Razib points me to a YouTube clip of a 18 y.o. girl promoting scientism over religious superstition. It seems like just a boring reminder of how cool you think you are at that age, as if it took balls to rail against organized religion -- a real iconoclast these days has bigger fish to fry. However, on watching several more of AngryLittleGirl's clips, my bullshit detector went screaming. If I had to bet, I'd say this is another lonelygirl15 hoax designed to appeal to geeky male loners aged 25 to 55. Her hook is blabbing about her atheism and civil libertarianism, so the target audience is the type who you'd expect to be a permanent student activist or work at an independent / used bookstore [1]. I'll review some suspicious features from several of her earliest clips -- when any giveaways are unlikely to have been anticipated or caught due to the inexperience of her and her crew in passing off fiction as reality [2].
For one thing, the content and form are so caricatured that it sounds like a hack's script of a smart but bratty teenager. I know plenty of those, and though they are annoying, this girl does appear to be very exaggerated. Furthermore, she typically talks for minutes on end without the awkward pauses, "ums," "ahs," "y'know," "the thing is," and so on, that anyone who isn't a professional actor or public speaker would have used repeatedly.
In A Rant Goes Wrong, she says that others have doubted whether she's real, since she tends to look off-camera (perhaps to read the script, get direction, etc.). Her lame response is that she's looking into the flip-out part of her video recorder, admiring how sexy she is. Bitch please. For an example of her repeatedly looking off-camera, see My Religious Background, where it does appear that she keeps looking to her left, and that her eyes make "I'm reading text" movements. In Now You Know, she makes up a statistic (that 95% of Americans are scientifically illiterate) and then chastises the viewer at length if they believed her, making liberal use of the word "gullible." That's exactly what someone is if they don't at least raise their eyebrows when watching her clips. Perhaps this is intended as an irony by her writers.
Another red flag is how the Back to the Future poster in her room is placed, and she is positioned, so that it's almost always entirely visible. It's announcing itself too loudly, again to get the attention of the nerd and geek audience -- "Wow, she likes Back to the Future too!" It's as subtle as a gun over the mantle in a Chekhov play. In general, the shooting and editing of the early clips are too professional to be the product of a high school senior. While the editing techniques aren't mind-blowing, they are noticeably the result of someone who's been to film school or had sufficient training. Since this may not be obvious, just compare such clips to any of those from a typical 18 y.o., or indeed of anyone without professional training. There is usually no editing at all, and if there is, it is very choppy rather than seamless. This is especially true in "confessional" clips, where the person usually just talks to the camera in an unbroken recording. It requires plenty of deliberate effort to break up her confessionals with gimmicky edits.
As an example, see The International Square Earth Society -- there are far too many edits for a high schooler to have bothered with, let alone achieve the level of at least a film school graduate. Evolution Revealed is another heavily edited and stylized clip. She's supposedly taking 4 or 5 AP classes, volunteers, etc., and yet has sufficient time to take on large film editing projects? C'mon! Now, in later clips, judging from the titles, she tries to prove that she isn't fake. I'm not going to waste any more time watching those as well, since the evidence from her early clips is overwhelming.
The internet, and YouTube in particular, have made possible a new trope: a naturally occurring Galatea. That is, it was always possible through literature, sculpture, film, and so on, for loners to create their image of the perfect woman, but it's hard to get past the awareness that she is a figment of the creator's imagination. YouTube confessionals are presumably only made by real-life people, so that a person can anticipate what qualities a loner would endow his perfect woman with and hire a convincing actress to play that part. Thus, those watching her videos get the satisfaction of spying on and being talked to by their dream girl, safe in their belief that she isn't a fiction. The trouble is that someone will always be willing to throw open the curtains and show that the whole thing was a hoax, leaving them adrift in disappointment. That happened with lonelygirl15, and it will happen with AngryLittleGirl as well.
[1] She's like a somewhat less ugly version of real-life Jacqueline Passey. Interestingly, the actress playing AngryLittleGirl is convincing due to her higher level of biological masculinity. She has a fairly masculine jaw-line for an 18 y.o. girl, and just watch An American Girl, where she makes her hands visible throughout. If you pause this clip at 4:27, you can see that on her right hand her ring finger is noticeably longer than her index finger. This is a masculine digit ratio. It would require a female that far into the right tail of the testosterone distribution to acquire an interest in male geek topics.
[2] There are, though, three major improvements since the lonelygirl15 hoax: 1) the lighting is more typical of someone's room, rather than professional; 2) there are no passing references to the supernatural; and 3) they hired a girl who is only semi-cute rather than breathtakingly beautiful.
Addendum: I already mentioned the "gullible" motif from "Now You Know," but I didn't mention that in that clip she says that if you're gullible enough to swallow any old factoid uncritically, then you're also gullible enough to be brainwashed by religion. Maybe that's the point of her persona: to show that atheists, who pride themselves on questioning everything, will eagerly eat up a fiction, no matter how egregiously affected her mannerisms, as long as it makes them feel good inside -- in this case, from the belief or faith that a girl like her exists somewhere out there and might even date them.
Whether or not her writers are that clever, there is another obvious instance of beating the atheists with their own stick: namely, that mass media outlets like YouTube are the opiate of the atheist male masses, lulling them into inaction as they give up on real-life girls, preferring daily fixes of AngryLittleGirl instead. In this respect, she far exceeds lonelygirl15 as voyeuristic geek porn.
June 23, 2007
June 16, 2007
Teen movies and human nature, part 2
Since my most popular post to date has been a review of Mean Girls, and since I've been meaning to partially review some of the movies in my Netflix queue, why not kill two birds with one stone and write a follow-up? I've been watching more teen movies than I should, but I was pretty clueless about adolescent girl life when I was a teenager myself, and given how important this time of your life is -- the tumultuous, transient state before you reach equilibrium -- the anthropologist in me wants to figure it all out and try to account for it. The guy part I pretty well understand from personal or one-degree-removed experience, though I note below some things that I was oblivious to at the time as well.
1) To get the superficial out of the way first, I've learned that of all the aspects of a female's physical appearance, the one that most honestly signals youthful fertility is tight skin, not just in the face but in general. For those who don't work with teenagers or college students, just remember: what items of clothing does almost every 18-ish girl wear, which disappear by just age 25? Hot pants and mini-skirts, which show off the legs -- not their length (females obsess over being tall later), but rather the tautness of the skin. This is true even if the girl isn't very athletic and so lacks muscular legs; it's mostly a bunch of fat that's nevertheless held tightly together. It looks like an inflated balloon stuffed with Jell-o, which as she ages resembles Saran wrap struggling to conceal porridge. The fatty deposits and lack of large muscles signal femininity, while the tight skin signals youth. After, I'd say, a girl's mid-20s (on average), only one of these becomes possible to maintain -- either a doughier appearance as the skin slackens around the fat, or a tightness achieved by vigorous exercise to burn off the fat and expand the muscle.
As a few examples, consider that three of the four main characters in Mean Girls are 18-22, and the exception (a 26 y.o. Rachel McAdams) opts for the treadmill junkie look to pass as younger. Also, in Never Been Kissed, only one of the main female characters (played by an 18 y.o. Jessica Alba) is of high school age, and she is the only one who wears hot pants consistently throughout the movie. The other three are 24-25 -- decades away from being old hags, but apparently past the point of being able to rock the hot pants. (Comparison here, and two more of Alba here and here.)
To prove that Alba's co-stars weren't always incapable of baring their legs, consider actress Marley Shelton: in the course of digging around for this post, I discovered that she played the red-hot lifeguard in The Sandlot. In case you haven't seen it, watch the famous pool scene here. It's clear that she too once had legs like Alba does in Never Been Kissed, but The Sandlot was made six years earlier, when Shelton was 19. Note that average human generation time is 20-25 years, so 25 does seem a natural point for the female body to begin its descent from maximum hotness, as it expects itself to have snagged a mate and born a child by then.
2) Related to a female's hotness during adolescence is her ability to enter a higher social stratum than that into which she was born. This is a pretty general pattern: guys are more locked into place since if they aren't already smart and/or wealthy, they won't be able to use their looks or charm to marry a smart / wealthy girl, whereas smart / wealthy guys are happy to take a trophy wife who may not be incredibly sharp or rich. Also, when more powerful groups invade and conquer another group, it's typically the native women (the best-looking among them, one assumes) who are absorbed into the conquering group, while native men are hung out to dry. For example, in Brazil about 97% of the male lineages are European (and 3% African), while those of females are fairly evenly split among European, Native, and African (39%, 33%, and 28%, respectively; the studies are listed at the end of Ch. 12 of Human Evolutionary Genetics).
Now, social strata in adolescence aren't distinguished so much by wealth as by popularity. There's a pretty easy way for a currently unpopular, newly arrived high school freshman to catapult herself into the popular peer group of senior jocks and other alpha males: assuming she's pretty, she can just put out. In contrast, it would be impossible for an unpopular freshman guy -- no matter how tall, hunky, and athletic -- to break into the ranks of hot senior cheerleaders.
3) As a result, adolescent females are locked in a much more bitter struggle with each other compared to male-male struggle. A senior cheerleader who's landed a hot popular boyfriend has to constantly chase away would-be trespassers in the three grade levels below her, many of whom would easily put out just to steal her glamorous boyfriend. This puts more pressure on the junior and senior girls to put out, lest their boyfriends leave them for more easily conquered freshman girls. A prediction is that Neuroticism in females will increase, or at least reach its peak, during adolescence, which recent personality trait studies have borne out (see here and here).
Adolescent males don't have to worry about this so much, as they don't really become desirable until they're juniors or seniors anyway. Thus, their would-be competitors -- guys who are 19-20 -- are nowhere to be seen. It's very rare for a high school girl to date a college guy, just because their social circles and daily routines don't overlap, so high school guys with girlfriends are well insulated from competition. Evolutionary psychologist Geoffrey Miller has made a similar point, although he went too far in suggesting that 18-23 y.o. girls would date guys in their late 20s or 30s if only there weren't artificial social barriers like socializing adolescents in a bubble (high school). Still, a watered-down version is tenable: if high school and college-aged people were socialized together, high school guys would suffer massively from college guy competition. This doesn't continue infinitely since, again, by about 25 you're clearly past the adolescent-adult threshold (excepting rareties like Jessica Alba or a younger Johnny Depp). And for those who think that being 25 doesn't make you a geezer, read this post to confirm that you are in fact an old fogey.
4) On a related note, though, an even greater effect of increased inter-generational mixing would, I think, be a decrease in caddish behavior among adolescent males. One of the girls I tutor (who just finished 11th grade) is the pretty, popular, cheerleader / athlete type. She's on the touchy side but is really a nice girl. One day before our session began, she had to get something off her chest, to warn me in case she wasn't able to focus very well: her boyfriend (the popular jock type) had been pressuring her to have sex, but she wouldn't budge. So, just one week before her birthday (!), her boyfriend dumps her, explaining that "I'm gonna see what I can get with someone else." I was praying that she wouldn't mention his name, because I would have hunted him down like the dog that he is (fortunately for him, she did not).
Even in a hunter-gatherer society, it doesn't require any brawn to kill someone -- just ballsiness and the element of surprise, such as a pre-dawn raid. And in a culture more advanced than an H-G one, populations are large enough for anonymity to protect an attacker from being found out. In our radical feminist society, though, guys aren't supposed to care about defending girls, since they supposedly can take care of themselves -- yeah right. In any civilized society, a girl's older brothers, cousins, and close male friends would give her cur of a boyfriend an intimidating warning, and failing any change on his part, beat him to within an inch of his life. But add to radical feminism the vast social chasm between adolescents and 25+ adults, and it becomes very difficult for would-be male protectors to stay in touch with who is dishonoring whom, and hence tough to keep the scoundrels in line.
When I casually mentioned around some 14-15 y.o. tutorees that if I ever have daughters, I'll punish their boyfriends should they misbehave, they responded like, "Oh my god, you're so clueless -- a girl would never tell her dad if something was going wrong with her boyfriend." It would seem to follow that any male of her dad's age wouldn't know either. So, it's crucial for brothers, cousins, and male acquaintances in their 20's and 30's to be part of an adolescent girl's social circle, so they can hear the bad news and straighten the offender out. In the present case, though, this responsibility falls on the shoulders of her adolescent male peers, who are too interested in her sexually to play the role of protector. And for the same reason, only the tiny minority of super-hot girls would receive protection (contra the definition of "White Knight Syndrome" above, which falsely states that White Knights try to help "any" girl in trouble, out of chivalry).
Ideally, older brothers and cousins would perform this role, since they are not interested in her sexually due to the incest taboo, as well as having a stronger protective instinct due to kin selection. If they aren't around, then males on the other side of the 25 y.o. age divide could step in, as the potential for mutual sexual attraction is minimal (unless he's Justin Timberlake), and assuming they are close acquaintances, they'll have a strong protective sense due to selection for reciprocal altruism. This is one case where neutral third parties, such as the police, could not be relied upon: cruelly mistreating a girl is not against the law. And that does make this case pretty dangerous, as it's easy to imagine feuds beginning with the male relatives of a girl ganging up on her degenerate boyfriend, whose relatives then retaliate. Still, I think if families were more in touch with what was going on in the community, the cad's family would be far outnumbered and would not be able to strike back, even if they wanted to.
5) Why are guys of any age far more likely to feel hyper-protective about the physical appearance of adolescent girls rather than older females (say, 30+), even if the older females are closely related (blood relative, wife, etc.)? In Mean Girls there is a hilarious scene in which Lindsay Lohan describes what Halloween parties are like for present-day American teenagers. Watch a 20-second clip that sums it up here. (Steve Sailer mooted this topic here, presumably without having seen the movie.) Note the expression of anxiety and helplessness on the father's face; obviously, any father would feel this way if his teenage daughter were going to a party in just lingerie, knee-high boots, and animal ears. (Also note how wimpy the modern American dad has become: if I were him, after I'd recovered from the nine heart attacks I'd had before I hit the floor, I'd lock her in her room until she dressed more tastefully.)
But throughout the movie, the script emphasizes his wife's desperate labors to act young and slutty. For example, she gets gigantic breast implants, wears a form-fitting pink tracksuit, offers her daughter a condom when she walks in on her making out with her boyfriend, and so on. Why doesn't the father weep in disgust at his wife's vulgar behavior? Well, her ship has already started to sink anyway, so why bother salvaging it if there is another intact ship that you see heading toward an iceberg? More to the point, though perhaps less tactfully, it's the same reason that we lock up our precious jewels in an airtight vault, while we leave our spare pennies just lying around in the open -- which ones are potential thieves interested in?
Take the example to the extreme: imagine your daughter was wearing a PVC bondage outfit a la Trinity from The Matrix. Your heart would race with anxiety about how she'd attract the filthy leers of every male in sight, and you'd send her off to boarding school. On the other hand, if your 35-40 y.o. sister were so dressed, you'd give her a calm but incredulous look like, "Who are you kidding, do you really think you can pull that off?" Or, "I just want to keep you from publicly embarrassing yourself." That is, unless she were a rare exception like 30-something Carrie-Anne Moss who played Trinity.
There are surely more lessons, but I've gone on long enough already. I'd just like to conclude this follow-up to the Mean Girls post, in which I griped about how I preferred Heathers, by noting that I've successfully managed to introduce the latter cult classic to some of my tutorees. Over the past several months, I've been tutoring kids for the SAT II Math 2C test, and in reviewing limits, I mentioned one of the final scenes of Mean Girls where Lindsay Lohan answers a limit question in a mathlete competition. The female students always react in a way indicating that this is one of their favorite movies, so I suggested that they watch the earlier Winona Ryder incarnation for comparison. One of my tutorees watched most of it on cable and said she liked it, even if dark humor wasn't her preference. On the last day I tutored her, she brought me a gift: a DVD of Heathers! I'd never received a gift from any student before, let alone one so awesome, so she is now officially the coolest student I've ever had. Now, if only I could get my male students to appreciate Nintendo and Super Nintendo video games, I'll have successfully passed on my generation's greatest cultural contributions.
1) To get the superficial out of the way first, I've learned that of all the aspects of a female's physical appearance, the one that most honestly signals youthful fertility is tight skin, not just in the face but in general. For those who don't work with teenagers or college students, just remember: what items of clothing does almost every 18-ish girl wear, which disappear by just age 25? Hot pants and mini-skirts, which show off the legs -- not their length (females obsess over being tall later), but rather the tautness of the skin. This is true even if the girl isn't very athletic and so lacks muscular legs; it's mostly a bunch of fat that's nevertheless held tightly together. It looks like an inflated balloon stuffed with Jell-o, which as she ages resembles Saran wrap struggling to conceal porridge. The fatty deposits and lack of large muscles signal femininity, while the tight skin signals youth. After, I'd say, a girl's mid-20s (on average), only one of these becomes possible to maintain -- either a doughier appearance as the skin slackens around the fat, or a tightness achieved by vigorous exercise to burn off the fat and expand the muscle.
As a few examples, consider that three of the four main characters in Mean Girls are 18-22, and the exception (a 26 y.o. Rachel McAdams) opts for the treadmill junkie look to pass as younger. Also, in Never Been Kissed, only one of the main female characters (played by an 18 y.o. Jessica Alba) is of high school age, and she is the only one who wears hot pants consistently throughout the movie. The other three are 24-25 -- decades away from being old hags, but apparently past the point of being able to rock the hot pants. (Comparison here, and two more of Alba here and here.)
To prove that Alba's co-stars weren't always incapable of baring their legs, consider actress Marley Shelton: in the course of digging around for this post, I discovered that she played the red-hot lifeguard in The Sandlot. In case you haven't seen it, watch the famous pool scene here. It's clear that she too once had legs like Alba does in Never Been Kissed, but The Sandlot was made six years earlier, when Shelton was 19. Note that average human generation time is 20-25 years, so 25 does seem a natural point for the female body to begin its descent from maximum hotness, as it expects itself to have snagged a mate and born a child by then.
2) Related to a female's hotness during adolescence is her ability to enter a higher social stratum than that into which she was born. This is a pretty general pattern: guys are more locked into place since if they aren't already smart and/or wealthy, they won't be able to use their looks or charm to marry a smart / wealthy girl, whereas smart / wealthy guys are happy to take a trophy wife who may not be incredibly sharp or rich. Also, when more powerful groups invade and conquer another group, it's typically the native women (the best-looking among them, one assumes) who are absorbed into the conquering group, while native men are hung out to dry. For example, in Brazil about 97% of the male lineages are European (and 3% African), while those of females are fairly evenly split among European, Native, and African (39%, 33%, and 28%, respectively; the studies are listed at the end of Ch. 12 of Human Evolutionary Genetics).
Now, social strata in adolescence aren't distinguished so much by wealth as by popularity. There's a pretty easy way for a currently unpopular, newly arrived high school freshman to catapult herself into the popular peer group of senior jocks and other alpha males: assuming she's pretty, she can just put out. In contrast, it would be impossible for an unpopular freshman guy -- no matter how tall, hunky, and athletic -- to break into the ranks of hot senior cheerleaders.
3) As a result, adolescent females are locked in a much more bitter struggle with each other compared to male-male struggle. A senior cheerleader who's landed a hot popular boyfriend has to constantly chase away would-be trespassers in the three grade levels below her, many of whom would easily put out just to steal her glamorous boyfriend. This puts more pressure on the junior and senior girls to put out, lest their boyfriends leave them for more easily conquered freshman girls. A prediction is that Neuroticism in females will increase, or at least reach its peak, during adolescence, which recent personality trait studies have borne out (see here and here).
Adolescent males don't have to worry about this so much, as they don't really become desirable until they're juniors or seniors anyway. Thus, their would-be competitors -- guys who are 19-20 -- are nowhere to be seen. It's very rare for a high school girl to date a college guy, just because their social circles and daily routines don't overlap, so high school guys with girlfriends are well insulated from competition. Evolutionary psychologist Geoffrey Miller has made a similar point, although he went too far in suggesting that 18-23 y.o. girls would date guys in their late 20s or 30s if only there weren't artificial social barriers like socializing adolescents in a bubble (high school). Still, a watered-down version is tenable: if high school and college-aged people were socialized together, high school guys would suffer massively from college guy competition. This doesn't continue infinitely since, again, by about 25 you're clearly past the adolescent-adult threshold (excepting rareties like Jessica Alba or a younger Johnny Depp). And for those who think that being 25 doesn't make you a geezer, read this post to confirm that you are in fact an old fogey.
4) On a related note, though, an even greater effect of increased inter-generational mixing would, I think, be a decrease in caddish behavior among adolescent males. One of the girls I tutor (who just finished 11th grade) is the pretty, popular, cheerleader / athlete type. She's on the touchy side but is really a nice girl. One day before our session began, she had to get something off her chest, to warn me in case she wasn't able to focus very well: her boyfriend (the popular jock type) had been pressuring her to have sex, but she wouldn't budge. So, just one week before her birthday (!), her boyfriend dumps her, explaining that "I'm gonna see what I can get with someone else." I was praying that she wouldn't mention his name, because I would have hunted him down like the dog that he is (fortunately for him, she did not).
Even in a hunter-gatherer society, it doesn't require any brawn to kill someone -- just ballsiness and the element of surprise, such as a pre-dawn raid. And in a culture more advanced than an H-G one, populations are large enough for anonymity to protect an attacker from being found out. In our radical feminist society, though, guys aren't supposed to care about defending girls, since they supposedly can take care of themselves -- yeah right. In any civilized society, a girl's older brothers, cousins, and close male friends would give her cur of a boyfriend an intimidating warning, and failing any change on his part, beat him to within an inch of his life. But add to radical feminism the vast social chasm between adolescents and 25+ adults, and it becomes very difficult for would-be male protectors to stay in touch with who is dishonoring whom, and hence tough to keep the scoundrels in line.
When I casually mentioned around some 14-15 y.o. tutorees that if I ever have daughters, I'll punish their boyfriends should they misbehave, they responded like, "Oh my god, you're so clueless -- a girl would never tell her dad if something was going wrong with her boyfriend." It would seem to follow that any male of her dad's age wouldn't know either. So, it's crucial for brothers, cousins, and male acquaintances in their 20's and 30's to be part of an adolescent girl's social circle, so they can hear the bad news and straighten the offender out. In the present case, though, this responsibility falls on the shoulders of her adolescent male peers, who are too interested in her sexually to play the role of protector. And for the same reason, only the tiny minority of super-hot girls would receive protection (contra the definition of "White Knight Syndrome" above, which falsely states that White Knights try to help "any" girl in trouble, out of chivalry).
Ideally, older brothers and cousins would perform this role, since they are not interested in her sexually due to the incest taboo, as well as having a stronger protective instinct due to kin selection. If they aren't around, then males on the other side of the 25 y.o. age divide could step in, as the potential for mutual sexual attraction is minimal (unless he's Justin Timberlake), and assuming they are close acquaintances, they'll have a strong protective sense due to selection for reciprocal altruism. This is one case where neutral third parties, such as the police, could not be relied upon: cruelly mistreating a girl is not against the law. And that does make this case pretty dangerous, as it's easy to imagine feuds beginning with the male relatives of a girl ganging up on her degenerate boyfriend, whose relatives then retaliate. Still, I think if families were more in touch with what was going on in the community, the cad's family would be far outnumbered and would not be able to strike back, even if they wanted to.
5) Why are guys of any age far more likely to feel hyper-protective about the physical appearance of adolescent girls rather than older females (say, 30+), even if the older females are closely related (blood relative, wife, etc.)? In Mean Girls there is a hilarious scene in which Lindsay Lohan describes what Halloween parties are like for present-day American teenagers. Watch a 20-second clip that sums it up here. (Steve Sailer mooted this topic here, presumably without having seen the movie.) Note the expression of anxiety and helplessness on the father's face; obviously, any father would feel this way if his teenage daughter were going to a party in just lingerie, knee-high boots, and animal ears. (Also note how wimpy the modern American dad has become: if I were him, after I'd recovered from the nine heart attacks I'd had before I hit the floor, I'd lock her in her room until she dressed more tastefully.)
But throughout the movie, the script emphasizes his wife's desperate labors to act young and slutty. For example, she gets gigantic breast implants, wears a form-fitting pink tracksuit, offers her daughter a condom when she walks in on her making out with her boyfriend, and so on. Why doesn't the father weep in disgust at his wife's vulgar behavior? Well, her ship has already started to sink anyway, so why bother salvaging it if there is another intact ship that you see heading toward an iceberg? More to the point, though perhaps less tactfully, it's the same reason that we lock up our precious jewels in an airtight vault, while we leave our spare pennies just lying around in the open -- which ones are potential thieves interested in?
Take the example to the extreme: imagine your daughter was wearing a PVC bondage outfit a la Trinity from The Matrix. Your heart would race with anxiety about how she'd attract the filthy leers of every male in sight, and you'd send her off to boarding school. On the other hand, if your 35-40 y.o. sister were so dressed, you'd give her a calm but incredulous look like, "Who are you kidding, do you really think you can pull that off?" Or, "I just want to keep you from publicly embarrassing yourself." That is, unless she were a rare exception like 30-something Carrie-Anne Moss who played Trinity.
There are surely more lessons, but I've gone on long enough already. I'd just like to conclude this follow-up to the Mean Girls post, in which I griped about how I preferred Heathers, by noting that I've successfully managed to introduce the latter cult classic to some of my tutorees. Over the past several months, I've been tutoring kids for the SAT II Math 2C test, and in reviewing limits, I mentioned one of the final scenes of Mean Girls where Lindsay Lohan answers a limit question in a mathlete competition. The female students always react in a way indicating that this is one of their favorite movies, so I suggested that they watch the earlier Winona Ryder incarnation for comparison. One of my tutorees watched most of it on cable and said she liked it, even if dark humor wasn't her preference. On the last day I tutored her, she brought me a gift: a DVD of Heathers! I'd never received a gift from any student before, let alone one so awesome, so she is now officially the coolest student I've ever had. Now, if only I could get my male students to appreciate Nintendo and Super Nintendo video games, I'll have successfully passed on my generation's greatest cultural contributions.
June 10, 2007
Visual: Countable union of countable sets is countable
Mark Chu-Carroll at Good Math, Bad Math has had a few interesting posts lately on the Axiom of Choice and the equivalent Well-Ordering Principle, which says that any non-empty set of natural numbers (i.e., {1, 2, 3, ... }) has a least element. As I demonstrated awhile ago, this is used to justify the soundness of the proof technique called mathematical induction. Part of Mark's post focused on providing a well-ordering for the integers (i.e., {... -3, -2, -1, 0, 1, 2, 3, ...}), whereby the least element is 0 and the sequence proceeds thus: {0, -1, 1, -2, 2, ...}.
Verbal descriptions are well and good, but many find visual approaches more intuitive. For some odd reason, writers of higher math textbooks tend to shun visualizations, which is irresponsible since most great thinkers conceive images before writing them down in logical propositions and proofs. Maybe this antipathy is some silly relic of Platonic thinking that loathes the sense of sight, as it can mislead, as opposed to the faculties of disembodied logical reasoning. Only thing is: we're human, so most math ideas begin as pictures. We may flesh them out in chains of logical reasoning, but it is a fraud to present an idea only by showing logically that it is true, without any visual or other motivation beforehand.
In any case, what's going on with this well-ordering of the integers? Well, we start with the number line composed of the integers, "pinch" it at 0, and fold the negatives onto the same side as the positives, shoving the positive neighbors apart and shoe-horning each negative between two non-negative neighbors (e.g., -1 is stuck between 0 and 1, -2 is stuck between 1 and 2, and so on). To make it clear how the shoe-horning works so that no negative is accidentally fused on top of a non-negative, let's look at the following diagram:
Imagine the individual integers as beads that rest on an infinitely long piece of string, which has tick marks where each integer is. To make room for the negative beads, we'll move each positive bead to the tick mark of its double -- that is, move 1 to 2, 2 to 4, 3 to 6, and in general move n to 2n, for positive n. Thus, the positive beads now rest only on the even tick marks on the positive side. Next, we move each negative bead to the tick mark of its double plus 1 -- that is, move (or keep) -1 to -1, move -2 to -3, -3 to -5, and in general move n to 2n + 1 for negative n. Now the negatives rest only on the odd tick marks on the negative side. We then send each negative n to its absolute value -- this where we rotate the set of negatives around the fulcrum 0, and since the positive odd spaces were cleared out in our first step, every negative bead has a place to fit in when folded onto the positive side. In this way, we get the well-ordering above: {0, -1, 1, -2, 2, ...}.
In reality, to well-order the integers, we could have "pinched" the number line at an arbitrary integer, not necessarily 0, and performed similar steps to zipper together the integers to the left of the fulcrum with those to the right. But using 0 makes the process clear.
In thinking up this visual, I came across another neat visual to show that the union of countably many countable sets is itself countable. Just now, we took 2 countably infinite sets -- the set of all positive integers and the set of all negative integers -- and formed a countable union. This is true because our beads occupy spaces whose tick marks are the natural numbers (and 0), and being able to bijectively map a set S to the natural numbers implies S is countable. But what if we wanted to do this for an arbitrarily large, though still countable, union of countable sets? This is why it's worth it to think in images, since we can instantly extend the visual analogy to this much larger, abstract case.
The problem now is how we zipper together this countably large union of countable sets? For simplicity, we'll consider each of these sets in the union as the natural numbers {1, 2, 3, ... }. I will frequently refer to the natural numbers both as just a countably infinite set, as well as an ordered sequence of terms (the obvious one -- (1, 2, 3, ...)). There is no potential for ambiguity since sets are not ordered while sequences are; this saves me from using roundabout ways of distinguishing what cannot be confounded.
Moving on, the trouble before was making room for every element of the set that we were attempting to shoe-horn in, and our solution was to assign one set to the even positives and the other set to the odd positives, so that no two distinct elements would ever overlap (and 0 would lie at the start, before the zippering). Suppose we have a third set to shoe-horn in -- on analogy with the simple case, we'll send the first set to {1, 4, 7, ...}, the second to {2, 5, 8, ...}, and the third to {3, 6, 9, ...}. Great, the trick works just as before!
In general, then, for the union of n countable sets, we partition the positive integers into n equivalence classes, where the members of a particular class are congruent to some integer a modulo n. That is, the members of a given class will have the same remainder when they are divided by n. In the example above, 1, 4, and 7 are all congruent to 1 mod 3 since they all leave a remainder of 1 when divided by 3. By assigning one countable set to the positives congruent to a mod n, another to the positives congruent to b mod n (where a =/= b), ad infinitum, we ensure that none of the elements overlap -- we can always shove the numbers aside to make room for the elements of yet another countable set. (A more formal proof concludes this post.) To help visualize this, remember before that we only had one set that we rotated around 0. In the general case, suppose that we had a countably large collection of arrows, along which the natural numbers were ordered (the numbers are only marked along one of the arrows to simplify the drawing, but all are so numbered):
We pick one arrow as the starting point and order the rest in clockwise fashion. If there are n arrows, we label them a_0, a_1, a_2, ... a_(n - 1). We then assign the arrow labeled a_k to the equivalence class of positives congruent to k mod n (k ranging from 0 to n-1). So, our first arrow, a_0, is assigned to the positives congruent to 0 mod n -- that is, the multiples of n: {n, 2n, 3n, ... m*n} for m a positive integer, stipulating that none are mapped to 0 (we'll add it in at the end). We assign particular elements of a_0 to elements of {n, 2n, 3n, ...} in the obvious order-preserving way: the natural number m in a_0 is sent to the mth term of the sequence (n, 2n, ...) -- 1 is sent to 1*n, 2 to 2*n, etc.
Now that we've spread out the natural numbers along arrow a_0, we move clockwise to the next arrow a_1. As before, we assign a_1 to the set of positives congruent to 1 mod n -- (1, 1 + n, 1 + 2n, ...) -- so that the mth natural number in a_1 is sent to the mth term of the latter sequence. We continue around through the final arrow a_(n-1), assigning it to the set of positives congruent to (n-1) mod n, again sending the mth element of a_(n-1) to the mth term of the sequence (n - 1, 2n - 1, ...). Having thus spread out the natural numbers within each arrow to occupy only the positives congruent to a mod n, for all a distinct from each other and ranging from 0 to n - 1, we then rotate all n arrows around 0 so they coincide with our starting arrow a_0. Since equivalence classes don't overlap, we've ensured that no two numbers will accidentally collide with each other. We might as well add in 0 to the start of the sequence since that makes the "least element" easy to remember (this doesn't affect countability). Below is an example where we have 4 sets of natural numbers to zipper together. This is shown up through the first 3 elements from each arrow, although each arrow contains all natural numbers. To make the procedure easy to follow, I've added a subscript to each number to indicate which arrow it originated from (click to enlarge).
Added: Note that not only is the union countable, but it's been ordered in a straightforward way.
More formal proof: We have bijectively mapped the elements of the union of n countable sets to the natural numbers (plus 0), which shows that this union is countable. Now add another arrow (i.e., another countable set, again assuming they're the natural numbers), so that there are now (n + 1) arrows. We translate the elements currently at non-zero multiples of n to non-zero multiples of (n + 1) in the obvious way -- the first element is sent to the first, and in general the mth element to the mth element between equivalence classes. Then we translate the elements currently at the positives congruent to (n - 1) mod n to the positives congruent to n mod (n + 1), again preserving the order. Continuing this way, we finally translate the elements currently at the positives congruent to 1 mod n to those congruent to 2 mod (n + 1). Having translated all elements from the case of only n arrows, we have now freed up spots at the set of positives congruent to 1 mod (n + 1), and we shoe-horn the elements of our new (n + 1)th arrow into these open spots in the order-preserving way.
So, assuming the procedure works for n arrows, it works for (n + 1) arrows. And we've already shown the procedure works for 2 arrows, so by induction we've proved that it works for any arbitrarily large, though countable, number of arrows. That is, the union of countably many countable sets is itself countable. (_)(_)
Verbal descriptions are well and good, but many find visual approaches more intuitive. For some odd reason, writers of higher math textbooks tend to shun visualizations, which is irresponsible since most great thinkers conceive images before writing them down in logical propositions and proofs. Maybe this antipathy is some silly relic of Platonic thinking that loathes the sense of sight, as it can mislead, as opposed to the faculties of disembodied logical reasoning. Only thing is: we're human, so most math ideas begin as pictures. We may flesh them out in chains of logical reasoning, but it is a fraud to present an idea only by showing logically that it is true, without any visual or other motivation beforehand.
In any case, what's going on with this well-ordering of the integers? Well, we start with the number line composed of the integers, "pinch" it at 0, and fold the negatives onto the same side as the positives, shoving the positive neighbors apart and shoe-horning each negative between two non-negative neighbors (e.g., -1 is stuck between 0 and 1, -2 is stuck between 1 and 2, and so on). To make it clear how the shoe-horning works so that no negative is accidentally fused on top of a non-negative, let's look at the following diagram:
Imagine the individual integers as beads that rest on an infinitely long piece of string, which has tick marks where each integer is. To make room for the negative beads, we'll move each positive bead to the tick mark of its double -- that is, move 1 to 2, 2 to 4, 3 to 6, and in general move n to 2n, for positive n. Thus, the positive beads now rest only on the even tick marks on the positive side. Next, we move each negative bead to the tick mark of its double plus 1 -- that is, move (or keep) -1 to -1, move -2 to -3, -3 to -5, and in general move n to 2n + 1 for negative n. Now the negatives rest only on the odd tick marks on the negative side. We then send each negative n to its absolute value -- this where we rotate the set of negatives around the fulcrum 0, and since the positive odd spaces were cleared out in our first step, every negative bead has a place to fit in when folded onto the positive side. In this way, we get the well-ordering above: {0, -1, 1, -2, 2, ...}.
In reality, to well-order the integers, we could have "pinched" the number line at an arbitrary integer, not necessarily 0, and performed similar steps to zipper together the integers to the left of the fulcrum with those to the right. But using 0 makes the process clear.
In thinking up this visual, I came across another neat visual to show that the union of countably many countable sets is itself countable. Just now, we took 2 countably infinite sets -- the set of all positive integers and the set of all negative integers -- and formed a countable union. This is true because our beads occupy spaces whose tick marks are the natural numbers (and 0), and being able to bijectively map a set S to the natural numbers implies S is countable. But what if we wanted to do this for an arbitrarily large, though still countable, union of countable sets? This is why it's worth it to think in images, since we can instantly extend the visual analogy to this much larger, abstract case.
The problem now is how we zipper together this countably large union of countable sets? For simplicity, we'll consider each of these sets in the union as the natural numbers {1, 2, 3, ... }. I will frequently refer to the natural numbers both as just a countably infinite set, as well as an ordered sequence of terms (the obvious one -- (1, 2, 3, ...)). There is no potential for ambiguity since sets are not ordered while sequences are; this saves me from using roundabout ways of distinguishing what cannot be confounded.
Moving on, the trouble before was making room for every element of the set that we were attempting to shoe-horn in, and our solution was to assign one set to the even positives and the other set to the odd positives, so that no two distinct elements would ever overlap (and 0 would lie at the start, before the zippering). Suppose we have a third set to shoe-horn in -- on analogy with the simple case, we'll send the first set to {1, 4, 7, ...}, the second to {2, 5, 8, ...}, and the third to {3, 6, 9, ...}. Great, the trick works just as before!
In general, then, for the union of n countable sets, we partition the positive integers into n equivalence classes, where the members of a particular class are congruent to some integer a modulo n. That is, the members of a given class will have the same remainder when they are divided by n. In the example above, 1, 4, and 7 are all congruent to 1 mod 3 since they all leave a remainder of 1 when divided by 3. By assigning one countable set to the positives congruent to a mod n, another to the positives congruent to b mod n (where a =/= b), ad infinitum, we ensure that none of the elements overlap -- we can always shove the numbers aside to make room for the elements of yet another countable set. (A more formal proof concludes this post.) To help visualize this, remember before that we only had one set that we rotated around 0. In the general case, suppose that we had a countably large collection of arrows, along which the natural numbers were ordered (the numbers are only marked along one of the arrows to simplify the drawing, but all are so numbered):
We pick one arrow as the starting point and order the rest in clockwise fashion. If there are n arrows, we label them a_0, a_1, a_2, ... a_(n - 1). We then assign the arrow labeled a_k to the equivalence class of positives congruent to k mod n (k ranging from 0 to n-1). So, our first arrow, a_0, is assigned to the positives congruent to 0 mod n -- that is, the multiples of n: {n, 2n, 3n, ... m*n} for m a positive integer, stipulating that none are mapped to 0 (we'll add it in at the end). We assign particular elements of a_0 to elements of {n, 2n, 3n, ...} in the obvious order-preserving way: the natural number m in a_0 is sent to the mth term of the sequence (n, 2n, ...) -- 1 is sent to 1*n, 2 to 2*n, etc.
Now that we've spread out the natural numbers along arrow a_0, we move clockwise to the next arrow a_1. As before, we assign a_1 to the set of positives congruent to 1 mod n -- (1, 1 + n, 1 + 2n, ...) -- so that the mth natural number in a_1 is sent to the mth term of the latter sequence. We continue around through the final arrow a_(n-1), assigning it to the set of positives congruent to (n-1) mod n, again sending the mth element of a_(n-1) to the mth term of the sequence (n - 1, 2n - 1, ...). Having thus spread out the natural numbers within each arrow to occupy only the positives congruent to a mod n, for all a distinct from each other and ranging from 0 to n - 1, we then rotate all n arrows around 0 so they coincide with our starting arrow a_0. Since equivalence classes don't overlap, we've ensured that no two numbers will accidentally collide with each other. We might as well add in 0 to the start of the sequence since that makes the "least element" easy to remember (this doesn't affect countability). Below is an example where we have 4 sets of natural numbers to zipper together. This is shown up through the first 3 elements from each arrow, although each arrow contains all natural numbers. To make the procedure easy to follow, I've added a subscript to each number to indicate which arrow it originated from (click to enlarge).
Added: Note that not only is the union countable, but it's been ordered in a straightforward way.
More formal proof: We have bijectively mapped the elements of the union of n countable sets to the natural numbers (plus 0), which shows that this union is countable. Now add another arrow (i.e., another countable set, again assuming they're the natural numbers), so that there are now (n + 1) arrows. We translate the elements currently at non-zero multiples of n to non-zero multiples of (n + 1) in the obvious way -- the first element is sent to the first, and in general the mth element to the mth element between equivalence classes. Then we translate the elements currently at the positives congruent to (n - 1) mod n to the positives congruent to n mod (n + 1), again preserving the order. Continuing this way, we finally translate the elements currently at the positives congruent to 1 mod n to those congruent to 2 mod (n + 1). Having translated all elements from the case of only n arrows, we have now freed up spots at the set of positives congruent to 1 mod (n + 1), and we shoe-horn the elements of our new (n + 1)th arrow into these open spots in the order-preserving way.
So, assuming the procedure works for n arrows, it works for (n + 1) arrows. And we've already shown the procedure works for 2 arrows, so by induction we've proved that it works for any arbitrarily large, though countable, number of arrows. That is, the union of countably many countable sets is itself countable. (_)(_)
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