Temporal Structures in Chinese Mathematics

Time and Methodos

The temporal structure of Chinese mathematics appears in at least two ways. One is its embedding into the well known cosmological and ontological dynamics which says the world is in a permanent change.
The second has a more a "praxeological" form and is discovered by an "ethno-methodological" approach to history. Jinmei Yuan is emphazising in her study "The role of time in the structure of Chinese logic" the double structure of temporality in the paradigm of Chinese Maths as the "now"-structure" of methodology and the dynamics of Ancient Chinese world-view.

Both should be understood as strictly different from the Greek approach of time and methodos as following a pre-given path/way.
From Ancient Greek μέθοδος (methodos) "pursuit of knowledge, investigation, mode of prosecuting such inquiry, system", from μετά, μέθ- (metα, meth-) "in the midst of, among, between, in common, along with, by aid of" + οδός (odos) "way, motion, journey".

But instead of denying the possibility of formalisms by Heraklit (panta rhei) and the dialecticians up to Hegel and dialectical materialism, the "now"-approach of Liu Hsiu shows an exciting possibility to do maths independently of axiomatics with its eternal truth and pre-given methodology (axioms+rules).

A striking similarity to the now-strategy is realized in ConTeXtures, a dynamic polycontextural programming language, I started a few years ago. The first step there is: design horizons! That means, the now tells, by analysis and experiences, situational, with which complexity and complication the method/strategy has to "start". A first sketch to model complex time-structures for programming can be found at:
www.thinkartlab.com/pkl/lola/From Ruby to Rudy.pdf
www.thinkartlab.com/pkl/lola/ConTeXtures.pdf

The question is not which philosophy mathematicians are supporting but what exactly are they doing when they are doing mathematics? Hence, how are they doing math is the question. This maybe called a "praxeological" or "ethnomethodological" approach (Garfinkel, Livingston). This, obviously, is in sharp contrast to ideology critical contemplations.
http://cseclassic.ucsd.edu/users/goguen/pps/real.pdf

A Western Summary of the Principles of Chinese Thinking

by Kaiping Peng, Richard E. Nisbett
Chinese ways of dealing with seeming contradictions result in a dialectical or compromise approach— retaining basic elements of opposing perspectives by seeking a “middle way.” European-American ways, on the other hand, deriving from a lay version of Aristotelian logic, result in a differentiation model that polarizes contradictory perspectives in an effort to determine which fact or position is correct.

Principle of change (Bian Yi Lu).

This principle holds that reality is a process. It does not stand still but is in constant flux. According to Chinese folk belief, existence is not static but dynamic and changeable. At the deepest level of Chinese philosophical thinking, "to be or not to be" is not the question because life is a constant passing from one stage of being to another, so that to be is not to be, and not to be is to be. Because reality is dynamic and flexible, the concepts that reflect reality are also active, changeable, and subjective rather than being objective, fixed, and identifiable entities.

Principle of contradiction (Mao Dun Lu ).

This principle states that reality is not precise or cut-and-dried but is full of contradictions. Because change is constant, contradiction is constant. Old and new, good and bad, strong and weak, and so on, co-exist in everything.
One of the first mandatory books for literate ancient Chinese was the Yi Jing /I-Ching (The Book of Changes), in which the principle of contradiction is clearly expressed. For example, its basic theme is that the world is simply a single entity, integrated over opposites.

Principle of relationship or holism (Zheng He Lu)

This principle probably constitutes the essence of dialectical thinking. It is a consequence of the principles of change and contradiction. It holds that nothing is isolated and independent, but everything is connected. If we really want to know something fully, we must know all of its relations -- how it affects and is affected by everything else. Or, to borrow a slogan from Gestalt psychology, the whole is more than the sum of its parts. Anything regarded in isolation is distorted because the parts are meaningful only in their relations to the whole, like individual musical notes embedded in a melody. [..]

The three principles of Chinese dialectical thinking are related. It is because of change that contradiction becomes inevitable; it is because change and contradiction are inevitable that it is meaningless to discuss the individual part without considering its relationships with other parts.

CULTURE, DIALECTICS, AND REASONING ABOUT CONTRADICTION
Kaiping Peng, Richard E. Nisbett
www-personal.umich.edu/~nisbett/cultdialectics.pdf

A discussion of the text is:
Brian Huss, Cultural differences and the Law of Noncontradiction: some criteria for further research, Philosophical Psychology, Vol. 17, No. 3, September 2004
www.tc.umn.edu/~huss0052/CPHP_17_3_03LORES.pdf
Some general informations about Cultural Geography:
http://www.apa.org/monitor/feb03/intelligence.html?

Brian Huss:
It is extremely difficult to provide a non-circular justification for the LNC (Law of Non-Contradiction), and yet the LNC seems to act as a basic standard for reasoning in the West. If non-Western cultures do not believe the LNC holds, then meaningful cross-cultural discussion and debate will be very difficult, to say the least. In this paper it is argued that the distinction between belief and acceptance is important in analyzing cross-cultural studies on the way people reason. [...] The distinction between belief and acceptance is used to demonstrate that the empirical data currently available fail to show that the LNC is not a universal of folk epistemology.

As a Westerner I have the feeling of reading a Western compilation about Chinese thinking (world-view, logic, ontology). I will not enter this discussion because too many assumption are made which have to be questioned. Maybe, American sociologists never have heard anything in the line of Heraklit, Hegel, Marx, Piaget and other Western dialecticians. As a base for educational and political consultation it seems to me extremely blind and hegemonistic.
Thus, I will start with only one simple question.
Obviously, my question will not deal with the problem if there is a contradiction for a Chinese farmer to be or/and not to be in the possession of $1000.-

What do we mean with "contradiction" (矛盾)?

I remember reading German, French and English translations of Mao Tse Tung’s study "On Contradiction". Most of his examples showed me that the term "contradiction" is misleading. The examples for contradiction are: polar, opposite, antagonism, struggle, etc. and logical contradiction was only a part of it.

"Contradiction and struggle are universal and absolute, but the methods of resolving contradictions, that is, the forms of struggle, differ according to the differences in the nature of the contradictions. Some contradictions are characterized by open antagonism and others are not. In accordance with the concrete development of things, some contradictions, which were originally non-antagonistic, develop into antagonistic ones, while others which were originally antagonistic develop into non-antagonistic ones."
"On Contradiction" (August 1937), Selected Works, Vol. I, p 344.
http://www.rrojasdatabank.org/mao11.htm

Mao's explanation is not easy to accept for non-dialecticians. First for Western philosophy and science there are no contradiction in the univere at all. Second, Mao's definition is in itself contradictionous. If contradictions are "universal and absolute", how do we have to understand the "but"? And the "absolute and universal" is changing all the time? Contradiction as a self-referential term, but not in Aristotelian logic. Neither in paraconsistent logics.

Then I learnt that the Chinese ideogram for contradiction, 矛盾, has absolutely nothing to do with the latin dictio and contra-dictio (speech and contra-speech). But about spear (矛)+shield (盾). Later I was told that there are not only two fighters with their spear+shield in a fighting position, but that the ideogram goes back to the hieroglyphs for sun and moon.

Not only that we are far away from any phono-logical terms of contradicting and contradiction with its logos-based duality of true and false, the structure of a fight between two fighters is not dual but 4-fold: 2 positions with spear+shield, i.e. in fact, spear vs. shield + shield vs spear.
And this is exactly the chiastic structure of change. Thus, change is not a simple continous floating Heraklitian flux but an interplay between different qualities.
In other words, the 3 principles mentioned above appear as a complex interacting pattern; "contradiction" and "change" are "one". Hence, the "speech act" of contradicting in a opponent/proponent game is a very small and specific layer, (for lawyers at court), of a "shield-盾+spear-矛"-interaction.

Therefore, I very much prefer the approach of studying what exactly Ancient Chinese mathematician did when the practised mathematics.
An important step to this kind of studies is done by Jinmei Yuan.
http://ccbs.ntu.edu.tw/FULLTEXT/JR-JOCP/jc106031.pdf

The Jinyou-Strategy of Chinese Math

"Chinese logicians in ancient times presupposed no fixed order in the world. Things are changing all the time. If this is true, then universal rules that aim to represent fixed order in the world for all time are not possible."

This sounds familiar to Heraklitian philosophy and the Western understanding of Chinese world-view. But suddendly there is something surprisingly different:

"Chinese logical reasoning instead foregrounds the element of time as now. Time, then, plays a crucial role in the structure of Chinese logic."

Because of the "mutual relations" and "bi-directional" structure of Chinese strategies I think the time mode of "now" is not the Western "now" appearing in the linear chain of "past–present–future". To understand "now" in a non-positivist sense of "here and now" it could be reasonable to engage into the adventure of reading Heidegger’s and Derrida’s contemplation about time. This seems to be confirmed by the term "happenstance" (Ereignis) which is crucial to understand the "now"-time structure.

The praxeological analysis discovers the patterns of "problem solving" before/beyond axiomatic deductions, i.e., beyond the linear pathway from problem to solution under an invariable method.

"To uncover the logical structure and presumption in Chinese mathematical art, I would like, first of all, to call attention to a few important and interesting features of the Nine Chapters:
1. None of the mathematical terms in the Nine Chapters have a given definition.
2. No demonstrations between a given problem and an answer are offered.
3. The 246 problems in the Nine Chapters mostly begin with the phrase Jinyou, which means “Now, there is . . .”
Jinyou is a general way to form patterns in the Nine Chapters.
Second, I would like to briefly summarize the patterns according to which the mathematical problems in the Nine Chapters are organized:
Pattern 1:
Now, there is ( jinyou) . . . Tell (qiu):
Answer (da):
Art/Method (shuyue):
Pattern 2:
The name of an art/method (shu) or a rule (fa)
Art/Method (shuyue):
Now, there is ( jinyou) . . . Tell (qiu):
Answer (da):
Art/Method (shuyue)
Pattern 3:
Now, there is ( jinyou) . . . Tell (qiu):
Answer:
Art/Method:
Another art/method:
Another art/method:
Another art/method:

Jinmei Yuan's comment

The first phrase here is “Now, there is . . .” ( jinyou).
If one takes a close look at the above pattern, one can easily see that “time” plays an important role in each mathematical problem-solving procedure. Almost all of the problems in the Nine Chapters start with the assumption, “Now, there is . . .” (jinyou), which is a good starting point for us to explore the logical space in these patterns.
To the extent that the time, “now” ( jin), is involved, the problems in which Chinese mathematicians are interested are particular ones, such as those that arise during a face-to-face conversation in the present.
In other words, Chinese logical space is structured in the time, “now.” Chinese people are only concerned with the logical relations that exist in the present practice, not something beyond the present time, such as “universal truth.”

The relevance of happenstance (Ereignis)

"The phrase jinyou is crucially important to understanding the patterns in the Nine Chapters. Having discussed the role of time, the now ( jin), in the patterns, the meanings of you in the phrase of “now, there is . . .”
(jinyou) should be clarified.

The character you in Chinese means that a happenstance exists or shows itself, or that something is possessed. The original character you is written in such a way that the top part is a hand and the bottom part is a moon.
In the Shuo Wen, an early Chinese lexicon, Xu Shen says, “You is the thing that does not always exist. Spring and Autumn has an explanation: [for example,] the happenstance of a solar eclipse or lunar eclipse.”

It is clear that you in the Nine Chapters does not hold the meaning of something that is given by mathematicians theoretically, but that it means a concrete problem that occasionally exists as a special event, in a particular time and space.

If one holds the presumption that there is a fixed order in this world and that things have their stable positions, then the notion of “given a problem” or “given a rule” can make sense in mathematical reasoning."

"Happenstance is the meeting between two strangers who have never met before, normally in a completely random situation."
http://en.wikipedia.org/wiki/Happenstance

Jinmei Yuan concludes

"Chinese mathematical art aims to clarify practical problems by examining their relations; it puts problems and answers in a system of mutual relation—a yin-yang structure for all the things in a changing world. The mutual relations are determined by the lei (kind), which represents a group of associations, and the lei (kind) is determined by certain kinds of mutual relations."