You appear to be changing the goalposts.
At first, you disagreed with me, by stating that I said religion was the entirety of culture.
I would argue that religion is an integral part of culture.
"The 'Enter' key is an integral part of a keyboard. Therefore the 'Enter' key is a keyboard."
That is what that you were first saying - and misrepresents my argument entirely.
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Now you're saying I stated that religion was an essential part of culture.
In which case, yes, he was making that argument. "Integral" (in this context, as opposed to maths) literally means "essential, fundamental", which is blatantly untrue in the context of religion. It can be (and often is) an important component of culture, but it is not a requirement. Either he/you were arguing that culture by definition must include religion as part of itself, or he/you were using words without understanding what they mean. If it's pedantry to point out someone making an untrue statement, I'll gladly call myself a pedant.
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So which one?
Besides, yes, human beliefs are and integral part of human culture. Name a human culture, throughout history, that has not been affected by religion, either inspiring it, being inspired by it, providing motivation or just been a part of it. I imagine you'll come up pretty short.
Funny, right after that bit about changing the goalposts? My point remains the same:
A thing being an important, common element of another thing does not mean that that the whole is identical to the part, or that the whole cannot exist without the part.I also think that you may be conflating "religion" with "spirituality". Religion implies belief in a god or gods, with some sort of organizing structure. So for starters there's the vast majority of Asia (at least pre-contact with Europe on a large scale); Buddhism has no gods and tends to have little or no organization. Cultures which "practice" animism would also count, as that's a completely informal belief without a real sense of gods (as opposed to spirits within various things).
On another note, if you're going to turn it in to "any culture which has in any way been influenced by the existence of religion", of course you're going to have an easier time of it. Inter-war Bolshevik culture was about as distanced from religion as possible, but under your terms it is still religious culture because the atheistic element of it was a rejection of religion. However, there are also modern post-religious cultures. A good example of this would be the Czech Republic, in which only 20.6% of people claimed to be religious, or Sweden, where only 23% of people believe in the existence of a god. But then, if you're going to argue that any culture which has been influenced in any way, shape, or form by religion is a religious culture, there's not much point in this discussion, because you can make the same argument about anything. For example, bigotry is an integral part of American culture because a decent percentage of the population has bigoted opinions.
Which brings us back to the point:
A thing being an important, common element of another thing does not mean that that the whole is identical to the part, or that the whole cannot exist without the part.