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How to acknowledge hypercomputation?
, 2007
"... We discuss the question of how to operationally validate whether or not a “hypercomputer” performs better than the known discrete computational models. ..."
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We discuss the question of how to operationally validate whether or not a “hypercomputer” performs better than the known discrete computational models.
Formal Proof: Reconciling Correctness and Understanding
"... A good proof is a proof that makes us wiser. Manin [41, p. 209]. Abstract. Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of pr ..."
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A good proof is a proof that makes us wiser. Manin [41, p. 209]. Abstract. Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical theorems formally proved. Formal proof is practically achievable. With formal proof, correctness reaches a standard that no penandpaper proof can match, but an essential component of mathematics — the insight and understanding — seems to be in short supply. So, what makes a proof understandable? To answer this question we first suggest a list of symptoms of understanding. We then propose a vision of an environment in which users can write and check formal proofs as well as query them with reference to the symptoms of understanding. In this way, the environment reconciles the main features of proof: correctness and understanding. 1
A ProgramSize Complexity Measure for Mathematical Problems and Conjectures
"... Abstract. Cristian Calude et al. in [5] have recently introduced the idea of measuring the degree of difficulty of a mathematical problem (even those still given as conjectures) by the length of a program to verify or refute the statement. The method to evaluate and compare problems, in some objecti ..."
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Abstract. Cristian Calude et al. in [5] have recently introduced the idea of measuring the degree of difficulty of a mathematical problem (even those still given as conjectures) by the length of a program to verify or refute the statement. The method to evaluate and compare problems, in some objective way, will be discussed in this note. For the practitioner, wishing to apply this method using a standard universal register machine language, we provide (for the first time) some “small ” core subroutines or library for dealing with array data structures. These can be used to ease the development of full programs to check mathematical problems that require more than a predetermined finite number of variables. 1
Chapter 1 Omega and the time evolution of the nbody problem
, 2007
"... The series solution of the behavior of a finite number of physical bodies and Chaitin’s Omega number share quasialgorithmic expressions; yet both lack a computable radius of convergence. 1.1. Solutions to the n–body problem The behaviour and evolution of a finite number of bodies is a sort of “rose ..."
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The series solution of the behavior of a finite number of physical bodies and Chaitin’s Omega number share quasialgorithmic expressions; yet both lack a computable radius of convergence. 1.1. Solutions to the n–body problem The behaviour and evolution of a finite number of bodies is a sort of “rosetta stone ” of classical celestial mechanics insofar as its investigation induced a lot of twists, revelations and unexpected issues. Arguably the most radical deterministic position on the subject was formulated by Laplace, stating that [1, Chapter II] “We ought then to regard the present state of the universe as the effect of its anterior state and as the cause of the one which is to follow. Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it an intelligence sufficiently vast to submit these data to analysis it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes.” In what may be considered as the beginning of deterministic chaos theory, Poincaré was forced to accept a gradual departure from the deterministic position: sometimes small variations in the initial state of the bodies could lead to huge variations in their evolution in later times. In Poincaré’s own words [2, Chapter 4, Sect. II, pp. 5657], “If we would know the laws of Nature and the state of the Universe precisely for a certain time, we would
Contents
, 2008
"... Different types of physical unknowables are discussed. Provable unknowables are derived from reduction to problems which are known to be recursively unsolvable. Recent series solutions to the nbody problem and related to it, chaotic systems, may have no computable radius of convergence. Quantum unk ..."
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Different types of physical unknowables are discussed. Provable unknowables are derived from reduction to problems which are known to be recursively unsolvable. Recent series solutions to the nbody problem and related to it, chaotic systems, may have no computable radius of convergence. Quantum unknowables include the random occurrence of single events, complementarity and value indefiniteness.