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>2 x 3 = 3 x 2 feels so obvious

You just got used to it because you learned it very young. I'm pretty sure it wasn't that obvious when you learned it.

If I remember correctly, in my school multiplication was explained in terms of rectangles made of unit squares. Commutativity was literally visible to the naked eye then.

I also think you define integers as successors. Z, S(Z), S(S(Z)) etc. you can prove that a * b = b * a, given that you have defined the + function and * functions.

It's both intuitively obvious geometrically, and algebraically is conceptually tricky to grasp. Or, more precisely it's trivial to decide that arithmetic operations are commutative or not, but tricky to decide correctly for all operations a child knows.

When my kids started school I was surprised to learn that at age 4 it is not obvious that the number of items on a whiteboard remains constant if all the items are moved to a different part of the board. So what seems obvious, probably is something we just don't remember learning.

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