This is due to the non-associative nature of FP. (a+b) + c != a + (b+c).
I disagree that the "axiom" as stated is fundamental. My main argument with it is that I don't see an easy way to go from the axiom to usable theorems about FP.
Happy to be wrong on this, but I am missing this at this time.
I'd say not only is fundamental, it's pretty much the only tool you have to start the analysis of an FP algorithm. If you want to analyze a naive sum algorithm for instance you do that by recursively applying the a ⊕ b = (a + b)(1 + ε) rule and figure out how the epsilons bubble up - the result being that the error is linear with the sum length.
