## Strong Determinism vs. Computability (1995)

Venue: | The Foundational Debate, Complexity and Constructivity in Mathematics and |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Hawkins95strongdeterminism,

author = {David Hawkins and Cristian Calude and Douglas I. Campbell and Karl Svozil},

title = {Strong Determinism vs. Computability},

booktitle = {The Foundational Debate, Complexity and Constructivity in Mathematics and},

year = {1995},

publisher = {Physics”, Kluwer}

}

### OpenURL

### Abstract

Are minds subject to laws of physics? Are the laws of physics computable? Are conscious thought processes computable? Currently there is little agreement as to what are the right answers to these questions. Penrose ([41], p. 644) goes one step further and asserts that: a radical new theory is indeed needed, and I am suggesting, moreover, that this theory, when it is found, will be of an essentially non-computational character. The aim of this paper is three fold: 1) to examine the incompatibility between the hypothesis of strong determinism and computability, 2) to give new examples of uncomputable physical laws, and 3) to discuss the relevance of Gödel’s Incompleteness Theorem in refuting the claim that an algorithmic theory—like strong AI—can provide an adequate theory of mind. Finally, we question the adequacy of the theory of computation to discuss physical laws and thought processes. 1

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Citation Context ...e concerning the computational status of the wave function 3 or continuous observables. Implicitly, the underlying sets are the classical (i.e., non-constructive) continua. 3 See Pour-El and Richards =-=[43]-=-, and the objections in Penrose [40], and Bridges [6]. 3s4 Is Description Possible? Can a system contain a description of itself? Of course, no finite system can contain itself as a proper part. What ... |

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Citation Context ...rding to classical logic, f is computable because there exists an algorithm that computes it, i.e. the algorithm that returns either one or zero, for all non-negative integers. Deep work due to Gödel =-=[33]-=- and Cohen[16] shows that neither the Continuum Hypothesis nor its negation can be proven within Zermelo-Fraenkel set theory augmented with the Axiom of Choice, the standard framework of classical mat... |

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Citation Context ...xist. 6 Uncomputability and Randomness: Two Examples Various physical problems lead to the question whether a function, in a certain a class, has a real root. Results due to Richardson [44], Caviness =-=[11]-=-, Wang [57] (see also Matijasevič [34]) show that for a large class of well-defined functions such a problem is not algorithmically solvable. Da Costa and Doria [18] have proven some undecidability re... |

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Citation Context ... theory might exist. 6 Uncomputability and Randomness: Two Examples Various physical problems lead to the question whether a function, in a certain a class, has a real root. Results due to Richardson =-=[44]-=-, Caviness [11], Wang [57] (see also Matijasevič [34]) show that for a large class of well-defined functions such a problem is not algorithmically solvable. Da Costa and Doria [18] have proven some un... |

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Citation Context ...idental? Does the incompleteness phenomenon have any relevance for a scientist’s daily life? This is a rather delicate question. If we adopt a topological point of view (see Calude, Jürgensen, Zimand =-=[9]-=-), then incompleteness is a rather common, pervasive phenomenon: the set of true, but unprovable statements is topologically “very large”, i.e. with respect to any reasonable topology the set of true ... |

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Citation Context ...s due to Richardson [44], Caviness [11], Wang [57] (see also Matijasevič [34]) show that for a large class of well-defined functions such a problem is not algorithmically solvable. Da Costa and Doria =-=[18]-=- have proven some undecidability results in physics using this tool. A different approach, based on Specker’s Theorem, was developed by Pour-El and Richardson [43]. In this chapter we shall build on t... |

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Citation Context ... physical motion has confronted the physics community with the theoretical question of whether or not to accept the classical (i.e., non-constructivist) continuum. As envisioned by Shaw [48] and Ford =-=[23]-=-, along with many others, “classical chaos” emerges by the effectively computable “visualization” of the incompressible algorithmic information of the initial values. Thereby, the classical continuum ... |

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Citation Context ...omputability and Randomness: Two Examples Various physical problems lead to the question whether a function, in a certain a class, has a real root. Results due to Richardson [44], Caviness [11], Wang =-=[57]-=- (see also Matijasevič [34]) show that for a large class of well-defined functions such a problem is not algorithmically solvable. Da Costa and Doria [18] have proven some undecidability results in ph... |

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Citation Context ...of Laplace’s scenario, 1 according to which the stage is set at the beginning and everything follows “mechanistically” without the intervention of God, without the occurrence of “miracles” (cf. Frank =-=[24]-=-). Strong determinism does not imply a computable Universe, as it says nothing about the computability of initial conditions or of physical laws. 2 Let us discuss this in the context of the computer s... |

10 | Are binary codings universal
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Citation Context ... that a total compression of the Universe is not actually possible as the existence of chaotic processes points out (Chaitin [12, 13], Rucker [46], Svozil [49, 50, 51], Calude [8], Calude and Salomaa =-=[10]-=-). How can we describe seemingly random processes in nature and reconcile them with supposed order? How much can a given piece of information be compressed? Calude and Salomaa [10] have suggested that... |

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Citation Context ...d Chaitin to show that two problems in elementary physics are undecidable and display pure randomness. 7 For an early investigation of a forecast inspired by recursion theory see Popper [42]. 8 Gandy =-=[27, 28]-=- has put forward related arguments imposing limitations to mathematical knowledge by the finiteness of physical objects. 5s6.1 Richardson-Wang and Chaitin Theorems An exponential Diophantine equation ... |

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Citation Context ...: Two Examples Various physical problems lead to the question whether a function, in a certain a class, has a real root. Results due to Richardson [44], Caviness [11], Wang [57] (see also Matijasevič =-=[34]-=-) show that for a large class of well-defined functions such a problem is not algorithmically solvable. Da Costa and Doria [18] have proven some undecidability results in physics using this tool. A di... |

8 |
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Citation Context ...dson, Wang, and Chaitin to show that two problems in elementary physics are undecidable and display pure randomness. 7 For an early investigation of a forecast inspired by recursion theory see Popper =-=[42]-=-. 8 Gandy [27, 28] has put forward related arguments imposing limitations to mathematical knowledge by the finiteness of physical objects. 5s6.1 Richardson-Wang and Chaitin Theorems An exponential Dio... |

6 |
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Citation Context ...tion 3 or continuous observables. Implicitly, the underlying sets are the classical (i.e., non-constructive) continua. 3 See Pour-El and Richards [43], and the objections in Penrose [40], and Bridges =-=[6]-=-. 3s4 Is Description Possible? Can a system contain a description of itself? Of course, no finite system can contain itself as a proper part. What we mean by “description” here is an algorithmic repre... |

6 |
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Citation Context ...stem can be derived from conduction of electricity on a conducting sheet covering the domain D. 16 The local existence of a potential Φ is described by the equality E = −dΦ; see Bamberg and Sternberg =-=[1]-=-. 8s7 Incompleteness In a remarkable paper entitled Intelligent Machines 17 ([54], 107-127) Turing investigates the possibility as to whether machines, i.e. computers, might show intelligent behaviour... |

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Citation Context ...e the answers can be independent in case the equation is constructed properly. Actually Chaitin has effectively constructed such an exponential Diophantine equation (see his last Lisp construction in =-=[14]-=-) with the property that the number of solutions jumps from finite to infinite at random as a certain fixed parameter is varied. Actually, saying that the “number of solutions jumps from finite to inf... |

4 |
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4 |
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Citation Context ...ble to predict the entire future. The puzzling consequence appears as soon as one tries to carry out this prediction! Gödel was interested in this problem as well. According to notes taken by Rucker (=-=[46]-=-, p.181) Gödel’s point of view is the following: It should be possible to form a complete theory of human behaviour, i.e. to predict from the hereditary and environmental givens what a person will do.... |

3 |
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Citation Context ...g the domain D. 16 The local existence of a potential Φ is described by the equality E = −dΦ; see Bamberg and Sternberg [1]. 8s7 Incompleteness In a remarkable paper entitled Intelligent Machines 17 (=-=[54]-=-, 107-127) Turing investigates the possibility as to whether machines, i.e. computers, might show intelligent behaviour. He considers the argument that machines are inherently incapable of exhibiting ... |

2 |
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Citation Context ... I anticipate something more subtle for the actual (non-algorithmic!) rules that govern the way that the world works! 2 From Boscovich to Gödel Perfect determinism was considered earlier by Boscovich =-=[4]-=-, Leibniz and Laplace (see Barrow [2]). The main argument is similar to the one used by Penrose: if all our laws, say, of motion, were in the form of equations which determine the future uniquely and ... |

1 |
Theories of Everything—The Quest for Ultimate Explanation
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Citation Context ...or the actual (non-algorithmic!) rules that govern the way that the world works! 2 From Boscovich to Gödel Perfect determinism was considered earlier by Boscovich [4], Leibniz and Laplace (see Barrow =-=[2]-=-). The main argument is similar to the one used by Penrose: if all our laws, say, of motion, were in the form of equations which determine the future uniquely and completely from the present, then a “... |