Definition: Mathematical way to define the "left" and the "right" How do we define the left and the right, (such as left/right handness) from a mathematical way of definition? How can we explain this definition to some inhabitant of a distant stellar system who has not yet learned the concept of left and right?
In particle physics, we have left-right asymmetry for weak interactions in particle physics: see http://www.nobelprize.org/nobel_prizes/physics/laureates/1957/press.html

"Let us assume that the magnetic field is created by means of a coil placed like a spool of thread on a table, and that the electric current is flowing counterclockwise in the wire. Then the north poles of the cobalt nuclei will be directed upwards. The experiment, now, gave the result that the electrons from the radioactive process with this arrangement were preferentially thrown downwards towards the floor. From this it follows unambiguously that the process lacks that right-left symmetry, which one had earlier assumed. Thus, by means of this experiment it could be explained to a person, who did not know it - let us say an inhabitant of a distant stellar system - what we mean by right and left. In fact, it would be sufficient to ask him to arrange the experiment so as to make the preferential direction of the electrons point downwards. The current will then have the same direction as that in which he has to turn at the command "left face". ...  In stating this we have tacitly made an assumption which is not quite confirmed as yet but which, as far as the experiments go, seems probable, namely that the results of all experiments performed with the opposite kind of elementary particles would be just such as to reestablish the right-left symmetry. "

Question: I am asking a mathematical definition or communication on the concept of the left and the right.
 A: A theoretical definition
In math, the most basic definition would be something like "If you move so that you're at the origin in 3-d, and you rotate so that 'up' is in the direction of (0,0,1) and 'forward' is in the direction of (1,0,0), then 'left' is in the direction of (0,1,0) and 'counter-clockwise' is turning from (1,0,0) to (0,1,0) through the shortest angle." If you know about vector algebra, then you can skip the rotation and translation: if $\vec u$ is up and $\vec v$ is forward (necessarily orthogonal to $\vec u$), then left is the cross product $\vec u\times \vec v$.
A real world caveat
The problem is that the definition above only matches the real world usage of "left" if you draw the coordinate axes in the conventional way. And if you aren't next to the alien to show them how you draw the axes, the definition above won't solve that unless you already agree on the meaning of "left". This is why the question of "is there a universal thing to identify the direction of left?" is one of physics.
A: There is no mathematical defination of LEFT and RIGHT.There is neither
a position called LEFT nor RIGHT in the real world. In a big space
your own position tells that what is right and what left.It is with
respect to you.So left and right positions are thought of with respect
to something.Think that you are in the space,then symmetric law of the
universe tells wherever you position is,you are the centre of the 
universe.There you can't decide what is right side or left side.So
there is no exact defination of LEFT and RIGHT without with respect to 
something.If there will exist a defination of what you have asked for
,it must be independent of any invented theory till date.But this can't
happen because we measure or think or calculate with respect to something.
