Our current leading theory of quantum physics is quantum field theory. For example this is what we use to describe the collisions at particle accelerators like the large hadron collider. Quantum field theory is experimentally tested to a very high precision.
But curiously we find that quantum field theory works in any number of dimensions. It's true that the physical behaviour it predicts is very different in different numbers of dimensions, but the theory itself works just fine. Indeed physicists often use quantum field theories in two space and one time dimension, or indeed just one space and one time dimension as models to help investigate various aspects of how the theory works. Alternatively, eleven dimensional supergravity is a quantum field theory that has been suggested as a way of including gravity in a quantum field theory.
The exception to this is string theory, where the theory requires that there be nine space and one time dimensions, though it remains to be seen whether this is physically relevant.
However in classical physics rather than quantum physics that we find arguments suggesting three space and on time dimensions are special. See for example Is 3+1 spacetime as privileged as is claimed?
