ABSTRACT
The notion of dimension is at the basis of human thinking, not in the least because our observations of reality around us necessitate an interpretation in terms of “dimensions”. This is absolutely obvious when we think about visual observations. Nevertheless if the human race would consist of blind beings, the notion of dimension would most probably be discovered and developed in the same way but perhaps based on “feeling” or motions of body parts. Anyway the mathematical notion would be the same. The dimension of a vector space is well-understood on an intuitive level but intuition may deceive us if we apply it to algebraic varieties that are not linear subspaces of Ank. In fact, an in-depth study of the theory we have developed so far quickly leads to the discovery that really we are dealing with objects having an algebraic, geometric as well as a topological side. Each of these aspects of the theory of varieties will have a specific notion of “dimension” associated to it. As H. Weyl once stated, the angel of Topology and the devil of abstract Algebra seem to be fighting for the soul of every discipline in Mathematics! No statement could be more to the point here; indeed, the notion of dimension in geometry is standing firmly at the crossroads 148of Algebra and Topology.
