The difference between the European concept of civilisation and other concepts of civilisation is that the European concept purports to be a universal category. Most cultures have a sliding scale of familiar-to-strange, and that often maps to a scale of superior-to-inferior. Most cultures, historically, have been happy to leave it at that: Rome is the most Roman, ergo, the best. China is the most Chinese, ergo, natural rulers of the whole Earth. Mongolia is the most Mongolia, so your stuff is now my stuff. Modern Europeans are no different in adopting that particular strain of arrogance.
What changes is, Europeans start framing this as a neutral, scientific principle. It's not that Europeans are definitionally more civilised, but that Europeans just so happen to be the most civilised. It naturally follows that civilisation is not a uniquely European achievement, so we can start placing other cultures on a scale from more to less civilised, and while our definitions invariably end up defining "civilisation" in terms of cultural proximity to eighteenth century Europe, we frame things such that this is just a question of historical happenstance rather than collective narcissism. Even when weirdo race theorists started to argue Europeans are the only people physically capable of developing higher civilisation, they still attempt to root their argument in material science: the superiority of European civilisation is still justified in reference to something outside of and prior to European civilisation as such.
All of which, ironically, comes out of the distinctly European tradition of self-criticism, because the self-justifying loop propounded by the Greeks, Chinese or even Medieval Europeans (for whom "Christendom" was the border between worthy and unworthy cultures) was not robust enough for early modern Europeans, at least not the intellectual strata, to rationalise their imperialism. It wasn't enough that Europeans are better just because, or even that God favours the white man just because; a rational world and a rational God must operate according to clear and scientific principals. Unfortunately, for a lot of us, this tradition of self-criticism goes precisely one layer deeper, and then grinds to an absolute halt.
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Part of our collective problem, I think, is that we've never really overcome the ancient tendency to think of "civilisation" in aesthetic and ethical terms, but we insist on defining it in political, economic or sociological terms. For the Chinese, the culture and values of the Han aristocracy were naturally the most aesthetically and ethically advanced, because they were taking that measure form inside a worldview structured by that culture. So the Romans, so the Babylonians, so even these self-critical Europeans. Because these sorts of aesthetic and ethical systems tend to require a fairly complex social and economic basis, we tend to up identifying the two over-directly, and making the mistake of thinking that because we live in a society of unprecedented social and economic complexity, we must also live in a society of unprecedented aesthetic and ethical sophistication.
Kyriakos inadvertently expresses this when he suggests that the relatively innocuous and really basically harmless practice of human cannibalism is enough to get one thrown out of the club of civilised peoples, even though this practice has absolutely no bearing on the social, economic or material complexity of a given society. It is "uncivilised" because it is deeply taboo within our culture, because it is so alien to our sense of elegant and righteous behaviour, because it appears to us so ugly and dishonourable, that we can't admit the possibility of them leading a life that is otherwise quite similar to our own. The fact that the cannibalism taboo, or at the very least the strength of this taboo in modern European culture, is basically an arbitrary one, is easily overlooked, because we've come to associated possession of our taboos directly with our ability to build roads and sewers and space shuttles.
Even much of our revaluation of "primitive" peoples seems similarly rooted in aesthetic and ethical concerns. Artwork and craftsmanship that was evidence of cultural inferiority to nineteenth century chauvinists have become highly regarded in a culture which places a high value on simplicity, authenticity and practicality. Egalitarian social structures and a wariness of authority have become admirable to a culture which is becoming extremely sensitive to status-differentiation. It's often not so much that people have come to appreciated that the lifestyle of the Maori or the Navajo or the Inuit are appropriate and human responses to a particular historical and geographical context, but that their aesthetic and ethical systems, or at least the simplistic interpretation we receive through popular culture, resonates more highly with the progressively minded than the gaudy tastes and brutal hierarchies of their own immediate civilisational procurers.