I'm trying to get into linear algebra, but more importantly, I'm trying to get into a topic of mathematics outside of school, and now my brain doesn't have to prioritize and filter questions for getting a grade.
I have in front of me this definition of associativity for a field:
For all $a, b, c ∈ F$, we have $a + (b + c) = (a + b) + c$ and $a · (b · c) = (a · b) · c$.
What I'm curious about is why it isn't this instead:
For all $a, b, c ∈ F$, we have $a + (b + c) = (a + b) + c = (a + c) + b$ and $a · (b · c) = (a · b) · c = (a · c) · b$.
In this more verbose definition I have explicitly added the third possible arrangement of arguments to the operations of addition and multiplication.
I suspect that this is not done because it would be redundant, but I felt like I needed to double check. Just before asking this question I had asked (of myself) why commutativity and associativity were different at all--they both were just about the order of things in notation. And the answer I arrived at it was that there must a difference between addition and multiplication the operations (which activates all the symbol manipulation procedures I've memorized) and addition and multiplication the functions.
Once I thought about them as functions it made more sense that they were different. One was about the individual arguments given to one invocation function, while the other was about the arguments given to two sequential invocations. So.. that's the kind of realization I'm trying to legitimize.
Thanks!
You are right, the "extra associativities" are redundant, at least in the context of a field where we also have commutativity. Specifically, suppose it is assumed that for all $x,y,z$ we have $$x+(y+z)=(x+y)+z\quad\hbox{and}\quad x+y=y+x\ .$$ Then $$(a+c)+b=(c+a)+b=c+(a+b)=(a+b)+c\ ,$$ same as one of your other expressions. Here the first equality uses commutativity (taking $x=a$ and $y=c$); the second uses associativity (taking $x=c$, $y=a$, $z=b$) and the third uses commutativity again (taking $x=c$ and $y=a+b$).
Commenting on your second last paragraph, the difference between commutativity and associativity is that in commutativity it is the order of the numbers (or other elements) which changes; in associativity the order of elements remains the same and it is the bracketing of the operations which changes.