Dimensions don't have to be 'xyz'. You can describe a position using polar coordinates just the same.

The point of having dimensions is that you can specify the position (and other conditions) of an object using a number of quantities which can be** independent** of each other. So you can change your x dimension without altering your y dimension by moving along a horizontal line on your ordinary graph paper. The x and y dimensions are independent (the posh word is orthogonal). If you used polar coordinates you still have independent dimensions so, for example, you can keep the radius constant and vary the angle and the points will lie on a circle. Using r and theta, you can specify any position on a flat sheet of paper (a plane). Radius and two (orthogonal) angles can specify any point in simple space.There are other sets of coordinates you can use but you still need three dimensions.

When you calculate hard things like the Universe and all that, you can arrive at equations which involve more than just x, y and z. The other dimension which you get from String Theory, for instance, have to be there in order to explain, in one fell swoop, how objects will interact.

So, on a simple level, xyz are enough to describe the position of objects. If you want to describe how they move about, you need to introduce time - a fourth dimension. If you need to describe the effects of electromagnetic fields and gravity all together, you can (they claim) do it using extra dimensions which, of course, we can't SEE. What they say is that moving along, say, the x dimension, we are, in fact moving through lots more dimensions. Although another object may be distant in the x dimension, its relation to us in its other dimensions may affect the way we move in the x dimension - i.e the other dimensions somehow explain how we experience the force of gravity from the Sun. String Theory *replaces* the theory which describes things in terms of Fields and masses. (Remember - Fields are not necessarily any more 'there' than anything else; we just use them as a model to predict and explain what goes on)

And, of course, when we talk of something with zero length, it may not have zero values for its other dimensions.

I am not too convinced that the two uses of 'dimension' are really that distinct from each other - except that Maths is a limited set of ideas (axioms). So, perhaps Mathematical Dimensions are just a sub set of Dimensions.

That's never stopped us from using all the rest of Maths (subset) to deal with the physical (complete set) World, though.