Innate mathematical ability

This morning I had the great pleasure of reading an article on LessWrong on innate ability by Jonah Sinick. Jonah has been one of my greatest influences and inspirations, having interacted with him substantially. He is unusual in one of the best ways possible. I would not be surprised if he goes on to do something extraordinary.

When I catch up with Jonah, I like to talk with him about math, mathematicians, and IQ, which happens to be what that article of his on LessWrong is about. 😉 That article resonates with me deeply because I myself had similar experiences as he did. It is hypothesized by me that I was also twice exceptional, albeit in different ways, with its effects compounded by my unusual background, all of which mediocrities within the American public school system are not good at dealing with in an effectual way.

This writing of Jonah has brought forth reflections in my own mind with regard to mathematical ability, development, and style. I’ll say that as a little kid under 6, I was very good at arithmetic and even engaged in it obsessively. However, by age 8, after two years of adjusting to life in America starting off not knowing a word of English, I had forgotten most of that. I was known to be good at math among the normal normal students; of course, that doesn’t mean much. In grade school, I was not terribly interested in math or anything academic; I was more interested in playing and watching sports, particularly basketball and baseball.

I didn’t have any mathematical enrichment outside of school other than this silly after school math olympiad program. Nonetheless, I managed to test into two year accelerated math once I reached junior high, not that it means anything. In junior high, we were doing this stupid “core math” with graphing calculators and “experiments.” I didn’t realize that I was actually a joke at math until I failed miserably at the state mathcounts contest, having not prepared for it, unlike all those other tiger mommed Asian kids, who to me seemed way beyond me at that time. It only occurred to me that I might have some real talent for math when I made the AIME in 10th grade, taking the AMCs for the first time, being one of four in my high school of about 2000 to do so. I thought it was fun solving some of those math contest problems, which were more g-loaded, with an emphasis on the pattern recognition side.

It was after that I started to read up on the history of mathematics and mathematicians. I taught myself some calculus and was fascinated by it, not that I understood it very well. But I could easily sense that this was much more significant than many of those contrived contest problems, and soon, I began to lose interest in the contest stuff. It was also after that that I learned about proving things, which the American public school math doesn’t teach. I finally realized what mathematics is really about.

Like Jonah, I had some difficulties with careless errors and mental organization. I don’t think my raw intellectual horsepower was very high back in high school, but fortunately, it has improved substantially since then that it is for the most part no longer the major impediment.

I took calculus officially in 11th grade, and it was a breeze for me. I could easily compute the areas and volumes and such but the entire time, I felt quite dissatisfied, because I could not actually understand that stuff at a rigorous, theoretical level as I poured through our textbook that went up to vector calculus during lecture, which was rather inane, expected if one considers the mismatch between cognitive threshold relative to the distribution of ability of the students. I knew from reading online the rich world of math far beyond what we were covering, most of which I was not intellectually mature enough to access at that time. However, I vividly remember during summer after 11th grade, while attending a math summer program, I was able to comfortably write out the delta epsilon definition of limit with understanding of why it was reasonably defined that way. Still, I would say I was still quite weak in terms of both my mathematical maturity and overall intellectual ability. There were too many things I wasn’t aware of, including the g factor, that I easily would have been had I been higher in verbal ability, which would have enabled me to read, absorb, and internalize information much more rapidly and broadly. In contrast, Jonah had discovered independently, or so he says, the lack of free will at the age of 7!

I made some incremental advances in my math knowledge from reading and thinking outside of school the next year. As for contest math, I almost made the USAMO. Though I had improved, I was still not terribly quick and careful with solving contest style problems and doing computations. I think close to graduation, I also solved some Putnam problems.

Only in undergrad did I learn real math more seriously, but even there, nothing too advanced. US undergrad is a joke, and I also was one, just to a lesser extent than most of my “peers.” Almost certainly, Jonah, based on he’s told me, had gained much deeper and broader knowledge at the same stage, from the reading works of giants like Euler and Riemann.

I’ve noticed how there are a lot of Chinese-(American) kids really into those high school math contests, and they now also dominate USAMO and Putnam (though careful, as in the latter, there you’ve got some of Chinese internationals drawn from the elite from China). I will say that at the lower levels, many of those kids have some pretty low taste and an inability to think outside the system that would enable them to discover the existence of real math, as opposed to this artificial math game that they enjoy playing or are pressured to doing so for college. Though those contests have a high pattern recognition component to them, there is not really much depth or substantial math knowledge. It is also my belief, with reference to Jonah’s article, that math contests are mostly M loaded while real math is more V loaded. So this behavior is consistent with the lopsidedness in favor of M and perhaps also short term working memory of Chinese students. It has also been Jonah’s belief that controlling for g, these contests select for low taste and value judgement, and I surely identify with that perspective. So maybe college admissions are somewhat fair to assess an Asian penalty?

Of the thesis of Jonah’s article, a representative figure is Terry Tao. There, Jonah also pointed out that Tao’s research in math is more concrete and problem solving oriented by pure math standards, in line with what appears to be the same lopsided (modulo the absolute level, as Terry is a far far outlier) cognitive profile of his based on testing at age 9 and 10. Again, people enjoy what they are best at, and though, Terry Tao is almost certainly at least +3 sigma at verbal, he is far more rare, at least +5 sigma, a real übermensch, in the (in some sense dual) pattern recognition component, which means he leans towards the areas of math more loaded on the latter. I have heard the saying that even other Fields medalists are intimidated by Terry Tao. The breadth and volume and technical power of his work is almost unrivaled and otherworldly. The media makes it seem like Terry is a league above even the other Fields medalists. However, Jonah seems to believe that the deepest and most leading of mathematicians are the ones who are more theory builders, who create through leaps of insight and synthesis new fields and directions that keep mathematicians busy for decades, and even centuries. That would be say Grothendieck or SS Chern, and an ability that is more loaded on verbal ability, crudely speaking. Again, I have felt the same. This might explain why the advantage of Chinese students is not anywhere near as pronounced in math research as in contests, and why some people say that generally speaking, the Chinese mathematicians are more problem solving and technical than theoretical, more analysis than algebra. Likewise, we can predict the opposite for Jews who are skewed in favor of verbal. A corollary of this would be that the Jews produce the deepest thinkers, adjusted somewhat for population, which is almost certainly the case, if you look at the giants of mathematics and theoretical physics.

I’ll conclude with the following remark. I used to revere somewhat those who placed very highly on those contests, until I realized that many of them are actually somewhat weak in terms of deep understanding and thinking at a more theoretical level. Yes, I have met MOSPers who got destroyed by real math and who are not very intellectually versatile, with glaring weaknesses; I was quite surprised initially that even I seemed to be smarter if not a lot than some of them. Once upon a time, I couldn’t understand those who appeared very strong at real math (and often also science and/or engineering and/or humanities) who struggled with more concrete math and/or contest-style problem solving, like Jonah, who has written on LessWrong of his difficulties with accuracy on the trivial math SAT. I’ve met this other guy, who I thought was an idiot for being unable to perform simple computations, who is leagues beyond me in the most abstract of math, who writes prolifically about partially V-loaded areas of math like model theory. Now, the more metacognitive me has awakened to the reality that I may never by deficit of my neurobiology be able to fathom and experience what they’re capable of. After all, there are plenty I am almost certain are and are essentially doomed to be very delusional by nature relative to me, and since I’m at the far tail but not quite so much, there are bound to be people who view me the same. I can only hope that I can become more like them through some combination of exposure and organic neurobiological growth, but I as a realist will not deem that very likely.


More on Asian stereotypes

I just stumbled upon this wonderful essay by Gwydion Madawc Williams on why the Ming voyages led by Zheng He (郑和) led to nothing. The quote of it particularly memorable to me was this:

The separation of craft and education as represented by China’s illiterate shipwrights was indeed a genuine weakness in the Chinese system.  Christian Europe always remembered that St Peter had been a fisherman and St Paul a tent-maker, and it was quite acceptable for learned people to also be involved in manufacturing.  The weakness of Confucianism was not so much that it rated agriculture and craft above merchant trade, but that it insisted on the educated being a learned caste distanced from all of these matters.

Again, it’s the Asian stereotype of being a study hard grind lacking in practical, hands-on skills and “well-roundedness” and “social skills” and all that that admissions officers use to justify denying Asian applicants. I’ll say that from what I know, that is still very limited, Confucianism was very much like that. The quote that epitomized this was: 劳心者治人,劳力者治于人, which translates to roughly “the worker of the mind governs, the physical worker is governed.” The whole imperial examination system essentially created an upper class of bookworms for whom any form of hands on labor was beneath. To be a true 君子, gentlemen, you were supposed to study the classics and write poetry and engage in all that Confucian bull shit. I myself don’t have a very high opinion of Confucianism. It’s too conservative for me, with all the emphasis on ritual and filial piety. It discouraged any form of innovation outside the system, outside what was already there, which is partly why China could not make the giant leaps in science that the West did. I’ve read some of the Analects of Confucius and know some of the quotes, and I don’t think Confucius was a deep philosopher at all; there is little actual substance in what he said. On the other hand, Mo Tzu was a much further reaching, more scientific, and surprisingly modern thinker, and had China followed his path instead of banishing his school of thought into obscurity, the world would be completely different now, with China likely having made many more leaps of progress than it had actually done. I’ll say that the West was able to escape the shackles of Christianity, but China could not by itself escape those of Confucianism, until its dire situation, with reached its nadir in 1900, forced it too.

Apparently, the elite college admissions officers aren’t terribly good at filtering out the real Asian grinds either, as I know one who went to Princeton, who I found ridiculous. He said that all he did in college was study, and even though he majored in math, he hardly knew any. Like, he didn’t know what a topological space is. When I went ice skating with him and some others, he was near the edge the whole time, and he characterized my skating backwards (not well at all) as “scary.” I told him I’m not very athletic and wasn’t even any good, unlike the girl he was dating at that time, who could do spins among other fancy “figure skating” things she was trying out. I did show him the video taken of this 360 somersault I did off a 15 feet cliff in Hawaii, into the water, which was the first time I had done anything like that. He was like: “that’s so scary.” I honestly didn’t know what to say. To justify himself, he was like: “Chinese parents only want their kids to study.” I told him that in China, there are some very athletic people who attend special sports schools. On that, he was like: “but those aren’t normal people.” I also remember when we went camping once, everybody else got drunk, so I got to drive that kid’s BMW back. He had told us that his father does business in Beijing, which might explain why he drives that kind of car. He came to US at age 4. His Chinese is absolutely awful though, and he doesn’t realize it. He will of course say: “I already know enough. Some people can’t even speak it.” 怎么说那,不仅是个书呆子,而且是个书都读不好的书呆子,连这样的sb还都被Princeton录取了。I’ve talked with one of my very smart Asian friends about this, and he was like: “but he’s socially normal, unlike us.” And more recently: “Maybe they do accept Asian grinds, just not the ones with bad social skills.”

From what I’ve seen, there are plenty of super conformist Asian grinds like him, but there are also many who aren’t, who are actually smart and interesting, like myself (or at least I hope). I think what he said about Chinese parents is somewhat true actually; after all, I saw many growing up. They do see academics as a way to get ahead more so than others, largely because in China, to get out of your rural village and/or not be stuck with a working class job, you had to do sufficiently well on the gaokao to get into a good major at a good university. It’s funny that I’ve actually seen a ton of ignorant, narrow-minded, and risk-averse uncool tiger Chinese parents. And I have also seen some extremely impressive ones, not just academically. There is again quite a wide range and variety.

There is a phenomenon I’ve witnessed, which is that if a person is extremely strong at X and merely above average at Y, then that person will seem weak at Y, even compared to another person about as good at Y but less lopsided. It seems a natural human cognitive bias to think this way. This is in fact applied rather perversely to Asians in stereotyping. For example, Asian students are perceived as weak at language and humanities because they are generally stronger at STEM. We all know that in fact math IQ and verbal IQ (which we can use crudely as proxies for STEM ability and humanities ability respectively) are highly correlated, which makes it highly unlikely that a STEM star is actually legitimately weak at humanities. He might not be interested in reading novels and such but that’s rather different. There is also that humanities is more cultural exposure loaded with a much higher subjective element to it, with much less of a uniform metric. It actually seems to me based on personal experience that is by no means representative that in terms of precise use of language and the learning of foreign languages, mathematicians and theoretical physicists are at or near the top in terms of ability. On this, I will give an opposing perspective that I identify with somewhat, which is that even if you’re very strong at Y, having an X that you are significantly more talented at is a weakness for Y, because engaging in Y deprives the joy derived from engaging in the X, which often leads to loss of interest over time. Maybe this is why employers shy from hiring people who they deem “overqualified?” On this, I have thought of how possibly the lopsided cognitive profile in East Asians (with what is likely at least 2/3 SD differential between math/visuo-spatial and verbal, normalizing on white European scores) predisposed the thinking of the elite (assuming that lopsidedness is preserved at the far tail) as well as the development of that society at large in certain ways, some of which may have been not the most conducive for, say, the development of theoretical science. This is of course very speculative, and I would actually hypothesize that the far tail cognitive elite among East Asians is more balanced in terms of the math/visuo-spatial and verbal split, given the great extent to which the imperial examination system, which tested almost exclusively literary things, selected for V at the tail instead of for M.

On the aforementioned bias, I’ll give another illustrative example. I once said to this friend of mine, a math PhD student, not Asian, how there’s the impression that people who are weaker academically tend to be better at certain practical things, like starting restaurants and businesses. We sure all know there are plenty who weren’t good at school but were very shrewd and successful at business, at practical things. That guy responded with reference to Berkson’s paradox. He said something like: “That’s because you are unlikely to see those who are bad at both. They tend to be in prison or in the lower classes.” I could only agree.

I’ll conclude with another more dramatic example. I used to, when I knew nothing about the subject, think that people who were really at math were weirdos and socially awkward. For one, there was this kid in my high school who was way better than me at math at the time, who was incredibly autistic. Also, summer after 10th grade, I saw Beautiful Mind, which depicts the mathematician as mentally crazy. Now I would bet the incidence of schizophrenia among the mathematically gifted is lower than it is in the whole population. It just happens that certain combinations of extreme traits are vastly more noticeable or exposed by the media to the public (a mathematician or physicist may think of this as weighing those with such combinations with a delta function, or something along that direction at least). I wasn’t quite aware of that at that time though. Only later, after meeting more math people did I realize that math people are not actually that socially out of it in general, far from it, at least once they’re past a certain age, by which they will have had the chance to interact with more people like them and form their own peer group.

It is my hope that people can be more cognizant of these biases described in this blog post.


Brainwashing in America

A friend of mine (white American, not Jewish) sent me this photo:


I never would have expected anything like this. A compilation of the rulers of the major media outlets in America, with a classification of their background, with Jewish as one of them. We all know that Jews in many racial classifications are not in their separate group, as they are considered white.

It reminds me of a guy who worked for Xinhua News Agency who switched to a technical field when he came to America, in particular a remark of him. Remember back a few years, the whole Jimmy Kimmel kill all Chinese episode? On that, he commented: in America, a TV host says “kill all Chinese,” and it’s like a joke. Just imagine if someone said that about the Jews. Of course, that would never happen because Jews control the media outlets in America. Now this is something that many people know is 100% true, but which our suffocating atmosphere of PC prevents us from talking about.

Now my question is how Jews managed to politic their way to such a position. Yes, we all know that they are very verbally and culturally gifted (they dominate film as well), but that can’t explain it all. There must be other major factors in play here.

Green in that diagram is for “goy”. With the murky resolution, I initially saw that as “gay”. Now that’s another minority group known to have been shunned, persecuted, historically that produces many a creative genius. Kolmogorov? Alan Turing? Robert MacPherson? But no, it refers to the complement of Semite minus non-white in this diagram, in other words, the white and non-Jewish. Check out this.

What I am bringing up here could not be more consequential. What people do is a function of what resides in one’s mind, which is influenced or even controlled in some cases by the cultural environment, by the information one accesses. Now, most people aren’t like me. Most people don’t have the awareness of their possibly being brainwashed that I do that leads them to learn foreign languages and contact alternative perspectives on matters. Most people don’t have the rigorous, critical mindset that brings them to question conventional modes of thought and taboos imposed by social and cultural norms. Most people don’t realize that information is often fed out of narrow self-interest of those who control it so that the people they rule can be controlled, so that they can worry less about the risk of being overthrown.

The US brought down the Soviet Union with ideas. It could not through military means. With economics, it is also tough. But it could by penetrating and influencing the elites and the masses with dangerous liberal ideology, or whatever one phrases it as, so that idiots like Gorbachev and Yeltsin may come to power there. That’s how powerful propaganda is. War is largely cultural. You win by winning someone’s heart.

Now, I’ve seen a girl born in Russia but raised in the US who actually believes that it was America who defeated Nazi Germany. That’s how successful America has been at brainwashing. As brilliant as Russians are technically, a bulk of the best of them are now in America working for the profit of American capitalists who are fundamentally at odds with them. This just goes to show how important cultural preservation and political organization are.

Kong Qingdong has noted that the West still controls international debate, and on that, perhaps the GFC is a form of self-protectionism? 扬长避短 (play up one’s strengths and down one’s weaknesses), as the Chinese would say. The Chinese communists did that in combat to great success. In China, I have seen many emphasize the importance of winning China’s international voice, as Chinese culture is still grossly misunderstood and misrepresented outside China, to the extent that even most Chinese kids reared in America learn a pseudo version of it. It should be obvious that there is another world on the other side of the curtain that one needs to learn a notoriously difficult foreign language in order to access. The gap here is no longer physical, with the internet. It is mental.

I’ll conclude by encouraging my readers to be more reflective and cognizant of the grasp one’s cultural environment has on oneself. You are not liberated until you tear its fetters away!



想起,我有一次观大美国数学家Robert MacPherson的采访,在此,他说在五十年代末六十年代初时,美国主流的教育哲学认为学(比如微积分)过早不良由此必导后正式学闲无聊,及他父母也如此认为。当然,他将此形容为”terrible educational theory”。其与苏联的恰恰相反,在苏联他们是鼓励在各种方面有超常天分的人从小纯粹追求卓越的发挥发展自己,并且政府公家会给这些人提供更多的教育及培养资源。想更细的了解他们怎么做的,可以看看Masha Gessen写的Perfect Rigor,是讲解决庞家来猜想的怪才Perelman,里面可预料有描述当时苏联高中数学尖子之圈的组织及文化。回到美国,真难以想象美国在冷战时期尽然有如此之愚昧的无科学根据的教育思想嵌入主流。

我自己是承受了十二年美国的垃圾公立教育,在一个普通的美国学校,与一些普通的美国孩子。老师实在太差,学生也笨得要命,我也是,只不过在某些地方没有像其他人那么差。美国老愿意指责某些别的国家没有言论自由,竟是政府的propaganda,但是在我眼里在教育洗脑,很难比美国做得更严重,美国教政治教历史是极端的不客观,忽略一切不符合美国统治阶级意识形态的,是建立在无知的基础之上。老师将荒唐讹谬熏陶在无辜单纯的孩子上,所以美国人长大了好多却认为希特勒是美国打败的。在数理化,美国学校不要求学生做证明题,所以美国人的严谨思维能力那么差,我都看到在美国技术公司当官的美国人愚昧到不记圆的面积公式,何况对诸如此类有任何理解。好多老师讲的这些我还都相信了,比如我的四年级老师说三角形内角度之和为一百八十,因为数学家每次以量角器测量所画的“稍有特殊的”三角依然会得一百八十。九年级,我们在历史和英语课,读的都是像Ayn Rand和Solzhenitsyn这样的人的作品,有明显的政治偏见。在这种环境之下,基本百分之百的中国孩子会对中国文化有极其稀奇古怪的眼观。孔庆东说过,现在主要战场是文化战场而在此,中国一大片土地都已经被占领了!因为我这个人智商比较高,不太受制于环境及社会压力的影响,所以我慢慢悟觉到,通过阅读网上的许多中文资料及客观的非畅销的英文资料,才晓得美国是如此之大的一个骗局。我这种人是不那么容易上当的,可惜,如我一般的人过少。我想强调,中国有墙,是,美国也有,它的墙是教育及文化自然导致的对外的不接受加上复杂的政治正确文化,达到人曰男女在理工科能力不平等都会遭到冲击,看看Larry Summers就知道了。对于某些体制,美国主流媒体也是扯淡回避,所以他们对中国的预测基本上都是错的。





A possible switch in focus from math to natural science

I find myself becoming more keen on natural reality over the last week or so, though my time has still been mostly concentrated on mathematics. It is possible that I am actually more suited to natural science than to mathematics, who knows. To estimate the expected extent of that finer, I’m going to go learn some natural science, like what else would I do.

I want to first talk about my experience with science in college, high school, middle school, and perhaps even earlier. In elementary school, in sixth grade, we had this science fair. My partner and I chose to do ours on wastewater treatment plants. There were some people who did solar power and even one who did population growth the previous year (social science is science I guess). I learned absolutely nothing from that; it’s like, how many kids at that age can actually learn science that isn’t bull shit?

In 7th grade, we had for our science course life science. It was mostly taking notes on various types of life, from fish to reptiles to plants. That’s when I first learned of the Linnaeus classification system. We didn’t do experiments really. Tests were mostly regurgitation of notes. There was nothing quantitative.

In 8th grade, it was earth science. The teacher was so dumb that in math class, this kid was like: how can you like Mrs.    ? She’s as dumb as a rock! To that, the math teacher, who later realized by me was a complete moron who didn’t even know what math was, was not terribly accepting, I’ll put it that way. We studied volcanos and earthquakes, watched documentaries on those types of things, and played around a bit with Bunsen burners and random equipment typical in chemistry laboratory, the names of which I know not. For names, I guess use this as a reference? I didn’t like that class and didn’t do well in it at all. My ADHD or what not was particularly severe in it.

In 9th grade, it was “physical science.” We did some problems in Newtonian mechanics, very simple ones, that’s when I first learned of Newtons, Joules, work, energy, those types of things. I actually found that pretty interesting. There was this project for making an elastic powered, or rubber band powered car to be more explicit. Really, there wasn’t much point in that other than as a way to pass time for kids. Having worked as a software engineer, I can guess that there are very systematic ways of designing and building that stuff. Of course, us kids just tinkered around in a way wherein we didn’t know at all what we were doing. I do remember there was a time when we were playing with this thing, called a crucible I think, that we were not supposed to touch, as doing so would smear black onto our hands, which I nonetheless still did, receiving, consequently, reprimand from the teacher.

In 10th grade, it was chemistry and then biology for the second half of the year. This was now at my grades 10-12 high school. The class was rumored to be impossibly hard, and the teacher was said to be a very demanding guy. There was, unlike the year previous, basically zero hands on. The chemistry part was very quantitative, I remember stoichiometry was a big part. There was nothing really hard about it; the students were simply too dumb to even perform very mechanical calculations. Kids would say: “it’s a lot of math.” At that time, I didn’t know the difference between math and science, and the other kids knew even less. Math is founded on the axiomatic system pioneered by the Greeks, about proving things in an a priori way, while science is about modeling natural phenomena and testing those models. Math in science is just a tool and not the focus. I recall we started off learning about uncertainties in measurement. There’s really nothing especially hard about that stuff, with a very systematic way of going about it, but the atmosphere and the way it was lectured about made it seem like such a grand thing to us. The second half the year as I said was biology. I wasn’t terribly engaged in that. I didn’t like the memorization involved. I liked math more. I made the AIME that year, taking the AMCs for the first time, and was one of four kids out of almost 2000 in our high school to do so, so that brought me to conclude that maybe I actually had some talent for math and science. I knew that physics was the hardest and brainiest of the sciences, with all that fancy math and Einstein, so I was rather keen to learn that. I checked out some physics books from the library I think, and the first thing I learned about was if I remember correctly centripetal acceleration, which confused me quite a lot at that time.

11th and 12th grade was physics with the same teacher. The class was rather dumbed down; it had to, especially on the math end, problem solving wise, since this is an American high school after all. There was quite a focus on phenomena, as opposed to formalism. I didn’t really like that much. I was more comfortable with formalism, with math being my relative forte at that time. We did some experiments, but I wasn’t good at them at all. I remember on the first day, when looking at some uniformly dense rod vertically situated, it occurred neither to me nor my partner to record its position at its center of mass. I didn’t really understand what I was doing throughout the whole time. The other kids, most of them, were worse. There were some who were confused about the difference between energy and power, the latter of which is the derivative of the former of course, after two years of it! I remember the whole time many kids would go: wow! physics! That kind of perspective, later understood by me, makes it almost impossible for one to really learn it. With just about everything, there is a right way of going about it. Discover it (mentally, with the aid of books, lectures, various resources) and you’ll do great. Be in awe of it, and you’ll never get it. The former is in line with the philosophy that you should focus solely on what is true, objectively, and not imagine anything that doesn’t aid in your convergence to the truth, and reminds me of the quote of Einstein that one should make something as simple as it can be but no simpler. Simplicity is gold in science and just about everything. Ability to recognize the redundant and superfluous and to generalize is the essence of intellectual ability, or to put it in more extreme terms, genius. The culture in American high school is the antithesis of that. Kids are always talking about how hard things, especially math loaded subjects, are, when they’re making it hard for themselves by imagining in their minds what is complete bogus from a scientific point of view. To digress, this holds as well as for subjects like history. Focus only and solely on the what are the facts and the truth they bare out. Don’t let political biases and personal wants and wishes interfere in any way. This is to my remembrance advice intellectual Bertrand Russell gave to posterity nearing his death. American history classes are particular awful at this. American teaching of history is very much founded on ignorance and American exceptionalism and a misportrayal of cultures or political systems it, or more like, its blood sucking elite, regards as evil for the simple reason that they are seen as threatening towards their interests. Math and science under the American public school system was pretty dismal. History (or social studies, as they call it) was perhaps more so, in a way more laughable and contemptible.

I hardly took science in college, being a math and CS major. I did take two quarters of physics and it was awful. Talking with some actual physics PhD students and physics PhDs gave me a more accurate idea of what physics really was, though I was still pretty clueless. It was evident to me at that time that physics, and probably also chemistry, was far more demanding in terms of cognitive ability as many of the CS majors, who could write code not badly, struggled with even very simple physics. Being in college, I had a closer look at the world of real science, of scientists, in America, which is very foreign. It dawned on me that science, as exciting as it sounds, is in America done mostly by underpaid ubermensch immigrant men, who are of a completely different breed both intellectually and culturally from most of the people I had encountered at that time. Yes, by then I had found my way to this essay by Greenspun. I’ll leave its interpretation up to the reader. 😉

You can probably guess that I think American science education is a complete joke, which is the truth. I felt like I only began to really learn things once I got out of the American school system, although for sure, the transition between high school and college in terms of content and depth and rapidity of learning was quite substantial. However, the transition from undergrad to out into the bigger world, where I could consider myself psychologically as more in the ranks of everyone, regardless of age or national origin, than in the ranks of clueless American undergrads at a mediocre program, was probably just as substantial in the same respect, albeit in a very different way.

Now let’s context switch to some actual science (that’s not pure math or artificial in any way).


A capacitor is made by taking some negative charge off a positive plate and transferring it to the negative plate. This obviously requires work. If the final voltage is \Delta V, then the average during the charging process is half of that. With the change in potential energy \Delta U_E as change times voltage (remember, voltage is potential energy per unit charge), we can write \Delta U_E = \frac{1}{2}Q\Delta V.

How to maintain charge separation? Insert an insulator (or dielectric) between the plates. Curiously, a dielectric always increases the capacitance (Q / \Delta V) of a capacitor. Its existence, via the charge on the plates, makes for a electrically polarized medium, which induces an electric field in the reverse direction that is in addition to the one induced by the capacitors alone. As you see, the negative charges in the dielectric lean towards the positive plate and same holds if you permute negative and positive. So if the plates, by themselves give rise to \mathbf{E}, the addition of the dielectric gives rise to some \mathbf{E_i} in the opposite direction. Call \kappa the coefficient of the reduction in the magnitude of the electric field with

E_{\mathrm{with\;dielectric}} = E_{\mathrm{without\;dielectric}} - E_i = \frac{E} {\kappa}.

We put that coefficient in the denominator so that

C_{\mathrm{with\;dielectric}} = \kappa C_{\mathrm{without\;dielectric}}.

To be more explicit, so that \kappa is proportional capacitance wise, which is reasonable since capacitance is what is more central to the current context. This kappa value is called dielectric constant, varying from material to material, under the constraint that it is always greater than 1.

Now one might ask if the capacitor is charging when the dielectric is inserted. If it isn’t, the voltage across will experience a sudden decrease, with the charge stored constant, and if it is, voltage will experience the same, but the charge on the plates will keep going up, as the voltage will too at a rate proportional to that of the increase of the charge, with the constant of proportionality the increased C. Needless to say, on taking derivative, a linear relation is preserved with the same coefficient of linearity.

The presence of a dielectric presents a potential problem, namely that if the voltage is too high, the electrons in the dielectric material can be ripped out of their atoms and propelled towards the positive plate. Obviously, this discharges the capacitor, as negative and positive meet to neutralize. It is said that this typically burns a hole through the dielectric. This phenomenon is called dielectric breakdown.