Dear Ernesto,
I enjoyed reading your essay, and I think that, while you modestly declare it "a play in speculation", you make some excellent points.
> To create a mapping onto reality beyond a simple model would require something else. Not to mention the 'true' nature of reality is up for debate, which is why I take as an assumption in this paper that "reality is mathematics".
These and other mathematical universe statements you make make sense to me, and I wrote about such things, for example here. You mention Tegmark, I believe that he proposes to take into account only computable mathematical structure, which I think makes his ideas digital philosophy rather than mathematical monistm as it seems. While I think both you and I find more appealing to go beyond this limitation. Which leads to the following.
> if and only if an example of Gödel's undecidability is found within nature, can we claim that there is something more fundamentally mathematical about reality, than the math simply being a useful tool. This is because a truly undecidable result would only be possible if there existed a true mathematical structure underlying reality. In fact there are a few instances of undecidable results being found within quantum mechanics (Cubitt, Moore).
I think this is idea that "a truly undecidable result would only be possible if there existed a true mathematical structure underlying reality" deserves more serious consideration. Usually people misuse Gödel's undecidability theorem in the complete opposite sense, which makes no sense. Too many still understand it as being proof of the limits of mathematics, not of the fact that it goes beyond the limits of logically consistent language.
> in this view, certain fundamental properties would also be high level properties.
This is another claim with which I agree, and it seems to me that a good example is the micro, quantum level of reality fails to determine the macro level, which makes me think that there's something fundamental at the macro level, although I don't consider it to be extra stuff than the wavefunction, just constraints of it. I wrote about this here sec. |7>, and here, example 10.
I also liked this one
> Because certain emergent properties cannot be explained from their constituent parts, from the point of view I am taking in this paper, they must be examples of Gödel truths [...] strong emergence.
which is something I think too, cf. my longer essay, def. 11. In fact, because "emergent" is sometimes used in completely opposite way by philosophers compared to physicists, I removed it everywhere in that essay and replaced the part about "emergence" in terms of "reducible", so "strongly emergent" became "weakly reducible" :). But the way I understand it is in terms of Gödel undecidability like yours.
Now, a good question I think may be whether for something to have Gödelian truths, it necessarily has to be a mathematical structure with no other ontology.
Cheers,
Cristi
