Historical Background and Physical Motivations

Abstract

Let us consider the free surface of a glass covering a table. And let us idealize it as being planar. What is its volume? Clearly zero since it has no height. An uninteresting answer to an uninteresting question. What is its length? Clearly infinity. One more uninteresting answer to another uninteresting question. Now, if we ask what is its area, we will have a meaningful answer, say 2 m2. A finite answer. Not zero, not infinity – correct but poorly informative features. A finite answer for a measurable quantity, as expected from good theoretical physics, good experimental physics, and good mathematics. Who “told” us that the interesting question for this problem was the area? The system did! Its planar geometrical nature did. If we were focusing on a fractal, the interesting question would of course be its measure in df dimensions, df being the corresponding fractal or Hausdorff dimension.