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Hi Alex,
Thank you for your post. We discussed this issue before at
http://www.astronomy.net/forums/god/messages/11070.shtml?show=top
but we didn't finish the conversation. I'm glad you brought it back up. Let me try another approach at answering your question.
In Differential Geometry, there is the concept of a manifold. A manifold is a space of, say, n dimensions that exists, or is embedded in a space of m dimensions where m>n.
All of the topological properties of the manifold can be the same as they would be if it weren't an embedded manifold.
The inverse square (cube, R, etc.) law, as you know, is a result of the topology of the space. In 2D, for example, you get an inverse R law. An inverse R law holds for figures on a sheet of paper because the paper is 2D. If the sheet of paper exists in a 3D room, the inverse R law still holds on the sheet of paper. It still holds because the sheet of paper is a 2D manifold in 3D space.
So in the same way, if our 3D space were a 3D manifold in 4D or 10D space, the inverse square laws would still hold just the same as if it weren't a manifold.
Everything in the manifold would still be 3D and no instrument or pointer could align itself to point in any direction that is not a linear combination of three basis vectors (e.g. one meter North, one meter East, and one meter Up). Therefore, we have no way of accessing anything outside of our manifold. Just the same as any figure drawn on a sheet of paper laying horizontally cannot point up.
This is completely consistent with the notion of dimensions being degrees of freedom of motion within the space. As long as everything in the manifold is 3D, any motion is confined to the manifold so there are still only three degrees of freedom. The same way as an ant crawling on a sheet of paper is confined to two degrees of freedom (unless it leaves the manifold and falls off). The ant has only two degrees of freedom even though the paper is a manifold in 3D space.
I think this easily, and mathematically, explains why the existence of large extra dimensions would not affect our inverse square laws.
Warm regards,
Paul
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