Understnading left vs. right-handed systems so I'm kind of unsatisfied with my knowledge about left and right-handed coordinate systems. I keep finding these hand-rules as an explanation whereas I'd have to like something which actually gives me a motivation for the definition and also gives me a more mathematical definition. And maybe some intuition. E.g. I'd like to actually understand what right-handed means, like what are the implications of it and also be able to determine if 2 given vectors in a 3D space are making up a left or right-handed system. 
Could someone explain it to me, without using one of the many hand rules? That's something one can do when I actually understood the basic concept, which I don't at the moment.
Thanks - Please throw as much math at me as you like :p
 A: Very short answer
Left and right-handed coordinate systems are related to what is called the orientation of a vector space.

Intuitive(?) explanation
Assuming you're already familiar with the standard Cartesian 2D coordinates, the normal orientation is to have the $x$-axis pointing to the right, and the $y$-axis pointing up. But this orientation is merely a convention.
If you start from just an oriented axis ($\mathbb R^1$) and add a dimension, you end up in the plane ($\mathbb R^2$). To obtain a proper orientation of $\mathbb R^2$, you have to decide, one way or another, which way is "up" and which one is "down". So either you adopt the standard convention, with the $y$-axis pointing up, or you could also decide that the normal orientation is to have the $y$-axis pointing down. (As a remark, it is common in image processing to have the $y$-axis pointing down, because the first pixel of a digital image is by default the top left pixel.)
Now, this problem of deciding of an orientation happens every time you increase the dimension by $1$. If you hold a sheet of paper horizontally (an $\mathbb R^2$ plane), it splits 3D space into two parts, and any of them could be the "upward" or "downward" orientation. More generally, any hyperplane in $\mathbb R^n$ splits $\mathbb R^n$ into two half-spaces, and you can choose either half-space to determine an orientation, but this choice is merely an arbitrary convention.
Something that is very specific to dimensions lower than 3, it is that you can easily observe or experience those spaces with your senses. So we just ended up applying common terminology to describe orientation in those spaces, such as up/down/left/right. I am not sure which fields used the terms left/right-handed first, but those can still be described using your hands, as you already know.
So, how did we decide which orientation would be called left-handed or right-handed? I honestly don't know. But from a very formal point of view, that is just a name for a certain convention, so there is little "maths" to explain on that level.

More details
If you had a look at the link I provided at the very start, the definition of orientation has to do with the determinant of a basis. In $\mathbb R^n$, you need $n$ linearly independent vectors to obtain a basis, and you can compute the determinant of these $n$ vectors. For a basis, the determinant is always non-zero, and when it comes to the notion of orientation we are particularly interested in whether the determinant is positive or negative.
Two basis are said to have the same orientation if their determinant have the same sign, and that property suffices to define and distinguish "left-handed" and "right-handed" orientations.
Note that in general, you compute a determinant after choosing a standard basis (often the standard Cartesian coordinates). This means that picking bases with a positive determinant as the standard orientation, is just saying that you want to match the orientation of that standard basis. But generally, I'd say the choice of that standard basis is guided by what is the simplest to use, given the data or the problem you're working with. So there is no intrinsic reason why one orientation is "better" than the other. It mostly comes down to consistently choosing one orientation that will not confuse you, and that everyone can recognize easily.
