Writing in the Quora Digest to which I subscribe, William La Chenal poses an interesting question: "Is there any difference between 4+5 and 5+4?" La Chenal then proceeds to answer his own question in an uniquely interesting way that only a fellow mathematician could fully appreciate or understand...I think.
He sets the stage for his rather convoluted explanation with the following story. "It's 9:51 a.m., and a mathematician has a train to catch, and an important phone call to make at 10:00. The train leaves in four minutes 35 seconds. That's four minutes to get to the train and board, t..hen five minutes to find a seat and get comfortable before reaching for his cell phone and making the important call."
"Or, it's five minutes to get to the platform in time see the the train vanishing in the distance, and four minutes to find a bench to make the phone call whilst waiting for the next train." It took a while for me to wrap my brain around that one.
In abstract algebra an Abelian group (after Norwegian mathematician Niels Henrik Abel, 1802-1829), also called a commutative group where the operation is invariant to the order in which the operands are written (commutative). Abelian groups generalize the arithmetic of addition of integers so the operation is commonly denoted by (+) plus.
If like me you are not a mathematician, it should be explained that integers are like whole numbers, including (0) zero, but they also include negative numbers -- but no fractions.
The Albelian group satisfies five axioms: closure, associativity, existence of an identity element, existence of an inverse element for each element of the group (the negative, or additive inverse), and of course commutativity -- that is, A+B=B+A for any A,B in the group.
In this context, which includes integer arithmetic, 5+4 has the same result as 4+5. Mathematicians are very keen on precise definition and context. Often altering conditions makes a big difference.
Meanwhile, a teacher in one of our North American schools is marking tests for common core maths. "The answer book says 4+5," our friend La Chenal astutely points out.
By the same token then, perhaps that is why they call a piece of lumber a two-by-four instead of a four-by-two? Then again, I may digress.
I don't know...Like I say, I'm not a mathematician! I still have trouble with grade school multiplication and fractions. Niels Henrik Abel may well have been my kind of guy.
My next assignment is to examine the conceptual integration process with respect to arithmetic word problems and how it compares to conceptual integration for sentences and other meaningful sequences. Arithmetic word problems are unique in that they combine elements of language and math and provide the opportunity for analogical alignment or misalignment between the semantic relations and the arithmetic relations in the problem. Know what I mean?
Bet you can't wait for another definitive explanatory expose in the down-to-earth, every day language for which I have gained a reputation.