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 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 minds 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 +4 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.

Math vs engineering

I am currently a full time software engineer. I don’t really like the work and I mostly find it draining though I guess I’m not bad at it, though I’m definitely not great. Much of it is process and understanding of requirements and the specific codebase (that includes the tools it uses), which is more often than not not fun at all though I find it more tolerable now. It pays well but is low status, as Michael O Church loves to say. The work is rather lowbrow by STEM standards. I was thinking that it loads not very highly on g (at least line of business engineering) but rather highly on conscientiousness and ability to grind. The people who excel are at it are those who can do that type of work for long hours and not feel tired, and often ones who have the genes to sleep 5 hours a day and still be fine. It’s not a very attractive or sexy ability, but it is a very useful and respectable one. One of my colleagues spent 4 years working on FPGAs just to design one chip and he said after that experience, he’s not gonna do anything related to chip design again. I know that chip design is much more technically involved, much higher barrier to entry, and is actually the hardest to replicate part of computing. Anybody can build a website but only a few places have the expertise and infrastructure to make a good CPU. The latter requires a sophisticated industrial process, the fabrication part, which involves much advanced applied physics, none of which I know. I’ve heard that because fabs are a physical constraint which run in cycle, it is imperative to meet deadlines, which means you need the types who can pull all-nighters, who can toil day in day out in the lab on very detail oriented work (that’s often grindy, not artsy or beautiful like math is) with little room for error. It also pays less than software engineering, for obvious economic reasons. On this note, I recall adults knowledgeable were telling me not to major in EE because there are few jobs in it now. Electronics is design once mass produce. So many of them have been outsourced.

Engineering is hard hard work. Not intellectually hard (though there is that aspect of it too in some of it), but grindily hard. Plumbing is inevitable, and you have to deal with some dirty complexity. You need a very high level of stamina and of some form of pain tolerance that I don’t regard myself as very high in, though I’ve improved substantially. It’s not a coincidence that engineering is what makes the big bucks, for individuals (somewhat) and for economies (or execs in them). Rich countries are the ones who can sell high end engineering products like cars and CPUs.

Mathematics, theoretical science, on the other hand, is much more about abstraction of the form that requires a higher level of consciousness. Math and theoretical physics are far more g-loaded than engineering is and attracts smarter people, a different breed of personality, those with a more intellectual upper class vibe that I see largely absent in software engineering. These are used in engineering, but in it, they are merely tools with the focus being on design and on practical application, with cost as a major consideration. It is like how in physics, there is much mathematics used, but because math is just a tool for it, physicists can be sloppy with their math. Pure theoretical science is much more deep and far less collective and organizationally complex, with a pronounced culture of reverence for individual genius and brilliance. There is also an emphasis on beauty and on some pure elevation of the human spirit in this type of pure thought.

I myself am by nature much more in the theoretical category though I am for now in the practical one, pressured into it by economic circumstances, which I am looking to leave. I will say though that I have derived some satisfaction and confidence from having some practical skills and from having done some things which others find directly useful, as well as having endured some pain so I know what that feels like. In the unlikely case that I actually make it as a mathematician, I can say that unlike most of my colleagues I didn’t spend my entire life in the ivory tower and actually suffered a bit in the real world. I can thereby be more down-to-earth, as opposed to the intellectual snob that I am. I will say though that I do genuinely respect those who are stimulated by engineering enough to do it 24-7 even in their spare time. I don’t think I will ever be able to experience that by my very makeup. However, I do at least suspect that I am capable of experiencing to some extent a higher world that most of those guys fail to, which should bring me some consolation.

STEM and pseudo-STEM

I will say pretty bluntly that I am the type of person who reveres most genius, theoretical genius in the brainiest disciplines, like math and theoretical physics and the likes. They are much more worth idolizing than people who obtain great success in other ways, such as through entrepreneurship. After all, mathematics represents in some sense the pinnacle of human civilization, the peak of human intelligence. Every civilization worth talking about developed seemingly effortlessly crafts, music, literature, engineering (of the non-modern kind), but it took so long for us humans to discover those fundamental theoretical truths. The pioneering of the axiomatic method, which planted the seed for modern science, of the Greeks, is in many ways as significant and as epoch-making in the long run of history as the controlled use of fire or invention of written language. This was something other great civilizations did not develop. The Chinese, for instance, as remarkable as they were in the practical arena, did not develop mathematics of that nature, and scholars in the area are more or less in consensus that such was why modern science could not spring in China as it had done in the West. It is worth noting that Euclid’s Elements was translated to Chinese jointly by Jesuit Matteo Ricci and Chinese scholar Xu Guangqi in the early 17th century, but it did not have the impact on Chinese thinkers and scholars that it should have had.

In virtually every nation, students who study math and theoretical physics are commonly seen as the smartest. Those are the fields that are widely seen as only for the geniuses. They are abstract in a way that many if not most people who excel in more practical, concrete disciplines, such as engineering, cannot handle. For instance, I’ve seen many computer science students struggle with the delta epsilon definition of limit, despite trying very hard and having had it explained to them by people who understand the subject matter well. From this, one can only hypothesize the high cognitive threshold associated. There must be something about their brain structure that renders it impossible or at least very difficult for those people to form the mental process for accurately understanding that abstract definition.

There is hard science, where there are 100% objectively correct answers, and there are softer sciences, where there is a lot of bull shit and much subjective judgement involved, and lots of people things and politics involved. I’d put computer science unambiguously in the latter category, especially the software engineering side of it. Computer science, as far as I see it, is a very marketing driven field. It is not a hard engineering. It is not making a chemical plant or sending a satellite into space where there is essentially no human component involved. Needless to say, software is much easier to get right than hardware. Just about any country can make a decent search engine (if they buy the hardware), but very few nations can make a decent CPU. In this respect, America is far far ahead, with Intel, AMD, Nvidia. Making a CPU requires learning not only of the VLSI but also mastery of the fabrication process, which has some pretty cutting edge applied physics. (Okay, I know nothing about that, just saying what seems to be true to me.)

It is rather odd that people, or the mass media, associate “innovation” and “technology” so much with these soft engineering companies, those who make products for regular end users. Google, Microsoft, Facebook. I recall this guy with a PhD in solid mechanics, who later did compiler development’s saying that EE is so much harder than software engineering but pays less for economic reasons. In contrast, Facebook is just a website, but it makes so much money! It is quite obvious (or at least it should be) that most of the most cutting edge technology is firstly developed for or by the military, and military use generally precedes civilian use.

It does appear though that nowadays the smartest people are staying away from the hard technology and science because there’s no money in that. They’d rather do some bull shit work at a SV firm or work in finance because it pays. In hard STEM, there are so many high IQ immigrants from China, Eastern Europe, India, driving down wages. Some of those people are so brilliant and intellectually powerful and know it all that it’s hard to imagine competing with them. This I find to be quite a pity, because in the ideal society the smartest people should be working on the hardest problems.

People who are into real STEM are a very small minority. They ignore those in SV tech who don’t know a thing about real math and science for instance. Yes, there are people in software engineering or computer science who don’t know what an eigenvalue is or what divergence or curl are. Many of the people in that field are more interested in the latest app and the latest IPO than in actual science and technology. This is especially so in the US, where the math and science education is quite dismal, and where the society is very money driven. It didn’t take me that long to realize that the undergraduate requirements are quite a joke. Students, who come in with minimal knowledge of math, physics, chemistry, take general courses for two years and major courses for the final two years. In contrast, education in Eastern Europe is 5 years and the students there take general STEM their first few years and specialized courses after that, and when they graduate they’re already at a very high level. Of course, at the top US undergraduate programs, the students are much better, and some extremely good, but even there, many of them are ill-prepared.

I really dislike marketing and faking it, though I see the need for it. There are some things that are simply impossible to fake, and the more you try to do so, the more pathetic you will seem. Nassim Taleb once said that it’s easier to buy and sell than to fry an egg. Steve Hsu has also said, in response to a comment on charity and service to the community with regard to college admissions, that one can fake that by volunteering in soup kitchens but on the other hand, one can’t fake the SAT or fake math contests. Richard Feynman once said that nature cannot be fooled. I myself look down on those who insist on denying what is objectively true. When one does that, it’s impossible to argue with him, and one should just let history prove him wrong and make a farce out of him. In Chinese, one can say that 事实胜于雄辩, which means that [objective] reality triumphs over oratory.

Since truth cannot be indefinitely hidden, let us strive for a culture of honesty and openness. Let us create a society where such is the norm, where one can speak the truth without fear of repercussions. Only then will we have genuine free speech, not the pseudo one granted to us by the constitution.