An expository article by the the algebraic geometer Yuri Manin always has something interesting in it, and his latest, entitled The notion of dimension in geometry and algebra is no exception.
In this article Manin discusses various ideas related to the notion of dimension, ranging over fractal geometry, non-commutative geometry and theoretical physics. He begins with a quote from Glenn Gould, which is quite amusing, but of obscure relation to the notion of dimension. Then he goes on to some history, from Euclid to Leibniz, finally veering off into a fascinating discussion of the relation of algebra and geometry, and ending with the sociological comment that visual mass media is leading to a dominance of right-brain mental faculties, and thus “projects us directly into dangerously archaic states of collective consciousness.”
The body of the article includes comments on Hausdorff dimension, dimensional regularization of path integrals, the theory of operator algebras, non-commutative geometry, a weird digression on databases, and supergeometry. He also discusses “Spec Z” (the “space” naturally associated to Z, the ring of integers) making various comments about it and giving arguments for its dimension being 1, 3 and infinity. Next there are some comments on modular forms, and finally a section on fractional dimensions in homological algebra.
Its not clear how seriously one should take all of this, but Manin’s article is definitely thought-provoking.