# Why the 1700 Japanese could have discovered calculus and modern science by 2200 or 2700

Originally published at 狗和留美者不得入内. You can comment here or there.

Some thoughts on Greek, Persian, Arab, Indian, and Chinese science

In the above piece, I omitted the accomplishments of Japanese mathematics based on the soropan (a Japanese version of the abacus imported from China in the 14th century via the Korean peninsula) and a Chinese 13th century mathematics text by Zhu Shijie [3]. These are well documented at this Japanese website [8] and in a book [4] by early 20th century math historian Yoshio Mikami. I have also uploaded here a PDF of that book: historyofjapanes00smitiala_bw.

I had known about Seki Takakazu [1] before, who was the first Japanese mathematician to do original work that was history making. There were Japanese mathematicians before him, but they mostly wrote books pertaining to the soropan as well as regurgitation of some results from 13th century Chinese math. I had long read that he discovered determinants and even Bernoulli numbers, and in addition to that, computed pi to 10 decimal places through a method rediscovered in the West in 20th century.  Today, I was shocked to learn that a Japanese mathematician 20 some years younger than Seki by the name of Takebe Kenkō [6] actually figured the infinite series for $\arcsin^2 x$. I skimmed thru the part on that in Mikami’s book, and it appears that he did in a computationally observational way. He also computed pi to 41 digits using some other mathematical method. What is of course more impressive of the two is how he managed to figure an infinite series for a trigonometric function, though of course it would have been much more so if he did so in deductive way, as Madhava in 14th century India perhaps or likely did. Mikami in his book notes that Seki never left, so it is unlikely that he got ideas from the West. And while the West and Arab world was leagues ahead at the theoretical level, China was ahead at the computational level, with relates to how Seki figured out determinants and resultants before the Western mathematicians did, not to mention Bernoulli numbers around the same time.

Before I easily noticed a certain relative absence of Chinese in mathematics, with an emphasis on lack of rigorous proof or axiomatic method. But in high school, I already knew of the likes of Liu Hui, Zu Chongzhi, Yang Hui (Pascal’s triangle in China is named after him), etc. Zu Chongzhi’s book wherein he obtained $3.1415926 < \pi < 3.1415927, \pi \approx \frac{355}{113}$ unfortunately was lost by the 12th century (they say the later students found it too difficult). Nonetheless, the result was recorded by Li Chunfeng and thus still preserved. The Chinese with the Confucian tradition, the imperial exam system well established around 700, and a large nation to govern later deemphasized mathematics. On the other hand, Japan, have gotten its culture, including its writing, much from China, were not burdened by these. So when the mathematics knowledge from China entered Japan, the tradition of wasan established itself quite quickly and solidly, and very quickly, the Japanese vastly exceeded what the Chinese had done in mathematics.

Given that in addition to $\arcsin^2$ series result, a different Japanese mathematician by name of Toshikiyo KAMATA [2] figured out the series for $\sin$ as well. So one can only imagine, given that the wasan tradition was already deeply embedded by then, that the Japanese, assuming left alone, would have by 2000 AD figured out the series of more trigonometric function. And given that those wasan mathematicians in Japan also figured out much later, presumably devoid of direct Western influence, a technique for finding minimums and maximums of polynomials that Fermat figured out in 17th century, it’s reasonable to expect that they would have discovered calculus at the level of Leibniz and Newton by 2500 AD at the latest.

It’s interesting how much Japan, after learning what it could from China, complemented China in an area where China was relatively weak in, relative to its size especially. Again, Japan was not burdened by imperial exams, exclusive use of logographic characters for writing, study of the classics, and an gigantic land and population to govern. As for the problem of Chinese characters, the Koreans introduced an alphabet in the 15th century, Hangul, and given that the first book printed with metal movable type happened in Korea, one can imagine that eventually, there would have been a printing explosion in Korea as well, notwithstanding some opposition to Hangul by the elite, which was revived in the late 16th century. There was significant interaction between Japan and Korea, especially in the late 16th and early 17th century when Japan tried and failed to conquer Korea, which according to Wikipedia due to military pressure had developed cannons more advanced than Chinese ones.

There’s a lot of anti-Japanese sentiment in China (much for Japanese conquests in China in modern times), much of which comes across as rather infantile. Some Chinese still make fun of Japan for having gotten its writing from China, viewing Japan as very derivative in general. However, the Japanese elite, despite their much lower start, noticed the fun and potential importance of mathematical research much more than did the Chinese elite did and made some serious progress in that regard. There are some Westerners who believe that without the axiomatic system of the Greeks and with Confucianism and an imperial exam system inhibiting innovation in science and technology, especially theoretical science, East Asians would have never gotten anywhere serious. I used to believe this too, but after seeing what the Japanese did, no longer so. It is true of course that the East Asian way of going about mathematics was a high M lower V type, while the Greek way, so abstract and theoretical, was the high V lower M type, which is consistent with psychometric results. Interestingly though, in the modern era after intellectual contact with the West, the Japanese have excelled more in algebra than in analysis.

Seeing how the Japanese after taking in some Chinese mathematics soon made the Chinese in mathematics look like idiots, my opinion of the Han Chinese can’t help but go down further. Relatedly, Japanese used the earlier Mongols and Manchus conquests of China to justify their conquest of China in the modern era as well. Speaking of which, during the Qing Dynasty, the only original discovery made in mathematics in China was actually done by a Mongol born in Inner Mongolia, who also helped the government with surveying in Xinjiang. In detail, after seeing that French Jesuit brought to China series for sine and cosine but not their derivations, the Mongol Ming Antu set out to find them and succeeded, discovering Calatan numbers and some other series or identities in the process. Of course, unlike with Seki, if not for direct Western influence, he almost certainly would not have worked on this problem.

In any case, seeing what the Japanese accomplished in mathematics in only  a few centuries drawing up only the work of earlier Chinese mathematicians, I believe that East Asians could have developed modern science by themselves. It might have taken a while, but I believe in Japan, it would have happened eventually. Given military competition between Japan and Korea, I can imagine that sooner or later, wasan would have been applied towards military matters as well, with hints towards physics, not to mention that according to [8], it had proliferated into the real world as well. Seki Takakazu was a contemporary of Newton, and although he achieved nowhere near as math, I believe he was just as talented. After all, unlike Newton, he and his Japanese colleagues had access to neither movable type printing on an alphabet nor all that scientific literature of the Greeks and Arabs, not to mention the Western European scientific community post-Renaissance. So in this regard, Seki really was a genius of first order, who did so much heaving lifting himself, and not only that, he founded the self-sustaining tradition of wasan in Japan. The giant whose shoulders he stood on directly was Zhu Shijie, whose works were translated and disseminated by a few of his Japanese predecessors.

Though Japan was not a civilization of its own, its strong discernment and interest in science, technology, and military, as well as ambitious and revolutionary improvement based upon what it took in from the outside world much contrasts it from China. In other words, the Japanese were quick to realize that knowledge and technology from the outside world were superior to their own, but without willing kowtow to the outsiders politically or intellectually, they single-mindedly and courageously developed their own school to eventually exceed those outsiders. One saw the same happening in Japanese theoretical physics and mathematics in the 20th century. The Japanese did not have the smug, self-centered, and narrow-minded attitude that the Chinese had with respect to the outside world, who even engaged in much self-delusion upon visibly superior European mathematics, astronomy, and military technology. There are too many spineless opportunist Chinese using American name schools and companies to engage in political, academic, and business comprador-ship and shenanigans in China; in general, I don’t think that Chinese actually inherently respect knowledge all that much, instead treating academics more as a way to gain status, climb up the ladder, and feed their narcissism and engaging in much intellectual posturing. If Chinese want to improve their culture and their country, they need to actively oust those people, and instead actively promote the likes of Seki Takakazu in their home country.

References

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