Universe is NOT a simulation!

Connecting the meta-mathematical and logical arguments of the Faizal et al. paper with the specific physical and geometrical proposals of Laurent Nottale's theory of scale relativity.  Is there a link?

The short answer is: Yes, scale relativity is not only in agreement with the publication but could be considered a potential physical manifestation of the very principles the paper discusses.

While the Faizal et al. paper doesn't mention scale relativity, their conclusions about the necessity of a non-algorithmic, undecidable component in a final theory find strong philosophical and structural resonance in Nottale's framework. Here’s a breakdown of the points of agreement and synergy:

1. Breakdown of Differentiability and Computability

• Scale Relativity's Core Idea: The fundamental postulate of scale relativity is that spacetime is not smooth and differentiable at a microscopic level. Instead, it is continuous but non-differentiable, possessing a fractal structure. This means that as you "zoom in" on a spacetime path (a geodesic), you find ever more detail, and the path's length between two points diverges to infinity.

• Connection to the Paper: A perfectly smooth, differentiable path can be described by a finite set of equations and parameters—it is fundamentally algorithmic. A non-differentiable, fractal path, however, contains infinite complexity. Describing such a path with perfect fidelity at all scales would require an infinite amount of information, making it inherently incomputable. Any algorithmic description would be an approximation at a certain resolution. This aligns perfectly with the paper's conclusion that a purely algorithmic "Theory of Everything" (FQG) is impossible. The fractal nature of spacetime in scale relativity provides a concrete physical reason for this incomputability.

2. Infinite Complexity and Chaitin's Incompleteness

• The Paper's Argument: Faizal et al. invoke Chaitin's theorem, which states that any formal system has a limit on the Kolmogorov complexity of the theorems it can prove. There will always be true statements whose complexity exceeds this limit.

• Scale Relativity's Manifestation: A fractal spacetime is a system of immense, arguably infinite, Kolmogorov complexity. The detailed state of a particle's trajectory within this fractal geometry could easily represent a "truth" (e.g., its precise position and momentum across all scales) that is too complex to be derived from any finite set of physical axioms. This provides a physical model for the "undecidable Gödel sentences" the paper describes. For example, predicting the exact outcome of a quantum measurement might be undecidable not just because of quantum randomness, but because the underlying spacetime geometry it traverses is incomputably complex.

3. Meta-Principles and Non-Algorithmic Structure

• The Paper's "MToE": The authors propose a "Meta-Theory of Everything" (MToE) that augments the computable part of physics with a non-algorithmic "external truth predicate" (T(x)). This represents a deeper level of structure or understanding that governs reality but isn't derivable from within the algorithmic system.

• Scale Relativity's Principle: The central principle of scale relativity—that the laws of physics must be covariant not only under changes in motion (standard relativity) but also under changes in scale (resolution)—is precisely such a meta-principle. It's a global constraint on the nature of reality that isn't derived from a calculation but is postulated as a foundational truth. This principle of scale covariance could be seen as a component of the non-algorithmic structure that the MToE seeks to describe.

4. Emergence and the Limits of Description

• The Paper's View: Spacetime and physical laws are emergent from a deeper, pre-geometric layer. The failure of algorithms occurs at this fundamental level.

• Scale Relativity's View: Classical, smooth spacetime is an emergent property. It is what we observe when we look at the universe at scales much larger than the fundamental Planck scale. The transition from the non-differentiable, fractal regime to the classical, smooth regime is a key part of the theory. This transition—much like the thermalization process mentioned by Faizal et al.—is likely a trans-computational step. One cannot algorithmically derive the smooth world from the infinite complexity of the fractal world; it is an emergent approximation.

Conclusion:

The theory of scale relativity provides a compelling physical model that embodies the abstract logical conclusions of the Faizal et al. paper. If spacetime were indeed fractal and non-differentiable as Nottale proposes, it would naturally lead to a universe where some truths are computationally undecidable. The "non-algorithmic understanding" required by the MToE would correspond to grasping the meta-principle of scale covariance and its consequences.

Therefore, scale relativity is in strong conceptual agreement with the argument that a complete and consistent theory of everything cannot be purely algorithmic.