TITLE: Why identity element is so important in abstract algebra?
QUESTION [3 upvotes]: Recently, I've started learning abstract algebra and all its algebraic structures that of sets, magmas, semigroups, groups etc. I didn't yet reach the ring-like structures but still. As we all know if there exists an element $e\in G$, called the identity element, then for any element $a\in G$, we have:
$e\circ a=a\circ e=a$
I understand, at least, this is the thing that has absolute importance in formulating some properties and obfuscated names as that of monoid. Which is the set having associative operation and identity element. Yes, that makes a set behave similar to $\mathbb{Z}^+$ under $+$ (well if we add commutative property as well). But is that the only use of the identity element? When and why did it come from in the first place? Or maybe even what happens if there was no identity element? Etc. As for now, this concept seems more like a hack to fit the theory into phenomena.
Don't get me wrong, I see the beauty of all these generalities but I want some historical/philosophical background that (I believe) let me grasp this concept more intuitively.

REPLY [3 votes]: Many algebraic structures are of the form: a set $X$ together with a binary operation $X\times X\to X$ satisfying some axioms. This is so commonplace that it may be difficult to spot a bias here; why binary operation? Why not a ternary operation? A bit later on in (mathematical) life one encountered algebraic structures defined by a ternary operation, so this question is certainly not moot. Moreover, we all know that from one binary operation we can form a ternary one and in fact one of any finite arity $n$, simply by repeating the binary operation $n-1$ times. If the binary operation is associative that the choice of precisely how to iterate is irrelevant, but otherwise one gets plenty of new functions. This may explain why associativity is such a common requirement on a binary operation. In any case, we may now seek to eliminate bias altogether and ask to be honest about all arities involved in any definition of an algebraic structure. Well, let's be a bit more modest and just consider a single associative binary operation. So, we know we actually have a whole family of operations, one for each $n\ge 1$. Well, that is very egalitarian, all finite numbers treated equally, except for $0$. Is there no $0$-ary operation to be obtained as well? Some thought will lead to the answer: the result of the $0$-ary operation is the good old identity element. 
So, the identity element is the final elimination of bias. This should not be seen is too philosophical a discussion. Universal algebra is a very respectable area of mathematics.
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