TITLE: Why do we need to calculate dimensions?
QUESTION [9 upvotes]: After studying Kenneth Falconer’s Fractal Geometry, I got some techniques for calculating the Hausdorff dimension. However I am a little bit disappointed because in the application part, the book is still only discussing calculating dimensions. Always calculating dimensions — in number theory, in complex analysis, in thermodynamics, etc.
So here come two questions:

Why do we need to calculate dimensions? I mean, by calculating dimensions, can we get anything other than the values of dimension, such as a theorem in complex analysis (of which the statement doesn’t include the concept of Hausdorff dimension) that cannot be proved without using the properties of Hausdorff dimension?
Does the study of dimension give us a lot of unexpected result in physics? I know that the chaotic nature of a system makes its evolution unpredictable. (An example is forecasting weather.) But can fractal analysis make the weather forecast more accurate? Or does the study of fractals just illustrates the error we have when forecasting weather without helping us to minimize it?

I am having a hard time to make this question specific, but maybe you still think it's too general. I hope you can help me to make it more specific.

REPLY [2 votes]: Consider the following problem in complex analysis: you have a compact set $K \subset \mathbb C$, and a bounded function that is holomorphic on $\Omega \backslash K$, where $\Omega$ is a domain containing $K$. Under which condition on $K$ can you automatically extend $f$ to a holomorphic function on $\Omega$? If it is the case for every such function $f$, $K$ is said to be removable.
Riemann's removable singularity says that you can do it if $K$ is finite. It is obvious that you cannot do it in general if $K$ has interior. 
It turns out that the Hausdorff dimension of $K$ plays a role. If its Hausdorff dimension is less than one, $K$ is removable. If it is more than one it is not. It if is one... then it's complicated. See https://en.wikipedia.org/wiki/Analytic_capacity for more details and references.
As for the second question, I am not a physicist and cannot answer. I will simply say that I would expect that in questions related to numerical simulations, knowing the Hausdorff dimension would tell you what mesh size to use for a given precision.