Why can ropes pull but not push? If I have object that is heavy, I can pull it with rope but cannot push it. Why? What breaks the symmetry of the system? I can push or pull anything if I choose, so why is the push possibility not possible?
 A: I think this is the most complex answer possible to OP's question; I don't know if I should be proud or ashamed of myself.

I can pull it with rope but cannot push it. Why? What breaks the symmetry of the system?

The system has symmetries, but not the one you are thinking about, you are confusing yourself. The true symmetries arise from the following facts:

*

*The robe has fixed lenght $l$

*The rope is bendable but not stretchable

Think about it: you have an object and a rope attached to it, this divides 3D space in two regions: the first one is a sphere of radius $l$ and center in the position of the object; if you have the other end of the rope, the one not attached onto the object, in this region then everything is fine! In fact the rope is bendable!
But you cannot have the other end of the rope outside the sphere, because this would mean that the rope has broken since it has a fixed lenght $l$! This implies that if you try to get the other end of the rope outside the sphere then the center of the sphere must move to prevent you from getting into that impossible configuration! (Impossible without breakage or deformation)
