Thursday, July 7, 2011

Zeno's Paradox in Ancient China

While I was having some High Mountain Oolong Tea with my parents, the topic of the diversity of the philosophies of ancient China made its way into our conversation. I grew up in Brazil so I missed out on most of the Chinese history that, as a student in Taiwan, I would have gotten in school. I've always enjoyed these topics whenever they came up in our conversations.

Our conversation started with the Hundred Schools of Thought, a Golden Age of Chinese philosophy near the end of the Zhou Dynasty (Eastern Zhou). The slow death of the Zhou Dynasty is divided into two periods: the Spring and Autumn period and Warring States period that ended with Qin Shi Huang's (the first emperor of China) unification of the Warring States.

The topic turned to the Chinese philosopher Zhuang Zi (4th century BCE) of Warring States period who, according to my parents, is considered to be Taoist in the Chinese tradition. In the eponymous work Zhuang Zi, he often engaged in dialectic with Hui Shi, who is more closely associated with the School of Names. As best they could remember, the major figures of the School of Names, by virtue of their verbal prowess, created paradoxes and exploited ambiguity in language that led to "ridiculous" results like "white horses are not horses" to confuse and defeat others who engaged them. It reminded me a little of the Sophists of ancient Greece. It piqued my curiosity and my parents urged me to do a little digging online.

I found this article on the life of Hui Shi and several translations of Zhuang Zi [1]. Looking for the original Chinese text and reference to Hui Shi, I came across this article on chapter 33 of Zhuang Zi. In it, Zhuang Zi criticized Hui Shi for his penchant for verbal sparring. According to him, Hui Shi delighted in his paradoxes and would engage with others in what Zhuang Zi considered to be pointless discussions. Among the many examples Zhuang Zi offered is this one:
which I translate clumsily word for word here to:
One meter/foot (or the ancient Chinese version of it) short stick, daily take half of it, ten thousand generations will not end
A little digging online I find the following translation of Zhuang Zi by Herbert Allen Giles (of Wade-Giles romanization system). His translation:
That if you take a stick a foot long and every day cut it in half, you will never come to the end of it.
There are some slight differences [2], but they essentially mean the same thing. After his translation, Giles gives the following suggestion on page 453:
Compare " Achilles and the Tortoise," and the sophisms of the Greek philosophers.
So I followed his suggestion and looked at the 3 versions of Zeno's Paradox listed in Wikipedia:
Achilles and the Tortoise
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.
Dichotomy Paradox
That which is in locomotion must arrive at the half-way stage before it arrives at the goal.
Arrow Paradox
If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.
I definitely noticed the similarities. So I read Giles' translation further to see if there was a solution provided in Zhuang Zi. Continuing with the translation by Giles :
And such was the stuff which dialecticians used to argue about with Hui Tzu, also without ever getting to the end of it.

Huan T'uan and Kung Sun Lung were of this class. By specious premisses they imposed on people's minds and drove them into false conclusions. But though they won the battle in words, they did not carry conviction into their adversaries' hearts. Theirs were but the snares of the sophist.

Hui Tzu daily devoted his intelligence to such pursuits, purposely advancing some preposterous thesis upon which to dispute. That was his characteristic. He had besides a great opinion of his own wisdom, and used to say, "The universe does not hold my peer."

Hui Tzu makes a parade of his strength, but is devoid of any sound system.
Just like in ancient Greece, this paradox wasn't resolved in ancient China, at least it seems that the solution was not known to Zhuang Zi (or author(s) of Zhuang Zi [4]). Zhuang Zi's citation of the paradox in his criticism of Hui Shi would seem to suggest that the paradox had no solution in his time. It wouldn't make sense for people to argue with Hui Shi incessantly on it if a solution were known, unless such were the dialectic prowess of Hui Shi that he could take a question with a known solution and twist it into a paradox.

The resolution of paradox using the mathematical solution by infinite geometric series would not come until the advent of limits and convergence of a geometric series, topics students study in high school Calculus today. I've read of a possible mathematical solution by Archimedes but I can't find a source, maybe someone can point me in the right direction. There is a reference in the Wikipedia article to the formula of a finite geometric series in Euclid's Elements Book IX Proposition 35. It seems likely that Archimedes, known for his use of method of exhaustion, would have been able to find a mathematical solution to the paradox with the formula for geometric series.

Hui Shi's version of Zeno's Paradox does allow us to ask a question that doesn't come up with the other versions. It brings up whether matter can be split forever and whether there is a smallest unit. My question is: What would happen if we divided a meter stick by half everyday for 10,000 years? Is it even possible? What other information would we need?

  1. Zhuang Zi is romanized in Pin Yin. It is romanized as Chuang Tzu in Wade-Giles.
  2. One difference you'll notice is that I used ten thousand generations where Giles says there's no end. This is actually typical of Chinese language. When describing a very long time, there is a tendency to use ten thousand (萬) to describe something large. The Japanese Banzai derives for this usage.
  3. Hui Shi who is sometimes referred to honorifically as Hui Zi, would be translated by Giles as Hui Tzu using Wade-Giles system instead of Hui Zi in Pin Yin.
  4. There seems to be some contention about the authorship of Zhuang Zi the book.


  1. My favorite modern Zenoism: In a typical job, you accrue paid vacation at a rate of, for example, one day per month of employment. While on paid vacation, you are employed, so you continute to accrue vacation.

    So once you leave on vacation, must you ever return?

  2. @Tim Someone oughta try it. It's now my new fave Zenoism. :)