> How do you gain associativity then? At a minimum, you can't throw information away.

That's an interesting perspective that I haven't considered before, thank you.

Now I'm wondering, could we throw away some information in just the right way and still maintain associativity? That is, it doesn't seem like throwing information away is fundamentally what's preventing us from having an associative operation, since we can throw information away and still maintain associativity by, for example, converting each summand to a 0 and adding them, and that operation would be associative. However, we would have thrown all information away, which is not useful, but we would have an associative operation.