Coincidentally, yesterday I was watching "Civilizations" on Netflix, and they brought up the 1986 discovery of these incredible Bronze Age statues from the 12th-11th century B.C.:
https://en.wikipedia.org/wiki/Sanxingdui
And they were saying this contradicts the "Middle Kingdom" idea, i.e. that all of Chinese culture came from one place. The book Sapiens reminds us that every civilization/government likes to make an argument for why it has inevitable "divine" authority, and China is no exception. They exaggerate their history to claim power.
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But I think there is probably a reasonable analogy to Europe. It's not inaccurate to say the "Europeans" conquered the Americas starting around the 1500's.
European states like Spain, Britain, and the Netherlands were constantly at war with each other. But they all made conquests to different places in the Americas at slightly different times (i.e. South and Central America for Spain, the Caribbean, North America, etc.)
These societies shared broad characteristics and influenced other. Not just agriculture, but also property rights, the rule of law, etc. Their spoken and written languages were mutually unintelligible, but they recognized each other's currencies and could trade with each other. They adopted each other's technologies.
Nobody would call them the same, but it is coherent to talk about them as "European". It's a matter of semantics, but I think you can also call a group of disparate people "Chinese" in the same way. Although in the latter case there is a much greater difference in time. The Europe of today has considerable cultural continuity with the Europe of 1500 (art, music, government to some extent, etc.). I don't know how true that is for pre-modern China, although I suspect that if you adjust for the time period, it's roughly similar.
I would be interested in pointers to more resources about that.
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Another point: what other term would I use besides "Chinese"? "Asian" does not seem accurate or what historians would use. I think historians would come up with a different term than "China" or "Sino" if the difference were extremely large.
For example, as far as I know Mycenaean society has basically nothing in common with modern Greece -- no continuity in art, music, government, etc. -- but it's still called "Mycenaean Greece" since the land area they occupied is the same.
https://en.wikipedia.org/wiki/Mycenaean_Greece
If there is another word for post-agricultural but pre-modern China, I would be happy to use it! "Chinese" is indeed a big and vague term with a lot of baggage to many people. (I am Chinese myself but born in America.)