Hi Alex,
Thank you for your post. It was interesting but it told me nothing I didn't already know. And, it didn't answer either of the questions I asked. I would still like to get answers to them.
You did, however, help me understand a little better what physicists mean by "small" dimensions. You also helped shed some light on the different points of view on these questions held by physicists, philosophers, and mathematicians.
I am an expert in none of these disciplines and I certainly can't speak for them. But I think that there is not enough overlap among them. There is too much disinterest, denial, and outright ignorance of other points of view.
Although mathematicians seem to choose terms that have some vernacular meaning, they stop there and make no formal connection between their concepts and reality.
Although physicists use the conceptual results of mathematics to great advantage, they (except for the few who come up with new theories) typically ignore or deny any aspect of mathematical theory that suggests the existence of something inaccessible to humans.
Although philosophers have a keen interest in all possibilities for existence, they typically do not learn enough math to follow physics in detail, much less to be able to understand the implications of the math that the physicists don't find useful.
For example, physicists limit their measurements of sheep's movements relative to the fence around the pasture. Mathematics suggests that the pasture may be bent, but mathematicians have no interest in commenting on whether the real pasture itself might be bent. In response, physicists say something like
"2D sheep universe can be bent, closed, etc  still being 2D. In this case sometimes it may be mathematically convenient to describe it as 2D boundary of 3D body (hyperboloid, sphere, etc) which does NOT mean that sheeps now have 3 degrees of freedom. Theit universe is still 2dimensional."
The philosopher should understand math well enough to be able to step in at this point and point out that, rather than being a mere mathematical convenience, the math and the physical observations imply that there is a real possibility for the existence of a third degree of freedom for the sheep's movement, if only they could fly.
Warm regards,
Paul
