Noncommutative geometry, which can rightfully claim the role of a philosophy in mathematicalstudies, undertakesto replacegoodoldnotionsofclassicalgeometry (suchas manifolds, vectorbundles, metrics, di?erentiable structures, etc. ) by their abstract operator-algebraic analogs and then to study the latter by methods of the theory of operator algebras. At ?rst sight, this pursuit of maximum possible generality harbors the danger of completely forgetting the classical beginnings, so that not only the answers but also the questions would defy stating in traditional terms. Noncommutative geometry...