Undecidability in Relational Biology as the Framework for Quantum Theory Baruch Garcia1
1no affiliation
baruchgarcia@gmail.com
baruchgarcia@utexas.edu
Abstract
According to relational biologists, following the example of Robert Rosen, living systems cannot be simulated on a Turing machine, since it is claimed that the self-referential nature of metabolic closure implies undecidability. If this is indeed true and living systems are described by undecidability and therefore there is no generalized algorithm to predict their behavior, then the mathematical description of living systems can provide a logical framework for quantum theory. This conclusion is based on the work of John Archibald Wheeler who believed (1) that the observer was integral to defining a quantum system and (2) since the observer is composed of quantum phenomena itself, this relationship is self-referential. This self-reference, according to Wheeler, also implies a mathematical description of undecidability.
Relational Biology and Undecidability
Robert Rosen, Francisco Garcia-Varela and Humberto Maturana, among others have attempted to define life as a self-referential system. Juan-Carlos Letelier among others, has catalogued these attempts (Letelier, 2011). For example, metabolism is a self-referential process; enzymes catalyze the products used in the construction of the enzymes themselves, a concept called “metabolic closure”. Self-reference is also the modus operandi of undecidability proofs (Davis, 1965) (van Heijenoort, 1967). Naturally, several biologists consider living systems not to have any generalized algorithm or as we might call them “undecidable”. This means we cannot simulate living systems on a Turing machine. Some biologists see this as a negative, an obstacle to work around.
In this paper, we will discuss a far more interesting implication of the undecidability of living systems. If this is in fact true, then we now understand why living systems appear to play such a crucial role in quantum mechanics, that is, why is an observer necessary? Despite the various alternatives to standard quantum interpretation, which claim no need for an observer, this is not a resolved issue (Weinberg, 2012).
What does self-reference and undecidability have to do with quantum theory at all? In fact, John Archibald Wheeler approached Kurt Gödel on the issue, yet the question remains an open one (Wheeler, 1998). There are
many parallels between quantum theory and undecidability. The similarities are difficult to ignore to a logician who understands quantum theory or a physicist who understands undecidability (Garcia, 2020).
Quantum Theory and Undecidability
Figure 1: Wheeler’s “U-Diagram”. Quantum Theory and the Universe come into being by the self-reference of the observer. The observer is not isolated from the quantum system, but an essential part of the quantum system. This is described mathematically by undecidability. Note the parallels between Wheeler’s self-referential observer and the self-reference as the definition of life (Wheeler, 1998). For further resources on Wheeler’s work, see jawarchive.wordpress.com.
Wheeler had a vision for quantum theory that did not belong to his generation. For those unfamiliar with his work, Wheeler was a student of Einstein and Bohr, and a mentor to Feynman, Everett, Thorne and countless other physicists who worked on the foundations of quantum theory and gravity. While quantum theory is battle-tested in the lab, and unambiguously described by linear algebra, Wheeler was not satisfied by textbook quantum
mechanics of the 1920’s (von Neumann, 1930) (Wheeler, 1998). His questioning of the foundations of quantum physics led, in part, to the development of the discipline of quantum information in the 1980’s (Misner, et al 2009). Even today, eminent physicists such as Witten and Maldacena have followed in Feynman’s footsteps and warned against ignoring Wheeler’s vision (Wolchover, 2019) (Maldacena, 2015). Wheeler himself was particularly interested in Garcia-Varela and Maturana’s work on Autopoiesis, and he wrote extensively in his notebooks which can be found at the American Philosophical Society in Philadelphia, but he never published on the subject.
The core concept of his “it from bit” approach to understanding the foundations of quantum mechanics was that the observer, who is part of the universe, brings elementary phenomena into existence, which in turn, makes up the observer (Wheeler, 1989). In other words, when one is trying to solve for a quantum description of the universe, as with the Wheeler-DeWitt equation, one must remember that the observer is inside of the universe, not outside of it. The observer must be included in the quantum description. Ed Witten emphasized that this aspect of quantum mechanics is still not understood. In an interview, he explained:
[Wheeler] drew a picture on the blackboard of the Universe visualized as an eye looking at itself. I had no idea what he was talking about. It’s obvious to me in hindsight that he was explaining what it meant to talk about quantum mechanics when the observer is part of the quantum system. I imagine there is something we don’t understand about that (Wolchover, 2017).
The mathematical language for this self-referential description of quantum theory is taken from undecidability proofs of Gödel, Turing and others (van Heijenoort, 1967). What does quantum theory, a theory clearly and thoroughly described by linear algebra, have to do with undecidability? Following Bohr’s example, the answer can be found in complementarity or as others might call it “wave-particle duality”. Von Neumann in his textbook “Mathematical Foundations of Quantum Mechanics” described two processes. Process 1 describes the measurement process or the collapse of the wave function. One might call this the “particle” description e.g. the position or momentum of the particle. Process 2 described the evolution process or the state vector evolve deterministically through various superpositions in a Hilbert Space e.g. the Schrödinger equation or Feynman’s Path Integral formalism.
For simplicity’s sake, let’s begin with a basic quantum equation everyone knows: E=hv. E can be thought of as a particle description, describing the energy of a particle, and v as a wave description, describing the frequency of a wave. In the particle description, we have a result which is measurable, but whose final conditions are inconsistent given the same exact initial conditions. What
do we mean by “inconsistent”? Quantum mechanics is a battle-tested theory, and the incredibly well-tested. This inconsistency does not mean that quantum mechanics gives contradictory results. In fact, the probability distributions are consistent among each experiment. What “inconsistency” means in this specific context, is that if an experimenter starts with one particular initial condition, the same final condition should not always be expected. Take for example, the classic Stern-Gerlach experiment. If one prepares the spin of an electron in a z+ state then measures the x-component, then sometimes the x- state will be measured and sometimes the x+ will be measured.
In contrast to this “inconsistency”, we see for the wave description that the evolution of the Schrödinger wave equation is consistent, that is, whenever the initial condition is chosen, the final condition of the wave equation will be known, deterministically, every time. Yet the state vector evolves through various linear combinations in a Hilbert space, superpositions which, by definition, cannot be measured.
What does this complementarity/duality have to do with undecidability? In undecidable proofs, e.g Rice’s Theorem, which is a generalization of the undecidability of the Halting Problem, we can say that a formal system is either provable by physical, mechanical means (as Gödel and Turing showed (Davis, 1965). But then it will be inconsistent, i.e. a statement and its negation can be proven. Or we can demand consistency, in which a statement is not provable using physical,mechanical means. Von Neumann’s Process 1 (particle description) corresponds to a statement which is provable, but inconsistent; its negation is provable. Von Neumann’s Process 2 (wave description) corresponds to a formal statement which is consistent but not provable by physical, mechanical means. This provides a logical framework for wave-particle duality, or which other physicists might call Bohr’s complementarity. For a further analysis see Undecidability as a Framework for Quantum Theory and Spacetime (Garcia, 2020)(Geroch, 1986).
Discussion
Can this abstract mathematical description of life provide a logical framework for the paradoxical dualities of quantum mechanics? If true, this would be the first step in a fundamentally new direction for quantum biology. Schrodinger’s question of “What is Life?” could be given a clearer answer. The answer would be that life is defined by quantum theory, and more specifically that life is defined by quantum theory as Wheeler imagined it - a self-generated structure.
Acknowledgments. The author would like to thank Noson Yanosky, the late E. C. G. Sudarshan, and a special thanks to Juan Carlos Letelier for introduction to mathematical biology.
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