  We propose an axiomatic approach to the concept of an intrinsic dimension of
a dataset, based on a viewpoint of geometry of high-dimensional structures. Our
first axiom postulates that high values of dimension be indicative of the
presence of the curse of dimensionality (in a certain precise mathematical
sense). The second axiom requires the dimension to depend smoothly on a
distance between datasets (so that the dimension of a dataset and that of an
approximating principal manifold would be close to each other). The third axiom
is a normalization condition: the dimension of the Euclidean $n$-sphere $\s^n$
is $\Theta(n)$. We give an example of a dimension function satisfying our
axioms, even though it is in general computationally unfeasible, and discuss a
computationally cheap function satisfying most but not all of our axioms (the
``intrinsic dimensionality'' of Ch\'avez et al.)
