[chii]

but then isn't this just a tautological argument? why does multiplying the sides of a rectangle give you the area?

[red75prime]

Multiplication in the form we know it arose from practical needs of area measurement, not as some abstract operation in semirings and rings.

Multiplying the sides gives you the area because multiplication was invented/discovered to do that.

[emilgouliev]

you don't need to invoke a concept of area to justify this. you can just think of a grid of dots. since multiplication is repeated addition, each row represents one step of the repeated addition.

2x3 = 3 + 3

...

...

3x2 = 2 + 2 + 2

..

..

..

since rotation doesn't change the number of dots, we expect 2x3 and 3x2 grid of dots to contain the same number of dots, which proves that multiplication is commutative.

[dboreham]

Because of little squares being added.