TITLE: What is the mistake with this proof that commutativity implies associativity?
QUESTION [6 upvotes]: I tried to show that commutativity of addition implies associativity. For this I assumed that there is no associative property and $ a + b + c  $  should be interpreted as $(a + b) + c $ .
$$(a + b) + c = a + b + c = b + a + c = c + b + a = (c + b) + a=(b + c) + a$$
$$ =a+(b + c).$$
I suppose this is incorrect but I am not sure of where exactly the mistake is. Does it have to do with wrong or lack of use of parentheses?
Am I required to use parentheses to apply the commutative property because the property is defined for only 2 elements so it should be
$ (b + a) + c = c +( b + a ) $ instead of $b + a + c = c + b + a$ ?

REPLY [3 votes]: You write:

Am I required to use parentheses to apply the commutative property
because the property is defined for only 2 elements so it should be
(+)+=+(+) instead of ++=++ ?

Yes. That is exactly it. I believe you have answered your question at least as well as any of the rest of us.