REPLY I TO LEBEL
> A good start for a new physical theory is an impossibility, either factual or postulated (see truth systems)
F: Yes, but is but no means the unique way.
>My conclusion is that no physical TOE is logically possible. Mathematical connection/description is still possible.
F: Wrong: Nature seems to have no problem combining relativity with quantum mechanics and gravity in a logical coherent way. Just because we are not yet able to do it is only a reflection of our limited intellectual power, and we should not be too arrogant to claim it cannot be done.
F: And Right: depends on what we mean by a TOE. Mathematics is infinite in complexity, and so is physics. We understand now how the human body works but we will never have a TOE of medicine (we are still battling diseases like cancer). Similarly, there will be no closed mathematical system describing a physics TOE like say the Euclidean geometry. (Axiomatizing physics is still possible.)
>I am not sure of your: physical reality vs real world vs Platonic world ...
F: Reality seems to be nothing but a collection of relational mathematical structures organized in a specific way. But this is fuzzy philosophy. What matters is that you can derive important uniqueness results about nature simply by comparing math and reality.
>I like you addressing the concept of truth: "The universal truth property can be shown to create a constraint on the event manifold that manifest itself as global hyperbolicity [13], typically called time." I would like your short explanation without having to dig into ref (13).
F: You cannot dig in ref 13 because it is not published yet. Basically you can translate Goedel's incompleteness proof almost 1-to-1 into the global hyperbolicity proof. This is very much related to your ideas: if something is true (like past history), then you have no freedom: you get only a frozen past (that cannot be rewritten by traveling back in time). No time travel to the past = global hyperbolicity. In Special relativity one has the Minkowski cone and LOCAL hyperbolicity which separates past from future, but in general relativity one can bend the space time a little bit at each point in such a ways that globally time can wind back to itself, and this violates GLOBAL hyperbolicity.
> You mention a metaphysical questions like the "Nature of time". I don't think that mathematical precision will answer that one.
F: mathematically, time=global hyperbolicity. Philosophically, time is the (unique) mechanism that prevents us from becoming inconsistent and separating existence from non-existence. Without time, there will be only frozen contextual truths (many at odds with each other) and also there will be inconsistent statements. Demanding universal consistency (or universal truth property) necessarily demands the existence of time.
>You say: "Instead on trying to solve the nature of reality, a much more useful approach is not to seek the similarities, but the diĀ®erences. And this is the main idea of this essay." My approach is exactly about "trying to solve the nature of reality" or the underlying reality.
F: Indeed. Initially I have gone your path as well. But what I try to do now is to extract mathematical consequences and prove that the real world has uniqueness. I believe we should be able to rigorously prove in the end that the universe must have 3 spatial and one time dimension, it must has quantum mechanics, it must has the gravitational, electromagnetic, weak, and strong forces, it must have the particles we observe and their properties, etc.
>"but I am trying to push the envelope and extend math's reach into a formerly philosophical domain" . Maths are based on logic and somehow they are most welcome in the truth system metaphysics. Simple logic only give me the initial "why"; after that I am pretty much lost; maths of explosive process evolution, maths explosive structure, etc.
F: But there are very important "why"s we do not have an answer yet: why are there 3 generations of matter? Why do we have the strong force? Why do we observe the value of the cosmological constant the way it is? Etc, etc. After all major "whys' are answered, then we can concentrate on secondary issues.
>LEBEL: Now, would you do me a favor and tell me where the second half of my essay start and list the fuzzy philosophy and speculation. I may have lost you there along with everyone else. This feedback would be appreciated.
F: you did not lose me; I was able to follow your essay to the end, although admittedly it became much harder to read after the first half. Your essay is losing the physics audience around page 5: "How many substances?" The (wrong) speculation begins fully on page 6: "We got ourselves a universe of time". I do understand fully what you mean, I used to think along similar lines myself, but I now know it to be mathematically incomplete. As much as I had wanted it so, I cannot obtain quantum mechanics only from my universal truth property. What you are missing is the second principle: composability. From composability and universal truth property one can mathematically derive the necessity of both quantum and classical mechanics.