Do negative dimensions make sense?

One way to define the dimension of a finite-dimensional vector space naturally extends to the definition of the Euler characteristic of a bounded chain complex of finite-dimensional vector spaces, and these can be negative. Somewhat relatedly, there is a notion of super vector space which also have a notion of dimension which can be negative. These explain, in some sense, the formula

$$\left( {n \choose d} \right) = (-1)^d {-n \choose d}$$

if you think of ${-n \choose d}$ as the dimension of the exterior power $\Lambda^d(V)$ where $\dim V = -n$. See this blog post for details.

Thinking in terms of negative dimensions also suggests some interesting dualities between Lie groups; for example, I think an inner product on a negative-dimensional vector space is a symplectic form, so in some sense the orthogonal groups of negative-dimensional vector spaces are the symplectic groups, or something like that. See, for example, this paper.

It is not a general notion of dimensionality, but homotopy theory and higher category theory (very closely related subjects) have found it useful to consider, for example, the $-1$-sphere: $$S^{-1}=\{x\in\mathbb{R}^0:\|x\|=1\}=\varnothing.$$ The nLab articles on negative thinking, the periodic table, and the sphere are relevant (I link to the Google cache because I can't get the pages themselves to work for me for some reason).