## 不知为何，突然想起测度论里的不可测度的维塔利集合

$\mu^*(E) = \inf\{\displaystyle\sum_{k=1}^{\infty} I_k : (I_k)_{n \in \mathbb{N}} \text{as open intervals}, \displaystyle\bigcup_{k=1}^{\infty} I_k \supset E\}$

$\displaystyle \mu ^{*}(A)=\mu ^{*}(A\cap E)+\mu ^{*}(A\cap E^{c})$

• 测度平移守恒，那就是 $\mu(S) = \mu(x+S), \forall x \in \mathbb{R}$
• $\{A_k\}_{k \in \mathbb{N}}$ 互相不交则 $\mu(\bigcup_{k=1}^{\infty} A_k) = \sum_{k=1}^{\infty} \mu(A_k)$
• $S \subset T$$\mu(S) \leq \mu(T)$

## A revisit of the drama behind the Poincaré

I recall back in 2008, when I first cared enough to learn about mathematicians, I read a fair bit of the media articles on the proof of the Poincaré conjecture. At that time, I was clueless about math, and these mathematicians seemed to me like these otherworldly geniuses. I do remember thinking once to myself that maybe it would be kind of cool to part of that world. Except at that time, I was way too dumb, and maybe I still am. However, now I actually have some idea of what math research is about, unlike back then, when my conception of math and mathematicians was more of a naive popular one.

Naturally, from that I learned about Shing-Tung Yau. I probably read that Manifold Destiny article by Sylvia Nasar and David Gruber that Yau was furious with, in response to which he hired a lawyer and had a PR site created for him to counter the libel (as perceived by him). That was pretty entertaining to read about.

The more I learned about math, about mathematicians, about how the world works, about the Chinese math establishment, and about Chinese language (which I’m pretty fluent with by now), the more accurately and deeply I could understand and thus appreciate all this. In particular, now that I know a little about Riemann surfaces, I feel closer to that rarefied world. I also read a fair bit in Chinese about that feud between Yau and Tian, which was also quite entertaining. If some of that stuff is actually true, then academia, even in its supposedly purest, hardest, and more meritocratic subject, is kind of fucked up.

Yesterday, I had the pleasure of talking with a Harvard math undergrad who is also an IMO gold medalist. And we both mentioned Yau. 😉

What can I say about all the politics and fight for credit over whole Poincaré conjecture? Surely, it was kind of nasty. It’s fair to say that Yau was pissed (or at least disappointed) that his school (of Chinese mathematicians) lost to this lone Russian Jew. Maybe in some years time, I’ll be able to judge for myself, but for now, it seems like Perelman’s proof was correct from the start and that what Cao and Zhu, along with the other two teams of two did were merely verification and exposition of Perelman’s result. Of course, attributing a proof entirely to an individual is somewhat misleading, because anyone who knows how math works knows that any proof of a big theorem employs sophisticated machinery and theory developed by predecessors. I’ve studied enough math now to recognize to some degree the actual substance, that is, what is genuinely original, versus what is merely derivative. In the case of Perelman, they say he was using the Ricci flow developed by Hamilton. I’ve encountered many times that in learning, it is much harder to learn about a topic I have little exposure to vastly different from anything I’ve seen before than to learn what is structurally similar (albeit different in its presentation and perhaps also level of generality) to something I had thought about deeply myself already, or at least seen.

Aside from the Poincaré, the focus of that New Yorker article, the authors of it also made it seem as if Lian, Liu, and Yau stole Givental’s proof of mirror symmetry as well about a decade earlier. After all, Givental published first. I suspected that might have been the case. The narrative even made it seem somewhat like Givental was this super genius whose arguments were somewhat beyond the comprehension of Lian, Liu, and Yau, who struggled to replicate his work. Maybe because I still see, or at least saw, Jews as deeper and more original than Chinese are. Again, I still know too little, but it does seem like Jews have contributed much more to math at the high end even in recent years, say, the past three decades.

Well, I found a writing on that doctoryau website by Bong Lian and Kefeng Liu documenting the flaws and deficiencies in Givental’s paper. It looked pretty thorough and detailed, with many objections. The most memorable one was

p18: Proposition 7.1. There was just one sentence in the proof. “It can be obtained by a straightforward calculation quite analogous to that in ‘[2]’.” Here ‘[2]’ was a 228-page long paper of Dubrovin.

And I checked that that was indeed true in Givental’s paper. This certainly discredited Givental much in my eyes. It’s like: how the fuck do you prove a proposition by saying it’s a straightforward calculation analogous to one in… a 228-page paper!!!!!!!!

Not just that. There is also

p27: Proposition 9.6. In the middle of its proof, a sentence read “It is a half of the geometrical argument mentioned above.” It’s not clear what this was referring to (above where? which half?)

and

p30: Proposition 9.9. This was about certain uniqueness property of the recursion relations. The proof was half a sentence “Now it is easy to check” But, again since we couldn’t check, it’s hard to tell if it was easy or not

So basically at least three times Givental proves with “it’s trivial,” once based on analogy with a 228-page paper.

There are far from all. There are many more instances of Givental’s arguing what Lian-Liu-Yau could not follow, according to that document, the list in which is also, according to its authors, who advise strongly the reader to “examine Givental’s paper make an informed judgment for himself”, “not meant to be exhaustive.” So they’ve listed 11 gaps in that paper, one of which is glaringly obvious of a rather ridiculous nature even to one who knows not the slightest about mathematics! And they suggest there is more that, to my guess, may be much more minor that they omitted in that document so as to avoid dilution.

I’ve noticed it’s often the Chinese scientists who have a bad reputation for plagiarism, made more believable by the dearth of first-rate science out of Chinese scientists in China, though that seems to be changing lately. On the latter, many Chinese are quite embarrassed about their not having won a homegrown Nobel Prize (until Tu Youyou in 2015 for what seemed to be more of a trial-and-error, as opposed to creative, discovery) or Fields Medal. On the other hand, I’ve also heard some suspicions that it’s the Jews who are nepotistic with regard to tenure decisions and prize lobbying in science, and what Givental did in that paper surely does not reflect well. I used to think that math and theoretical physics, unlike the easier and more collaborative fields in STEM (with many working in a lab or on an engineering project), revere almost exclusively individual genius and brilliance, but it turns out that to succeed nowadays typically involves recommendations from some super famous person, at Connes attests to here (on page 32), not surprising once one considers the sheer scarcity of positions. Now I can better understand why Grothendieck was so turned off by the mathematical community, where according to him, the ethics have “declined to the point that outright theft among colleagues (especially at the expenses of those who are in no position to defend themselves) has nearly become a general rule.” More reason why I still hesitate to go all out on a career in mathematics. It can get pretty nasty for a career with low pay and probability of job security, and I could with my talents make much more impact elsewhere. One could even say that unequivocally, one who can drastically increase the number of quality math research positions (not ridden with too many hours of consuming duties not related to the research) would do more to progress mathematics than any individual genius.

I’ll conclude with some thoughts of mine on this Olympiad math that I’ve lost interest in that many mathematicians express low opinion of, though it clearly has value as a method of talent encouragement and selection at the early stage, with many Fields Medalists having been IMO medalists, usually gold. I recall Yau had criticized the system of Olympiad math in China, where making its version of MOSP gives one a free ticket to Beida and Qinghua, as a consequence of which many parents force or at least pressure their kids into Olympiad math prep courses as early as elementary school. Even there, several of the IMO gold medalists have become distinguished mathematicians. I have in mind Zhiwei Yun, Xinyi Yuan, and Xuhua He, all speakers at this year’s ICM. So the predictive power of IMO holds for the Chinese just as well as for the non-Chinese. I personally believe that Olympiad math is beneficial for technical training, though surely, the actual mathematical content in it is not that inspiring or even ugly to one who knows some real math, though for many gifted high schoolers, it’s probably the most exciting stuff they’ve seen. I do think though that one seriously interested in mathematics would have nothing to lose from ignoring that stuff if one goes about the actual math the right way.

It’s kind of funny. A few days ago when I brought up on a chat group full of MOSP/IMO alumni that now, almost half of the top 100 on the Putnam (HM and higher) are Chinese, one math PhD quite critical of math contests was like: “ST Yau would weep.” Well, I don’t think ST Yau actually regards Olympiad math as a bad thing (half tongue-in-cheek, I even remarked on that chat that doing math contests (as a high schooler) is much better than doing drugs). Many of the Olympiad/Putnam high scorers do quite well, and in some cases spectacularly so, in math research. One point I shall make about them is that they are, unlike research, a 100% fair contest. Moreover, the Putnam, which I placed a modest top 500 on, solving three problems, has problems which do not require specialized technical training as do the inequalities and synthetic geometry problems in Olympiad math that have elegant solutions. On that, I have wondered based on their current dominance of those contests: could it be that at the far tail, the Chinese (who did not actually create the scientific tradition themselves) are actually smarter than the others, including the Jews? Could it be that the Chinese are actually somewhat disadvantaged job placement and recognition wise in math academia out of a relative lack of connections and also cultural bias? What I saw in that sound and unobjectionable rebuttal of Givental’s paper, in contrast to what was presented in the media, only makes this hypothesis more plausible. I am not denying that Givental did not make a critical contribution to the proof of mirror symmetry. That he did, along with some other predecessors, seems to be well acknowledged in the series of papers by Lian-Liu-Yau later that actually gave the first rigorous, complete proof of mirror symmetry. Idea wise, I read that Lian-Liu-Yau did something significant with so called Euler data, and though not qualified to judge myself, I have every reason to believe that to be the case for now.

## Luboš Motl, and some thoughts on monopolies

I had the pleasure of reading some blog posts of Luboš Motl on present day academia. I first learned of him when I was a clueless undergrad. He seemed like this insanely smart theoretical physicist. Of course back then I was dumb and in awe of everything, so what else could I think? I know that he pissed off so many people that he was forced from resign from his tenure track position at Harvard physics in string theory. His academic work I am of course nowhere close to qualified to comment on, but people have said it’s first rate, and I’ll take their word. I even thought the guy was crazy. My very smart friend, in some online interaction with him, was scoffed off with: “You don’t understand vectors!” That guy later characterized the hypothetical combination of Luboš and this other guy I know, a PhD student in string theory, who is quite academically elitist and also so in terms of expecting good values and a fair degree of cultural/historical knowledge, as “a match made in heaven.” I also recall a commenter on Steve Hsu’s blog remark that Luboš has Aspergers syndrome or something like that. Anyhow, this time when reading the blog of Luboš, I no longer felt a sense of awe but rather a strong sense of clarity and reasonability in his thinking. He can be quite abrasive in some other contexts maybe, such as in his campaign against the climate change advocates (oh, on that I recently learned Freeman Dyson is also on the same side as Luboš on this one), but I believe it arises purely out of positive intentions on his part for the future of humanity, which many view as on a course of decline.

So the blog posts of Luboš read by me most memorable were on Scott Aaronson and feminism, a proposal for more political brainwashing requirements at Harvard, and Terence Tao’s silly mathematization of why Trump is not fit to be president respectively. On the first, I never knew Scott had followed the current feminist bandwagon. On the second, I’ve become more repulsed by and concerned with what I would characterize as absurd political notions (not matching with objective facts) held by many of elite school credentials, a sign that our elite selection is failing. On the third, I can’t believe Tao, a mathematician, would try to artificially mathematize a political matter. I would think that a mathematician would know better that substance trumps presentation in science.

Another friend of mine with a math PhD told me to my surprise a few years ago that now, we see many great scientists marginalized. I used to have the naive view that hard science fields like math and theoretical physics were almost entirely meritocratic and of a culture tolerant and supportive of independent, rational thinking and dismissive of the disingenuous marketing the norm in the business world, but now I increasing am doubting that, not that I deny at all that those fields are far better than the softer, less g-loaded areas of STEM, let alone non-technical subjects. It’s kind of sad that even mathematicians in high places like Aaronson and Tao are promoting such behavior with their prominent positions. So that friend of mine might be right on his somewhat of a verdict that the scientific community is in a catastrophic state right now.

I would say this is much owing to the scarcity of positions. Tenure is such a rare commodity nowadays that one who obtains it so often uses it to advance their political agenda, and sadly on that, it seems the bad guys are winning. Direct, honest, objective guys like Steve Hsu are few and fewer. Of course, different groups fighting for their own interests, for advancement of their own, be it their ethnic group, their political party, or their field of study, is deeply embedded in human nature and a necessity for survival. We now see in academia what in hyperbole are religious wars between different fields, different schools of thought, often in a manner that defies the so-called freedom of expression and thought that the university is in its ideal supposed to be for.

What I have just written holds within the theme of civilizational decline. On the matter of preservation of Western (white) civilization, my white American friend raised Christian remarked:

IDK the new divide is not “white vs nonwhite” it’s more like “people who have civilization worth preserving vs everyone else”

On that I asked with a chuckle: “what about Jews?” And he was like:

They have a country they should go there where they can’t parasitize everyone else

On that I recalled that my friend, another math PhD student, regards Jews more as a social class than as an ethnicity. He does have a point since as far as I know, the distinctiveness of Jews as an ethnic group is blurry in that they were this group in the Middle East with a religious culture of their own their seldom mixed with others despite often living amongst them. There, the leaks were more outwards with Jews converting to Christianity and thereby leaving permanently.

However, upper classes, especially ones in intellectual ability, within an ethnic group are still largely identified with and respected by the majority as emblematic of the group at large in some sense, which would contradict the aforementioned interpretation. I see that ordinary whites still view upper class whites as their own, as do ordinary Chinese with respect to intellectually elite Chinese, yet no other group really identifies with Jews the same way as far as I can tell.

Let me reiterate again that I, with many Jews I much respect and also some I talk to who have been major influences on me, am not anti-Semitic. Not that anti-X can be viewed as a binary variable. Lobos also said that in contrast, sex can be because there are X and Y chromosomes, so wise men think alike. 😉

I have commented before that

“Anti-Semitism” has become this political buzzword now. It basically is equivalent to anti-Jewish. So what? Many people in the world are also anti-Chinese, or anti-American, or anti-German, or anti-(any ethnic group or country), so what, they have the right to be, so long as they do not infringe too much. Also, keep in mind that anti-X is not binary; it’s very complex. Just like you almost never like or dislike everything about a person, you also can like certain things about a particular culture or people or country, and not like certain things.

I heartily believe that every group can be openly examined for their behavior as a collective. There is nothing wrong with that, and racist stereotypes are there for a reason after all. Pertaining to a specific one, Anti-Semitic conspiracy theorists (or most like cynical realists) might think that Jews want to absorb every competent group into their order so that they can have smart people working for them instead competing against them, and of course they will share power mostly amongst themselves.

Obviously, if you want to gain leverage over someone absorb him into your system make him dependent on you. We see this in international relations all the times. For example, in military technology, US and USSR created their own independent ecosystems, and many smaller countries had to more or less choose one or the other. There is a similar phenomenon in the software industry, with a very small number of widely used languages and frameworks. We’ve seen that many businesses are stuck with Microsoft once they use it for a while, and then there is a chain effect across the entire market.

We also see that Jews are also on top of arguably the premier credentialist hierarchy that is the Ivy League, with their accounting for arguably half or more of its presidents and senior administrators, and now people sort of need it to advance their career in America and even some other places, from which comes inevitably owing to our nature the political game of allotment of these scarce credentialist resources. Lately, Asians have realized by now that they can’t let Jews control too much of its distribution, favoring groups it fears not at the expense of those who pose more of a threat to themselves. On this, I have written that US higher education was and still is somewhat of a tool for cultivating (pseudo)-elite Chinese within an ecosystem wherein Jews have disproportionate influence. Chinese are a unique group in that they are intelligent, large, and a civilization and culture that emerged and evolved almost entirely independently of outsiders. (On the other hand, it is the modern science that Chinese are increasingly excelling at that is, in contrast, purely a product of Western civilization.) For this reason, Chinese have been very difficult if not impossible to absorb into any other system. Historically, even though the Mongols and Manchus had conquered China militarily, culturally they were much more absorbed into China than the other way round.

I believe cultural diversity (globally, not within every single country) is beneficial if not necessary for the overall health of human civilization. Referring back to the putative degraded state of US academia, Alain Connes, a French Fields Medalist, thinks the collapse of the Soviet science system, was catastrophic for science, since the USSR was a crucial counterweight to America. It was during the Cold War that was the golden period for STEM in America too, with Apollo 11 a climax. Now, with everybody absorbed into the American system sociologically, people are far less inclined to work on new things and instead play it safe in existent research programs, especially with grants and tenure-track, whereas in USSR in the research institutes, which he believes produced the best science, everyone basically had tenure from the start. That was quite an new and interesting perspective when I first saw it, and now, knowing more, I can see why he thinks that. Also, I think with China and Chinese, the mentality used to be, from the beginning of the reform and opening up, primarily one of how to gain approval from and integrate into what is globally prestigious along the (US-led) status quo, with say a sizable contingent obsessed with Ivy League, but that is taking a turn in the recent years now that China is far richer and more advanced than before. Still, one can say there was still back then a minority but one large enough to produce effect of talented people in China who thought all that prestige worship was silly and persisted in what they were doing to the extent that they gradually built more critical mass that while formerly much ignored by outsiders is now attracting ever more attention.

I’ve noted that different political factions and ethnic groups competing for resources for themselves will always be a thing, and one can think of scientific disciplines and schools of thought as political factions in some sense, which are in some cases even largely segregated by ethnic groups, with different countries having their own distinctive schools in various scientific disciplines. Sometimes, being too influenced by what others are doing and how others are thinking detracts from independent inquiry. Science in the long-term historical perspective values those who create new fields which turn out to be important. I have certainly seen the perspective that problem solvers in existent fields are a dime a dozen and it’s the theory builders who blaze new trails who are the real geniuses, one that resonates with me. For instance, the Greeks were the founders of the pure mathematics, and it was the step they took that was the more difficult and revolutionary, with Chinese civilization’s not having done so.

Politically in analogy, I admire the USSR for their having blazed a radically new trail that though ultimately unsuccessful, drastically altered the course of the 20th century and gave much to humanity in science and technology and the arts. Since China very successful today is in some sense an inheritor of the Soviet legacy, it surely hasn’t died out and is even rejuvenating. In contrast, I read on the Chinese QA site Zhihu an answer stating the proposition that after Qin Shihuang unified China in 212 BC, he forcibly made everything uniform across the whole country, burning books and burying scholars not in order with the official line of thought, enough that China as a civilization made little headway in intellectual thought for the next two millennia. Intellectuals only followed what was already there and could not escape it to create any tradition radically different, until superior forces without eventually forced change within.

The conclusion we can draw from all this is that monopoly of a form that discourages radically new ideas and development of alternative systems is detrimental to the advancement of human civilization.

## Understanding Human History

I had the pleasure to read parts of Understanding Human History: An Analysis Including the Effects of Geography and Differential Evolution by Michael H. Hart. He has astrophysics PhD from Princeton, which implies that he is a serious intellectual, though it doesn’t seem like he was quite so brilliant that he could do good research in theoretical physics, though an unofficial source says he worked at NASA and was a physics professor at Trinity University who picked up a law degree along the way. I would estimate that intellectually, he is Steve Hsu level, perhaps a little below, though surely in the high verbal popularization aspect, he is more prolific, as evidenced by that book, among many others, such as one on the 100 most influential historical figures. He is active in white separatist causes (heh) and appears to have had ties with the infamous and now deceased Rushton.

Lately, with pardon for possible hindsight bias from reading, I have been more inclined to look at the world from a long term historical perspective. I have always had some inclination to believe that to judge an intellectual fully in terms of impact take decades and often generations, especially political ones. As a derivative to this, I feel I am, relative to most, less susceptible than most to fads and trends and care less about short term recognition and credentialism. The ideal is to let history be the judge, which it will be eventually and inevitably.

In this post, I’ll give a summary of what I would regard as some of the most prominent points in that book. Keep in mind though that I won’t strictly refer to the book and will instead draw from various sources online, with the book as more of an inspiration. To start, I recall reading as a kid that the Euphrates and Tigris rivers in Mesopotamia (modern day Iraq, Syria, Turkey) are cradles of civilization. On that, Hart was somewhat elaborate on the development of agriculture that took place there at least as early 11,000 BC. This was not soon after the last glacial period which many speculate vastly enhanced the intelligence of peoples in the more northern latitudes, particularly in Northeast Asian and in Europe, through brutal elimination of those unable to survive under the harsh demands brought forth to them in the cold winters. The earliest well-accepted evidence of writing appears to be again in Mesopotamia around 3100 BC. Around the same time, independent writing systems also arose in Egypt, but with that, historians and archaeologists cannot be sure whether it was truly independent, as the geographic proximity between Egypt and Mesopotamia was not large.

An independent civilization arose in China too, which was geographically isolated from the larger part of world. On its east (and to a less extent, south) is the Pacific Ocean, on its West are some of the world’s highest mountain ranges, and on its north are relatively barren lands. Respectively, agriculture and writing emerged in China not long after in Mesopotamia. The body of inscriptions on oracle bones from the late Shang dynasty gives the earliest evidence for what consensus would regard as genuine writing, which was around 1200 BC. There has been, though, an excavation dating back to as early as 6600 BC, of some form of proto-writing of the Peiligang culture. One ought to keep in mind that here we are talking about confirmed upper bounds in time, which will hopefully become tighter and tighter with time as more archaeological discoveries emerge and emerge. While we cannot definitely rule out that Mesopotamia influenced the development of writing in China, it is extremely unlikely that such was the case, due to the great geographic barriers.

I have had the pleasure of skimming through parts of the most classic of Chinese classics, including the I Ching, which are difficult to understand as one would expect. Those are the Chinese biblical equivalents. Unfortunately for history, the first emperor of China who unified all of China in 221 BC, preserving such unity by enforcing uniform weights and measures, ordered an infamous burning of books and scholars, which means that many priceless artifacts of Chinese civilization were forever lost, but of course, many books were able to escape his decree.

The Chinese did not develop an alphabet, as we all know. This was obviously disadvantageous in many ways, but it also enabled China to remain as one culturally, as languages with alphabets can more easily evolve. In China, there are mutually unintelligible dialects (such as Mandarin and Cantonese, which are still very similar in their oral form), but they all employ the same writing system unalterable. One can observe that the legacy of this persists deeply today with China unified and Europe very fragmented culturally and politically with the EU somewhat of a farce as a political organization according to many.

You are now aware [‘you’ being King Gelon] that the “universe” is the name given by most astronomers to the sphere the centre of which is the centre of the earth, while its radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account (τά γραφόμενα) as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the “universe” just mentioned. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun on the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface.

The Greek were too rich and too farsighted in their scientific thinking and achievements, and I shall give no more concrete examples here for the sake of time.

This is in stark contrast to the Chinese civilization that Hart claims is the only one that can overall rival Western European civilization. Whatever scientific schools of thought, such as that of Mo Tzu, that existed were not well-preserved and eventually lost prominence to Confucianism, which did not emphasize rigorous scientific thinking, instead with an overemphasis on social relations of a more conformist nature that came with it an imperial examination system focused on literary topics for selecting people to govern the country. The ancient Chinese did not display much curiosity in the logical and natural world. Hart notes how even in 1600 AD, the Chinese knew far less than the Greeks in mathematics, and there is still as far as I am aware not of any evidence of widespread recognition of the round earth among Chinese scholars.

There is reason for a geographic explanation to this. Hart brings up the advantageous geographic position of Greece for its development of civilization. It was, on the Mediterranean, a maritime culture. It was, being further east than Italy, and thus in much closer cultural contact with the Mesopotamians, the cradle of civilization on the larger, non-Chinese part of the world. Additionally, it was close with Egypt. On the other hand, Chinese civilization was basically all to itself, contributing very crudely to somewhat of a less adventurous spirit, less curiosity about the outside world, and by extension, less curiosity about the natural world. Of course, what appears to be the lack of emphasis on theoretical matters of the ancient Chinese also has deep and far from well understood, owing to lack of complete picture due to loss of artifacts, roots. The location of the Greeks is not alone though. Hart also believes that the Greeks, being in a colder climate, had a higher IQ (or biological intelligence), which was what enabled them to surpass both the Mesopotamians and the Egyptians.

The Chinese brought to the world two major inventions that radically altered the course of history, which were uniquely and definitely Chinese. They were paper making and gunpowder. The papermaking process was invented by court eunuch Cai Lun in 105 AD. It was the first inexpensive medium for writing, as opposed to papyrus and bamboo, that enabled for China a great leap forward culturally. In 751 AD, some Chinese paper makers were captured by Arabs after Tang troops were defeated in the Battle of Talas River, and from that, the techniques of papermaking then spread to the West gradually, reading Europe in the 12th century. This is so impactful and impressive, because Western civilization was not able to uncover this critical process for over a millennia when they finally learned of it from outsiders. For this very reason, Hart put Cai Lun as number 7, right ahead of Gutenberg, inventor of the printing press in the 15th century in German. To justify that, he claims that Gutenberg would not have invented the printing press if not for paper, and that this invention being purely one of Chinese civilization that was transmitted to the West over a millennia later in addition to its history altering impact was not one that was inevitable in the sense of being a product of the historical epoch in which it came about. The Chinese also invented printing, with woodblock printing in the 8th century Tang dynasty and movable type (one for each character) by Bi Sheng in the 11th century. However, because of the thousands of Chinese characters as opposed to the tens of letters of the alphabet, movable type did not have anywhere as near of an impact. There is little if any evidence that Gutenberg was influenced in his invention by the one from China.

The importance and again pure Chineseness in invention of gunpowder is also without question. It revolutionized combat and was what enabled Europeans, with their improved guns, to later conquer the New World. Gunpowder was invented by Chinese alchemists in the 9th century likely by accident in their search for an elixir of life. The first military applications of gunpowder were developed around 1000 CE, and in the following centuries various gunpowder weapons such as bombs, fire lances, and the gun appeared in China. Gunpowder was likely transmitted to the Western world gradually via the Mongol invasions, which extended as far as Hungary.

The final of the so called Four Great Inventions of China not yet mentioned is the compass, which facilitated the voyages to Africa of Zheng He in the early 15th century. For that though, while very possible, there seems far from any conclusive that it spread to the Islamic World and Europe as opposed to be having been reinvented there.

Transitioning from China to the medium between China and the West, the Islamic world, we must delve into the Islamic Golden Age, traditionally dated from the 8th century to the 13th century, during which many important scientific discoveries were made. Though my knowledge of Islamic cultures is scant, I do know of Alhazen, Omar Khayyam, and Al-Khwārizmī. In particular, his seven-volume treatise on optics Kitab al-Manazir, while perhaps questionable on his theories of light, was notable for its emphasis on empirical evidence that combined inductive reasoning, which was relatively neglected by the Greeks, with the rigorous deductive reasoning that the Greeks championed to the extremes. We do know with certainty that this magnum opus was translated to Latin, greatly influencing later European scientists and thinkers as important as Leonardo Da VinciGalileo GalileiChristiaan HuygensRené Descartes, and Johannes Kepler. Moreover, Al-Khwārizmī’s work on arithmetic was responsible for introducing the Arabic numerals, based on the Hindu–Arabic numeral system developed in Indian mathematics, to the Western world. There is evidence of solid knowledge of trigonometry, with for instance the law of sines pervasive in the scientific literature from Islamic scholars of that time. With reference to Hindu, I shall note that Indian mathematics and astronomy were quite impressive, certainly more so than Chinese mathematics, which though calculating pi to 7 digits as early as the 5th century, which held a 900+ year record, among many other applied and computational achievements, was severely lacking in its theoretical foundations, was, with AryabhataBrahmaguptaBhāskara I, among others who did work close or on par with those of Islamic scholars mathematically but much earlier, between the 5th and 7th centuries. Because many foreign words are contained within their texts, we can be relatively sure that there was Greek and Mesopotamian influence. Relating to that, Hart does not see Indian or Islamic mathematics as terribly original and more as derivative of Greek works, with significance more in the nature of preservation, though with Western European civilization having been the dominant, and often entirely so, for so long, one ought to be careful of Eurocentric bias. The achievements of Indians and Arabs to math and science ought to be more thoroughly investigated and fairly acknowledgment, in particular how they may have influenced later developments in the West. On that note, I shall say that I was super impressed that in the 14th century, the school of Madhava of Sangamagrama managed to discover infinite series for trigonometric functions of sine, cosine, tangent and arctangent. As a special case of arctangent, we have that

${\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +{\frac {(-1)^{n}}{2n+1}}+\cdots,$

which was later rediscovered by Leibniz. This of course hints or indicates that Madhava already knew at that time some form of proto-calculus, with as a concrete example Rolle’s theorem, which his predecessor from the 12th century Bhāskara_II had already stated. It’s possible that knowledge of these results were transmitted to Europe, but online sources stay that no evidence for that has been found. This probably influenced Hart’s verdict that Indian/Hindu civilization, while superior to China’s in theoretical science, was far less influential, with of course, India’s having received some knowledge of the Greeks, whereas the Chinese developed independently, with Euclid’s Elements only translated to Chinese in the early 17th century, where it, unfortunately for China, did not have the impact it should have had.

We all know that the West created the modern world, with the Renaissance, the scientific revolution, and the industrial revolution, and discovering, conquering, and colonizing more and more of it with their superior ships and guns, white Europeans virtually ruled the entire world by the late 19th century, ushering in unprecedented growth revolutionary in its quality and exponential in its quantity. It has continued to the point of air travel and internet communications that has drastically reduced the distance between cultures and peoples, with racial intermixing and immigration ever more common and accepted, though of course, the majority still live and mix with their own, in their ancestral homeland.

For the 21st century, Hart also predicts that the breakthroughs will be achieved mostly by white Europeans (that includes Jews) and East Asians, and we already see that happening. I do not recall his stating that the East Asian civilization represented mostly by China and Japan have been on rapid rise lately, and I shall surely point that out, out of what I regard as both its reality and significance (as opposed to any ethnic chauvinism on my part). It is the formerly weak but now strong and still rapidly strengthening other side of human civilization that is less fairly acknowledged, though with its rise, that will gradually change, just as the rise and later sheer dominance of the West enabled it to easily impose its standards and culture on others regardless. With mathematics again as the representative for the pinnacle of human civilization, we can see how very recently Yitang Zhang stunned the world by proving infinite bounded gaps between primes and Shinichi Mochizuki is receiving ever more press for the inter-universal Teichmüller theory that claims to solve the abc-conjecture, one of the most important problems in number theory, the queen of mathematics (according to Gauss), that could possibly becoming one of the most important new mathematical theories of the 21st century. On that, my friend once remarked: “Mochizuki could be the 21st century Grothendieck!” It is quite remarkable and also surprising that the culture and civilization for which theoretical science had been a glaring weakness historically is now verging on its apex, though the surprising part is less so when one takes IQ into account, with now the cultural factors more controlled for owing to the near universal access to information provided by the Internet. Additionally, China is excelling at and amazing, with some effect of unease, the world at what it has traditionally been strong at, namely large-scale engineering projects, but this time, of a nature guided by the modern science of the West. As examples, we see the world’s fastest trains in a nation-wide network, the world’s largest genome sequencing factory, and a great wall of sand dredged on the South China Sea. They are modern Chinese parallels of the Great Canal, the Great Wall, and the mega ships of Admiral Zheng He an order of magnitude larger than those of Columbus. Comes unity comes strength, or so the saying goes. It is one that persists in Chinese civilization today that is enabling more in China what the West cannot do, in practice.

There are scholars and advocates who lament that Western civilization, threatened by dysgenic immigration among other things, is in decline, and that its culture and civilization, which includes a certain purity of its people, ought to be preserved, which includes Hart himself. Given the overwhelming contribution of the West to human civilization, with Greek and Latin roots, has contributed to human civilization, one cannot not identify somewhat with this point of view. On this note, Rushton has even hypothesized that the Black Death precipitated the Great Divergence by suddenly and drastically enhancing the gene pool through killing off a quarter and as much as a half in some places of the European population via more or less a freak accident, one that has been regressing ever since to its natural level. It is somewhat unfortunate in some sense that the horrific legacy of Nazism, which was such that many Western peoples began to outwardly oppose ideas of racial superiority, has developed up to today towards a form of irrational racial egalitarianism and SJW culture that denies any honest, scientifically objective discourse on race differences, which are patently there, which we have the ability now to examine vastly more closely, powerfully, and scientifically than in Hitler’s time that is so politically obstructed for the aforementioned reason. Having referred to dysgenics, I shall also note that the technology and globalization we have today we are rather evolutionarily maladjusted to. Foremost of all, with reference to modern medicine, evolution does not let the weak live or spread its seed, and moreover, evolution is not terribly suited for vastly multi-ethnic societies either. The world now exhibits so much more mercy than before, often at the expense of the advancement of civilization. Yes, we know and have much more than our ancestors, but are we biologically superior to them? Perhaps we are at the far far tail, which increasingly breeds assortatively, but overall, I would say almost certainly not.

As for the 21st century, how it will pan out, only time will tell. However, if I were to bet, I would say that its winner and its legacy, viewed from the long term historical perspective, say a millennia from now, will be whoever musters the courage to control our own evolution to take us beyond the confines of Homo sapiens, so extraordinary and yet so limited in its might, and also at times also so foolish in its wisdom.

To conclude, my message to my generation and the future of humanity, inspired partly by Bertrand Russell:

## Why mathematics

I had the pleasure of chatting briefly with a math PhD student, with the conversation largely centered on what kind of math are you interested in. He is doing discrete probability and combinatorics, something along the lines of that. He said that he spent a year studying commutative algebra during undergraduate, but eventually decided that he would not do math that deep and instead is concentrating on an area with less requirement in terms of acquired knowledge and more low-hanging fruit to pick, the parts of math of a more problem solving nature. He went on to say that of the math undergraduates at his top (but not Beida or Qinghua) institution in China, by junior year, only five were studying the purist of pure math, and later during graduate school, all but one of them, who is now doing research in string theory, have given up, instead choosing not pure PDEs but PDEs for biology and the likes, to illustrate the low rate of success for pure pure math. I told him that I still want to do really deep math (of which we can use algebraic geometry) and see the parts of math not requiring deep knowledge as not as meaningful to do research in (of course, I don’t expect to succeed, realistically gauging that I am, while highly talented, not a genius). On that, he more or less said that you should try and that you never know unless you try. Of course, he did more constructively say that learning commutative algebra requires knowing deeply thousands of definitions, and just going through ten of them a day is already very good. Maybe attempting this is not terribly wise when I see people objectively smarter than I am who eventually chose easier fields, like theoretical statistics.

Now this brings me to reflect on why I am doing pure mathematics? Why am I devoting so much time and energy (with overall enjoyment and satisfaction at this point still pretty high) on this arcane, useless subject? How much of it is out of an ego to prove how smart I am versus the intrinsic thirst for the knowledge? Of course, the two are somewhat intertwined, as you’ll see in what I’m about to say.

As for my background, I studied some CS in college and also spent some years in the software industry, which I’ve grown very distasteful of. I don’t like CS people very much in general. They make a big deal out of low-hanging fruit. Like, MapReduce is trivial theoretically; it’s more about the engineering, in particular the locality to minimize network IO, which in distributed systems is usually the bottleneck. There is nothing deep about it. Algorithms is cool, and I enjoyed them, doing okay in some coding contests, solving say plenty of TopCoder 500s (but not quickly enough during the short 75 minute time frame of the contest). However, algorithms I view as more of a game, full of clever little tricks but of little substance, recreational math at best, at least the type of algorithms I did. Engineering wise, I see the value, but I don’t see myself as naturally inclined to it at all, and in fact, among the strong folks in that, I’m probably rather weak. I don’t think those people are terribly smart from an IQ point of view. They’re not as cultured in some sense. (That top MIT math major (though he works in combinatorics heh) says the same, that science is for high math high verbal people with refined intellectual tastes while engineering is for high math (note that this often does not even hold for software engineering) lower verbal folks of a dronish nature.) In any case, I don’t think I’m in the same species as all these people in software engineering who know absolutely nothing about continuous math, the type of math you see in physics, like I think that’s just bad, or at least different, taste, or simply lower IQ enough that they cannot even understand it. I thought at one point that I might want to do CS theory. Not anymore. I think that’s a cool field with many good problems, but again, much of it lacks depth and importance, often with little connection to the mainstream of mathematics.

I see mathematics as in some sense the pinnacle of human civilization and of human intelligence. I’ve probably said before that humans discovered literature, music, crafts,  and engineering (non-modern) long long time ago, but mathematics took so long, which just goes to show how unnatural it is for the human brain. It is a pursuit of truth in the rigorous and absolute sense that one sees not in natural science either, though of course, the deductive method that underlies math is thoroughly used in natural science. Moreover the structures investigated in mathematics are of such a fundamental and pure nature which often appear in reality, though of course the purists, with the Greeks as the pioneers of that, view mathematics as a Platonic ideal to be investigated for its own sake independent of reality. What the Greeks did I would say is rather unnatural, because I recall early on, it did not feel so natural for me to disentangle mathematics with the reality, having seen it more as a tool for reality.

Mathematics is so full of substance, unlike almost all other subjects. It emphasizes high quality, with often deep, fundamental ideas explained in a few pages, in austere, terse language. It is a scientific study that tolerates absolutely no bullshit and aims for the simplest possible explanation of pure, strictly incontrovertible truth by logic. It is an escapism from the mediocrity and nonsense we see in much of the world and most humans too intellectually dazed for the clear thinking necessary to perceive mathematical truth.

I see my ever greater interest and appreciation, and of course, ability and knowledge, for mathematics as an inevitable consequence of my neurobiological maturation, which is fortunately to an extent far enough that I am able to experience as much of this world of truth invisible to most humans around me, though of course, I can only admire those true geniuses, those far superior brains, who can fathom so much deeper and more rapidly than I can. On this, I shall say that mathematics may well be what separates homo sapiens from whatever species eventually evolves beyond it. I would bet that in another millennia, we will have people for whom mathematics is as natural a language as natural language is to humans. Just as humans have evolved their brain and also their anatomy of throat and mouth such that learning (non-formally) and articulating language is instinctive, humans may evolve their brains further such that that holds for mathematics as well.

Over time, I’ve come to realize more so that mathematics is about the right mental perception. Ideally, one can see the mathematics in one’s head. Text is but a medium of transmission (with reading the fastest bandwidth in terms of information transmission to the human brain), but without a well-formed brain rational and composed, there is basically nothing one can do to genuinely absorb the truth that exists independent of one’s perception of it. It is often that one intuitively feels like one can understand certain mathematics one hears or reads, but looking more closely, one finds such is not the case, being unable to visualize it with enough clarity that one can independently explain it.

My learning of mathematics has been far from entirely smooth. I have despaired much about simply not being smart enough, especially upon seeing another seemingly effortlessly master what was utterly perplexing for me. Fortunately, that all improved over time. Though of course, as the Dunning-Kruger effect would say, the better you become the more can see your incompetence and your limitations. The experience of being able to experience the life of mind with ever more clarity, fine grain of control, and awareness has been an internally exhilarating experience.

Mathematicians are in some spiritual aristocrats, and mathematics arguably has more of an intellectual upper class air to it than any other subject. What is aristocracy? It is to many a relation by blood to those politically important or foundational. But is political power really the pinnacle of human experience? I say no, and I would say that it is the experience of the deepest scientific truths, one which requires both biological genius as well as the substantial cultural exposure that naturally comes with it, especially in today’s day and age of universal access to information. Human experience in any case hinges on consciousness, and one’s subjective conscious experience is always the product of neurons. Thus, mathematics has to it an aristocracy that no amount of money or political title or physical appearance or dress can buy; there is no royal road to mathematics, as Euclid said. So in some sense, mathematics is the greatest gift of God to a human he conceived on earth.

What are other characteristics of non-trivial engagers of mathematics that one easily associates with aristocracy? First comes to mind language and literacy. In virtually every culture, literacy was in the old days a sign of class, of privilege. In the West, it was the Catholic priests and in the East, it was the Confucian scholars. In virtually every religion or ideology or culture, the masters of that culture through literacy were highly esteemed. For example, in Jewish culture, there were the rabbis. Those with the most mastery of language where often the ones of authority, much owing to their exclusive access of certain information that facilitates political and mind control of plebs. From this, emerged learned aristocracies which developed their distinctive elite cultures, along with to some degree a distinctively evolved genetic line. These aristocrats evolved an ability to parse and memorize text far greater than the masses who had to labor in the fields. They developed and evolved a certain form of refinement and manners and self-control, as well as physical appearance, that came to be characterized as one of an aristocratic nature.

With this said, in the West, during the Renaissance and the subsequent scientific revolution, the men of science were often ones from a learned religious background of deep conviction in their religious faith who were intellectually courageous enough to go beyond it, to go about to discover scientific truth often with inspiration from the God they held deep in their hearts. They conceived of a much more rational and accurate world that turned out had been there all along without their knowing. All this eventually ushered in a new age of human history of exponential human discovery, of fundamental scientific truths, of unseen lands, of modern machines, that has culminated in the globalization we have today. All of this has much of its roots in mathematics.

To say all this would imply my yearning to become an aristocrat, which brings to another point, namely, that mathematics, while aristocratic, is more or less coldly meritocratic, and thus is aristocratic mostly in its intellectually noble content. For a brilliant kid from a poor background, mathematics is the most straightforward means of social mobility. Mathematics does not require expensive equipment or facilities or elite social connections. Provided a sufficiently high caliber mind, excelling in mathematics is relatively natural, since one can read on one’s own and solve mathematical problems on one’s own, starting with olympiad style problems at the secondary school level. Though we see plenty of mathematical families, mathematics is not grossly nepotistic as is say acting or offices of political power. In its purist essence, the culture of mathematics reveres genius from wherever he hails and despises any form of ascension based on social connections.

I have observed in those of high mathematical talent a propensity for what I would regard as refined taste in other areas as well, in music, in literature, in politics, and in aesthetics of human beauty as well. Speaking of which, math is widely considered as having the smartest people and being the most g-loaded subject (along with its nearest neighbor theoretical physics), because there is some evidential truth to that, that it is often the mathematicians who are the most versatile. Mathematicians are well known (at least to me) for their often extraordinary foreign language ability, along with what is not infrequently talent in engineering and music as well. So there really is much to suggest towards the bold hypothesis that the man of mathematics is the most ideal of man evolved on earth.

To conclude, I will note that I sincerely empathize with those who have had genuine struggles with mathematics or more extremely, who hate it, let alone appreciate it. By no means should one consider oneself as lesser if one is not good at mathematics as tempting as it may be. Though it is an intellectual pursuit achievements of which lie in the pinnacle of human civilization, there is almost no direct use in it, and the world does not need many mathematicians. In fact, there is, economically based on the very dismal job situation, quite a glut of mathematicians now, which makes it prudent for one to be discouraged from pursuing it as a career if one has not displayed extraordinary gift in the subject. Doing mathematics helps no one directly, but doing engineering or carpentry or nursing surely does, and as someone who has indulged so much in mathematics, I do feel guilty at times from my lack of contribution to the real world. Again, this is why I say that to go into mathematics, one ought to have a really good reason, part of why I have been inspired to write this post.

## My awesome roommate

The guy that is the topic of this post himself did up to high school, as far as I know, in Hong Kong, so we have some more in common than usual culturally I guess. He was just telling me about how he had read 矛盾论, which I haven’t even read, at least not in detail, myself. He was saying, on the putative connection between scientific talent and Marxism, perhaps how dialectical materialism is inherently a very scientific way of thinking. I myself know basically nothing about dialectical materialism and even think it’s kind of high verbal low math bullshit, but I can tell that the materialist side of it is very scientific in its very nature, and similarly, dialectics is a very analogies/relationships way of thinking, which is something that high IQ people are by definition good at. Surely, there is much more I can learn from this guy, especially about Chinese language and culture and politics.

On this, I am reminded of another amateur (but professional, or better, level for sure) Marxist scholar, who is genuinely encyclopedic in his historical and cultural knowledge, in particularly a perceptive quote of him that made a deep impression on me:

Europe has always been in rebellion against itself, and continues to be so.  There was nothing but futility in the attempt by superficially Westernised Chinese to be authentically Westernised Chinese by being imitative and reverential of the current embodiment of those values.  You could only be an authentically Westernised Chinese by being a rebel against the current embodiments of Western values, at least in as far as they hampered China or seemed to be irrelevant.  And that’s why Mao was China’s best Westerniser to date, despite his very limited experience of the mundanities of Western life.

As I’ll detail in a future article, visitors to the Chinese Communist bases at Bao’an and later Yen’an noticed that these were the only Chinese in China who behaved more or less as Westerners would have behaved in a similar situation.  Other Chinese might speak good English, wear Western suits and sometimes show considerable knowledge of Western culture: but it was all imitation and the inner core was different and ineffective.  Western-trained engineers and geologists who returned to China kept their distance from hands-on practical work, because anything resembling manual labour would have lost them status in the eyes of Chinese intellectuals.  They were imprisoned by a tradition stretching back to Confucius and beyond.  Only a few broke these ancient taboos, mostly the Communists and some scattered left-wingers in the weak middle ground.  And it was the modernised Chinese in the Communist Party who chose to raise up Mao as the prime teacher of this new understanding.

I remember when my obsessively talented Russian friend once said to me that sometimes he feels like he’s another Pavel Korchagin, I thought he was ridiculous. Well, I’ll be equally ridiculous and say that I feel like I very much exhibit what Gwydion described in Mao that is “authentically Westernized Chinese,” which is very much the antithesis of what I see in most ABCs, despite being half an ABC myself.

If only more people could be like me…

## Oleg

Oleg is one of my ubermensch Soviet (and also part Jewish) friends. He has placed at (or at least near) the top on the most elite of math contests. He is now a math PhD student with an advisor even crazier than he is, who he says sometimes makes him feel bad, because he has done too little math research wise. However, this persona alone is not that rare. Oleg’s sheer impressiveness largely stems from that on top of this, he is a terrific athlete, extremely buff and coordinated, enough that he can do handstand pushups, to the extent that he regards such as routine. Yes, it is routine for a guy contending for a spot on a legit gymnastics team, but you wouldn’t expect this from a math nerd huh?

Today, I was talking to him and some others about gym. In particular, I was saying how I could at one point do 10 pullups but dropped down to 2 after a long hiatus. The conversation went as follows:

Me: Oleg I’m back to 5 pull-ups now
Oleg: that’s good although make sure you’re doing them for real
i still don’t believe you could do 10 but then dropped down to 2
Me: Oh I’m very sure they’re full pullups
Okay maybe it was 8
Oleg: i’d like to see evidence
Me: Alright I’ll have someone videotape me do pullups today in gym

And so I did.

Later, Oleg suggested something pretty funny:

i still think you should get tattoos and gain 25 lb of muscle, that would be hilarious
and say “oh, that’s too low, i don’t want to breed babies with you”
followed by a cackle
i’d watch that show

Not surprisingly, Oleg, as buff as he is, has had some success with girls, though he regards himself as shy and struggling in that regard. I keep telling him that he needs to marry a girl who’s both super smart and attractive like he is, so that he can optimize his chance of making superhuman babies. His only disadvantage now is that he’s a poor math PhD student, but he can easily change that by, say, joining DE Shaw, from what I’ve read is full of uber nerdy macho Eastern European men. He’s not very interested in money though, and expresses content with his graduate student stipend, which I find laughable.

I find it regrettable that most ubermensch men smart enough for legit doctoral programs in math and physics are unable to find a mate who is commensurate with them, ability wise, even with some adjustments, even when they’re well-rounded like Oleg is. Why is this? Excessive Aspergers? On that, I know someone who will say along the lines of

in an actual long-term relationship you have to share most of your life with the person, and if they don’t understand the way you look at the world then it creates friction
sure, the girl doesn’t need to understand high energy physics, I have other friends for that

Maybe some females could give us some advice, other than the cliche “hit the gym” that you’ll often hear from males. Such would be much appreciated! 😉

## 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 +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 sunday

I had a chill day thinking about math today without any pressure whatsoever. First I figured out, calculating inductively, that the order of $GL_n(\mathbb{F}_p)$ is $(p^n - 1)(p^n - p)(p^n - p^2)\cdots (p^n - p^{n-1})$. You calculate the number of $k$-tuples of column vectors linear independent and from there derive $p^k$ as the number of vectors that cannot be appended if linear independence is to be preserved. A Sylow $p$-group of that is the group of upper triangular matrices with ones on the diagonal, which has the order $p^{n(n-1)/2}$ that we want.

I also find the proof of the first Sylow theorem much easier to understand now, the inspiration of it. I had always remembered that the Sylow $p$-group we are looking for can be the stabilizer subgroup of some set of $p^k$ elements of the group where $p^k$ divides the order of the group. By the pigeonhole principle, there can be no more than $p^k$ elements in it. The part to prove that kept boggling my mind was the reverse inequality via orbits. It turns out that that can be viewed in a way that makes its logic feel much more natural than it did before, which like many a proof not understood, seems to spring out of the blue.

We wish to show that the number of times, letting $p^r$ be the largest $p$th power dividing $n$, that the order of some orbit is divided by $p$ is no more than $r-k$. To do that it suffices to show that the sum of the orders of the orbits, $\binom{n}{p^k}$ is divided by $p$ no more than that many times. To show that is very mechanical. Write out as $m\displaystyle\prod_{j = 1}^{p^k-1} \frac{p^k m - j}{p^k - j}$ and divide out each element of the product on both the numerator and denominator by $p$ to the number of times $j$ divides it. With this, the denominator of the product is not a multiple of $p$, which means the number of times $p$ divides the sum of the orders of the orbits is the number of times it divides $m$, which is $r-k$.

Following this, Brian Bi told me about this problem, starred in Artin, which means it was considered by the author to be difficult, that he was stuck on. To my great surprise, I managed to solve it under half an hour. The problem is:

Let $H$ be a proper subgroup of a finite group $G$. Prove that the conjugate subgroups of $H$ don’t cover $G$.

For this, I remembered the relation $|G| = |N(H)||Cl(H)|$, where $Cl(H)$ denotes the number of conjugate subgroups of $H$, which is a special case of the orbit-stabilizer theorem, as conjugation is a group action after all. With this, given that $|N(H)| \geq |H|$ and that conjugate subgroups share the identity, the union of them has less than $|G|$ elements.

I remember Jonah Sinick’s once saying that finite group theory is one of the most g-loaded parts of math. I’m not sure what his rationale is for that exactly. I’ll say that I have a taste for finite group theory though I can’t say I’m a freak at it, unlike Aschbacher, but I guess I’m not bad at it either. Sure, it requires some form of pattern recognition and abstraction visualization that is not so loaded on the prior knowledge front. Brian Bi keeps telling me about how hard finite group theory is, relative to the continuous version of group theory, the Lie groups, which I know next to nothing about at present.

Oleg Olegovich, who told me today that he had proved “some generalization of something to semi-simple groups,” but needs a bit more to earn the label of Permanent Head Damage, suggested upon my asking him what he considers as good mathematics that I look into Arnold’s classic on classical mechanics, which was first to come to mind on his response of “stuff that is geometric and springs out of classical mechanics.” I found a PDF of it online and browsed through it but did not feel it was that tasteful, perhaps because I’m been a bit immersed lately in the number theoretic and abstract algebraic side of math that intersects not with physics, though I had before an inclination towards more physicsy math. I thought of possibly learning PDEs and some physics as a byproduct of it, but I’m also worried about lack of focus. Maybe eventually I can do that casually without having to try too hard as I have done lately for number theory. At least, I have not the right combination of brainpower and interest sufficient for that in my current state of mind.