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.