Finding another theory as well-developed as Scale Relativity that aligns so directly with the arguments in the Faizal et al. paper is challenging. This is primarily because most fundamental physics theories, even the most speculative ones, implicitly aim to be "computable" or algorithmic.
However, there is another major candidate. While its approach is markedly different, it arrives at similar philosophical conclusions about the non-algorithmic nature of reality. This is the body of work by Roger Penrose, particularly his concept of Objective Reduction (Orch-OR) and his research on twistor theory and spin networks.
Let's compare these two perspectives.
Roger Penrose and Non-Computable Physics
For decades, Penrose has been one of the few leading physicists to explicitly argue that fundamental physics must contain a non-computable element. In fact, his arguments are cited in the Faizal et al. paper.
The Core Argument: Gödel's Theorem and Consciousness
Penrose's starting point is Gödel's incompleteness theorem. He argues that human mathematicians can grasp the truth of mathematical statements (like Gödel's own unprovable statement) that no formal algorithmic system can prove. From this, he concludes that human consciousness itself must be non-algorithmic. He then posits that this non-algorithmic capability must arise from a physical process in the brain that is, itself, non-computable.
The Source of Non-Computability: Quantum Gravity and Wave Function Collapse
To find a physical source for this non-computability, Penrose focuses on the measurement problem in quantum mechanics. The evolution of a quantum system is perfectly deterministic and computable (via the Schrödinger equation) as long as it is not measured. The act of measurement, the "collapse of the wave function," is probabilistic and poorly understood.
Penrose proposes a solution called Objective Reduction (OR). He suggests that this collapse is not an effect of measurement but a real, objective physical phenomenon triggered by gravity. When a quantum superposition reaches a certain mass threshold or creates a significant enough spacetime curvature, it spontaneously collapses.
Crucially, Penrose speculates that this process of Objective Reduction is not algorithmic. It isn't merely a matter of rolling a probabilistic die. Instead, it is a fundamental physical process that operates in a way that transcends Turing computation. This is where the "truth predicate" discussed by Faizal could be physically manifested.
The Mathematical Frameworks: Spin Networks and Conformal Cyclic Cosmology
Penrose utilizes several advanced mathematical tools. Spin networks, which he co-developed and which are also central to Loop Quantum Gravity, describe spacetime as a discrete network of quantum relationships. While these are often treated algorithmically, Penrose suggests their global dynamics and the emergence of a smooth spacetime could involve a non-computable complexity.
Furthermore, his cosmological theory, Conformal Cyclic Cosmology (CCC), posits an infinite succession of universes, or "aeons." The transition from an old, cold, empty universe to the Big Bang of a new one involves a mathematical transformation that could effectively "erase" information and introduce a fundamentally non-computable element into the foundations of physics.
A Comparative View: Penrose vs. Nottale
While Nottale's Scale Relativity and Penrose's theories are very different, they share a profound philosophical alignment with the conclusions of Faizal et al.
Nottale locates non-computability in the static structure of spacetime itself—its intrinsic fractal and non-differentiable texture. For him, the infinite complexity is a geometric property of the arena in which physics takes place.
Penrose, on the other hand, finds non-computability in the fundamental dynamics of physics—specifically, in the physical act of wave function collapse. For him, reality is not just a complex structure but also involves an active, non-algorithmic process.
Both theories, despite their different origins and mechanisms, converge on the same radical conclusion: the ultimate laws of the universe are not a simple algorithm that we could one day program into a computer. They contain a form of complexity or a type of process that fundamentally transcends computation.
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