Earlier quoted context omitted.
A better definition then would be: Assuming each dimension is infinite, how many dimensions do you need to uniquely address a bit. The infinite requirement is there so the answer can't be reduced to "1 dimension" every time (if you had a 10x10 matrix you could just turn it into an array of 100 elements).
Slight nit-pick, but any number of dimensions can be reduced to one, using a pairing function https://en.wikipedia.org/wiki/Pairing_function or a space-filling curve https://en.wikipedia.org/wiki/Space-filling_curve
Clearly some other limitations are required to make "dimensions" a useful metric in this context.