Why are multiplication and addition associative?

Question

Without workin in a rigorous formal system, how can one intuitively establish that multiplication and addition are associative operations on the real line (including negative numbers)?

Answer 1

Let's use the following analogy for the multiplication case; suppose we have a block of wood in the corner of the room, dimensions $a \times b \times c$. It doesn't matter whether you lift the side with dimension $a \times b$ up $c$ or if you take a wood-line $a$ and have it fill out the area $b \times c$; it will fill the same volume. You can extend this to a single negative number if you "drill through walls" with a sort of wood-debt; multiple negatives will require a kind of checkerboard pattern for the corner of the room.

For addition, think of it like a line-map, and you are taking directions. It doesn't matter if you stop at $a$, sleep, and then go further $b + c$, versus stopping at $a + b$, sleep, then go further $c$. You'll end up at the same place, no matter what.

If you would like a more formal derivation, you'll probably want to have associativity of addition and multiplication for rational numbers, then derive the real numbers with your favorite method (Dedekind cuts, Cauchy sequences are two examples), and then find that associativity remains.

Answer 2

In a more general setting, associativity of addition is linked to Desargues' property. See for example (http://arxiv.org/pdf/1305.6851)