Dear Cristi,
You refer to "...does the spacetime need a mathematics to exist?" I am not sure how they can be separated, so that spacetime can exist without math...
I will try to explain: the spacetime can be defined only as a mathematical entity created by observers (Platonic) but it is not necessary. It can be defined as a purely physical entity (real) that does not need observers. As you see the approach to spacetime can be dual: one mathematical and one physical (Einstein also attributed an elasticity property to the spacetime regarding it as physically existing entity but not the ether). I don't know if my view is clear. It is a lot of language intricacies and philosophy involved in the issue. It is very hard and maybe impossible to stay independent from our language and culture notions.