## The Broad Conception Of Computation (1997)

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Venue: | American Behavioral Scientist |

Citations: | 11 - 2 self |

### BibTeX

@ARTICLE{Copeland97thebroad,

author = {B. Jack Copeland},

title = {The Broad Conception Of Computation},

journal = {American Behavioral Scientist},

year = {1997},

volume = {40},

pages = {690--716}

}

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### Abstract

A myth has arisen concerning Turing's paper of 1936, namely that Turing set forth a fundamental principle concerning the limits of what can be computed by machine - a myth that has passed into cognitive science and the philosophy of mind, to wide and pernicious effect. This supposed principle, sometimes incorrectly termed the 'Church-Turing thesis', is the claim that the class of functions that can be computed by machines is identical to the class of functions that can be computed by Turing machines. In point of fact Turing himself nowhere endorses, nor even states, this claim (nor does Church). I describe a number of notional machines, both analogue and digital, that can compute more than a universal Turing machine. These machines are exemplars of the class of nonclassical computing machines. Nothing known at present rules out the possibility that machines in this class will one day be built, nor that the brain itself is such a machine. These theoretical considerations undercut a numb...

### Citations

1157 |
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(Show Context)
Citation Context ...ch-Turing) universality of computation ... provides perhaps the best reason for believing in the (functional) AI thesis to begin with' (1996: 390). 9. In point of fact the alphabet of the machines of =-=Turing 1936-=- was not binary. 10. With thanks to Ned Block, Sean Broadley, Martin Davis, Dan Dennett, Jon Doyle, Bert Dreyfus, Robin Gandy, Allen Hazen, Andrew Hodges, Kevin Korb, Donald Michie, Chris Mortensen, R... |

833 | The modularity of the mind - Fodor - 1983 |

686 |
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- 1952
(Show Context)
Citation Context ... We shall usually refer to them both as Church's thesis, or in connection with that one of its ... versions which deals with 'Turing machines' as the Church-Turing thesis. (Kleene 1967: 232; see also =-=Kleene 1952-=-.) In mathematical logic this is the standard use of the term 'Church-Turing thesis'. However, it is nowadays not uncommon for a very different proposition, that whatever can be calculated by any mach... |

650 | Quantum theory, the Church-Turing principle, and the universal quantum computer
- Deutsch
- 1985
(Show Context)
Citation Context ...ering. A further reason to revise traditional terminology in the manner suggested here is provided by the recent emergence of the field of quantum computation. Algorithms for quantum Turing machines (=-=Deutsch 1985-=-, Solovay and Yao 1996) are not effective procedures, since not all the primitive operations of a quantum Turing machine can be performed by a person unaided by machinery. (While the algorithms execut... |

580 | Consciousness Explained - Dennett - 1991 |

469 |
The Emperor’s New Mind
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(Show Context)
Citation Context ...be Uncomputable physical processes stretches back over at least four decades (see, for example, Da Costa and Doria 1991, Doyle 1982, Kreisel 1967, 1974, Pour-El 1974, Pour-El and Richards 1979, 1981, =-=Penrose 1989-=-, 1994, Scarpellini 1963, Stannett 1990, Vergis et al. 1986; Copeland and Sylvan 1997 is a survey). A notional oracle can certainly be implemented by means of a machine similar to M1. For example, rel... |

454 |
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- Minsky
- 1967
(Show Context)
Citation Context ... 9sIn the literature the word 'algorithm' is rarely used in the way in which it is used here. Standardly, 'algorithm' is used as a synonym for 'effective procedure' (see, for example, the glossary in =-=Minsky 1967-=-: 311), there being no accepted term for machine-executable procedures that are not effective (i.e. which cannot also be carried out by an obedient human clerk who is devoid of any imagination or tale... |

390 | Minds, brains, and programs
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- 1980
(Show Context)
Citation Context ...puter program, or in a later version, upon the occupants of the Chinese gym being able in principle 12sto enact the steps of the algorithm that determines the behaviour of some connectionist network (=-=Searle 1980-=-, 1990). Thus in both cases the argument demands that the algorithm in question be an effective procedure. Indeed, one way of explaining the notion of an effective procedure is as a machine program th... |

363 |
On the proper treatment of connectionism
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- 1988
(Show Context)
Citation Context ... of computing with real numbers is sometimes taken seriously is connectionism (for example, Garzon and Franklin 1989, MacLennan 1994, Rumelhart and McClelland 1986: 45-49, Siegelmann and Sontag 1994, =-=Smolensky 1988-=-, section 8, Wolpert and MacLennan 1993). Korb raises a doubt: to suppose 'neural networks to be computationally superior [to Turing machines] ... is to suppose that the analog device is sensitive to ... |

349 | nie Rediscovery of the Mind - Searle - 1994 |

279 | The conscious mind: In search of a fundamental theory - Chalmers - 1996 |

263 |
An unsolvable problem of elementary number theory
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(Show Context)
Citation Context ... this is a thesis concerning the limits of what a human mathematician can compute, not a thesis concerning the limits of machine computation. Church put forward an equivalent form of the same thesis (=-=Church 1936-=-). Since Church published first, the thesis is usually called 'Church's thesis'. The term 'Church-Turing thesis' seems to have been first introduced by Kleene (with a small flourish of bias in favour ... |

256 |
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(Show Context)
Citation Context ... of the theorem. 18sOne area of computer science in which the idea of computing with real numbers is sometimes taken seriously is connectionism (for example, Garzon and Franklin 1989, MacLennan 1994, =-=Rumelhart and McClelland 1986-=-: 45-49, Siegelmann and Sontag 1994, Smolensky 1988, section 8, Wolpert and MacLennan 1993). Korb raises a doubt: to suppose 'neural networks to be computationally superior [to Turing machines] ... is... |

195 | Physical Symbol Systems
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- 1980
(Show Context)
Citation Context ...well, for example, appeals to the Church-Turing thesis improperly so called in the course of his famous argument for the crucial sufficiency-part of his and Simon's physical symbol system hypothesis (=-=Newell 1980-=-). A physical symbol system is a universal Turing machine, or any equivalent system, situated in the physical - as opposed to the conceptual - world (1980: 147-50, 154-55, 161). (The tape of the unive... |

181 | Computation and cognition: Toward a foundation for cognitive science - Pylyshyn - 1986 |

171 |
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(Show Context)
Citation Context ... a bounded amount of memory, and (b) the Cray's speed of operation is limited by various real-world constraints. (It is often said, incorrectly, that a Turing machine is necessarily slow (for example =-=Haugeland 1985-=-: 140). A Turing machine is an idealised device and has no real-world constraints on its speed of operation.) A program or 'machine table' for a Turing machine is a finite collection of instructions e... |

130 | What computers still can't do: A critique of artificial reason - Dreyfus - 1992 |

125 | Shadows of the Mind: A Search for the Missing Science of Consciousness - Penrose - 1994 |

123 |
Systems of Logic Based on Ordinals
- Turing
- 1939
(Show Context)
Citation Context ...chine can compute the halting function may be found in Minsky (1967: 148-9).) Turing himself was the first to consider nonclassical digital computing machines, in his Ph.D. dissertation (published as =-=Turing 1939-=-). This work seems to be little known among cognitive scientists. Turing called his machines O-machines, or 'oracle' machines (1939: 172ff). An O-machine consists of an ordinary Turing machine augment... |

87 | Analog computation via neural networks
- Siegelmann, Sontag
- 1994
(Show Context)
Citation Context ...er science in which the idea of computing with real numbers is sometimes taken seriously is connectionism (for example, Garzon and Franklin 1989, MacLennan 1994, Rumelhart and McClelland 1986: 45-49, =-=Siegelmann and Sontag 1994-=-, Smolensky 1988, section 8, Wolpert and MacLennan 1993). Korb raises a doubt: to suppose 'neural networks to be computationally superior [to Turing machines] ... is to suppose that the analog device ... |

74 | an Eternal Golden Braid - Hofstadter - 1979 |

66 |
Abstract computability and its relations to the general purpose analog computer
- Pour-El
- 1974
(Show Context)
Citation Context ...In brief summary her results are: (1) there are Turing-machine-computable functions that cannot be computed by the GPAC; and (2) the action of the GPAC can always be approximated by a Turing machine (=-=Pour-El 1974-=-, theorems 3 and 7). There was a crucial lacuna in the proof of Shannon's principal theorem and Pour-El is able to obtain her results only because of a significant modification she makes to Shannon's ... |

56 | Church’s thesis and principles for mechanism, in - Gandy - 1980 |

55 | The wave equation with computable initial data such that its unique solution is not computable - Richards - 1981 |

40 |
The Oxford Companion to the Mind
- Gregory
- 1998
(Show Context)
Citation Context ...nd' states: 'Turing showed that his very simple machine ... can specify the steps required for the solution of any problem that can be solved by instructions, explicitly stated rules, or procedures' (=-=Gregory 1987-=-: 784). Dennett maintains that 'Turing had proven - and this is probably his greatest contribution - that his Universal Turing machine can compute any function that any computer, with any architecture... |

38 |
Recursive real numbers
- Rice
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(Show Context)
Citation Context ...nctions, to be written E(x,y). Where x and y are any Computable numbers, E(x,y)=1 if and only if x=y, and E(x,y)=0 if and only if x≠y. (A proof to the effect that E(x,y) is not Computable is given b=-=y Rice 1954-=-: 786 and Aberth 1968: 287.) Turning from concepts that Turing himself explicitly presented, I will now introduce the general notions of a computing machine and an algorithm. The general requirements ... |

38 | The complexity of analog computation
- Vergis, Steiglitz, et al.
- 1986
(Show Context)
Citation Context ...r at least four decades (see, for example, Da Costa and Doria 1991, Doyle 1982, Kreisel 1967, 1974, Pour-El 1974, Pour-El and Richards 1979, 1981, Penrose 1989, 1994, Scarpellini 1963, Stannett 1990, =-=Vergis et al. 1986-=-; Copeland and Sylvan 1997 is a survey). A notional oracle can certainly be implemented by means of a machine similar to M1. For example, relative to some given ordering of the arguments of the haltin... |

33 |
and L A Rubel, A differentially algebraic replacement theorem
- Lipshitz
(Show Context)
Citation Context ...and 1993b I explain why the Chinese room argument enjoys no success against position (1). 5. In fact Pour-El's own proof is also incomplete. The situation has now been improved by Lipshitz and Rubel (=-=Lipshitz and Rubel 1987-=-, Rubel 1988, 1989). 6. Geroch and Hartle (1986) make a similar point in connection with deriving predictions to within given accuracies from physical theories some of whose constants are Uncomputable... |

30 |
X-Machines and the Halting Problem: Building a super-Turing Machine
- Stannett
- 1990
(Show Context)
Citation Context ...etches back over at least four decades (see, for example, Da Costa and Doria 1991, Doyle 1982, Kreisel 1967, 1974, Pour-El 1974, Pour-El and Richards 1979, 1981, Penrose 1989, 1994, Scarpellini 1963, =-=Stannett 1990-=-, Vergis et al. 1986; Copeland and Sylvan 1997 is a survey). A notional oracle can certainly be implemented by means of a machine similar to M1. For example, relative to some given ordering of the arg... |

29 | Computer Models of Mind - Boden - 1988 |

28 | Mathematical theory of the differential analyser - Shannon - 1941 |

27 | Is the Brain’s Mind a Computer Program - Searle - 1990 |

27 | The Representational Theory of Mind - Sterelny - 1990 |

23 |
Intelligent Machinery’, National Physical Laboratory Report
- Turing
- 1948
(Show Context)
Citation Context ...dure previously mentioned, writing the resulting digits on the blank squares of the tape. Equivalently one may consider a Turing machine that has been modified by the addition of two input lines (see =-=Turing 1948-=-: 17ff, Copeland and Proudfoot 1996). This arrangement removes the need to inscribe the representations of x and y on the tape prior to the commencement of the computation. If a digit is applied to an... |

21 | Artificial intelligence: A philosophical introduction - Copeland - 1993 |

21 | Computability and Physical Theories - Geroch, Hartle - 1986 |

20 | A notion of mechanistic theory - Kreisel - 1974 |

20 |
Continuous symbol systems: The logic of connectionism
- MacLennan
- 1992
(Show Context)
Citation Context ...utside the ambit of the theorem. 18sOne area of computer science in which the idea of computing with real numbers is sometimes taken seriously is connectionism (for example, Garzon and Franklin 1989, =-=MacLennan 1994-=-, Rumelhart and McClelland 1986: 45-49, Siegelmann and Sontag 1994, Smolensky 1988, section 8, Wolpert and MacLennan 1993). Korb raises a doubt: to suppose 'neural networks to be computationally super... |

18 | Could a machine think - Churchland, Churchland - 1990 |

16 |
I.: A computable ordinary differential equation which possesses no computable solution
- Pour-El, Richards
- 1979
(Show Context)
Citation Context ...as to whether there may actually be Uncomputable physical processes stretches back over at least four decades (see, for example, Da Costa and Doria 1991, Doyle 1982, Kreisel 1967, 1974, Pour-El 1974, =-=Pour-El and Richards 1979-=-, 1981, Penrose 1989, 1994, Scarpellini 1963, Stannett 1990, Vergis et al. 1986; Copeland and Sylvan 1997 is a survey). A notional oracle can certainly be implemented by means of a machine similar to ... |

16 |
Some mathematical limitations of the general-purpose analog computer
- Rubel
- 1988
(Show Context)
Citation Context ...he Chinese room argument enjoys no success against position (1). 5. In fact Pour-El's own proof is also incomplete. The situation has now been improved by Lipshitz and Rubel (Lipshitz and Rubel 1987, =-=Rubel 1988-=-, 1989). 6. Geroch and Hartle (1986) make a similar point in connection with deriving predictions to within given accuracies from physical theories some of whose constants are Uncomputable numbers. 7.... |

16 | Digital simulation of analog computation and Church’s thesis - Rubel - 1989 |

15 | The mind–body problem - Fodor - 1981 |

15 |
A Companion to the Philosophy of Mind
- Guttenplan
- 1994
(Show Context)
Citation Context ...tween input and output are functionally well-behaved enough to be describable by ... mathematical relationships ... we know that some specific version of a Turing machine will be able to mimic them' (=-=Guttenplan 1994-=-: 595). These various quotations are typical of current writing on the foundations of the computational theory of mind. Turing had no result entailing that a Turing machine can 'display any systematic... |

13 |
The confluence of Ideas in 1936
- Gandy
- 1988
(Show Context)
Citation Context ...that the existence of such procedures has commonly been overlooked. Their existence tends to be obscured by the fact that the word 'mechanical' is standardly used as another synonym for 'effective'. (=-=Gandy 1988-=- outlines the history of this use of the word 'mechanical'.) On this way of speaking, the issue of whether there are machine-executable procedures that are not effective becomes hidden, for the questi... |

13 |
Mathematical logic: what has it done for the philosophy of mathematics
- Kreisel
- 1967
(Show Context)
Citation Context ...g in the black boxes. Speculation as to whether there may actually be Uncomputable physical processes stretches back over at least four decades (see, for example, Da Costa and Doria 1991, Doyle 1982, =-=Kreisel 1967-=-, 1974, Pour-El 1974, Pour-El and Richards 1979, 1981, Penrose 1989, 1994, Scarpellini 1963, Stannett 1990, Vergis et al. 1986; Copeland and Sylvan 1997 is a survey). A notional oracle can certainly b... |

8 | Computability and Logic, 2nd edition - Boolos, Jeffrey - 1980 |

8 | What is computation - Copeland - 1996 |

8 |
Zwei Unentscheitbare Probleme der Analysis. Zeitschrift für mathematische Logik und Grundlagen der Mathematik
- Scarpellini
- 1963
(Show Context)
Citation Context ...ical processes stretches back over at least four decades (see, for example, Da Costa and Doria 1991, Doyle 1982, Kreisel 1967, 1974, Pour-El 1974, Pour-El and Richards 1979, 1981, Penrose 1989, 1994, =-=Scarpellini 1963-=-, Stannett 1990, Vergis et al. 1986; Copeland and Sylvan 1997 is a survey). A notional oracle can certainly be implemented by means of a machine similar to M1. For example, relative to some given orde... |

7 |
Reflections on Church’s Thesis’, Notre Dame
- Kleene
- 1987
(Show Context)
Citation Context ...ntly meets the Kleene-Hofstadter requirement that an algorithm be capable of being 'communicated reliably from one sentient being to another of reasonable mathematical aptitude by means of language' (=-=Kleene 1987-=-: 493-4, Hofstadter 1980: 562). (To anticipate a little, this remark applies also to algorithms executed by the nonclassical digital machines introduced in section 3.) The action of M1 can be approxim... |