Dear Irek,
I found your essay to be most thought provoking. I enjoyed reading it, and I would like to make a few comments.
Early on you state "Maybe at the deepest level there are no algorithms, infinity is common, real numbers are basic? The Uncomputability signpost looks then not so surely pointing into a blind alley but maybe to something foundational. At least it fits perfectly into the Wheeler line of thinking which makes its exciting to exploration." If you consider a material Universe made entirely of particles (matter and aether) then what you say may be true, although I suspect 'infinity' is may not be needed (or true).
You state: "If that is not strange enough we are now able to tell something about nothing. Nothingness would typically be thought of like an empty set in the category of sets. This sounds like an algorithmic description but nothing should not be algorithmic because algorithms describe 'something'. Nothingness thus should be somehow non-algorithmic." Zero is a number that has an important role in many algorithmic structures, and I think you can apply this to your 'nothingness' argument as well. I would then ask why should nothingness be non-algorithmic?
"Where is the TOE lurking? Or, you can ask, where is the physics here? If anything, we could only see fleeting appearance of some basic mathematical structures on top of uncomputable sequences. Where is then the usual stuff of physics, fields, forces, waves, particles, matter and the Universe? The answer is that we only covered the foundational TOE level from nothingness to simple mathematical structures which are indispensable for physics. We also see that there are colossal permutation groups in action which makes possible emerging of mathematical structures with much higher complexity." It is a good question that you pose. We need very special structures to create the stuff of physics. Our best approach, whether top-down or bottom-up, is to use some or all of the physical framework that we have already created through centuries of scientific experimentation. We cannot do that by sifting randomly though endless mathematical permutations, although Wolfram shows us the path to this methodology. I truly believe all mathematical structures are created by intelligent entities, and that there is no Platonic realm, as envisaged by many.
In your conclusion you argue "We indicate here that mathematics exists, mathematical structures emerge virtually out of nothingness tied with uncomputability. Due to the richness of possibilities in our framework physics appears then as a very special mathematical structure which provides strong support to the notion that physics can be completely reduced to mathematics and all mathematical structures exist." I support some of this statement, but I do not agree that all mathematical structures exist. I am not even sure that a computational TOE is a mathematical structure.
I cover some of this ground in my essay "Wandering towards a 'Theory of Everything' and how I was stopped from achieving my goal by Nature", where I look at my own physical TOE from a computability point of view.
It may be that the mathematical structures you seek are the proverbial "Emperor's new clothes'.
Best wishes on your well constructed essay.
Lockie Cresswell