* Ancient Chinese mathematics was completely sterile, and never really got anywhere. Even if they had disconnected mathematical "proofs", they never realized proof as a general principle (to handle the more intricate notions - more and more intricate logical relations and functions), and so instead the mathematics just remained as astronomy or surveying. The myth that they had a general "proof" of the Pythagorean theorem is
- the claim originates in a faulty translation by Needham. Of course, the Greeks never quite realized the full power that proof might have, but they nevertheless applied this principle in geometry, whereas the Chinese avoided geometry.
* Chinese "philosophy" seems to be mostly some sort of elaborate but feeble baby-talk. They did not have any general conception of the syllogism, nor the formal analogy between inference and algebra (Boole), nor quantification (or even the implicit idea that a "general" concept - every time an algebraist writes down a symbol or diagram - consists of varying every possible particular case, of changing one part of the diagram while another part is unchanged). Their neo-Confucian "synthesis" (Zhou Dunyi, Shao Yong, the Chengs, Zhu Xi [Chu Hsi], even the polymathic Wang Fuzhi [Fu-chih]) was wholly "metaphorical" instead of logical: see this and this , this and especially this . Very early on, the Chinese had a brief period of military engineers (the Mohists ) who were engaged in dialectical analysis, but that still remained trivial since it was unfertilized by (sufficiently formal) mathematics - the entire point of mathematics was the perfect generality of its concepts, that one does not refer to any particular thing (e.g. particular triangle) but every possible object with a certain property. And of course the puzzle of how purely "deductive" inference can be non-trivial is a starting point for much of meaningful "philosophy" (Leibnitz, Kant, Peirce, Frege, Husserl, Wittgenstein, Ramsey etc. all started from the philosophy of mathematics).
Zhou Dunyi's chart of the "supreme ultimate"
The other branch of Chinese "philosophy" is really just Buddhism - an interesting recent attempt to synthesize Whitehead with the Hua-yen Buddhists can be found here . Both formal philosophy and Husserlian phenomenology seems to have caught on in China. Chen Qiwei wrote an interesting article on C. S. Peirce.
* The Chinese discovered the magnetic compass by pure trial-and-error. Nevertheless it was not the compass itself, but the combination of it and sufficiently accurate clocks (the most difficult feat of engineering before the industrial age) that enabled the accurate determination of longitude (that only European countries were able to do). This is common knowledge among specialists in the history of technology (though apparently not among "general historians").
* Chinese "martial arts" is of very dubious value, except maybe as eye candy in martial arts movies.
* Positive aspects are: the calligraphy , classical gardens , the very long novels, elaborate food , ceramics , jade , ink painting, furniture (they have these special wooden construction techniques) and cinema (movies like "Crouching Tiger" etc. are interesting) - of which they are completely unmatched in terms of the absolute quantity of high-quality work, requiring the finest technique. Calligraphy is their unique, "abstract" art form, just as classical (instrumental, tonal and atonal) music is for Western Europe.
CIA Documentary on China (more relevant today than ever):