Mathematical formulation of Leibnizian world: a theory of individual-whole or interior-exterior reflective systems

Time and Life in the Relational Universe: Prolegomena to an Integral Paradigm of Natural Philosophy

Relational ideas for our description of the natural world can be traced to the concept of Anaxagoras on the multiplicity of basic particles, later called “homoiomeroi” by Aristotle, that constitute the Universe and have the same nature as the whole world. Leibniz viewed the Universe as an infinite set of embodied logical essences called monads, which possess inner view, compute their own programs and perform mathematical transformations of their qualities, independently of all other monads. In this paradigm, space appears as a relational order of co-existences and time as a relational order of sequences. The relational paradigm was recognized in physics as a dependence of the spatiotemporal structure and its actualization on the observer. In the foundations of mathematics, the basic logical principles are united with the basic geometrical principles that are generic to the unfolding of internal logic. These principles appear as universal topological structures (“geometric atoms”) shaping the world. The decision-making system performs internal quantum reduction which is described by external observers via the probability function. In biology, individual systems operate as separate relational domains. The wave function superposition is restricted within a single domain and does not expand outside it, which corresponds to the statement of Leibniz that “monads have no windows”.

Physical limits of computation and emergence of life

The computational process is based on the activity linking mathematical equations to a materialized physical world. It consumes energy which lower limit is defined by the set of Planck's values, i.e. by the physical structure of the Universe. We discuss computability from the quantum measurement framework. Effective quantum computation is possible via the maintenance of a long-living cold decoherence-free internal state, which is achieved by applying error-correction commands to it and by screening it from thermal fluctuations. The quantum Zeno effect enables coherent superpositions and entanglement to persist for macroscopic time intervals. Living systems maintain coherent states via realization of their own computing programs aiming them to survive and develop, while their non-computable behavior corresponds to a generative power that arises beyond combinatorial capabilities of the system. Emergence of life brings in the Universe a creative activity that overcomes the limits of computability.

Quantum computation, non-demolition measurements, and reflective control in living systems

Internal computation underlies robust non-equilibrium living process. The smallest details of living systems are molecular devices that realize non-demolition quantum measurements. These smaller devices form larger devices (macromolecular complexes), up to living body. The quantum device possesses its own potential internal quantum state (IQS), which is maintained for a prolonged time via reflective error-correction. Decoherence-free IQS can exhibit itself by a creative generation of iteration limits in the real world. It resembles the properties of a quasi-particle, which interacts with the surround, applying decoherence commands to it. In this framework, enzymes are molecular automata of the extremal quantum computer, the set of which maintains highly ordered robust coherent state, and genome represents a concatenation of error-correcting codes into a single reflective set. The biological evolution can be viewed as a functional evolution of measurement constraints in which limits of iteration are established, possessing criteria of perfection and having selective values.

Quantum monadology: a consistent world model for consciousness and physics

The NL world model presented in the previous paper is embodied by use of relativistic quantum mechanics, which reveals the significance of the reduction of quantum states and the relativity principle, and locates consciousness and the concept of flowing time consistently in physics. This model provides a consistent framework to solve apparent incompatibilities between consciousness (as our interior experience) and matter (as described by quantum mechanics and relativity theory). Does matter have an inside? What is the flowing time now? Does physics allow the indeterminism by volition? The problem of quantum measurement is also resolved in this model.