[hpcjoe]

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.

[augustt]

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.