These Hausdorff-Besikovitch dimensions are really mathematical concepts, in the limits of very special objects, and one of the major advances was to see this playing a role in real objects that do not fit mathematical constraints.
Very true. Which means that to make a major advance one still had to take a step back. In mathematics, Hausdorff defined his dimension in a very particular fashion, which includes a limit. Limits actually are not so bad in physics but also an inf, which is an impossible alteration - he wanted to obtain a measure, an outer character measure. It's a mathematical requirement of no significance outside of his field. Besides, Hausdorff being a mathematician, a very pure one, wanted to have the finish applicable to all shapes, at least all shapes in metric space. Therefore the search for generality, the search for a proper measure with which everything could be done with, was paramount in his mind. These two searches are totally irrelevant to physics and to my concern. In my discussions with Henry McKean about the Hausdorff Besikovitch dimension, I always kept telling him, "But why do you do something so complicated, so unnecessarily complicated?" And he would say, "We need it for mathematical purposes." And I always kept in mind that this may not be necessary, and the very important feature of my work from the very beginning was that different, how should I say, pre-asymptotic concepts, which are very much in the spirit of the Hausdorff dimension but very different and finite, are much more important than the limit itself. When I started talking about using the Hausdorff dimension in physics my friends, my mathematical friends, were making fun of me. They were saying, "You? That can't be used. It is, first of all, esoterica; secondly it is only a limit and it does not distinguish between shapes having otherwise very big differences. I said, " Yes, yes, yes. Except, that I'm not interested in arbitrary shapes, I'm interested in self-similar shapes. For self-similar shapes all these considerations appear. There's no local, there's no middle, there's no global. Everything is at the same scale." So in a way by walking back, by destroying some generality that the concept had from the very beginning, I was able to make it useable. And this is a lesson which got lost; very often since then, my work has consisted mathematically speaking in not going to a limit, of looking at what happens before the limit, and by doing so all kinds of very beautiful notions become meaningful which otherwise become meaningless. I would just mention in passing the idea of negative dimension as a measure of a degree of emptiness, which sounds almost like a ridiculous notion in itself, but a dimension can be negative, it measures it very effectively, and numerically, measurably, usefully.
