## 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.

## Weierstrass products

Long time ago when I was a clueless kid about the finish 10th grade of high school, I first learned about Euler’s determination of $\zeta(2) = \frac{\pi^2}{6}$. The technique he used was of course factorization of $\sin z / z$ via its infinitely many roots to

$\displaystyle\prod_{n=1}^{\infty} \left(1 - \frac{z}{n\pi}\right)\left(1 + \frac{z}{n\pi}\right) = \displaystyle\prod_{n=1}^{\infty} \left(1 - \frac{z^2}{n^2\pi^2}\right)$.

Equating the coefficient of $z^2$ in this product, $-\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2\pi^2}$, with the coefficient of $z^2$ in the well-known Maclaurin series of $\sin z / z$, $-1/6$, gives that $\zeta(2) = \frac{\pi^2}{6}$.

This felt to me, who knew almost no math, so spectacular at that time. It was also one of great historical significance. The problem was first posed by Pietro Mengoli in 1644, and had baffled the most genius of mathematicians of that day until 1734, when Euler finally stunned the mathematical community with his simple yet ingenious solution. This was done when Euler was in St. Petersburg. On that, I shall note that from this, we can easily see how Russia had a rich mathematical and scientific tradition that began quite early on, which must have deeply influenced the preeminence in science of Tsarist Russia and later the Soviet Union despite their being in practical terms quite backward compared to the advanced countries of Western Europe, like UK and France, which of course was instrumental towards the rapid catching up in industry and technology of the Soviet Union later on.

I had learned of this result more or less concurrently with learning on my own (independent of the silly American public school system) what constituted a rigorous proof. I remember back then I was still not accustomed to the cold, precise, and austere rigor expected in mathematics and had much difficulty restraining myself in that regard, often content with intuitive solutions. From this, one can guess that I was not quite aware of how Euler’s solution was in fact not a rigorous one by modern standards, despite its having been noted from the book from which I read this. However, now I am aware that what Euler constructed was in fact a Weierstrass product, and in this article, I will explain how one can construct those in a way that guarantees uniform convergence on compact sets.

Given a finite number of points on the complex plane, one can easily construct an analytic function with zeros or poles there for any combination of (finite) multiplicities. For a countably infinite number of points, one can as well the same way but how can one know that it, being of a series nature, doesn’t blow up? There is quite some technical machinery to ensure this.

We begin with the restricted case of simple poles and arbitrary residues. This is a special case of what is now known as Mittag-Leffler’s theorem.

Theorem 1.1 (Mittag-Leffler) Let $z_1,z_2,\ldots \to \infty$ be a sequence of distinct complex numbers satisfying $0 < |z_1| \leq |z_2| \leq \ldots$. Let $m_1, m_2,\ldots$ be any sequence of non-zero complex numbers. Then there exists a (not unique) sequence $p_1, p_2, \ldots$ of non-negative integers, depending only on the sequences $(z_n)$ and $(m_n)$, such that the series $f (z)$

$f(z) = \displaystyle\sum_{n=1}^{\infty} \left(\frac{z}{z_n}\right)^{p_n} \frac{m_n}{z - z_n} \ \ \ \ (1.1)$

is totally convergent, and hence absolutely and uniformly convergent, in any compact set $K \subset \mathbb{C} \ {z_1,z_2,\ldots}$. Thus the function $f(z)$ is meromorphic, with simple poles $z_1, z_2, \ldots$ having respective residues $m_1, m_2, \ldots$.

Proof: Total convergence, in case forgotten, refers to the Weierstrass M-test. That said, it suffices to establish

$\left|\left(\frac{z}{z_n}\right)^{p_n}\frac{m_n}{z-z_n}\right| < M_n$,

where $\sum_{n=1}^{\infty} M_n < \infty$. For total convergence on any compact set, we again use the classic technique of monotonically increasing disks to $\infty$ centered at the origin with radii $r_n \leq |z_n|$. This way for $|z| \leq r_n$, we have

$\left|\left(\frac{z}{z_n}\right)^{p_n}\frac{m_n}{z-z_n}\right| < \left(\frac{r_n}{|z_n|}\right)^{p_n}\frac{m_n}{|z_n|-r_n} < M_n$.

With $r_n < |z_n|$ we can for any $M_n$ choose large enough $p_n$ to satisfy this. This makes clear that the $\left(\frac{z}{z_n}\right)^{p_n}$ is our mechanism for constraining the magnitude of the values attained, which we can do to an arbitrary degree.

The rest of the proof is more or less trivial. For any $K$, pick some $r_N$ the disk of which contains it. For $n < N$, we can bound with $\displaystyle\max_{z \in K}\left|\left(\frac{z}{z_n}\right)^{p_n}\frac{m_n}{z-z_n}\right|$, which must be bounded by continuity on compact set (now you can see why we must omit the poles from our domain).     ▢

Lemma 1.1 Let the functions $u_n(z) (n = 1, 2,\ldots)$ be regular in a compact set $K \subset C$, and let the series $\displaystyle\sum_{n=1}^{\infty} u_n(z)$ be totally convergent in $K$ . Then the infinite product $\displaystyle\sum_{n=1}^{\infty} \exp (u_n(z)) = \exp\left(\displaystyle\sum_{n=1}^{\infty} u_n(z)\right)$ is uniformly convergent in $K$.

Proof: Technical exercise left to the reader.     ▢

Now we present a lemma that allows us to take the result of Mittag-Leffler (Theorem 1.1) to meromorphic functions with zeros and poles at arbitrary points, each with its prescribed multiplicity.

Lemma 1.2 Let $f (z)$ be a meromorphic function. Let $z_1,z_2,\ldots = 0$ be the poles of $f (z)$, all simple with respective residues $m_1, m_2,\ldots \in \mathbb{Z}$. Then the function

$\phi(z) = \exp \int_0^z f (t) dt \ \ \ \ (1.2)$

is meromorphic. The zeros (resp. poles) of $\phi(z)$ are the points $z_n$ such that $m_n > 0$ (resp. $m_n < 0$), and the multiplicity of $z_n$ as a zero (resp. pole) of $\phi(z)$ is $m_n$ (resp. $-m_n$).

Proof: Taking the exponential of that integral has the function of turning it into a one-valued function. Take two paths $\gamma$ and $\gamma'$ from $0$ to $z$ with intersects not any of the poles. By the residue theorem,

$\int_{\gamma} f(z)dz = \int_{\gamma'} f(z)dz + 2\pi i R$,

where $R$ is the sum of residues of $f(t)$ between $\gamma$ and $\gamma'$. Because the $m_i$s are integers, $R$ must be an integer from which follows that our exponential is a one-valued function. It is also, with the exponential being analytic, also analytic. Moreover, out of boundedness, it is non-zero on $\mathbb{C} \setminus \{z_1, z_2, \ldots\}$. We can remove the pole at $z_1$ with $f_1(z) = f(z) - \frac{m_1}{z - z_1}$. This $f_1$ remains analytic and is without zeros at $\mathbb{C} \setminus \{z_2, \ldots\}$. From this, we derive

\begin{aligned} \phi(z) &= \int_{\gamma} f(z)dz \\ &= \int_{\gamma} f_1(z) + \frac{m_1}{z-z_1}dz \\ &= (z-z_1)^{m_1}\exp \int_0^z f_1(t) dt. \end{aligned}

We can continue this process for the remainder of the $z_i$s.      ▢

Theorem 1.2 (Weierstrass) Let $F(z)$ be meromorphic, and regular and $\neq 0$ at $z = 0$. Let $z_1,z_2, \ldots$ be the zeros and poles of $F(z)$ with respective multiplicities $|m_1|, |m_2|, \ldots$, where $m_n > 0$ if $z_n$ is a zero and $m_n < 0$ if $z_n$ is a pole of $F(z)$. Then there exist integers $p_1, p_2,\ldots \geq 0$ and an entire function $G(z)$ such that

$F(z) = e^{G(z)}\displaystyle\prod_{n=1}^{\infty}\left(1 - \frac{z}{z_n}\right)^{m_n}\exp\left(m_n\displaystyle\sum_{k=1}^{p_n}\frac{1}{k}\left(\frac{z}{z_k}^k\right)\right), \ \ \ \ (1.3)$

where the product converges uniformly in any compact set $K \subset \mathbb{C} \ \{z_1,z_2,\ldots\}$.

Proof: Let $f(z)$ be the function in (1.1) with $p_i$s such that the series is totally convergent, and let $\phi(z)$ be the function in (1.2). By Theorem 1.1 and Lemma 1.2, $\phi(z)$ is meromorphic, with zeros $z_n$ of multiplicities $m_n$ if $m_n > 0$, and with poles $z_n$ of multiplicities $|m_n|$ if $m_n < 0$. Thus $F(z)$ and $\phi(z)$ have the same zeros and poles with the same multiplicities, whence $F(z)/\phi(z)$ is entire and $\neq 0$. Therefore $\log (F(z)/\phi(z)) = G(z)$ is an entire function, and

$F(z) = e^{G(z)} \phi(z). \ \ \ \ (1.4)$

Uniform convergence along path of integration from $0$ to $z$ (not containing the poles) enables term-by-term integration. Thus, from (1.2), we have

\begin{aligned} \phi(z) &= \exp \displaystyle\sum_{n=1}^{\infty} \left(\frac{z}{z_n}\right)^{p_n} \frac{m_n}{t - z_n}dt \\ &= \displaystyle\prod_{n=1}^{\infty}\exp \int_0^z \left(\frac{m_n}{t - z_n} + \frac{m_n}{z_n}\frac{(t/z_n)^{p_n} -1}{t/z_n - 1}\right)dt \\ &= \displaystyle\prod_{n=1}^{\infty}\exp \int_0^z \left(\frac{m_n}{t - z_n} + \frac{m_n}{z_n}\displaystyle\sum_{k=1}^{p_n}\left(\frac{t}{z_n}\right)^{k-1}\right)dt \\ &= \displaystyle\prod_{n=1}^{\infty}\exp \left(\log\left(1 - \frac{z}{z_n}\right)^{m_n} + m_n\displaystyle\sum_{k=1}^{p_n}\frac{1}{k}\left(\frac{t}{z_n}\right)^k\right) \\ &= \displaystyle\prod_{n=1}^{\infty}\left(1 - \frac{z}{z_n}\right)^{m_n} \exp \left(m_n\displaystyle\sum_{k=1}^{p_n}\frac{1}{k}\left(\frac{t}{z_n}\right)^k\right).\end{aligned}

With this, (1.3) follows from (1.4). Moreover, in a compact set $K$, we can always bound the length of the path of integration, whence, by Theorem 1.1, the series

$\displaystyle\sum_{n=1}^{\infty}\int_0^z \left(\frac{t}{z_n}\right)^{p_n}\frac{m_n}{t - z_n}dt$

is totally convergent in $K$. Finally, invoke Lemma 1.1 to conclude that the exponential of that is total convergent in $K$ as well, from which follows that (1.3) is too, as desired.     ▢

If at $0$, our function has a zero or pole, we can easily multiply by $z^{-m}$ with $m$ the multiplicity there to regularize it. This yields

$F(z) = z^me^{G(z)}\displaystyle\prod_{n=1}^{\infty}\left(1 - \frac{z}{z_n}\right)^{m_n}\exp\left(m_n\displaystyle\sum_{k=1}^{p_n}\frac{1}{k}\left(\frac{z}{z_n}^k\right)\right)$

for Weierstrass factorization formula in this case.

Overall, we see that we transform Mittag-Leffler (Theorem 1.1) into Weierstrass factorization (Theorem 1.2) through integration and exponentiation. In complex, comes up quite often integration of an inverse or $-1$ order term to derive a logarithm, which once exponentiated gives us a linear polynomial to the power of the residue, useful for generating zeros and poles. Once this is observed, that one can go from the former to the latter with some technical manipulations is strongly hinted at, and one can observe without much difficulty that the statements of Lemma 1.1 and Lemma 1.2 are needed for this.

References

• Carlo Viola, An Introduction to Special Functions, Springer International Publishing, Switzerland, 2016, pp. 15-24.

## 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.