Re: how many dimensions do physists say space-time has?

In the everyday world we perceive space to have just 3 dimensions.
Frequently, physicists also call time a dimension, making a 4-D
"spacetime".  This simple sounding idea is actually quite subtle:
the concept of spacetime arose because Einstein's theories of relativity
actually say that, in some sense, space can be converted into time
and vice-versa.  To give an example, inside the so-called event horizon
around a black-hole, what "outside" observers consider to be space, the
unfortunate "inside" observer will consider to be time.  Thus the "inside"
observer is doomed because s/he can't resist the onward flow of time
taking them to the centre of the hole!

However, we shouldn't fall into the trap of considering space and time
to be identical.  Although we understand them to be intimately related,
clearly there is something special about time in as much as each observer
is constrained to move steadily forward according to their own direction
of time. For this reason physicists often say spacetime has 3+1 dimensions 
rather than 4.

Anyway, moving on to the possibility of yet more dimensions.  A number
of the theories which have been developed in recent years, as part of
the attempt to unify the forces of nature, have required extra dimensions.
For example, the particle physicists current favourite line of attack,
so-called  superstring theory, is formulated in 9+1 dimensions (the
"record" for any theory to date is 26!).  Of course, these extra
dimensions aren't just invented willy-nilly, instead they seem to
be required by the mathematical formulation of the theories.

But, at this point, you might well ask where this extra dimensions
could be hiding?  In these theories, there is nothing special about
the extra dimensions, so in principle you or I or photons of light,
or whatever, should be able to move through them if they are there.
The suggestion is (and this is of course speculation) that the extra
dimensions must be "compactified", in other words they curve round
and meet themselves behind.  To picture what this means, if a dimension 
was curved on the scale of a few meters, we would see the world rather 
as if we were standing in a room with large mirrors on opposite walls - 
as the light rays went round and round the compactified dimension, we 
would see every object repeated endlessly. (Strictly speaking, the
mirror analogy is not quite right since we would see both the front and 
back of our heads etc.)

In the case of these higher dimensional theories, the idea is that
the extra dimensions would be curled up on scales of only 10 to
the power -33 meters: rather too small for us to observe directly!