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TheOtherDave comments on Polyhacking - Less Wrong
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This assumes context-insensitivity. If I'm in a triad and my relationship with #2 is different depending on whether #3 is around or not, then 3 people have six relationships.
Of course, once I acknowledge context as mattering, I'm very close to acknowledging that even dyads aren't simple. If my relationship with my husband is different depending on whether his dad is around or not, then 2 people have an uncountable number of relationships.
That seems more consistent with my experience with relationships.
I conclude that the kind of relationship that can be counted with the kind of math you propose here is fairly irrelevant to my actual relationships.
Uncountable? Really? You have as many relationships as the cardinality of the real line? In that case you could end infinitely-many relationships and still have the same number left.
Snark aside, you're just redefining what a relationship is. My friend may not behave exactly the same in various contexts, but he's not a different person and it's not a different friendship. I don't have a thousand parents (or a thousand "parentships") just because my two parents interact with me in different contexts.
A much better point to make would be that people manage O(N^2) friendship relations without apparent difficulty. Yet it seems pretty clear to me that a romantic relationship requires much more effort (more "emotional resources" we might say) than all but the closest of friendships.
I endorse setting snark aside.
I agree that we understand relationships differently. Whether that's due to me "redefining" relationship away from some default baseline that previously existed, I'm less clear about, but I don't suppose it matters much.
I agree that you don't have a thousand parents. Neither are there twelve people in a quad. Whatever it is you're counting, it isn't people.
I agree that people manage lots of friendships without apparent difficulty, and I agree that most romantic relationships require more effort than most friendships. Whether that's a better point to make, I'm less clear about, but I don't suppose it matters much.