## Definability as hypercomputational effect

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Venue: | Applied Mathematics and Computation |

Citations: | 5 - 4 self |

### BibTeX

@ARTICLE{Cooper_definabilityas,

author = {S. Barry Cooper},

title = {Definability as hypercomputational effect},

journal = {Applied Mathematics and Computation},

year = {},

pages = {72--82}

}

### Years of Citing Articles

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### Abstract

The classical simulation of physical processes using standard models of computation is fraught with problems. On the other hand, attempts at modelling real-world computation with the aim of isolating its hypercomputational content have struggled to convince. We argue that a better basic understanding can be achieved through computability theoretic deconstruction of those physical phenomena most resistant to classical simulation. From this we may be able to better assess whether the hypercomputational enterprise is proleptic computer science, or of mainly philosophical interest.

### Citations

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(Show Context)
Citation Context ... since Turing’s [15] discussion of ‘unorganised machines’ (see Jack Copeland and Diane Proudfoot’s article [16] On Alan Turing’s Anticipation of Connectionism), and McCulloch and Pitts’ seminal paper =-=[17]-=- on neural nets. But despite the growth of computational neuroscience as an active research area, putting together ingredients from both artificial neural networks and neurophysiology, something does ... |

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Citation Context ... intelligence as something that resides purely within the autonomous brain. The richness of mental activity appears as an extension of an equally complex world with which it interacts. Here is Brooks =-=[37]-=- again, drawing out this inclusiveness in a particulary dramatic way: “Real computational systems are not rational agents that take inputs, compute logically, and produce outputs ...It is hard to draw... |

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Citation Context ...f this extended Turing model — what is often called the Turing universe, familiar to us via its associated Turing degree structure — in most current computability theory texts (such as [39], [40], or =-=[41]-=-). The intention now is to use this mathematical model, in conjunction with our scientific understanding of the workings of the neural setting, to, on the one hand, give precision to interpretations o... |

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Citation Context ...details of this extended Turing model — what is often called the Turing universe, familiar to us via its associated Turing degree structure — in most current computability theory texts (such as [39], =-=[40]-=-, or [41]). The intention now is to use this mathematical model, in conjunction with our scientific understanding of the workings of the neural setting, to, on the one hand, give precision to interpre... |

165 | The Fabric of Reality: The - Deutsch - 1998 |

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(Show Context)
Citation Context ...ssibility of hypercomputation “may be akin to proofs that fairies are logically impossible: damn hard to be convincing.” In 1998 Jack Copeland [25] claimed to have rediscovered in Turing’s 1939 paper =-=[7]-=-, based on his 1938 thesis at Princeton, a previously unsuspected hypercomputational agenda, based on the familiar oracle Turing machine — an exegesis which seemed to throw the embattled classical com... |

133 | The Feeling of What Happens - Damasio - 2000 |

70 | NonTuring computations via MalamentHogarth space-times Int.J.Theor.Phys
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Citation Context ...tion of whether hypercomputation, if it actually occurs, can be successfully harnessed to human computational needs. This is how Martin Davis [10] puts it in relation to Etesi and Nemeti’s discussion =-=[11]-=- of relativistic behaviour near black holes: “Of course, even assuming that all this really does correspond to the actual universe in which we live, there is still the question of whether an actual de... |

51 |
Computability Theory
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(Show Context)
Citation Context ...d the details of this extended Turing model — what is often called the Turing universe, familiar to us via its associated Turing degree structure — in most current computability theory texts (such as =-=[39]-=-, [40], or [41]). The intention now is to use this mathematical model, in conjunction with our scientific understanding of the workings of the neural setting, to, on the one hand, give precision to in... |

34 |
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(Show Context)
Citation Context ...utational neuroscience as an active research area, putting together ingredients from both artificial neural networks and neurophysiology, something does seem to be missing. This 4leads Rodney Brooks =-=[18]-=- to allude to the fact that “neither AI nor Alife has produced artifacts that could be confused with a living organism for more than an instant.” Again: “...neural networks alone cannot do the job” op... |

28 |
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(Show Context)
Citation Context ... and Wiedermann’s [2] observation 2 that “the classical Turing paradigm may no longer be fully appropriate to capture all features of present-day computing.” In contrast to this, one has Martin Davis =-=[3]-=- confidently asserting that: “The great success of modern computers as all-purpose algorithm-executing engines embodying Turing’s universal computer in physical form, makes it extremely plausible that... |

27 | The Turing machine paradigm in contemporary computin
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(Show Context)
Citation Context ...vices, intelligent agents, operating systems, and graphical user interfaces, cannot be modeled by Turing machines; alternative models are needed.” This echoes and extends van Leeuwen and Wiedermann’s =-=[2]-=- observation 2 that “the classical Turing paradigm may no longer be fully appropriate to capture all features of present-day computing.” In contrast to this, one has Martin Davis [3] confidently asser... |

27 | Narrow versus wide mechanism: Including a re-examination of Turing’s views on the mind-machine issue - Copeland - 2000 |

25 |
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(Show Context)
Citation Context ...gnetic resonance imaging scan (fMRI), bring us much closer to obtaining useful models. Connectionist models of computation based on the workings of the human brain have come a long way since Turing’s =-=[15]-=- discussion of ‘unorganised machines’ (see Jack Copeland and Diane Proudfoot’s article [16] On Alan Turing’s Anticipation of Connectionism), and McCulloch and Pitts’ seminal paper [17] on neural nets.... |

23 |
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(Show Context)
Citation Context ...ichness of structure we observe is matched by the as yet unsolved problems of showing that aspects of, for instance, the Mandelbrot and certain Julia sets are computable (see [31], and, for contrast, =-=[32]-=-, [33], [34]). Of course, such mathematical examples provide a metaphor for the way real-world complexity is generated by the iteration of simple algorithmic content, and nicely illustrate how the ana... |

17 | Church’s Thesis: a kind of reducibility axiom for constructive mathematics - Kreisel - 1970 |

16 | Incomputability in nature
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(Show Context)
Citation Context ...t the theoretical question is certainly of interest.” 4 Davis’ particular doubts here are addressed on a number of levels in the István Németi and Gyula Dávid article [12] in this volume. 3argued in =-=[5]-=-, is ‘damn hard to be convincing’, but which we would expect to provide an important element in the establishment of a larger paradigm shift. 2 Neural mapping and real-world hypercomputation If one is... |

16 | A fast algorithm for Julia sets of hyperbolic rational functions
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(Show Context)
Citation Context ...tructure we observe is matched by the as yet unsolved problems of showing that aspects of, for instance, the Mandelbrot and certain Julia sets are computable (see [31], and, for contrast, [32], [33], =-=[34]-=-). Of course, such mathematical examples provide a metaphor for the way real-world complexity is generated by the iteration of simple algorithmic content, and nicely illustrate how the analysis of app... |

15 |
The Church–Turing Thesis: Breaking the myth
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- 2005
(Show Context)
Citation Context ...ared to be a largely reactive one. Reflective of the growing body of literature openly questioning the relevance of classical models of computation is the following quotation from Goldin and Wegner’s =-=[1]-=- paper from CiE 2005: 1 The author wishes to thank István Németi for a number of helpful comments on a first draft of this paper, which have led to a number of improvements. Preparation of this articl... |

15 |
Turing’s O-machines
- Copeland
- 1998
(Show Context)
Citation Context ... Welch [4] points out, because logical proofs of the impossibility of hypercomputation “may be akin to proofs that fairies are logically impossible: damn hard to be convincing.” In 1998 Jack Copeland =-=[25]-=- claimed to have rediscovered in Turing’s 1939 paper [7], based on his 1938 thesis at Princeton, a previously unsuspected hypercomputational agenda, based on the familiar oracle Turing machine — an ex... |

15 | Computations via experiments with kinematic systems
- Beggs, Tucker
- 2004
(Show Context)
Citation Context ... has a precise parallel in the difficulty we have knitting together these two types of computational model. If you come up with a model which elevates information content (such as in Beggs and Tucker =-=[36]-=-), you will probably be told by someone that it is unrealisable. If you admit a rich pre-existent information content in your model to give scope to the interactivity we identified earlier (as do most... |

12 | Alan Turing: The Enigma. Vintage - Hodges - 1992 |

10 | Is the Mandelbrot set computable
- Hertling
(Show Context)
Citation Context ...ions. The extraordinary richness of structure we observe is matched by the as yet unsolved problems of showing that aspects of, for instance, the Mandelbrot and certain Julia sets are computable (see =-=[31]-=-, and, for contrast, [32], [33], [34]). Of course, such mathematical examples provide a metaphor for the way real-world complexity is generated by the iteration of simple algorithmic content, and nice... |

7 |
Computation beyond Turing Machines: Seeking appropriate methods to model computing and human thought
- Wegner, Goldin
- 2003
(Show Context)
Citation Context ... inclusive — to shepherd the proposed new paradigm back into the classical fold. There are limits to this though, determined by those on the success of hypercomputational modelling. Goldin and Wegner =-=[27]-=- certainly have the modelling of more than internal connectivity in mind when they quote from Robin Milner’s 1991 Turing Award lecture [28, p.80]: “Through the seventies, I became convinced that a the... |

7 | Hyperbolic Julia Sets are Poly-Time Computable
- Braverman
- 2005
(Show Context)
Citation Context ...s of structure we observe is matched by the as yet unsolved problems of showing that aspects of, for instance, the Mandelbrot and certain Julia sets are computable (see [31], and, for contrast, [32], =-=[33]-=-, [34]). Of course, such mathematical examples provide a metaphor for the way real-world complexity is generated by the iteration of simple algorithmic content, and nicely illustrate how the analysis ... |

5 |
On the possibility, or otherwise, of hypercomputation
- Welch
- 2004
(Show Context)
Citation Context ...estion ‘What is a computation?’, and, by itself, makes the existence of any more general form of computation extremely doubtful.” This has become a hard position to maintain. If only, as Philip Welch =-=[4]-=- points out, because logical proofs of the impossibility of hypercomputation “may be akin to proofs that fairies are logically impossible: damn hard to be convincing.” In 1998 Jack Copeland [25] claim... |

5 | Solvable and unsolvable problems - Turing - 1954 |

5 | On Alan Turing’s anticipation of connectionism
- Copeland, Proudfoot
- 1996
(Show Context)
Citation Context ...ectionist models of computation based on the workings of the human brain have come a long way since Turing’s [15] discussion of ‘unorganised machines’ (see Jack Copeland and Diane Proudfoot’s article =-=[16]-=- On Alan Turing’s Anticipation of Connectionism), and McCulloch and Pitts’ seminal paper [17] on neural nets. But despite the growth of computational neuroscience as an active research area, putting t... |

4 |
The Incomputable Alan Turing. In Turing 2004: A celebration of his life and achievements
- Cooper
- 2004
(Show Context)
Citation Context ...ivistic computers and non-uniform complexity theory. In: Calude et al (eds.) UMC 2002. Lecture Notes in Computer Science Vol. 2509, Springer-Verlag, Berlin, 2002. pp. 287–299. 2computability (and in =-=[8]-=- I have indicated that these were not as simple as sometimes thought), Copeland has effectively spotlighted the potential for mathematically analysing the computationally complex in terms of its algor... |

3 | Beyond Gödel’s Theorem: Turing nonrigidity revisited
- Cooper
- 1998
(Show Context)
Citation Context ...e not as simple as sometimes thought), Copeland has effectively spotlighted the potential for mathematically analysing the computationally complex in terms of its algorithmic content. As we argued in =-=[9]-=-: “If one abstracts from the Universe its information content, structured via the basic ...fundamental laws of nature, one obtains a particular ...manifestation of the Turing universe ..., within whic... |

3 |
to appear) Computability, computation and the real world
- Davis
(Show Context)
Citation Context ...tational effects. The other is the more down-to-earth question of whether hypercomputation, if it actually occurs, can be successfully harnessed to human computational needs. This is how Martin Davis =-=[10]-=- puts it in relation to Etesi and Nemeti’s discussion [11] of relativistic behaviour near black holes: “Of course, even assuming that all this really does correspond to the actual universe in which we... |

3 |
Computability and emergence
- Cooper
- 2006
(Show Context)
Citation Context ...tics governing the emergence of new relations from chaotic environments. This leads us to turn the picture of re-representations of mental imaging as a describable mapping on its head, and think (see =-=[21]-=-) in terms of descriptions in terms of a structure defining, and hence determining, the mental re-representations. Looking at this more closely, what seems to be happening is that the brain stores awa... |

3 | Elements of interaction:Turing award lecture - Milner - 1993 |

2 |
Relativistic computers and the Turing barrier. This volume
- Németi, Dávid
- 2005
(Show Context)
Citation Context ...In which case, the answer to the second question will give us a better idea of whether the hypercomputational enterprise is proleptic computer science, or philosophy. The approach of Németi and Dávid =-=[12]-=- is a rather different one from that taken here. Theirs is based on what they describe as ‘a major paradigm shift in our physical world-view as well as our cosmological one’. They seek to give scienti... |

2 |
Solvable and Unsolvable Problems. Penguin Science News 31, 7–23. Turing 2004: A celebration of his life and achievements 16 The Incomputable Alan Turing
- Turing
- 1954
(Show Context)
Citation Context ...s both widely suspected of transcending the standard Turing model, and of whose inner workings we have a high level of detailed knowledge, we need look no further than the human brain. Turing himself =-=[13]-=- talks about “the inadequacy of ‘reason’ unsupported by common sense”, and in his 1939 paper [7] says: “Mathematical reasoning may be regarded ...as the exercise of a combination of ...intuition and i... |

1 | Turing’s connectionism. In Alan Turing: Life and legacy of a great thinker - Teuscher - 2004 |