MathNovatory/Pure Maths/Numerical Structure/COMMUTATIVITY

COMMUTATIVITY

This seems to be a very misunderstood concept of elementary mathematics which limits itself to stating that "addition and multiplication are commutative but subtraction and division are not". We are temped to believe that these statements are questionable and that another, far more relevant, remains unsuspected.

False

The habitual definition of commutativity runs something like this. "Addition is commutative because 2 + 3 = 3 + 2, but subtraction is not commutative because 2 - 3 /= 3 - 2". Multiplication and division are treated the same way.

The problem here seems to be one of misinterpreting the syntax of the mathematical symbols. The "+" sign is interpreted as being a conjunction, like the word "and", when in reality it is the imperative verb "add" with complement 3, creating a phrase "to the value 2, add the value 3". The value 2 is not there by chance, but is the result of "to the value 0, add the value 2", for a complete phrase "to the value 0, add the value 2, then add the value 3". This is evidently commutative with "to the value 0, add the value 3, then add the value 2". Now if the same procedure were applied to subtraction, we would have "to the value 0, add the value 2, then subtract the value 3", which would be perfectly commutative with "to the value 0, subtract the value 3, then add the value 2".

In this light, addition, subtraction, multiplication, and division seem all perfectly commutative.

True

Lost in a jungle of misinterpreted syntax, the veritable enigma of commutativity seems to have been completely overlooked, and that is the commutativity of
(a) the "times" an operation is made
with
(b) the "base value" of the operation.

How come
2 + 2 + 2 = 3 + 3
when
2 x 2 x 2 /= 3 x 3
?

What is the basic difference between addition and multiplication
     which makes one commutative and not the other?
All are welcome in the task of finding a satisfactory explanation.