Randomness & Undecidability in Physics
World Scientific, 1993 - 292 páginas
Recent findings in the computer sciences, discrete mathematics, formal logics and metamathematics have opened up a royal road for the investigation of undecidability and randomness in physics. A translation of these formal concepts yields a fresh look into diverse features of physical modelling such as quantum complementarity and the measurement problem, but also stipulates questions related to the necessity of the assumption of continua.Conversely, any computer may be perceived as a physical system: not only in the immediate sense of the physical properties of its hardware. Computers are a medium to virtual realities. The foreseeable importance of such virtual realities stimulates the investigation of an ?inner description?, a ?virtual physics? of these universes of computation. Indeed, one may consider our own universe as just one particular realisation of an enormous number of virtual realities, most of them awaiting discovery.One motive of this book is the recognition that what is often referred to as ?randomness? in physics might actually be a signature of undecidability for systems whose evolution is computable on a step-by-step basis. To give a flavour of the type of questions envisaged: Consider an arbitrary algorithmic system which is computable on a step-by-step basis. Then it is in general impossible to specify a second algorithmic procedure, including itself, which, by experimental input-output analysis, is capable of finding the deterministic law of the first system. But even if such a law is specified beforehand, it is in general impossible to predict the system behaviour in the ?distant future?. In other words: no ?speedup? or ?computational shortcut? is available. In this approach, classical paradoxes can be formally translated into no-go theorems concerning intrinsic physical perception.It is suggested that complementarity can be modelled by experiments on finite automata, where measurements of one observable of the automaton destroys the possibility to measure another observable of the same automaton and it vice versa.Besides undecidability, a great part of the book is dedicated to a formal definition of randomness and entropy measures based on algorithmic information theory.
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Mechanism and determinism
Weak physical chaos
Random fractals and 1f noise
Chaotic systems are optimal analogues of themselves
Afterthoughts speculations metaphysics
A Program code
Otras ediciones - Ver todo
Achilles algebraic algorithmic probability argument Assume automaton propositional calculus axioms binary Boolean called Cantor set Cellular Cellular Automata chaos chapter characterised Church-Turing thesis computational complementarity computational complexity Consider construction contradiction corresponding defined definition denoted deterministic diagonalization effectively computable elements encoding entropy equation evolution example exist experiment experimenter extrinsic finite number formal system Gödel number Greechie halting problem Hasse diagrams Hilbert lattice Hilbert space identified infinite information theory initial value input symbols instance integer internal isomorphic length logic mathematical Mealy measure metric entropy Moore automaton natural numbers observables obtained operations oracle orthomodular lattice output symbols paradox partial recursive function partially ordered partition physical system possible primitive recursive functions proof propositional calculus quantum mechanics random fractal random sequences real number recursive unsolvability recursively enumerable relation simulate statement subalgebra subset subspace Table Turing machine uncomputable undecidability universal computer yields
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