I think the other replies are overcomplicating this.

+ is a binary operation, and a+b+c can’t be interpreted without knowing whether one treats + as left-associative or right-associative. Let’s assume the former: a+b+c really means (a+b)+c.

If + is commutative, you can turn (a+b)+c into (b+a)+c or c+(a+b) or (commuting twice) c+(b+a).

But that last expression is not the same thing as (c+b)+a. Getting there requires associativity, and floating point addition is not associative.