How many dimensions of space are there?
We cannot possibly say. All our science is an observation model: a game we play on paper and computers, that just models a projection into our senses, not the underlying reality.
By analogy our science is the red circle seen on a TV screen, where the reality could be a sphere, a spiral end on, or an infinite number of others.It also models only those things that interact with our senses. Red says as much about our senses as it does about the phenomenon. So non interacting particles , forces, dimensions are not accounted at all.
That is why the shrinking God of the gaps paradox is demonstrably false. Science explains nothing of what is at afundmental level. It models and uses the predictable behaviour of things that repeat in our observation space. It models what it normally does, not is.
Even hawking accepted the context of science when he spoke of "model dependnent reality" . That is , no unique model can account for the universe, there are multilpe models, that have to be used whwere they work, the models sometimes conflict.. He did not say so in those terms but "goodbye the theory of everything"
It is true that string theory sees a harmony in using a number 10: but the equations cannot resolve what cannot be observed, so are not determinate at anythign other than special cases.
So dimenssions could be finite, they could be infinite. We cannot possibly say.
Ask God.