## Cayley-Hamilton theorem and Nakayama’s lemma

The Cayley-Hamilton theorem states that every square matrix over a commutative ring $A$ satisfies its own characteristic equation. That is, with $I_n$ the $n \times n$ identity matrix, the characteristic polynomial of $A$

$p(\lambda) = \det (\lambda I_n - A)$

is such that $p(A) = 0$. I recalled that in a post a while ago, I mentioned that for any matrix $A$, $A(\mathrm{adj}(A)) = (\det A) I_n$, a fact that is not hard to visualize based on calculation of determinants via minors, which is in fact much of what brings the existence of this adjugate to reason in some sense. This can be used to prove the Cayley-Hamilton theorem.

So we have

$(\lambda I_n - A)\mathrm{adj}(\lambda I_n - A) = p(\lambda)I_n$,

where $p$ is the characteristic polynomial of $A$. The adjugate in the above is a matrix of polynomials in $t$ with coefficients that are matrices which are polynomials in $A$, which we can represent in the form $\displaystyle\sum_{i=0}^{n-1}t^i B_i$.

We have

\displaystyle {\begin{aligned}p(\lambda)I_{n} &= (\lambda I_n - A)\displaystyle\sum_{i=0}^{n-1}\lambda^i B_i \\ &= \displaystyle\sum_{i=0}^{n-1}\lambda^{i+1}B_{i}-\sum _{i=0}^{n-1}\lambda^{i}AB_{i} \\ &= \lambda^{n}B_{n-1}+\sum _{i=1}^{n-1}\lambda^{i}(B_{i-1}-AB_{i})-AB_{0}.\end{aligned}}

Equating coefficients gives us

$B_{n-1} = I_n, \qquad B_{i-1} - AB_i = c_i I_n, 1 \leq i \leq n-1, \qquad -AB_0 = c_0I_0$.

With this, we have

$A^n + c_{n-1}A^{n-1} + \cdots + c_1A + c_0I_n = A^nB_{n-1} + \displaystyle\sum_{i=1}^{n-1} (A^iB_{i-1} - A^{i+1}B_i) - AB_0 = 0$,

with the RHS telescoping and annihilating itself to $0$.

There is generalized version of this for a module over a ring, which goes as follows.

Cayley-Hamilton theorem (for modules) Let $A$ be a commutative ring with unity, $M$ a finitely generated $A$-module, $I$ an ideal of $A$, $\phi$ an endomorphism of $M$ with $\phi M \subset IM$.

Proof: It’s mostly the same. Let $\{m_i\} \subset M$ be a generating set. Then for every $i$, $\phi(m_i) \in IM$, with $\phi(m_i) = \displaystyle\sum_{j=1}^n a_{ij}m_j$, with the $a_{ij}$s in $I$. This means by closure properties of ideals the polynomial coefficients in the above will stay in $I$.     ▢

From this follows easily a statement of Nakayama’s lemma, ubiquitous in commutative algebra.

Nakayama’s lemma  Let $I$ be an ideal in $R$, and $M$ a finitely-generated module over $R$. If $IM = M$, then there exists an $r \in R$ with $r \equiv 1 \pmod{I}$, such that $rM = 0$.

Proof: With reference to the Cayley-Hamilton theorem, take $\phi = I_M$, the identity map on $M$, and define the polynomial $p$ as above. Then

$rI_M = p(I_M) = (1 + c_{n-1} + c_{n-2} + \cdots + c_0)I_M = 0$

both annihilates the $c_i$s, coefficients residing in $I$, so that $r \equiv 1 \pmod{I}$ and gives the zero map on $M$ in order for $rM = 0$.     ▢

## 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 disease, 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.

## Implicit function theorem and its multivariate generalization

The implicit function theorem for a single output variable can be stated as follows:

Single equation implicit function theorem. Let $F(\mathbf{x}, y)$ be a function of class $C^1$ on some neighborhood of a point $(\mathbf{a}, b) \in \mathbb{R}^{n+1}$. Suppose that $F(\mathbf{a}, b) = 0$ and $\partial_y F(\mathbf{a}, b) \neq 0$. Then there exist positive numbers $r_0, r_1$ such that the following conclusions are valid.

a. For each $\mathbf{x}$ in the ball $|\mathbf{x} - \mathbf{a}| < r_0$ there is a unique $y$ such that $|y - b| < r_1$ and $F(\mathbf{x}, y) = 0$. We denote this $y$ by $f(\mathbf{x})$; in particular, $f(\mathbf{a}) = b$.

b. The function $f$ thus defined for $|\mathbf{x} - \mathbf{a}| < r_0$ is of class $C^1$, and its partial derivatives are given by

$\partial_j f(\mathbf{x}) = -\frac{\partial_j F(\mathbf{x}, f(\mathbf{x}))}{\partial_y F(\mathbf{x}, f(\mathbf{x}))}$.

Proof. For part (a), assume without loss of generality positive $\partial_y F(\mathbf{a}, b)$. By continuity of that partial derivative, we have that in some neighborhood of $(\mathbf{a}, b)$ it is positive and thus for some $r_1 > 0, r_0 > 0$ there exists $f$ such that $|\mathbf{x} - \mathbf{a}| < r_0$ implies that there exists a unique $y$ (by intermediate value theorem along with positivity of $\partial_y F$) such that $|y - b| < r_1$ with $F(\mathbf{x}, y) = 0$, which defines some function $y = f(\mathbf{x})$.

To show that $f$ has partial derivatives, we must first show that it is continuous. To do so, we can let $r_1$ be our $\epsilon$ and use the same process to arrive at our $\delta$, which corresponds to $r_0$.

For part (b), to show that its partial derivatives exist and are equal to what we desire, we perturb $\mathbf{x}$ with an $\mathbf{h}$ that we let WLOG be

$\mathbf{h} = (h, 0, \ldots, 0)$.

Then with $k = f(\mathbf{x}+\mathbf{h}) - f(\mathbf{x})$, we have $F(\mathbf{x} + \mathbf{h}, y+k) = F(\mathbf{x}, y) = 0$. From the mean value theorem, we can arrive at

$0 = h\partial_1F(\mathbf{x}+t\mathbf{h}, y + tk) + k\partial_y F(\mathbf{x}+t\mathbf{h}, y+tk)$

for some $t \in (0,1)$. Rearranging and taking $h \to 0$ gives us

$\partial_j f(\mathbf{x}) = -\frac{\partial_j F(\mathbf{x}, y)}{\partial_y F(\mathbf{x}, y)}$.

The following can be generalized to multiple variables, with $k$ implicit functions and $k$ constraints.     ▢

Implicit function theorem for systems of equations. Let $\mathbf{F}(\mathbf{x}, \mathbf{y})$ be an $\mathbb{R}^k$ valued functions of class $C^1$ on some neighborhood of a point $\mathbf{F}(\mathbf{a}, \mathbf{b}) \in \mathbb{R}^{n+k}$ and let $B_{ij} = (\partial F_i / \partial y_j)(\mathbf{a}, \mathbf{b})$. Suppose that $\mathbf{F}(\mathbf{x}, \mathbf{y}) = \mathbf{0}$ and $\det B \neq 0$. Then there exist positive numbers $r_0, r_1$ such that the following conclusions are valid.

a. For each $\mathbf{x}$ in the ball $|\mathbf{x} - \mathbf{a}| < r_0$ there is a unique $\mathbf{y}$ such that $|\mathbf{y} - \mathbf{b}| < r_1$ and $\mathbf{F}(\mathbf{x}, \mathbf{y}) = 0$. We denote this $\mathbf{y}$ by $\mathbf{f}(\mathbf{x})$; in particular, $\mathbf{f}(\mathbf{a}) = \mathbf{b}$.

b. The function $\mathbf{f}$ thus defined for $|\mathbf{x} - \mathbf{a}| < r_0$ is of class $C^1$, and its partial derivatives $\partial_j \mathbf{f}$ can be computed by differentiating the equations $\mathbf{F}(\mathbf{x}, \mathbf{f}(\mathbf{x})) = \mathbf{0}$ with respect to $x_j$ and solving the resulting linear system of equations for $\partial_j f_1, \ldots, \partial_j f_k$.

Proof: For this we will be using Cramer’s rule, which is that one can solve a linear system $Ax = y$ (provided of course that $A$ is non-singular) by taking matrix obtained from substituting the $k$th column of $A$ with $y$ and letting $x_k$ be the determinant of that matrix divided by the determinant of $A$.

From this, we are somewhat hinted that induction is in order. If $B$ is invertible, then one of its $k-1 \times k-1$ submatrices is invertible. Assume WLOG that such applies to the one determined by $B^{kk}$. With this in mind, we can via our inductive hypothesis have

$F_1(\mathbf{x}, \mathbf{y}) = F_2(\mathbf{x}, \mathbf{y}) = \cdots = F_{k-1}(\mathbf{x}, \mathbf{y}) = 0$

determine $y_j = g_j(\mathbf{x}, y_k)$ for $j = 1,2,\ldots,k-1$. Here we are making $y_k$ an independent variable and we can totally do that because we are inducting on the number of outputs (and also constraints). Substituting this into the $F_k$ constraint, this reduces to the single variable case, with

$G(\mathbf{x}, y_k) = F_k(\mathbf{x}, \mathbf{g}(\mathbf{x}, y_k), y_k) = 0$.

It suffices now to show via our $\det B \neq 0$ hypothesis that $\frac{\partial G}{\partial y_k} \neq 0$. Routine application of the chain rule gives

$\frac{\partial G}{\partial y_k} = \displaystyle\sum_{j=1}^{k-1} \frac{\partial F_k}{\partial y_j} \frac{\partial g_j}{\partial y_k} + \frac{\partial F_k}{\partial y_k} = \displaystyle\sum_{j=1}^{k-1} B^{kj} \frac{\partial g_j}{\partial y_k} + B^{kk}. \ \ \ \ (1)$

The $\frac{\partial g_j}{\partial y_k}$s are the solution to the following linear system:

$\begin{pmatrix} \frac{\partial F_1}{\partial y_1} & \dots & \frac{\partial F_1}{\partial y_{k-1}} \\ \; & \ddots \; \\ \frac{\partial F_{k-1}}{\partial y_1} & \dots & \frac{\partial F_{k-1}}{\partial y_{k-1}} \end{pmatrix} \begin{pmatrix} \frac{\partial g_1}{\partial y_k} \\ \vdots \\ \frac{\partial g_{k-1}}{\partial y_k} \end{pmatrix} = \begin{pmatrix} \frac{-\partial F_1}{\partial y_k} \\ \vdots \\ \frac{-\partial F_{k-1}}{\partial y_k} \end{pmatrix}$.

Let $M^{ij}$ denote the $k-1 \times k-1$ submatrix induced by $B_{ij}$. We see then that in the replacement for Cramer’s rule, we arrive at what is $M^{kj}$ but with the last column swapped to the left $k-j-1$ times such that it lands in the $j$th column and also with a negative sign, which means

$\frac{\partial g_j}{\partial y_k}(\mathbf{a}, b_k) = (-1)^{k-j} \frac{\det M^{jk}}{\det M^{kk}}$.

Now, we substitute this into $(1)$ to get

\begin{aligned}\frac{\partial G}{\partial y_k}(\mathbf{a}, b_k) &= \displaystyle_{j=1}^{k-1} (-1)^{k-j}B_{kj}\frac{\det M^{kj}}{\det M^{kk}} + B_kk \\ &= \frac{\sum_{j=1}^k (-1)^{j+k} B_{kj}\det M^{kj}}{\det M^{kk}} \\ &= \frac{\det B}{\det M^{kk}} \\ &\neq 0. \end{aligned}

Finally, we apply the implicit function theorem for one variable for the $y_k$ that remains.     ▢

References

• Gerald B. Folland, Advanced Calculus, Prentice Hall, Upper Saddle River, NJ, 2002, pp. 114–116, 420–422.

## A nice consequence of Baire category theorem

In a complete metric space $X$, we call a point $x$ for which $\{x\}$ is open an isolated point. If $X$ is countable and there are no isolated points, we can take $\displaystyle\cap_{x \in X} X \setminus x = \emptyset$, with each of the $X \setminus x$ open and dense, to violate the Baire category theorem. From that, we can arrive at the proposition that in a complete metric space, no isolated points implies that the space uncountable, and similarly, that countable implies there is an isolated point.

## On questioning authority

A couple years ago, my friend who won high honors at the Intel Science Talent Search told me that he was talking this guy who created some app that allows you to schedule a Uber ride for later, who was also at/near the top of the same science competition, who is extraordinarily versatile and prolific. I watched a little of a video of a TED talk he gave, wherein he explained what one can learn from ancient Hebraic texts. Overall, I wasn’t terribly terribly impressed by it, though it was quite eloquently delivered. Mostly because with those types of things, one is too free to interpret and thus, the lessons/messages given were overly generic so as to make them almost meaningless, one of which was how the Bible teaches the importance of questioning authority, with reference to the refusal to bow to the golden image of King Nebuchadnezzar by Shadrach, Meshach, and Abednego as an exemplary.

Here, Joshua like many from the same cultural root portrays questioning authority as a pillar of the Jewish moral and intellectual spirit. I would say that this has already gotten to the point of cliche. There is also, again, that people have different ideas of what it means to question authority.

First of all, what is an authority? An authority can manifest itself in many forms. It can be a political authority. It can be a government, especially a “dictatorship,” as much as I hate the usage of that word. It can be a boss at work. It can be a distinguished professor. It can be an adult when you’re a child. It can be an official or not moral, religious, or political code/ideology, or commonly accepted versions of history and its verdicts, by which I mean judgments of history as opposed to hard facts more or less incontrovertible, such as what exactly happened on X day with documentation abound. It can be the tradition we are all taught to abide by growing up with little question of their rationale and relevance, especially as times pass and change.

A corollary of my last paragraph is that to talk about questioning authority alone is almost utterly meaningless. You absolutely need some context, and Joshua did provide some. In the specific example of his I regurgitated, it is standing up against a dictator, and I’ll elaborate my thoughts on that.

Growing up in America, in my social studies classes and in the media, the mantra of dictatorship vs democracy with the latter morally superior and in many cases with its defense by virtually any means justified was heard again and again that it has itself become an authority taboo to challenge by our political norms. First of all, I want to clarify that here by democracy I am referring to a political system where elect representatives from among themselves to form a governing body. There is another form of more general democracy where the government does what is, or is at least perceived as, in the best interest of the entire nation or populace. What American political culture fails to discuss sufficiently is the vital matter of to what extent the former democracy implies the latter one, with the latter’s being, hopefully, the end goal.

In contrast, dictatorships are portrayed as one lone, usually brutal dictator having absolute power, being able to order virtually anything, and thus, leading often to genocidal regimes with mass murderers such as Hitler, Stalin, Mao, etc. This image may be tempting to many but it is in reality rather ridiculous. Yes, a dictator has enormous power and stays at the top often for decades, in contrast to the four year term system in America, which is very frowned upon in our culture, but surely, a dictator is not politically omnipotent. He has plenty of people underneath that he needs to satisfy, and though he may have a cult of personality within the propaganda, people are basically free to ignore him and go about their own business. He is also a human too, just like you, with very human interests, though sure, he may be a psychopath of some sort. There is also a vital point that almost always for a dictator to come to power, he must have a high degree of support from a large number of people, and thus, dictators in practice have little incentive to work against people’s interests, with getting people to like him being largely in his interest. Ironically, dictatorships can be very good at motivating people to achieve great things and providing certain continuity and long-term perspective difficult within a system where the people can easily choose to elect a new leader. In fact, if I have someone pressuring or forcing me when I don’t want to to do what is good for me (like waking up early on a weekend) and good for the society at large (like not being a parasite), I consider that to be a very positive thing. On this note, talking with someone in China recently, that guy was like: China now has 10 year terms for leaders, and maybe it should be gotten rid of, because it’s too little time for a leader to do anything serious, as he would have to pass the torch before he can be finished. Maybe Xi Jinping should try to extend his presidency past his 10 year term. Even in America, during WWII, Roosevelt was president for 16 years.

I personally love reading and watching controversial and sensitive material that most people dare not to. I’ve read plenty of material in Chinese banned in the mainland (but of course, still easily obtainable there if one really wants), most memorable of which was the very well-written, of high literary quality, autobiography by 巫宁坤 (Wu Ningkun). I’ve watched an anti-Semitic Nazi movie and also a North Korean movie out of sheer curiosity of certain places so smeared by our media. I also think that Soviet music is some of the most beautiful music out there. I have also, not surprisingly, watched some PRC (propaganda) movies from the 50s and 60s, which I felt were very well-made. The scariest and most grotesque movie I watched was one on the WWII Japanese human experimentation camp, Unit 731. A few weeks ago, I also had the pleasure of watching Saving Private Ryan, which I also much enjoyed, though surely it’s, as a Hollywood movie, more or less well-accepted here on our soil, unlike some of the previous ones, for which many would think I’m crazy, which I’m obviously not, for watching. I would say that this is out of a combination of my political intellectual curiosity and a distaste for certain oppressive, intolerant mainstream views and norms in America. Shaped by these explorations, I am of the belief that people should be more tolerant of differences and more politically and culturally open-minded. Be emotionally insensitive and let others be who they are. Also, be reasonable, precise, and stick to the facts. This is a concrete and substantive characterization of how throughout my life, I have challenged and questioned authority in the political intellectual domain.

Joshua is obviously promoting his own Jewish culture in that TED talk. On this, I’ve come to note that Jews in America are for the most part entirely unashamed, if not eager, to display and extol their culture. This is in contrast to Chinese who grow up here, many of whom try to distance themselves from their roots. Well, I guess there are self-hating Jews (like Bobby Fischer, who I feel I can understand much more now, with where he’s coming from) as well, but overall, they seem far less conspicuous. I believe the latter is out of a combination of their lack of self-confidence, the gross bastardization of Chinese culture in America, and the difficulty of learning the Chinese language in an American environment even when parents speak it at home, especially the written aspect.

There is the cliche saying that Chinese people in general, due to certain elements deep-rooted in Chinese culture, are very deferential to authority, which stifles creativity and innovation. I’ve surely thought about this and my views have evolved over time the more I’ve learned and seen. It is obviously too simplistic a notion presented by those of meager and often incorrect understanding. I do believe that Confucianism had and still has a strong element of the phenomenon described, but so did Christianity, just of a very different character.

Personally, I have to say that the more I learn, the more impressed I am with the fearless and pure spirit Chinese people have displayed in questioning and challenging authority, especially in the 20th century. I have written here before that I believe China has the richest revolutionary history of the 20th century of any nation or culture, with that of course much owing to the circumstances. China in the 20th century, being in deep trouble, had a dire need for revolutionaries, martyrs, and heroes. With this, the Chinese led by the communists essentially created a new Chinese culture on top of the traditional Chinese culture that had Confucianism as the guiding ideology. There is now a rich tradition and culture of Chinese communism, especially in military and social science, that has become holy in some sense, as is Jerusalem, which became so also out of certain formational historical events, that is very revolutionary in its essential spirit. However, the Chinese being materialists view all this as a force of nature rather than a force of God, a key contrast to holiness in the Abrahamic religions.

Another essential difference is that while Jews have more or less based themselves upon the Western system, having taken great advantage for themselves of the Western imperialism that came out of the discovery of modern science in the West, which they are also in service to politically, with reliance on it, the Chinese have more or less created an independent system from the West without kowtowing to pressures to conform, which has proven to be a correct decision, one that took much political courage and belief in oneself. The foundation for modern China was built largely in the 50s and 60s with little direct exchange with the West, if one excludes the Soviet Union from that, and in certain cases direct confrontation, with the freeze in relations owing to that in the Korean War, the Chinese challenged the Western authority successfully in a military setting in a way unimaginably shocking. It is only now very much in hindsight that while that inability to trade with the West for a few decades very much delayed China’s economic growth in certain respects, it brought about the creation of a very distinctive political culture and system deeply embedded that remains distanced from the mainstream in spite of reform and opening up, of a nature that may well be an advantage for China in the long term if not already. In this respect, Chinese culture has produced a feat and tradition of questioning authority that will forever live in our historical memory.

Another that I have noticed is the upright dedication to truth exhibited at large by Chinese scholars in the often corrupt and political social sciences that become authoritative, relative to those in the West. It is a reflection of good judgment of the Chinese people on who to promote in that arena. It does have much to do that China has in modern times been humbled by and learned so much from the West, the source of the most unprecedentedly radical and explosive growth in human history, but I also dare say that it is an indicator of very high moral character of Chinese civilization. In Chinese intellectual and media circles, bullshitting and falsifying history for political motives seems much more frowned upon. I believe that in this respect, history will eventually look at what the West led by America, that is heavily influenced by Jews in the social sciences, has done with utter disgrace, with various facades unlikely to continue indefinitely.

Speaking of truth, in terms of scientific truth, Chinese civilization has, however, contributed very little in comparison, though surely, Chinese produced a good number of revolutionary scientific breakthroughs in the 20th century, especially later in it. I find it somewhat odd how it is seldom said directly in the West that modern science is a product almost entirely of Western civilization with Greek roots and later Islamic preservation and expansion. Because scientific achievement requires so much in the way of the quality that is the subject matter of this article, surely the Confucianism based Chinese civilization has experienced a dearth of it of a nature that was only learned from the West later on. Now, Chinese are indeed quite relieved and also proud that in STEM, they have been increasingly successful and are now on the verge of reaching a world leading position, with much more to contribute to the world.

I’ll conclude with the following message. If Jews value questioning authority so much, they should let their authorities in media in America be freely and openly challenged. They should let their majority representation among Ivy League presidents and senior administrators be questioned too. In anything that is not terribly meritocratic and more connections and reputation based, their gross overrepresentation often well over 30%, so long as is objectively there, ought to be seriously questioned.