You are aparently ignorant of the consistency checks that string theorists do to ensure that they "write down dimensions on a piece of paper" in a consistent manner.

Well, the extra dimensions are required by various consistency checks.

For example, suppose you have a string---as it moves through time, they sweep out a two dimensional surface.  The question is, are there any properties of the two dimensional surface (called the world-sheet) that tell us about the background in which the string propogates?

Well, classically no.  There are no constraints on the number of dimensions coming from the world-sheet.

When one quantizes the theory, however, one finds that the consistency of the theory REQUIRES the number of dimensions to be greater than four---specifically, string theory is only consistent in ten dimensions.  What does consistency mean?  Basically, one must make sure that the theory is unitary---this means that probabilities add up to one, and such things.

Note that this is a significant improvement over the case in regular quantum field theory, which can be formulated in any dimension.  It is only the phenomenological constraints that we put on the theory (chiral fermions, etc.) that tell us we live in four dimensions.

Are the first three dimensions real things?  Do they have any influence on the physical universe?

If you say they allow motion or are degrees of freedom, then are you saying that they can disallow motion, prevent it?


Steve