Of course non-commutative can be a part of a mathematical model; no, I'm not showing the opposite point. For clarification, my point is this: if some operation

Of course non-commutative can be a part of a mathematical model; no, I'm not showing the opposite point. For clarification, my point is this: if some operation can be proven to be commutative, but you don't acknowledge that it is, you've just made things more difficult for yourself (and maybe others). It would be absurd if someone worked really hard to evaluate a_1+a_2+...a_n strictly left to right, when there might a more convenient (from a human grey-matter standpoint) order to evaluate those numbers. That's because we know that addition here is associative and commutative. Uncovering these properties is what math is all about -- discovering these truths and exploiting them to good effect. If you can prove that something is not commutative given your axioms, you can attempt to revise your axioms -- maybe one axiom was redundant and only served to impose unnecessary constraints on your model. If you can't find a revised set of axioms that satisfy what you need, congrats -- you've found the simplest system you could come up with -- though someone might, down the road, come along and show you that there was, indeed, a simpler set of axioms that you couldn't envision. That's the process of mathematical development. If you're good at math, you'll discover the axioms you need to get the generality you're looking for, and if commutativity is to be had, you'll find a way to get it. If you find no way to get you some commutativity, you at least avoid pretending that you have it (that is, you don't write buggy software). Good math skills will either get you commutativity (which is intuitive) or prove non-commutativity (intuitive again, because you're avoiding bugs) -- it's win-win.