Where does the commutative property of addition originate from?

I think I'm probably missing out something pretty obvious here, but is there a reason why addition is a commutative operator?

I've tried thinking about it as "because it's defined this way", but I don't see how this is consistent with how matrix multiplication is not commutative. To be a little more precise, isn't the non-commutative property of matrix multiplication a result of how we have defined the operation of matrix multiplication, instead of "because matrix multiplication is defined to be non-commutative" (i.e. a property following how the operation is defined vs how the operation is defined) ?

• Matrix multiplication is more like composing two functions than actual multiplication, and composition is not commutative. It's also a ring axiom. – Anadactothe Mar 7 at 20:26
• Addition is not necessarily commutative either. It is over the set of integers and reals, etc.... – Brad S. Mar 7 at 20:26
• It comes from commuting along the sides of a rectangle. To go from southwest to northeast, it doesn't matter if I go east first or north. Thus, $e+n=n+e$. I can build a rectangle for any two positive numbers and so we don't need to define that real numbers commute. – John Douma Mar 7 at 20:29
• Yes, I would agree with you. Commutativity or associativity or whatever is a property of an operation that depends on the definition of that operation. An exception might be the case where you have an operation the takes two different kind of operands -- multiplication of a vector by a scalar say. Then you might define $\lambda \mathbf{X}$ explictly, and then define $\mathbf{X}\lambda$ to make the operation commutative, but I think this is nit-picking. – saulspatz Mar 7 at 20:32
• The commutative property of addition of integers comes from direct observation of reality. It doesn't matter if you pick two apples and then three apples or if you pick three apples and then two apples. In either case you will have five apples. In a similar manner one may observe that two baskets containing three apples each have the same number of apples as three baskets containing two apples each. Having observed two instances $a+b=b+a$ and $ab=ba$ one forms the abstraction: commutative property. The abstraction is confirmed as useful when it is found to apply in other contexts. – John Wayland Bales Mar 7 at 22:18