Why is the word associative used to represent the concept of the associative property?

Question

For the commutative property ...

According to wikipedia:

The word "commutative" is a combination of the French word commuter meaning "to substitute or switch" and the suffix -ative meaning "tending to" so the word literally means "tending to substitute or switch."

Therefore the choice of the word commutative to represent the concept of commutative property makes sense.

if you switch the order of the operands, you get the same result

a * b = b * a

What is the corresponding story for the associative property?

Answer 1

In French, associer means making links and connections. Therefore, associative literally means tending to make links and connections. If $\star$ is an associative law, one has: $$(a\star b)\star c=a\star(b\star c).$$ With an associative law, you get the same result regardless of the pairwise associations.

Answer 2

I don't know if this is the origin, but I like to make an analogy to when two people are associates -- they're business partners (or have some similar relationship).

So when we write $(a + b) + c$, then $a$ and $b$ are hanging out together -- they're associates. But the associative property tells us that this is the same as $a + (b + c)$, when $b$ and $c$ are the ones hanging out together.

So colloquially, the associative property tells us that it doesn't matter how we group our terms (or factors) into pairs of associates -- who we associate with whom -- to perform the individual additions (or multiplications).

Answer 3

I think Hamilton came up with the term "associative," probably as a result of thinking about the (now-called) octonions that Cayley was writing to him about. If you'll allow me to speculate, I think it's likely it's because in parenthesized expressions such as $a(bc)$, two terms that are immediately concatenated together can be called "associated" in the standard English meaning of the term.