I found that interesting, and it 'fits' into his framework, but it is not clear to me why that specific relationship predominates. Intuitively, I would seek a group theoretic explanation. Back to numbers and our favorite formula. Regardless, that made me do a quick search and fished this out: [that law] has a very low kolmogorov complexity i.e.
few ALU operations of add and shift and multiply
(taylor series).
So LOW kolmo complexity = 1/r^2 or any other
simple law
source I would like to interrupt here to make a remark. The fact that
electrodynamics can be written in so many ways - the differential
equations of Maxwell, various minimum principles with fields, minimum
principles without fields, all different kinds of ways, was something
I knew, but I have never understood. It always seems odd to me that
the fundamental laws of physics, when discovered, can appear in so
many different forms that are not apparently identical at first, but,
with a little mathematical fiddling you can show the relationship. An
example of that is the Schrödinger equation and the Heisenberg
formulation of quantum mechanics. I don't know why this is - it
remains a mystery, but it was something I learned from experience.
There is always another way to say the same thing that doesn't look at
all like the way you said it before. I don't know what the reason for
this is. I think it is somehow a representation of the simplicity of
nature. A thing like the inverse square law is just right to be
represented by the solution of Poisson's equation, which, therefore,
is a very different way to say the same thing that doesn't look at all
like the way you said it before. I don't know what it means, that
nature chooses these curious forms, but maybe that is a way of
defining simplicity. Perhaps a thing is simple if you can describe it
fully in several different ways without immediately knowing that you
are describing the same thing.