A collection of infinite-dimensional subspaces of a vector space is almost disjoint if distinct subspaces have finite-dimensional intersection; it is maximal almost disjoint, or mad, if it is maximal with respect to this property. We consider the possible sizes of such families, particularly the minimum (infinite) size, and the spectrum of all possible sizes. The former cannot be decided in ZFC, it is small in some models of set theory and large in others, while the latter can be made arbitrarily large.