The notation is from math; the point of it is that you can manipulate it like a normal expression, but it's "anonymous"; o(1) means some function that goes to z

The notation is from math; the point of it is that you can manipulate it like a normal expression, but it's "anonymous"; o(1) means some function that goes to zero, but we don't bother with a name, just recording the asymptotic behavior. For example, f'(x) = lim (f(x+h)-f(x))/h can be rewritten with an error term f'(x) = (f(x+h)-f(x))/h + o(1), and then you can manipulate it more freely, say like f(x+h) = f(x) + hf'(x) + o(h). There's no need to drag out a bunch of useless names, each qualified by a set membership, to do this. I mean e(h) in o(1), e2(h) := h e(h) in o(h), etc. The failure of "reflexivity", eg O(f) = O(f), is because the anonymity hides whether the two O(f)s are referring to the same function (ie. exactly what a name would tell us).