Watching a less than glamorous documentary that I've already seen about John Nash while doing math...I know it's kind of a silly question but I'm trying to answer the question WHY is addition commutative, implying things like it won't matter what order you put your groceries on the conveyer belt; you could try putting the cheaper items first or last or in any order and the total will be the same (at least in this parallel--in others, addition need not be commutative).  The question is why.  I am happy to report that the commutativity is just another consequence of the research I have shared with some of you about what I call grammatical systems.  The lattice grid with the taxicab metric is a grammatical system.  I can use the general grammatical system induction principle which works in all grammatical systems to prove that the distance between (p,q) and the origin equals the distance between (q,p) and the origin, hinting that one possible answer to my question lies within the symmetry of rectangles.  Of course, proving the commutativity of addition isn't all that interesting but another consequence of grammatical system induction is that a similar principle applies to all of set theory which forms the basis of the majority of math.  In the back of my mind, the big question I have which I may never solve is whether or not axiom independence can be proven this way.