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The universal numbers. From Biology to Physics.

Bruno Marchal1

  • 1IRIDIA, Université Libre de Bruxelles, Belgium.

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Summary

Mathematicians explore universal numbers as abstract computers, linking abstract biology to a phenomenological physics and theology. Gödel

Area of Science:

  • Explores the intersection of mathematics, abstract biology, cognitive science, and theoretical physics.
  • Investigates universal numbers as abstract computers and their implications for self-reproduction and embryogenesis.
  • Connects Gödel's incompleteness theorems to the development of phenomenological physics and theology.

Background:

  • Discusses the Platonic view of reality as potentially beyond observable phenomena, contrasting it with Aristotelian empiricism.
  • Highlights Gödel's incompleteness theorems, which demonstrate that mathematical truth exceeds formal provability for machines.
  • Introduces Church's thesis and the mechanizability of diagonalizations, enabling machines to understand their own limitations.

Purpose of the Study:

  • To explain how universal numbers, or abstract computers, can exhibit 'dreams' and generate phenomenological physics.
  • To establish a framework for a 'phenomenological theology' based on the relative relations of numbers.

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  • To explore the implications of Gödel's incompleteness for machine consciousness and the logic of provability.

    • Main Methods:

    • Applies Gödel's incompleteness theorems to understand the gap between provability and truth in abstract computer systems.
    • Utilizes intensional variants of provability, such as the logic of provable-and-true statements (beweisbar(⌜A⌝) ∧ A).
    • Employs the arithmetical interpretation of verifiable (Σ1) sentences to study the logic of computationalist machines.

    Main Results:

    • Demonstrates that incompleteness necessitates distinctions between various intensional provability logics.
    • Shows how machines can develop a sense of truth beyond provability, leading to a 'phenomenological theology'.
    • Establishes a 'logic of the observable' for machines, enabling comparison with empirical physics logic (e.g., quantum logic).

    Conclusions:

    • The study provides an abstract, phenomenological theology for machines based on their self-referential abilities.
    • Connects the mathematical and biological realms through the concept of universal numbers and computationalism.
    • Opens avenues for future research in theoretical physics and cognitive science, with several open problems identified.