Dimensions Defined without Coordinates???

[Antonio Lao]

Mathematicians, I'm sure, have solidly clear idea of dimension but not to me. So allow me to ask some questions for all who have a clear understanding of the concept of dimension.

Can we define dimension without the use of so called multidimensional reference frames with their equal number of coordinates? 3 coordinates in Cartesian system. 4 coordinates in Einstein's relativities. 10 and 11 coordinates(?) in superstring and M-theory.

What is the true relational correspondence between dimension and direction?

How can direction be defined without a coordinate system or a reference frame?

What is the physical meaning of angles (plane, solid and abstract phase angle)? Can direction be defined without angles?

[matt grime]

The answer to all your questions is yes - there is nothing to require you to have a phyisical model in defining any of these things.

Take C^n as affine n space over the complex numbers, or any other algebraically closed field. n, the dimension is the krull dimension - the maximal length of a chain of ideals satifying certain properties.

Any space with a real inner product may have angles between (position) vectors defined by the angle between a and b is given by (a,b)=(a,a)^{1/2}(b,b)^{1/2}cos{theta}

and all these are coordinate free objects.

[Antonio Lao]

Many thanks for this great enlightenment!