Abstract
It is difficult to prove that something is not possible in principle. Likewise it is often difficult to refute such arguments. The Lucas-Penrose argument tries to establish that machines can never achieve human-like intelligence. It is built on the fact that any reasoning program which is powerful enough to deal with arithmetic is necessarily incomplete and cannot derive a sentence that can be paraphrased as “This sentence is not provable.” Since humans would have the ability to see the truth of this sentence, humans and computers would have obviously different mental capacities. The traditional refutation of the argument typically involves attacking the assumptions of Gödel’s theorem, in particular the consistency of human thought. The matter is confused by the prima facie paradoxical fact that Gödel proved the truth of the sentence that “This sentence is not provable.”
Adopting Chaitin’s adaptation of Gödel’s proof which involves the statement that “some mathematical facts are true for no reason! They are true by accident” and comparing it to a much older incompleteness proof, namely the incompleteness of rational numbers, the paradox vanishes and clarifies that the task of establishing arbitrary mathematical truths on numbers by finitary methods is as infeasible to machines as it is to human beings.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Baker, S., Ireland, A., Smaill, A.: On the use of the constructive omega-rule within automated deduction. DAI Research Paper 560, University of Edinburgh, Edinburgh, United Kingdom (1991), Also available as: http://www.dai.ed.ac.uk/papers/documents/rp560.html
Bundy, A., Giunchiglia, F., Villafiorita, A., Walsh, T.: Gödel’s incompleteness theorem via abstraction. IRST-Technical Report 9302-15, IRST, Povo, Trento, Italy (February 1993)
Chaitin, G.J.: The Limits of Mathematics. Springer, Heidelberg (1998)
Dedekind, R.: What are numbers and what are they for? (1888)
Gödel, K.: Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatshefte für Mathematik und Physik 37, 349–360 (1930)
Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38, 173–198 (1931)
Grzegorczyk, A., Mostowski, A., Ryll-Nardzewski, C.: The classical and the ω-complete arithmetic. The Journal of Symbolic Logic 23(2), 188–206 (1958)
van Heijenoort, J. (ed.): From Frege to Gödel – A Source Book in Mathematical Logic, 1879-1931. Havard University Press, Cambridge (1967)
Hilbert, D.: Über das Unendliche. Mathematische Annalen 95, 161–190 (1926)
LaForte, G., Hayes, P.J., Ford, K.M.: Why Gödel’s theorem cannot refute computationalism. Artificial Intelligence 104, 265–286 (1998)
Lucas, J.R.: Minds, Machines and Gödel. Philosophy 36, 112–127 (1961)
Lucas, J.R.: A paper read to the Turing Conference at Brighton on April 6 (1990), http://users.ox.ac.uk/~jrlucas/Godel/brighton.html
Lucas, J.R.: Minds, Machines and Gödel: A Retrospect. In: Millican, P., Clark, A. (eds.) Machines and Thought – The Legacy of Alan Turing, ch. 6, vol. 1, pp. 103–124. Oxford University Press, Oxford (1996)
Penrose, R.: The Emperor’s New Mind. Oxford University Press, Oxford (1989)
Penrose, R.: Mathematical intelligence. In: Khalfa, J. (ed.) What is Intelligence?, ch. 5, pp. 107–136. Cambridge University Press, Cambridge (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kerber, M. (2005). Why Is the Lucas-Penrose Argument Invalid?. In: Furbach, U. (eds) KI 2005: Advances in Artificial Intelligence. KI 2005. Lecture Notes in Computer Science(), vol 3698. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11551263_30
Download citation
DOI: https://doi.org/10.1007/11551263_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28761-2
Online ISBN: 978-3-540-31818-7
eBook Packages: Computer ScienceComputer Science (R0)