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.
There is another extension of this that comes to mind, and that is the convergence of the form of equations that describe vastly different systems (beyond inverse square laws, I mean). For example, Newtonian oscillators, in some circumstances, form solutions to population models in certain ecological systems. It seems that every time that Equation X satisfies problem 1 and (unrelated) problem 2, the complexity of the system (in a descriptive sense, anyway) has decreased. If there is some generalized string out of which these common forms of equations can be backed, I would be speechless, stunned, but not really surprised. Maybe there is hope for a unified field theory in information theory, instead of quantum (or the wretched string theory).
It has to be Number. It must be N - simple act of partitioning and counting. I'm convinced of it, but how to prove it! :(If there is some generalized string out of which these common forms of equations can be backed, I would be speechless, stunned, but not really surprised. Maybe there is hope for a unified field theory in information theory, instead of quantum (or the wretched string theory).
- Its the geometry that's important
Agreed. But is geometry more foundational than number? (I feel like we're back in ancient Greece.) There is a very interesting tension between number and geometric form. And addressing this will drag in Cantor as well. This question has been giving headaches to the pointy head set for thousands of years. Great insight and question. I wonder if answering that is beyond our ken. I am open to the future possibility, given the root words earth-measure, of a demonstration that provides a number theoretic basis for geometry. But indeed, why would it cap at 3-D. Per ancient lore, scripture, and even recent musings of string theory, there are additional dimensions that are 'unseen'. So I would read your "We should search for why our world is 3D" as "why our perception of the world is 3D".We should search for why our world is 3D, and not some other space, if we want to know why we have inverse square laws.
- I would read your "We should search for why our world is 3D" as "why our perception of the world is 3D".
I actually considered writing the reply that way, but I shy away from speculation; we know the universe has at least three large dimensions (four if we count time). I would love to hear a number theoretical reason for three. I can't accept that it is arbitrary. I can generally see why we can't exist in 1 or 2D, as movement around other objects would be impossible for anything solid. But I have never been able to imagine a reason why more didn't occur. I suppose that's because we can't dream in 4D. We can imagine--and therefore reasonably reject--lower dimensions, but only math can tell us about higher ones.
- Is his generalized thesis that the inverse square law itself, that is so common in physics, is a result of the low pass Kolmogorov filter?
I'll pass the question along, but yes, that is the general thrust of the paper. The def. of causality was imo brilliant. Also note the remark regarding the threshold of broken symmetries is the most tantalizingly clue informing the pervasive phenomena of phi/fibonacci in natural systems I have found to date.
Have you come across D'Arcy Thompson? He spent great effort in his career trying to generalize convergent trends in evolution based on simple mathematical models (well, as simple as he could make them). He was writing before information theory was discovered. I suspect that if physical laws are generalize-able from information theory that evolutionary trends must be, as well, given that species are forged as one possibility among seemingly infinite states. (Interestingly, I first came across Kolmogorov when studying mathematical biology; there's a strange convergence!)
(Yes. I love On Growth and Form. [arch school days and my obsession to create a morphological system for generating buildings. Remember that drawing?])