Alan Turing and the mathematical objection

Minds and Machines

Volumn 13, Issue 1, 2003, Pages 23-48

  Piccinini, Gualtiero ^a  

^a UNIVERSITY OF PITTSBURGH   (United States)

Author keywords

Artificial intelligence; Church Turing thesis; Computability; Effective procedure; Incompleteness; Machine; Mathematical objection; Ordinal logics; Turing; Undecidability

Indexed keywords

ARTIFICIAL INTELLIGENCE; COMPUTABILITY AND DECIDABILITY; DIGITAL COMPUTERS; FORMAL LOGIC; THEOREM PROVING;

CHURCH-TURING THESIS; MATHEMATICAL OBJECTION; ORDINAL LOGICS;

TURING MACHINES;

EID: 0037317005     PISSN: 09246495     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1021348629167     Document Type: Article

References (102)

1. Benacerraf, P. and Putnam, H. (1964), Philosophy of Mathematics, Englewood Cliffs, NJ: Prentice-Hall.
2. Bernays, P. (1935a), 'Sur le platonisme dans les mathématiques', L'enseignement mathématique 34, pp. 52-69, reprinted in P. Benacerraf and H. Putnam (1964), Philosophy of Mathematics, Englewood Cliffs, NJ: Prentice-Hall, pp. 274-286.
3. Bernays, P. (1935b), 'Quelques points essentiels de la métamathématique', L'enseignement mathematique 34, pp. 70-95.
4. Bernays, P. (1967), 'Hilbert, David', in P. Edwards, ed., Encyclopedia of Philosophy, Vol. 3, New York: Macmillan and Free Press, pp. 496-504.
5. Boolos, G. and Jeffrey, R. (1974), Computability and Logic, New York, NY: Cambridge University Press.
6. Chalmers, D., ed. (1995), 'Symposium on Roger Penrose's Shadows of the Mind', Psyche 2, http://psyche.cs.monasch.edu.au.
7. Church, A. (1936), 'An Unsolvable Problem in Elementary Number Theory', The American Journal of Mathematics 58, pp. 345-363, reprinted in M. Davis (1965), The Undecidable, Ewlett, NY: Raven Press, pp. 88-107.
8. Church, A. (1937), 'Review of Turing (1936-1937)', Journal of Symbolic Logic 2, pp. 42-43.
9. Colvin, S.I. (1997), Intelligent Machinery; Turing's Ideas and Their Relation to the Work of Newell and Simon, unpublished M.S. Dissertation, Carnegie Mellon University.
10. Copeland, B.J. (1996), 'The Church-Turing Thesis', in E.N. Zalta, ed., Stanford Encyclopedia of Philosophy, http://plato.stanford.edu as consulted in January 1999.
11. Copeland, B.J. (1999), 'A Lecture and Two Radio Broadcasts on Machine Intelligence by Alan Turing', in K. Furukawa, D. Michie, and S. Muggleton, eds, Machine Intelligence 15, Oxford: Oxford University Press.
12. Copeland, B.J. (2000), 'Narrow Versus Wide Mechanism: Including a Re-Examination of Turing's Views on the Mind-Machine Issue', The Journal of Philosophy XCVI, pp. 5-32.
13. Copeland, B.J. (forthcoming), The Essential Turing, Oxford: Oxford University Press.
14. Copeland, B.J. and Proudfoot, D. (1996), 'On Alan Turing's Anticipation of Connectionism', Synthese 108, pp. 361-377.
15. Davis, M. (1958), Computability and Unsolvability, New York: McGraw-Hill.
16. Davis, M. (1965), The Undecidable, Ewlett, NY: Raven Press.
17. Davis, M. (1987), 'Mathematical Logic and the Origin of Modern Computers', in E.R. Phillips, ed., Studies in the History of Mathematics (The Mathematical Association of America). pp. 137-165, reprinted in R. Herken (1988), ed., The Universal Machine: A Half-Century Survey, New York: Oxford University Press, pp. 149-174.
18. Dennett, D.C. (1978), Brainstorms: Philosophical Essays on Mind and Psychology, Cambridge, MA: MIT Press.
19. Detlefsen, M. (1995), 'Wright on the Non-Mechanizability of Intuitionist Reasoning', Philosophia Mathematica 3, pp. 103-119.
20. Eddington, A.E. (1928), The Nature of the Physical World, Cambridge: Cambridge University Press.
21. Feferman S. (1988), 'Turing in the Land of O(z)', in R. Herken, ed., The Universal Machine: A Half-Century Survey, New York: Oxford University Press, pp. 113-147.
22. Folina, J. (1998), 'Church's Thesis: Prelude to a Proof', Philosophia Mathematica 6, pp. 302-323.
23. Gandy, R. (1980), 'Church's Thesis and principles for mechanisms', in J. Barwise, H.J. Keisler, and K. Kunen, eds., The Kleene Symposium, Amsterdam: North-Holland, pp. 123-148.
24. Gandy, R. (1996), 'Human versus Mechanical Intelligence', in P.J.R. Millican and A. Clark, eds., Machines and Thought: The Legacy of Alan Turing, Oxford: Clarendon, pp. 125-136.
25. Gödel, K. (1931), 'On Formally Undecidable Propositions of Principia Mathematica and Related Systems I', Monatschefte für Mathematik und Physik 38, pp. 173-198, reprinted in M. Davis (1965), The Undecidable, Ewlett, NY: Raven Press, pp. 5-38.
26. Gödel, K. (1934), 'On Undecidable Propositions of Formal Mathematical Systems', in M. Davis (1965), The Undecidable, Ewlett, NY: Raven Press, pp. 41-71.
27. Gödel, K. (1951), 'Some Basic Theorems on the Foundations of Mathematics and Their Implications', edited text of Josiah Gibbs Lecture delivered at Brown University, in S. Feferman et al., eds., (1995) Collected Works, Vol III, Oxford: Oxford University Press, pp. 304-323.
28. Gödel, K. (1958), 'On a Hitherto Unutilized Extension of the Finitary Standpoint', Dialectica 12, pp. 280-287, reprinted in S. Feferman et al., eds., (1990), Collected Works, Vol II, Oxford: Oxford University Press, pp. 241-251.
29. Gödel, K. (1965), 'Postcriptum to 'On Undecidable Propositions of Formal Mathematical Systems", in M. Davis (1965), The Undecidable, Ewlett, NY: Raven Press, pp. 71-73.
30. Gödel, K. (1972), 'Some Remarks on the Undecidability Results', in S. Feferman et al., eds., Collected Works, Vol. II, Oxford: Oxford University Press, 1990, pp. 305-306.
31. Grush, R. and Churchland, P.S. (1995), 'Gaps in Penrose's Toilings', Journal of Consciousness Studies 2, pp. 10-29; reprintned in P.M. Churchland and P.S. Churchland (1998), On the Contrary; Critical Essays, 1987-1997, Cambridge, MA: MIT Press, pp. 205-229, 324-327.
32. Herbrand, J. (1971), Logical Writings, W.D. Goldfarb, ed., Dordrecht: Reidel.
33. Herken, R., ed. (1988), The Universal Machine: A Half-Century Survey, New York: Oxford University Press.
34. Hilbert, D. and Ackermann, W. (1928), Grundzüge der theoretischen Logik, Berlin: Springer.
35. Hodges, A. (1983), Alan Turing: The Enigma, New York: Simon and Schuster.
36. Hodges, A. (1988), 'Alan Turing and the Turing Machine', in R. Herken, ed., The Universal Turing Machine: A Half-Century Survey, New York: Oxford University Press, pp. 3-15.
37. Hodges, A. (1997), Turing; a Natural Philosopher, London: Phoenix.
38. Ince, D.C., ed. (1992), Collected Works of A.M. Turing: Mechanical Intelligence, Amsterdam: North Holland.
39. Kálmar, L. (1959), 'An Argument Against the Plausibility of Church's Thesis', in A. Heyting, ed., Constructivity in Mathematics, Amsterdam: North-Holland, pp. 72-80.
40. Kleene, S. C. (1935), 'General Recursive Functions of Natural Numbers', American Journal of Mathematics 57, pp. 153-173, 219-244.
41. Kleene, S.C. (1938), 'On Notation for Ordinal Numbers', Journal of Symbolic Logic 3, pp. 150-155.
42. Kleene, S. C. (1973), 'Recursive Predicates and Quantifiers', Transactions of the American Mathematical Society 53, pp. 41-73, reprinted in M. Davis (1965), The Undecidable, Ewlett, NY: Raven Press, pp. 254-287.
43. Kleene, S.C. (1952), Introduction to Metamathematics, Amsterdam: North-Holland.
44. Kleene, S. C. (1987a), 'Gödel's Impression on Students of Logic in the 1930s', in P. Weingartner and L. Schmetterer, eds., Gödel Remembered, Napoli: Bibliopolis, pp. 49-64.
45. Kleene, S.C. (1987b), 'Reflections on Church's Thesis', Notre Dame Journal of Formal Logic 28, pp. 490-498.
46. Kleene, S.C. (1988), 'Turing's Analysis of Computability', in R. Herken, ed., The Universal Machine: A Half-Century Survey, New York: Oxford University Press, pp. 17-54.
47. 1-paths Through O', Journal of Symbolic Logic 37, pp. 311-334.
48. Kreisel, G. (1987), 'Gödel's Excursions Into Intuitionistic Logic', in P. Weingartner and L. Schmetterer, eds., Gödel Remembered, Napoli: Bibliopolis, pp. 65-179.
49. Leiber, J. (1991), An Invitation to Cognitive Science, Cambridge, MA: Basil Blackwell.
50. Lucas, J.R. (1961), 'Minds, Machines and Gödel', Philosophy 36, pp. 112-127.
51. Lucas, J.R. (1970), The Freedom of the Will, Oxford: Clarendon Press.
52. Lucas, J.R. (1996), 'Minds, Machines, and Gödel: A Retrospect', in P.J.R. Millican and A. Clark, eds., Machines and Thought; The Legacy of Alan Turing, Oxford: Clarendon Press, pp. 103-124.
53. Mendelson, E. (1963), 'On Some Recent Criticism of Church's Thesis', Notre Dame Journal of Formal Logic 4, pp. 201-205.
54. Mendelson, E. (1990), 'Second Thoughts about Church's Thesis and Mathematical Proofs', Journal of Philosophy 88, pp. 225-233.
55. Michie, D. (1982), Machine Intelligence and Related Topics; an Information Scientist's Weekend Book, New York: Gordon and Breach Science.
56. Mundici, D. and Sieg, W. (1995), 'Paper Machines', Philosophia Mathematica 3, pp. 5-30.
57. Newman, M.H.A. (1954), Obituary Notice for Alan Turing, The Times, 16 June.
58. Newman, M.H.A. (1955), 'Alan Mathison Turing,' in Biographical Memoirs of Fellows of the Royal Society, London: The Royal Society, pp. 253-263.
59. Odifreddi, P. (1989), Classical Recursion Theory, Amsterdam: North-Holland.
60. Penrose, R. (1994), Shadows of the Mind; A Search for the Missing Science of Consciousness, Oxford: Oxford University Press.
61. Post, E. (1936), 'Finite Combinatorial Processes. Formulation I', Journal of Symbolic Logic 1, pp. 103-105, reprinted in M. Davis (1965), The Undecidable, Ewlett, NY: Raven Press, pp. 289-291.
62. Post, E. (1944), 'Recursively Enumerable Sets of Positive Integers and their Decision Problems', Bulletin of the American Mathematical Society 50, pp. 284-316, reprinted in M. Davis (1965), The Undecidable, Ewlett, NY: Raven Press, pp. 305-337.
63. Post, E. (1965), 'Absolutely Unsolvable Problems and Relatively Undecidable Propositions; Account of an Anticipation', in M. Davis (1965), The Undecidable, Ewlett, NY: Raven Press, pp. 340-433.
64. Pratt, V. (1987), Thinking Machines; The Evolution of Artificial Intelligence, Oxford: Basil Blackwell.
65. Proudfoot, D., and Copeland, B. J. (1994), 'Turing, Wittgenstein and the Science of the Mind', Australasian Journal of Philosophy 72, pp. 497-519.
66. Rogers, H. (1967), Theory of Recursive Functions and Effective Computability, New York: McGraw-Hill.
67. Shanker, S.G. (1987), 'Wittgenstein versus Turing on the Nature of Church's Thesis', Notre Dame Journal of Formal Logic 28, pp. 615-649.
68. Shanker, S.G. (1995), 'Turing and the Origins of AI', Philosophia Mathematica 3, pp. 52-85.
69. Sieg, W. (1994), 'Mechanical Procedures and Mathematical Experience', in A. George, ed., Mathematics and Mind, New York: Oxford University Press, pp. 71-117.
70. Sieg, W. (1997), 'Step by Recursive Step: Church's Analysis of Effective Calculability', Bulletin of Symbolic Logic 3, pp. 154-180.
71. Sieg, W. and Byrnes, J. (1996), 'K-graph Machines: Generalizing Turing's Machines and Arguments', in P. Hájek, ed., Gödel '96, Berlin: Springer, pp. 98-119.
72. Siegelmann, H. (1995), 'Computation Beyond the Turing Limit', Science 268, pp. 545-548.
73. Tamburrini, G. (1988), Reflections on Mechanism, unpublished Ph.D. dissertation, Columbia University.
74. Turing, A.M. (1936), 2pp. Typescript of précis of Turing (1936-1937), made for Comptes Rendues. AMT K4, Contemporary Scientific Archives Centre (CSAS), King's College Library, Cambridge.
75. Turing, A.M. (1936-1937), 'On Computable Numbers, with an Application to the Entscheidungs-problem', reprinted in M. Davis (1965), The Undecidable, Ewlett, NY: Raven Press, pp. 116-154.
76. Turing, A.M. (1937), 'Computability and λ-definability' Journal of Symbolic Logic 2, pp. 153-163.
77. Turing, A.M. (1939), 'Systems of Logic Based on Ordinals', Proceedings of the London Mathematical Society, Ser. 2 45, pp. 161-228, reprinted in M. Davis (1965), The Undecidable, Ewlett, NY: Raven Press, pp. 155-222.
78. Turing, A.M. (1940a), Letter to Newman, dated "early 1940?" by R. O. Gandy, AMT D/2, Contemporary Scientific Archives Centre, King's College Library, Cambridge. Printed in Copeland, forthcoming.
79. Turing, A.M. (1940b), Letter to Newman, dated 21 April, "1940" added by R.O. Gandy, AMT D/2, Contemporary Scientific Archives Centre, King's College Library, Cambridge, Printed in Copeland, forthcoming.
80. Turing, A.M. (1945), 'Proposal for Development in the Mathematical Division of an Automatic Computing Engine (ACE)', reprinted in D.C. Ince (1992), ed., Collected Works of A.M. Turing: Mechanical Intelligence, Amsterdam: North Holland, pp. 20-105.
81. Turing, A.M. (1947), 'Lecture to the London Mathematical Society on 20 February 1947', reprinted in D.C. Ince (1992), ed., Collected Works of A.M. Turing: Mechanical Intelligence, Amsterdam North Holland. pp. 87-105.
82. Turing, A.M. (1948), 'Intelligent Machinery', reprinted in D.C. Ince (1992), ed., Collected Works of A.M. Turing: Mechanical Intelligence, Amsterdam: North Holland, pp. 87-106.
83. Turing, A.M. (1950), 'Computing Machinery and Intelligence', in D.C. Ince (1992), ed., Collected Works of A.M. Turing: Mechanical Intelligence, Amsterdam: North Holland.
84. Turing, A.M. (1951a), 'Intelligent Machinery, a Heretical Theory,' Lecture given to '51 Society' at Manchester (c) 1951, AMT B.4, Contemporary Scientific Archives Centre, King's College Library Cambridge. Printed in E.S. Turing (1959), Alan M. Turing, Cambridge: Heffer & Sons, pp. 128-134, and in B.J. Copeland (1999), 'A Lecture and Two Radio Broadcasts on Machine Intelligence by Alan Turing', in K. Furukawa, D. Michie, and S. Muggleton, eds., Machine Intelligence 15, Oxford: Oxford University Press.
85. Turing, A.M. (1951b), Programmers' Handbook for the Manchester Electronic Computer. University of Manchester Computing Laboratory. Also available at .
86. Turing, A.M. (1951c), 'Can digital computers think?' Typescript of talk broadcast in BBC Third Programme 15 May 1951, AMT B.5, Contemporary Scientific Archives Centre, King's College Library, Cambridge. Printed in B.J. Copeland (1999), 'A Lecture and Two Radio Broadcasts on Machine Intelligence by Alan Turing', in K. Furukawa, D. Michie, and S. Muggleton, eds., Machine Intelligence 15, Oxford: Oxford University Press.
87. Turing, A.M. (1952), 'Can automatic calculating machines be said to think?' Typescript of broadcast discussion in BBC Third Programme, 14 and 23 January 1952, between M.H.A. Newman, A.M. Turing, Sir Geoffrey Jefferson, R.B. Braithwaite, AMT B.6, Contemporary Scientific Archives Centre, King's College Library, Cambridge, Printed in B.J. Copeland (1999), 'A Lecture and Two Radio Broadcasts on Machine Intelligence by Alan Turing', in K. Furukawa, D. Michie, and S. Muggleton, eds., Machine Intelligence 15, Oxford: Oxford University Press.
88. Turing, A.M. (1953), 'Digital Computers Applied to Games', in B.V. Bowden, ed., Faster Than Thought, London: Pittman, pp. 286-310, reprinted in D.C. Ince (1992), ed., Collected Works of A.M. Turing: Mechanical Intelligence, Amsterdam: North Holland, pp. 161-185.
89. Turing, A.M. (1954), 'Solvable and Unsolvable Problems', reprinted in D.C. Ince (1992), ed., Collected Works of A.M. Turing: Mechanical Intelligence, Amsterdam: North Holland, pp. 187-203.
90. Turing, A.M. (undated), Latter to Ashby, NPL M11, Archieves of the Science Museum Library, London. Also available at .
91. Turing, E.S. (1959), Alan M. Turing, Cambridge: Heffer & Sons.
92. van Heijenoort, J. ed., (1967) From Frege to Gödel, Cambridge, MA: Harvard University Press.
93. von Neumann, J. (1931), 'The Formalist Foundations of Mathematics', Erkenntnis 2, pp. 91-121, reprinted in P. Benacerraf and H. Putnam (1964), Philosophy of Mathematics, Englewoods Cliffs, NJ: Prentice-Hall, pp. 50-54.
94. von Neumann, J. (1951), 'The General and Logical Theory of Automata', in A. H. Taub, ed., (1963), Collected Works; Vol V, London: Pergamon Press, pp. 288-328.
95. Wang, H. (1974), From Mathematics to Philosophy, New York: Humanities Press.
96. Watson, A.G.D. (1938), 'Mathematics and Its Foundations', Mind 47, pp. 440-451.
97. Webb, J.C. (1980), Mechanism, Mentalism, and Metamathematics, Dordrecht: Reidel.
98. Whitehead, A.N. and Russell, B. (1910-1913), Principia Mathematica, Cambridge: Cambridge University Press.
99. Wilkes, M.V. (1985), Memoirs of a Computer Pioneer, Cambridge, MA: MIT Press.
100. Wittgenstein, L. (1976), Wittgenstein's Lectures on the Foundations of Mathematics Cambridge, 1939, in C. Diamond, ed., Ithaca, NY: Cornell University Press.
101. Wittgenstein, L. (1980), Remarks on the Philosophy of Psychology, Vol. 1, G.E.M. Anscombe, and G.H. von Wright, eds., Chicago: University of Chicago Press.
102. Wright, C. (1995), 'Intuitionists Are Not (Turing) Machines', Philosophia Mathematica 3, pp. 86-102.