## The Extended Turing Model As Contextual Tool

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@MISC{Cooper_theextended,

author = {S. Barry Cooper},

title = {The Extended Turing Model As Contextual Tool},

year = {}

}

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### Abstract

Abstract. Computability concerns information with a causal – typically algorithmic – structure. As such, it provides a schematic analysis of many naturally occurring situations. We look at ways in which computabilitytheoretic structure emerges in natural contexts. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. The adequacy of the classical Turing model of computation — as first presented in [18] — is in question in many contexts. There is widespread doubt concerning the reducibility to this model of a broad spectrum of real-world processes and natural phenomena, from basic quantum mechanics to aspects of evolutionary development, or human mental activity. In 1939 Turing [19] described an extended model providing mathematical form to the algorithmic content of structures which are presented in terms of real numbers. Most scientific laws with a computational content can be framed

### Citations

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(Show Context)
Citation Context ...cibility to this model of a broad spectrum of real-world processes and natural phenomena, from basic quantum mechanics to aspects of evolutionary development, or human mental activity. In 1939 Turing =-=[19]-=- described an extended model providing mathematical form to the algorithmic content of structures which are presented in terms of real numbers. Most scientific laws with a computational content can be... |

80 |
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(Show Context)
Citation Context ...’s work was formative in established a more intimate relationship between mathematics and science, and one which held the attention of Turing, in various guises, throughout his short life (see Hodges =-=[10]-=-). Just as the history of arithmetically-based algorithms, underlying many human activities, eventually gave rise to models of computation such as the Turing machine, so the oracle Turing machine sche... |

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Citation Context ...hisms. Despite decades of work by a number of leaders in the field, the exact status of the conjecture is still a matter of controversy. For those of us who have grown up with Thomas Kuhn’s 1962 book =-=[13]-=- on the structure of scientific revolutions, such difficulties and disagreements are not seen as primarily professional failures, or triggers to collective shame (although they may be that too), but r... |

49 |
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Citation Context ...ientifically important is at stake. A far more public controversy currently shapes developments around important issues affecting theoretical physics — see, for example the recent books of Lee Smolin =-=[17]-=- and Peter Woit [21]. As Peter Woit [21, p.1] describes, according to purely pragmatic criteria particle physics has produced a standard model which is remarkably successful, and has great predictive ... |

36 |
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Citation Context ...nt is at stake. A far more public controversy currently shapes developments around important issues affecting theoretical physics — see, for example the recent books of Lee Smolin [17] and Peter Woit =-=[21]-=-. As Peter Woit [21, p.1] describes, according to purely pragmatic criteria particle physics has produced a standard model which is remarkably successful, and has great predictive power: By 1973, phys... |

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(Show Context)
Citation Context ... Citing Penrose as an early champion of the role of causality, he also mentions Rafael Sorkin, Fay Dowker, and Fotini Markopoulou, known in this context for their interesting work on causal sets (see =-=[1]-=-), which abstract from causality relevant aspects of its underlying ordering relation. Essentially, causal sets are partial orderings which are locally finite, providing a model of spacetime with buil... |

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Citation Context ...s a summary of some of the main features of the Turing interpretation, drawing out parallels between scientific activity and what the Turing model provides. For further discussion of such issues, see =-=[2]-=-, [4], [5], [6] and [7]. Science Turing landscape Physical entities treated as information Structures information Theories describing relations over the Functionals over the reals modelled on reals, e... |

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Citation Context ...be composite processes, in which the underlying physical principles are understood, but the mathematics of their workings outstrip available analytical techniques. Overforty years ago, Georg Kreisel =-=[11]-=- distinguished between classical systems and cooperative phenomena not known to have Turing computable behaviour, and proposed [12, p.143, Note 2] a collision problem related to the 3-body problem, wh... |

13 |
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Citation Context ...hat of Turing, based as it is on a careful analysis of the characteristics of algorithmic computation. This brings us to a well-known research programme, initiated by Hartley Rogers in his 1967 paper =-=[16]-=-, in which he drew attention to the fundamental problem of characterising the Turing invariant relations. Again, the intuition is that these are key to pinning down how basic laws and entities emerge ... |

11 |
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(Show Context)
Citation Context ...machine, which made its first appearance in Turing [19], should be familiar enough. The details are not important, but can be found in most reasonable introductions to computability (see for instance =-=[3]-=-). The basic form of the questioning permitted is modelled on that of everyday scientific practice. This is seen most clearly in today’s digital data gathering, whereby one is limited to receiving dat... |

11 | How Can Nature Help Us Compute
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(Show Context)
Citation Context ...y of some of the main features of the Turing interpretation, drawing out parallels between scientific activity and what the Turing model provides. For further discussion of such issues, see [2], [4], =-=[5]-=-, [6] and [7]. Science Turing landscape Physical entities treated as information Structures information Theories describing relations over the Functionals over the reals modelled on reals, enabling ca... |

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(Show Context)
Citation Context ...ummary of some of the main features of the Turing interpretation, drawing out parallels between scientific activity and what the Turing model provides. For further discussion of such issues, see [2], =-=[4]-=-, [5], [6] and [7]. Science Turing landscape Physical entities treated as information Structures information Theories describing relations over the Functionals over the reals modelled on reals, enabli... |

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4 |
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(Show Context)
Citation Context ...some of the main features of the Turing interpretation, drawing out parallels between scientific activity and what the Turing model provides. For further discussion of such issues, see [2], [4], [5], =-=[6]-=- and [7]. Science Turing landscape Physical entities treated as information Structures information Theories describing relations over the Functionals over the reals modelled on reals, enabling calcula... |

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Citation Context ...te its mathematical attractions, there are growing misgivings aboutits claimed status as “the only game in town” as a unifying explanatory theory. Here is how one time string theorist Daniel Friedan =-=[9]-=- combatively puts it: The longstanding crisis of string theory is its complete failure to explain or predict any large distance physics. . . . String theory is incapable of determining the dimension, ... |