "clotted" look. (I also had a picture of the Crab nebula on my wall as a kid.) So my little toy explains, in a very visual way, why nebulae look the way they do.
Using this description
I should be able to estimate the "mean kinematic viscosity" and "Atwood number" of my little toy; it might even make it clear how to build one. ( Or maybe not.Collapse )
Every time I tell people about things like this I had as a kid they tell me "no wonder you're like that".
|Monday, October 9th, 2006|
|Egg-free banana bread
I don't like cooking with eggs, so I went looking for a banana bread recipe without them. After all, bananas are a common egg substitute in vegan cooking, so how essential can the eggs be in banana bread? All the recipes I could find, though, used either eggs or some weird vegan goop (flax seed paste?). So I experimented, and I seem to have a fairly good banana bread recipe for which you can lick the bowl without trepidation:
- 6 bananas, ripe, frozen
- 1 tsp vanilla
- 1 tbsp molasses
- 1/4 c butter
- 1 tbsp water
- 1 c sugar
- 2 c flour
- 1 tsp baking powder
- 1/2 tsp baking soda
- 1 tsp cinnamon
Mix the wet ingredients thoroughly. The bananas hurt to peel when still frozen but tend to liquefy if melted. Running them under hot water can help. I also usually need to microwave it a bit to melt the butter in so you can mix it.
Mix the dry ingredients thoroughly in another bowl.
Grease two medium loaf pans, and preheat the oven to 350 degrees F.
Mix the two bowls until smooth (but not longer) and immediately put the batter in the loaf pans and put them in the oven. (The production of carbon dioxide starts upon mixing, and the liquid is not really goopy enough to capture it until baking starts.) Bake for 45 minutes (really until the kitchen smells of banana bread and the texture isn't too sticky).
Makes two medium loaves.
|Saturday, September 16th, 2006|
I was at McGill when the news about the Dawson shootings started coming in. Watching the secretaries try to contact their children (who were Dawson students), wondering if any of the victims was a student I'd taught over the summer, really brought it close to home. It's an apalling event, and I can't imagine what those who are even closer to it must be feeling.
What happens now? We know who did it, we know how he did it (with a legally-owned rifle). We mourn the dead, hope for the injured, and offer compassion and support as best we can to the families of those shot. We try to understand why it happened, and we try to prevent anything like it from happening again.
Why did it happen? In this modern age, for good or ill, we can read the writings of the killer and try to get some idea of what he was thinking. What we find is that he seems to have been in love with the idea of guns, violence, death, and evil. I use that last word advisedly in this morally relativistic age as I can find no better - he was in love with what I see as evil, and he seems to have been in love with what he saw as evil. He sought out movies, video games, and "sports entertainment" that glorified guns, violence and death, and which treated evil at best neutrally and at worst in a positive light. Of course he found many such things in popular culture; you need not look far. Finally he sought to make real what he was dreaming about, to use a gun, to be violent, to kill and be killed, and to commit evil.
How did he come to such a state? I don't know. He alludes to betrayal, rejection by a girl, and hatred of everyone, but none of those is really enough to explain how the idea took hold of his mind.
Did the video games, the movies, the culture on the websites he visited, contribute to the violence? This is a touchy question. I think, before trying to answer it, we should be clear that there are two questions here. One: Do violent video games (or whatever) make people more likely to be violent? and two: If they do, what should we do about it? Advocates of free speech oppose the banning or restriction of such materials; fair enough, they may have a case. But they seem often to go about it by trying to argue the first point, by claiming that the materials aren't a problem in the first place. To avoid having to say "yes these things lead to violence, but it's important that they be permitted anyway", they produce politically-motivated studies designed to show that the materials are not dangerous. Bending the truth in order to dodge hard questions is not okay.
In fact, that's my first example in the dangerous-materials discussion: the custom, in movies and television, of setting up a moral choice between the easy but wrong and the difficult but right, having the character choose the difficult but right path, and then by magic melting the difficulty away. This is moral cowardice on the part of the writers, and it contains a poisonous message: that the right way is, though it may be disguised, always the easy way. That just making a choice and hoping for a miracle to occur is actually a feasible way to solve problems.
Who cares, it's just fiction anyway, right? Well, no. How often do you run into difficult moral decisions in your daily life? How much personal experience do you have with life-or-death decisions? We learn our ideas about how the world works, about how the world should work, from the world itself where possible, but fiction, and possibly all of art, exists to fill in where our own experience falls short. Some religions formalize this process, using fiction (in the form of parables, cautionary tales, and holy books) to describe a moral code. But good fiction can take a penetrating look at a way of thinking and show it to a reader. The reader may or may not agree with the author, but their understanding shapes their thinking afterwards.
For example, "The Godfather" describes a certain code of honour, of family loyalty and even of ethics, in extreme, turbulent circumstances. It's not a code of ethics I want to live by in its entirety - protection rackets are nasty, for example - but the movie is an interesting study of how it affects the characters who live by it. Understanding it does affect how I think about honour, loyalty, and ethics.
But fiction is malleable, and people do not often take it in critically. You can lie to people in fiction as thoroughly as you can in non-fiction. Not, of course, in the characters, who are obviously not real, but by quietly ignoring consequences, emphasizing improbable circumstances, and a host of other devious tactics. It need not be intentional; in an effort to create a charming, harmless children's story, you can polish away any troublesome negative conesquences of any decision. Perhaps that's why some of the old fairy tales are so dark: written to warn people against a hostile world.
So I think that, at least in principle, fiction can change the way you think, and sometimes for the worse. Did the killer's choice of fiction turn him into a killer? I don't think so - or at least, he chose it in the first place. He sought out films that told him what he wanted to hear.
Telling people what they want to hear can be a very profitable business - ask your favorite con man, faith healer, televangelist, or recruiter for the madrassas that teach terrorism. People are so willing to believe, sometimes, that all they need is a little encouragement to go completely beyond the pale.
Why did this killer fall so in love with guns, violence, death, and evil? Why was he so eager to hear that message, so ready to believe? I don't think we know. But we do know that there are many people, for many reasons, who wind up in that place; they seem to be more numerous where conditions are more desperate (some Palestinian refugee camps, for example).
How can we prevent such a thing from happening again?
We could try further restricting gun ownership. After all, he legally owned the rifle he used; perhaps if it had been illegal he would not have owned it? Unfortunately, in this country, banning rifles is a problem. Some people really do need rifles - not for hunting, which is in most cases just a hobby, but because they need to go where there are dangerous animals. If you are working in the field in the high arctic, you really do need to worry about being eaten by polar bears. You need to carry a gun. Coming up with some kind of gun law that permits this sort of necessary guns but makes it impossible to use a legal gun to go on a shooting spree is going to be exceedingly difficult. I think our current law (roughly: rifles OK if kept appropriately locked up, automatic weapons and handguns basically illegal) does a credible job, even if it failed to prevent the current atrocity.
We could try banning or resticting access to kinds of material that might cause people to think in ways we disapprove of. I think that's dangerous - that same material is often the best argument for why we disapprove of those ways of thinking. I think it's a good thing that it's possible to go and read Mein Kampf in Hitler's own words.
We could try putting security guards, metal detectors, and guard dogs outside all our schools. After all, nothing's too much if it keeps our children safe. But what does that do to the children, growing up in an environment like that? It makes them afraid, it tells them they're not trusted, it tells them it's all right to bring in weapons they can sneak in, it tells them to expect violence, and perhaps it encourages them to commit violence to preempt their fears. I think it's a terrible idea.
I very much fear that we have to accept that terrible incidents like this one are a rare and awful but unavoidable part of our way of life, a side effect of having a society as free as ours. We already so accept the much greater daily risk of being killed in a car accident or dying of AIDS. We can condemn the man, curse his memory, and spit on his grave, but I do not think we dare make certain this never happens again.
|Sunday, September 10th, 2006|
|My God, it's full of stars!
What do you see when you fall into a black hole?
Probably not this. This is a simulation of what you would see falling into a primordial black hole, so you can see an alternate universe through it (boringly, the alternate universe is just a mirror image of our own). The black regions are the past singularity, a "white hole"; I've decided not to have it emit anything. A real black hole would have matter inside it (the infalling star) which you would see, and within which the Schwarzschild solution does not apply.
You can see the whole video (of which this is a frame) on archive.org
. (There's a version on Google Video
but they rather butcher uploaded videos.)
If you want to carefully approach the black hole without falling in, things look rather different
(this is on my home computer so please be gentle). As you approach R=2M, the hole appears to close around you. Hovering at or within R=2M is of course impossible. In these panoramas I have turned off redshift; everything would be uniformly very blueshifted (so you would be being fried by gamma rays as you accelerated frantically trying to hold your position). For a clearer view of the weird optical distortions, I generated versions with a checkerboard sky
. At R=3M
, light can travel in circular orbits around the black hole, and it appears to fill half the sky.
Thanks to Jasmine Strong
for the use of a fast machine
to compute these on.
|Thursday, September 7th, 2006|
HTML email is one of those subjects that generate massive irrational flame wars whenever mentioned, particularly to people who've been using email for a certain number of years. In spite of the fact that I've been using email long enough, I actually think HTML email can be appropriate. And I have a particular application I'd like to be able to use it for. But first, the usual objections
to HTML email:
- HTML e-mail is dangerous: Everybody receives so much HTML spam these days that computers that trust HTML mail get rapidly virus-infested and then wiped. Just use a mail client that does a credible job (even Thunderbird will do).
- HTML e-mail wastes bandwidth: I think my ISP cares more about my downloading the entire Ubuntu live DVD (it took me about 45 minutes!) than about the size of my email. Maybe it'll fill my 3 GB gmail mailbox...
- HTML e-mail doesn't always work: So don't send it to people without asking first. A decent mail client keeps track for you.
- HTML e-mail can connect to the internet by itself: Not in any sane modern client. Even Thunderbird qualifies as sane in this regard.
- HTML e-mail renders slowly: It sure doesn't slow my 1.6 GHz Pentium-M down much, and it's no worse than practically every web page out there.
- HTML e-mail is not always reader-friendly: So don't write shoddy HTML mail. ASCII is not always very reader-friendly either - how many emails have you received with butchered vertical alignment, all caps, no punctuation, swarms of inappropriate apostrophes...? Semiliterates can make themselves unintelligible in any format you choose.
- Digested lists hate HTML mail: So don't send it to digested lists.
Most of these appear to be objections to the early adoption of HTML mail, when clients were still buggy, slow, and insecure, and when the standards of bandwidth waste were much lower. But a still reasonable objection is: “What do we need HTML mail for? Flat ASCII is good enough for anybody. Why, when I started using email, you had to enter it on 80-column punch cards and walk six miles in the snow to the computer centre to submit it...” All right, I'm being silly. Why isn't ASCII good enough for everybody?
- International users. Not everybody uses the Latin alphabet, and even those who do often use non-ASCII extension characters such as è, ç, «, and, um, €.
- Funny output devices. People who read their email in odd-shaped windows don't necessarily like scrolling horizontally to read 80-column ASCII mail.
- Links. Why on earth should I need to put “
http://www.flickr.com/photos/afternoon_sunlight/157536846/in/set-72057594091041603/” in my email and hope the reader's mail client successfuly turns that into the right link?
- Semantic markup. Often my emails are just dashed off, but once in a while it'd be nice to be able to use section headers and suchlike to indicate what was what.
- Images. Just sticking an attachment on the end is fine and normal, but sometimes it'd be nice to be able to attach a caption or explanation, possibly involving all the above features.
- Math. Currently I have to include things in my email like “
I_\nu = I_0 \delta'(\mu-1)” in order to convey what I'm talking about. MathML works now, for lots of people with not much effort, at least on the Web. Since software reuse is not a total fiction, this means it can easily be turned on in many email clients.
You can reasonably argue that many of these can be achieved without the use of HTML mail. There's UTF-8 mail (which has to be encoded somehow to pass through legacy mail servers, and which breaks almost the same mail clients). There's format=flowed
, which also has its entrenched opponents. There are people who don't mind the inconveniences I list above, and almost nobody cares about being able to email integral equations. I'm not asking everybody to switch. But I do wonder whether the people who hate HTML mail have actually thought about their position any time in the last five years.
|Back to school
For a change from p-adic modular forms, I thought I'd take a physics class
I wasn't sure how solid my physics background really was, particularly after having spent years buried in algebraic geometry, but it looks all right. Well, we did more or less casually derive the Planck function
for the emission from a black-body radiator. I experienced a moment of alarm when he asked “everyone remembers Bose-Einstein statistics, right?” but the sea of blank looks reassured me.
Fortunately, I'd been thinking about black body radiation recently anyway; the colours of black body radiators at a range of temperatures from 650K to 65000K (on a logarithmic scale) are visible on the left. Generating this led me into a maze of documents about colour; I now know much more about it than I did before. Unfortunately, this has had the effect of making me more dissatisfied with my (digital) camera and my screen. All those beautiful colours I take pictures of are being jammed down into a little triangle
I'm also curious, now, about all those colours that aren't in the usual RGB gamut - what do they look like? How, I wonder, would you go about actually producing light from all over the space of perceptible colours?
Two tunable monochrome lights would do it, but tunable monochrome lights are not exactly readily available, particularly not if you want them calibrated.
I suppose you could build a gadget you peer into with a light bulb, a diffraction grating, and a slit. Well, two, I suppose, with brightness controls.
Conceivably you could use a tunable dye laser, but this is not exactly a pocket gadget.
Perhaps an arrangement where you separate some broad-spectrum light (a halogen lamp, say) onto a white diffuse reflector, cover some pieces with black paper, and condense the rest onto another diffuse reflector. Actually, with a beamsplitter and a retroreflective background you could do it quite efficiently. (Figuring out which segments to block will be a challenge...)
And then there are the colours which our cones could in principle signal but which nobody can ever see, like "psychedelic aquamarine
|Tuesday, August 15th, 2006|
I went to a talk yesterday about neutron star equations of state. Well, perhaps that should be “neutron star”. It got me thinking about how scientists view (and feel about) yet-unproven theories.( The long story.Collapse )
In short, there are a variety of theories about what happens inside a neutron star, but they fall into two camps: those who think it’s just neutrons, and those who think there’s some exotic stuff in there. Each theory predicts a curve relating the mass and radius of neutron stars. If you could measure the mass and radius of even one neutron star, you could eliminate many theories immediately.
Unfortunately, no neutron star has a well-known mass or radius. Various tricks allow one to get approximate values, by making various assumptions, but there’s definitely a need for more observational data.( The long storyCollapse )
Currently we can’t really rule out any of the theories. In fact, even if you trust all the experimental data, all the theories can be adjusted (by changing the assumptions and approximations slightly) to accomodate the known data. So really, the answer to “what goes on inside a neutron star?” is “we don’t know”.
If you ask an astrophysicist who works on this stuff, though, you’ll probably find that she believes one of the theories and not the others. Of course she’ll acknowledge that the evidence isn’t in yet, but she believes one of them. And she’ll go looking for data that eliminates the others. This isn’t just a question of abstract speculation, she’ll spend scarce grant money based on her belief. And she’ll try to come up with models that support her belief. Differences in beliefs about these things can make astrophysics a much more personal kind of competition than one might expect. And that’s a good thing, for the most part, I think, because people are more easily motivated by competition on a personal scale (and astrophysics is often very very far from a human scale).
Problems can arise, though. When the press gets hold of a scientist, often the doubts and qualifications don’t come through, and what gets reported is something like “neutron stars are actually made of strange quarks”. And sometimes the scientist holds onto their belief long after evidence opposing it has become overwhelming.
|Monday, August 7th, 2006|
|Why do we believe what we believe?
I support gun control. I’m not interested, just now, in arguing about whether it’s a good idea or not. I’m interested in looking at why I support it. It’s tempting to tell myself that I support it because I’ve considered the question deeply and with an open mind, and the balance of scientifically-determined evidence supports my view. This is, of course, bollocks.
I’ve never liked guns, though I don’t think I’ve ever touched a real one. The little air rifles I’ve fired never made much of a positive impression on me, certainly less so than the bows. I was brought up in a society where guns were essentially reserved for police. (Of course I know other Canadians have guns, but I had no contact with them, nor was I particularly aware that they existed.) Add to that my father’s horror stories of gun-related accidents an the army, and you get a pretty negative attitude towards guns as a kid. Crime was also totally invisible to me, and the police visible only as vague helpful figures. So gun control seemed like a pretty good idea.
What about as an adult, as someone who (in principle) might be able to vote for or against gun control? Surely I did a great deal of research, digging up evidence for it? Well, no. Mostly I’ve just kept my childhood attitude. Occasionally some piece of news will manage to attract my attention — school shootings, say, or the Polytechnique massacre
. These are of course interpreted as further evidence that guns are bad news, and that it’s better if they aren’t in circulation. I also hear some of the more extreme “cold dead hands
” anti-gun-control people on the news; it’s easy to dismiss them as nuts, and their arguments as irrelevant in a country that’s not run according to the same constitution. In other words, my attitudes towards guns shape the way I perceive all this data.
What about hard statistics? Surely they should be unambiguous enough to support or contradict my opinions, right? Well, not really. For example, I remember reading about a study that showed that violent crime increased when guns were banned. But of course, you can make a single study show almost anything, if you’re clever. Besides, it was on a pro-gun website, so why should I trust it? In fact, almost all the statistics I’ve seen have shown that banning guns doesn’t help make a safer society. Almost all of them have been from pro-gun people. And none of it was convincing. Why? Shouldn’t I be convinced by statistics?
Well, to some extent there’s a difference in values here. What if violent crime — fistfights and so on — increased but murders decreased? Is that a good thing or a bad thing? What about the attitudes — does banning guns save us from having to live in fear of being shot? Does allowing guns make the threat of violence a factor in all social activities? If you’re determined, you can render irrelevant almost any study of such a difficult issue.
Even with my values it’s not at all clear that banning guns helps. For example, it’s important to me that physically weaker people shouldn’t have to live in fear; a woman with a gun is on an equal footing to a man with a gun, in a violent situation, but without a gun the same can’t normally be said. “Gays with guns don’t get bashed
.” Tough, isn’t it?
What about politics? Normally, wanting gun control comes in a package with other unrelated issues like supporting women’s right to abortion, human rights for gay and transgendered people, health care for the poor, and so on. So part of the reason I support gun control is because I support many of those other things (not necessarily for any better reason).
Is there some general attitude that provides the difference between pro-gun and anti-gun people? I think perhaps it’s a question of who you trust to protect you. If you feel like you can trust the police and the government to protect you, of course you want them to be the only people with guns. If you feel like you have to handle your own protection, of course you want to have a gun to do it with.
What’s the point of all this? Tempting as it is to claim that I have a reasoned, rational position on gun control, it’s just not true. Attitudes I picked up as a child have shaped what I read and how I interpret it, in a way that reinforces those attitudes. Without a conscious reevaluation, I would probably have simply gone on thinking that gun control was the only sensible approach. Now, I still think it’s a good idea but I’m aware of just how unsupported that feeling is.
So here’s a challenge for you: pick a controversial subject on which you have an opinion. There are plenty, I don’t really need to list them here, do I? Now ask yourself, honestly:
How did you come to have that opinion in the first place?
What have you done to check it out?
How much time have you spent reading the opposition’s point of view? (Not the extreme nutcases — it’s fun to laugh at them, they’re not threatening, and seeing how stupid they are makes you feel better for disagreeing with them — but the reasonable, well-argued points of view. If you don’t think there are any reasonable people who oppose your point of view, either you didn’t pick a controversial topic, or you’re not looking for them.)
Look at the real differences of opinion there. How much is a disagreement on actual facts, and how much on values?
What attitude underlies this difference of opinion?
|Thursday, August 3rd, 2006|
|“We Keep a Light”
I’ve just finished reading “We Keep a Light” by Evelyn Richardson
. She and her husband bought Bon Portage island
(it’s only a few miles long) and moved there to raise a family. The lighthouse keeper’s job didn’t pay well enough to support them, so they did some farming (although conditions there are difficult).
It’s an interesting book; there’s no dramatic tension, it’s just a charming picture of life as a lighthouse keeper around the time of the second world war. She talks about everything from the history of the island to educating her children to listening with apparent interest to her husband’s duck-hunting buddies. Every household task was more difficult than we’d expect — for a while they had to get fresh water through a trapdoor in the middle of the kitchen floor, getting onto and off of the island involved racing the little dory up the dock before the waves could batter it to pieces, they had to have an evacuation plan for every big storm in case it undermined the lighthouse — but she captures the joys of life too. The beauty of the sea, the little rituals of birthdays and Christmas presents, even homemade music — somehow they managed to get a piano onto the island, hauling it out of the dory and up the beach on a calm day.
The book particularly interested me because I visited Bon Portage earlier this year. Alas, Canada no longer maintains any manned lighthouses, but the island is now owned by Acadia University
and used for biology field courses. We walked all the way around the island, so when Mrs. Richardson describes parts of the island I can often picture them.
|Tuesday, August 1st, 2006|
|+C considered harmful
There's a joke
Two math professors are sitting in a café, talking. The subject turns to education, and one of them starts bemoaning the generally parlous state of mathematical education today. The other maintains it's not so bad. When the first goes to the bathroom, the second calls over the waitress. “Here's $10. When I call you over and ask you a question, just answer ‘x squared over two’. All right?”
The waitress agrees. The first math professor returns, and the second says “Here, I'll bet you $20 that that waitress knows some calculus!” The first agrees, and they call the waitress over. “What's the integral of x dx?” the second math professor asks.
“x squared over two,” the waitress replies, as instructed. The first math professor, surprised, pays up. Under her breath, as she walks away, the waitress mutters “... plus a constant.”
Haw, haw. I know, it's a bad math joke. And you've probably heard it before. The reson I bring it up is, we teach generation after generation of student to write “+C” after all their antiderivatives:
Why do we do this to them? The usual explanation is that it's the most general antiderivative. It's not. The most general antiderivative, in this case, is:
Since the original function isn't defined at 0, the domain isn't connected and we can add a separate constant to each one. The most general antiderivative of (say) the tangent function is much worse, allowing an infinite number of additive constants. So mechanically adding “+C” to every answer doesn't give the most general antiderivative.
I think the real reason you do it is because you're abusing mathematical notation one way or another. The problem is that the antiderivative is not a well-defined operator - it only gives you an answer up to one or more constant offsets; put another way, the answer's not a function but a function modulo locally constant functions. Of course that's not a reasonable thing to say to (most?) beginning calculus students. The problem is that if you simply write down equalities you get things like:
However, simply having a new variable appear without comment is also problematic. The students need to remember to use a different variable every time; for that matter, they need to know what it means (and they often don't). Stewart
points out that simply writing “+C” is inadequate for 1/x, but then blows it in the next problem by claiming it's adequate for x^n. To be fair, concealed within the list of problems he asks the reader to find an antiderivative of 1/x^2 that is zero at both 1 and -1, but otherwise he seems to use “+C” regardless of whether it's the most general solution or not. Spivak
makes some disparaging comments about the indefinite integral, states that he doesn't like to write “+C” but that others do, and then says “Although it is possible (Problem 13) to obtain contradictions if this point is disregarded, in practice such difficulties do not arise, and concern for this constant is merely an annoyance.”
Granted Spivak's audience is prospective mathematicians, I'm still inclined to side with him for a general calculus class.
|Sunday, July 30th, 2006|
What happens when you pass through the event horizon of a black hole?
If it's big enough, there won't be enough tides to feel. If there's not too much stuff falling in, you won't be burned up be superheated plasma falling in with you. Would you even know it had happened? There's nothing special about that piece of spacetime. Well, except that there's presumably some Hawking radiation being produced there, so you'll see a blackbody glow. (What does that look like from the inside?) But if the hole is big enough to have small tides, it's so cold it'll barely matter.
There may be some photons on the horizon itself; it's an equilibrium point for photons — not orbiting, just sitting there — but it's an unstable one, so there probably won't be many (any?). If nothing new is falling into the hole, it should sweep its neighbourhood clean pretty effectively; photons from just outside the hole will work their way further away, getting redshifted in the process.
Looking out at the distant stars, you can expect to see something, though. Light does fall into the hole, and you'll see it as you fall towards the singularity; it'll be very blueshifted (how much?). When you're outside the hole, no light rays come to you from the direction of the hole itself; in fact, light is bent towards the hole, so you'll see a black circle that looks bigger than the horizon should. When you approach the horizon, eventually half the sky gets blacked out. Not by being obscured, though; you'll see stars by gravitational lensing that are behind the hole. Do you actually see several copies of the sky? Can a light beam wrap around the hole more than once?
Once you've actually passed through the hole, I think the circle where you see photons from the outside world narrows, like falling into a well, again without blocking anything. I don't think anything, inside or outside the hole, can come to you from the directions outside that circle; in a sense the inward direction becomes timelike, in that you can only see things coming from your “past”, that is, outside. Does the direction that was originally time become spacelike, allowing you to move around at will? Does this allow time travel, in some sense? I don't think it does, in any useful sense. Certainly you can't make any closed timelike cycles, since the radial direction is now timelike (one-way).
Of course, all this is for a non-rotating hole. And I wouldn't volunteer to actually do the experiment, since not only will you not survive, you can't report anything back. But it's an interesting question to think about, and mostly answerable using general relativity and cleverness. John Wheeler has a cute applet
that lets you explore trajectories around a black hole (although only outside, I think, and in Schwarzschild coordinates). A similar one in Kruskal-Szekeres coordinates would be more interesting, but unfortunately there's really no adequate coordinate to map to real time.