Is associativity implicated by commutativity I definitely feel like an idiot because of the fact that I have to ask you the following question, but anyway I just don't know any other option.
If just proven two things which are the opposites of each other, so at least one of my proofs have to be wrong.
I tried to solve the following problem:  

Definition: $a \boxplus b = 2a+2b$. $\boxplus$ is commutative.
  Is $\boxplus$ assosiativ?

So I found two ways to solve the problem:
1. per definition
We want to show $(a \boxplus b) \boxplus c = a \boxplus (b \boxplus c)$.
$(a \boxplus b) \boxplus c = (2a+2b)*2+2c = 4a+4b+2c.$
And on the other side
$a \boxplus (b \boxplus c) = 2a+2(2b+2c)= 2a+4b+4c$
$4a+4b+2c \neq 2a+4b+4c$. So $\boxplus$ is not assosiative.  
2. Show that commutativity implicates associativity
We want to proof $
(a \circ b) \circ c = a \circ (b \circ c)$ [$\circ$ shall be commutative]
$
(a \circ b) \circ c = a \circ b \circ c = b \circ a \circ c = b \circ c \circ a = (b \circ c) \circ a.
$  
I know that my mistake have to look stupid and obvious, but I just can't see where I'm wrong.
Thank you all for helping me!
 A: Your first proof is correct, other than the fact that you write (twice) “$=2c$” instead of “$+2c$”.
The other proof is wrong from the start. The expression $a\circ b\circ c$ is ambiguous if you don't assume associativity. A commutative operation doesn't have to be associative.
A: In your second way to solve the problem, you write the expression $a\circ b\circ c$ which is not defined.

We usually only define $a\circ b\circ c$ if we already know that $\circ$ is associative, in which case we can shorten $(a\circ b)\circ c$ and $a\circ(b\circ c)$ to simply $a\circ b\circ c$ because we already know that both of those expressions are equal. If we do not know that, then the expression $a\circ b\circ c$ is not yet defined, because it could mean either one or the other, and there is no guarantee that they are the same.
Certainly, we can define $a\circ b\circ c$ to mean $a\circ(b\circ c)$, but if you define it like that, then your claim that 
$$a\circ b\circ c = b\circ a \circ c$$
becomes (using the definition) the claim that 
$$a\circ(b\circ c) = b\circ(a\circ c)$$
which is not a claim that can be proven by simply claiming commutativity.

Similarly, if you define $a\circ b\circ c=(a\circ b)\circ c$, you hit a problem because then you actually can claim that $$a\circ b\circ c= (a\circ b)\circ c=(b\circ a)\circ c=b\circ a\circ c$$
but then the claim
$$b\circ a\circ c=b\circ c\circ a$$
is equivalent to
$$(b\circ a)\circ c = (b\circ c)\circ a$$
which can again not be proven usinc commutativity alone.
A: You are using associative property in your proof. $$(a \circ b) \circ c = a \circ b \circ c = b \circ a \circ c = b \circ c \circ a = (b \circ c) \circ a.$$
Without associativity the following $$a \circ b \circ c $$ simply does not make sense.
