Seems to me “three dimensions” is simply a mathematical notion, defined as three directions which are separated by 90 degree angles. The 90 degrees mean that these dimensions are independent and that one can vary without effecting the others. This strikes me as a cultural construction, not an insight into reality. Why? Because there are many parallel “directions” in which the properties of objects can vary without effecting the others. What about colour or orientation for example?
I suppose the physicist would say that orientation is reduced to 3 dimensional movement if the object is considered a group of related objects moving through space, and the same for colour as the bouncing of light rays off of objects. So the solution seems to be to break objects into smaller and smaller units (sound familiar?) where these smaller and smaller units have nothing but positions in space and time. Regarding 9 dimensional space, it seems obvious that 9 is the number of dimensions required to satisfy the mathematical models to explain the phenomena in question. So how many dimensions are there really? Well, it depends on the level of abstraction in the model/description.