I was actually just reading Cajori's history of mathematical notations when he made some comment about how Chinese algebra (Tien Yuan, is it? I am not sure about his romanization) was severely lacking in any notion of theoretical proof but was in turn dominated by associative thought, and how strange it is that while Chinese algebra was comparatively advanced, the equational form remained implicit. I suppose he was trying to explain the absence of a general theory of equations by the existence of an inelastic notation, which is assumed to be a mathematical shorthand of Chinese philosophical trends. I used to read these old math histories -- by Smith and Needham and various Oxford people, say -- and they would always point to the stratification of Chinese mathematical (specifically algebraic) thought by the thirteenth century or thereabouts. I have no idea if this is indeed true, but I guess it would all depend on what sort of algebra they're talking about and how they define it. I mean, if you designate algebra as an art which allows you to solve an equation like a2 + bx + c = 0 expressed in these symbols, it's a 16th century development. If you allow other less convenient symbols, you jump back to, what, the 3rd century BC?? And if you are to class algebra as any problem which would now be solved by algebraic methods, then 2000 BC people might be acquainted with it. But -- based for instance on how Cajori states his proposition and on even a cursory reading of excerpts from various math treatises (like "Nine Chapters on Mathematical Art," if memory serves) -- it would seem that Chinese algebra doesn't really fall into any of these categories. It's something rhetorical -- or should I say positional? -- using symbols (as we generally understand it) only rarely and late. In other words there's a whole lot of abstract monosyllabic technical ideograms indicating generalized quantities and operations. And then the counting board was laid out in such a way that certain places were occupied by specific kinds of quantities (unknowns, powers, etc). So something like a permanent filing system of mathematical patterns was established, which ought to explain all these generalizations about organic thinking and obsessions with concrete problems etc. (Well, such a complete positional notation should render unnecessary most of present fundamental symbols... and yet would possibly lead to a position from which no further advance was possible).
Anyway, what I am really wondering about is how this dovetails with the fact (??) that the main importance of mathematics in Chinese history always seemed to be in relation to the calendar, and the implications thereof. In "Lives of the Mathematicians," for example, you find all these mathematicians who were at one point or another in their lives called upon to remodel the calendar. The establishment of the calendar was, as per a long-established corpus of cosmological beliefs, the prerogative of the emperor, and its acceptance on the part of tributary states signified loyalty to him. When famines or rebellions occurred, it was often concluded that something must be wrong with the calendar, and mathematicians were asked to reconstruct it. This preoccupation might have fixed them irretrievably to concrete number (which I was mentioning earlier), being that in the calendrical field mathematics would have been considered socially orthodox and Confucian. Though unorthodox Taoist connections are sort of assumed de facto but you're still missing thought-connections between, hm, alchemists like Ko Hung and mathematicians like Sun Tzu. And were there Chinese mathematicians who wrote their theories and equations in verses, as what mostly happened in the codification of Indian mathematical knowledge (Lalivati, hail ^^)?
... Yes, right. This is all because Kristina and I were talking about fortunetelling and onmyoudo, and I asked her whether onmyouji constructed the orthodox court calendars, and what sort of calendars they made (if someone says Chinese-of-course, I'll -- do something), and whether they had more to do with otogi-zoshi notations and astrological metaphysics (... the twenty-odd volume Japanese Luni-Solar Calendar, anyone?) than, uh, a specific (that is: strictly) mathematical canon. With the fortunetelling reeling somewhere in between, because it should depend.