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?)


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

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

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

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

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

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

Luboš Motl, and some thoughts on monopolies

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

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

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

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

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

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

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

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

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

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

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

I have commented before that

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

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

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

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

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

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

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

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

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_is 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_is.      ▢

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_is 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.


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
























中国人要敢于大胆宣传更客观正确的事实和价值观,纠正批判错误的观念和荒谬的心态,在这一点,中国觉得能做的比西方国家好,我个人也写到过,中国人更尊重历史事实,更尊重实际依据,不善于西方习惯而置之等闲之无耻之流氓扯淡宣传行为。作为特例,美国大学文化及体质,此包括其录取制度,尤其是本科生的,已经出了不少大的问题,有好多不良非准确文化之宣传,此包括对亚裔之排斥,诬蔑,及嘲笑,所以赵宇空这样的勇士才要对它们进行斗争(当然,美国顶尖大学研究绝对是最牛的,教授是完全不同类人)。这些问题当然也延续到了美国的中学小学,导致现在美国学生吸收了不少错误的观念,在美国的华裔孩子也有受此之感染,而如上所述,即使在某些精英界,服从的人得以鼓励,不接受或排斥的人,如我,有得以过非人化,虽然我比他们远远更清醒,读到的知道的远远更多,也远远更多元化,而且考虑的也远远更客观。为什么在美国华裔不太爱学文科,因为美国的文科说白了好多都是垃圾,文化界也很垃圾,埋没于荒唐愚昧脱离现实之白左反共自由主义之政治正确文化。即使好多学理工科的人也都受到了这些影响,像我提到的那个人。 看到好多中国孩子,包括那个人,在美国受到这些的影响,真的让我感到很遗憾。这样的人我会偶尔建议他们读读中文,增加一下他们的知识,认识到美国所宣传的极其片面,读不懂可以用谷歌翻译,但是他们是听不进去这些的。




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.

Hart shies away not from emphasizing the deep and revolutionary contributions to human civilization of the ancient Greeks totally merited. By far the most prominent and eternal of these was the development of the rigorous scientific method in its deductive form. The magnum opus of this is Euclid’s Elements, which was a compilation of propositions rigorously proven by his predecessor Greek mathematicians such as Thales and Pythagoras, who were pioneers of this great intellectual tradition that Western civilization and to a lesser extent Islamic civilization later on created and successfully preserved. Additionally, most certainly influenced by the Pythagorean mathematical tradition, the Greeks achieved substantially in geodesy and astronomy, with Erathosthenes calculating with an error of 2% to 15% the circumference of the earth using the differing angles the shadows from the sun made as the basis of his trigonometric calculations. From this, one can infer that by then, the Greeks already had well-established the sphericity of the earth. We even have evidence from The Sand Reckoner of Archimedes that Aristarchus of Samos (c. 270 BC) had proposed a heliocentric model in a work Archimedes had access to but has now been unfortunately lost. The English translation of that is as follows:

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.

So, despite being non-white, I shall out of my respect for reason and reality publicize my well-justified view that white supremacy is, or at least was, too manifest not to be believed in. Not too long ago, white European civilization has essentially been in a completely different league from the rest, miles ahead in its content enough to give an appearance of white man’s being a higher species than the rest, with the rest of the world more or less compelled to learn the ways of the West. Of course, being ahead in terms of accumulation of culture, knowledge, and technology does not imply biological superiority, of which IQ is the best proxy. On that, it is well established within the scientific community on the matter that East Asians have a slightly higher IQ than white Europeans, with the advantage largely being in math and visuo-spatial. This is solidly evidenced by the success of Japan and later China, and to a lesser extent South Korea, Taiwan, Hong Kong, and Singapore. The Japanese of the late 19th century were uncertain with regard to whether they could do modern science and compete with Westerners, but not long after, they came to the realization that they were not bad, with their decisive defeat of the Russians in 1905 referenced in Hart’s book. By WWII, Japan was basically an advanced country and had also produced some truly groundbreaking work in pure science at home with Takagi and Yukawa as their pioneers for mathematics and physics respectively. The Chinese students who studied in the West in some mass after China’s defeat in the Boxer Rebellion in 1900 also did quite well, though China internally was only able to modernize rapidly after the establishment of the PRC ended the instability and destruction of war at home that had plagued China for over a century, which it did very rapidly and successfully. By the 1940s, there were already quite a handful of Chinese doing revolutionary or at least first-rate work in science, particularly in mathematics, exemplified by Hua Luogeng and Shing-Shen Chern. By 1970, in spite of starting from near ground zero a few decades ago, China already had thermonuclear weapons and a satellite in orbit, notwithstanding little trade with the West following the Korean War and a later break with the Soviet Union. Now, not even 50 years after that, many people in the West are quite scared of what appears to be China’s supplanting the US as the world’s number one and thereby bringing legitimacy to a civilization with cultural values and political systems very different that evolved independently of the rest of civilization, and this is in fact what the infamous race realist scholars like Rushton and Lynn have predicted would happen largely on the basis of the higher IQ of East Asians that they to some extent popularized. Of course, this is far from absolute, with for example that the Jews (who are basically white, Western) have measured an IQ higher than East Asians of a greater magnitude than the difference between East Asians and (non-Jewish) whites. Hart, being Jewish himself, shies away not either from citing the plethora of world-changing Jewish contributions to science and culture in Europe, the United States, and Russia/Soviet Union from the 19th century on. We can see that the two superpowers, the US and the USSR both depended tremendously on the Jews for solving their hardest technical problems. For instance, the nuclear weapons programs of both countries, especially in theory, were filled with Jews, with Hans Bethe, Edward Teller, Yakov Zel’dovich, and Vitaly Ginzburg as examples. It is even fair to say that to some extent the 20th century was the Jewish century.

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:

Be rational! Be tolerant, but not of mindless PC! Dare to create new heights! Dare to improve the human race!