I’ve heard it’s been mathematically proven that there are six more dimensions besides the four we already know about, but what I’d like to know is how in the world is it possible to prove something like that? What are these dimensions? Are they some kind of subspace or hyperspace or are they beyond our ability to imagine? And how do the physicists know what to look for anyway?
You are referring to superstring theory which is far from “proven”. The additional dimensions are said to be “curled up” to a fantasically small size which is why we would not notice them. Gravity would propagate through these extra dimensions which would allow experimental evidence for the theory given precise enough measurements.
You may want to read The Elegant Universe by Brian Greene to learn more. You can also get info online from here: http://www.superstringtheory.com/index.html
rsa beet me to it. Exelent book I also suggest it.
This is something I’ve often wondered about: why is it common to use the term “curled up” when describing this? I’ve always found it puzzling and confusing. I recently read a book (I think it was Flatterland by Ian Stewart) which explained something like it in the following way: We describe a line as one dimensional, even though it has a second (and even a third). It is just negligibly small the the other dimension(s). Likewise with a piece of paper. We use it as a model of two dimensions, even though it has a very small component in the third dimension. So is this what is meant in String Theory when the additional dimensions are described as “curled up” – that they just stretch for very tiny distances in those additional directions?
-b
The “extra” dimensions are presumed to be very small (possibly as large as a millimeter, if the experiments haven’t ruled that out yet, but probably about Planck’s length, which is much smaller than we can hope to measure). The thing is, though, things get messy very quickly if you just have them ending: What’s past the edge? The simplest solution is to suppose that they loop around, so you end up back where you started: Hence, curled up.
Basically, how they decided on ten (or more, depending on the theory) is that string theory won’t “fit” into any less than ten dimensions, and string theory looks like our best bet yet for a quantum theory of gravity, which physicists have been seeking for the past 80 years or so.
Ah. Makes more sense. Thanks Chronos. 
-b
Just to add to what Chronos said, the mathematics of string theory limits the shape of the extra dimensions to what are called Calabi-Yau spaces. So for each point in our everyday 3D space there would also be an additional six dimensions curled up into these tight little spaces like these:
http://www.cs.indiana.edu/l/www/hyplan/hanson/smoothN5.jpg
http://www.lactamme.polytechnique.fr/Mosaic/images/CAYA.41.0129.D/display.html
Re curled up. First imagine that flatlanders actually occupied the surface of a torus. If it were very large, they wouldn’t realize it wasn’t a plane. Toruses can be given a flat metric so that no local measurement can tell you it isn’t a plane. The way to see this is imagine the torus as a square in which if you go past say the left vertical edge you appear at the same place on the right vertical edge and same for the horizontal edge. (This is described by saying you identify both pairs of opposite edges.) This is what is meant by saying that a dimension is curled up. Now what string theorists seem to be saying is that we have four ordinary dimensions (including time) and then six more that are not only curled up, but very tiny, so tiny to be imperceptible at ordinary scales.
To think of 2 dimensions with one curled up,think of an ant crawling inside a hose pipe. He can crawl along the length in a stright line and never come back to the same place. But if he crawls along the width he wont go far till he’s right back where he started.
You are confusing the concept of a line with a representation of a line. The same with a plane, although the paper has width and depth, try to just think of the surface of the paper. Just as the surface of the earth is two dimensional (you only need two coordinates longitude and laditude to locate something on its surface) the earth its self is three dimensional.
No, I am not, although perhaps I didn’t explain myself very well. Of course an idealized line, by definition, has only one dimension. And, of course, any physical line that we create with a pencil on paper will have three. I was just attempting to create an analogy (or, to be honest, steal an analogy from Mr. Stewart) to describe how, even though space may extend to infinity in some directions, it may be small in others. So, while I may travel infinitely far in three mutually perpendicular directions (assuming an open universe, which is another discussion), it may be possible to travel a very small distance–a Plank length, say–in six additional mutually perpendicular directions. It is even nicer if these additional directions are “curled up”–wrapped around like a torus or the inside of a hose–because then you don’t have to worry about messy things like edges and boundaries.
-b
I’m sure when I read about it they thought there was 11 dimensions?
I believe the 11-dimensions figure includes time as a dimension while the 10-dimensions figure does not.
String Theorists used to think there had to be 10 dimensions (our 3 space and 1 time, plus 6 space curled up) until they made some amazing discoveries about a decade ago and figured out that there should be 11 instead (1 more curled up space dimension). That was called the second superstring revolution. The extra dimension was “hiding” in the strings themselves, which were really 2-dimensional membranes (“branes”) that looked like 1-dimensonal strings if one of the dimensions was small enough. This discovery allowed them to start to unify half a dozen different string theories into one big “M Theory” (work in progress). You can read all about it the Brian Greene book mentioned earlier.
So, 11 is the more up-to-date number, although ten, eleven, what’s the big deal? All but 4 are just curled up anyways.