The title is the question. But I don't really know why our real world has 3 and not more or less dimensions. Are there some mathematical considerations about this fact?
String theory uses some quantum mechanical computations and has found out that the critical dimension is 10. If 6 dimensions are compactified one has 3 dimensions plus one time dimension. The 6 compactified dimensions mathematically should induce our 4 fundamental forces of nature. However, I don't really think that String theory is true.
Are there different kind of explainations of this fact?
The real world is the Space of the Time. It has many dimensions. Our material body has 3 dimensions in the Space, because we consider the "real" World as the 3-dimensional space. The General Relativity is the first step to understanding that the World has many dimensions. The base of the GR is the space-time where the time has only one dimension. It is possible that there exist many variants of different worlds, where and the space and the time have different values of dimensions.
We perceive space as a 3-dimensional world, because our physical body has 3 dimensions. The Special Theory of Relativity explains that the world possesses three spatial dimensions and one time dimension. The Minkowski 3-space is flat, and time flows uniformly therein. The 3-space of the General Theory of Relativity is curved, and time does not flow uniformly. Moreover, time can be stopped therein (on the surface of a black hole). It can also flow in the opposite direction, from the future to the past (inside the mirror world): so GR adds a new dimension --- outlook beyond the mirror. We cannot see other spatial dimensions, but we can assume that time has many dimensions. Let every body has own flow of time, as own particular dimension of time. It is possible to accept by means of the fractal theory, where the space-time of the Universe is a fractal, and all the bodies are its fragments (planets, stars, galaxies..). This is the multi-dimensionality of the Space-Time of the Universe. These fractals possess fractional dimensions, therefore all the fragments interact with each other.
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