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Gödel, Minds, and Machines - Web links

For the past fifty years there has been a continuing argument as to whether Gödel’s incompleteness proof shows that minds are more than mere machines (albeit complex ones). A common line of argument is that a human can ‘see’ that the ‘Gödel sentence’ is ‘true’ but a machine could not have this ‘insight’, since a machine must always follow definite rules, just like a formal system. And even after fifty years of argument neither side is prepared to concede defeat and lay aside their cherished beliefs. Those advocating this viewpoint are commonly called ‘anti-mechanists’, while those opposing it are called ‘mechanists’.

 

As you can see by the number of links below, there is a plethora of articles on this subject, of various degrees of quality. It seems that you can say almost anything about Gödel’s proof, provided you avoid any mention of the ultimate taboo – that is, you never suggest that there might be an error in Gödel’s logic.

 

It doesn’t seem to occur to anyone that the continuing conflict of viewpoints and the heated debate over the issue might indicate a problem in Gödel’s proof, rather than a problem in its interpretation. It is somewhat ironic that the mechanists are as definitive as the anti-mechanists in their deeply held belief that there cannot possibly be any error in Gödel’s proof. The demonstration that there is actually a fundamental flaw in the logic of Gödel’s proof (see The flaw in Gödel’s proof: here), and also in the logic of various other incompleteness proofs (see Analysis of Incompleteness Proofs: here) settles the question at once, since, at a stroke, it pulls out the principal cornerstone of the anti-mechanist argument.

 

Below are links to various articles on the Gödel, Minds, and Machines question. Links to articles on other aspects of Gödel’s theorem can be found on this page: Gödel Links. If you think an item should be included on this page please let me know.

 

Please note: Inclusion in this list does not mean I recommend any of the items in this list.

 

Online material on the Gödel, Minds, and Machines argument

See further down the page below for books and non-free online material

 

Note: The principal anti-mechanists are John Randolph Lucas, Roger Penrose, Michael Redhead and Storrs McCall. There are many more pro-mechanists. The articles by the principal anti-mechanists are listed first - this is for the convenience of the reader only, and does not indicate any partiality towards these authors.

 

The principal anti-mechanist authors

Lucas, J R Minds, Machines and Gödel Philosophy 36 (1961) pp 112-127
Lucas, J R Satan Stultified: A Rejoinder To Paul Benacerraf The Monist, vol.52, 1 (1968) - a response to Beancerraf’s paper God, the devil, and Gödel (MS Word doc file)
Lucas, J R Human and Machine Logic: a Rejoinder British Journal for the Philosophy of Science, 19 (1968) pp 155–156 - a response to Good’s paper Human and machine logic (MS Word doc file)
Lucas, J R Mechanism: A Rejoinder Philosophy 45, (1970) pp 149-151 - a response to Coder’s paper Gödel’s theorem and mechanism (MS Word doc file) and Lewis’ paper Lucas against mechanism (MS Word doc file)
Lucas, J R This Gödel is Killing Me: a Rejoinder Philosophia, 6, no.1 (1976) pp 145-148 - a response to Hutton’s paper This Gödel is killing me (MS Word doc file)
Lucas, J R Review of Mechanism, Mentalism and Metamathematics (Judson Webb) The British Journal for the Philosophy of Science, 33 (1982) pp 441-444
Lucas, J R The Gödelian argument: Turn over the page a talk by J R Lucas (1996)
Lucas, J R A Simple Exposition of Gödel’s Theorem a talk by J R Lucas, (1997)
Lucas, J R The Implications of Gödel’s Theorem a talk by J R Lucas, (1996 - 1998)
Lucas, J R The Implications of Gödel’s Theorem (more technical) a talk by J R Lucas, (1998)
Lucas, J R The Implications of Gödel’s Theorem - handout handout for the above talk
Lucas, J R A paper read to the Turing Conference at Brighton on April 6th, 1990  
Lucas, J R Commentary on Turing’s ‘Computing Machinery and Intelligence’ a comment (2005) on Turing’s paper Computing Machinery and Intelligence
Lucas, J R A response to a paper by Professor Feferman PDF a response to Feferman’s paper, Gödel’s incompleteness theorems, free will and mathematical thought PDF
Penrose, Roger Beyond the Doubting of a Shadow Psyche (1996) 2 (23) pp 89-129 - a reply to the various reactions (see Baars, Bernard: P1, Chalmers, David: P2, Feferman, Solomon: P3, Klein, Stanley: P4, Maudlin, Tim: P5, McCarthy, John: P6, McCullough, Daryl: P7, McDermott, Drew: P8, Moravec, Hans: P9 below) to his book Shadows of the Mind (see below for details of the book).

 

 

Other authors

Baars, Bernard J [Return to link: P1] Can physics provide a theory of consciousness?  
Benacerraf, Paul God, the devil, and Gödel (MS Word doc file) The Monist (1967) 51 pp 9-32
Bojadziev, Damjan Article in Mind versus Gödel Book: Mind Versus Computer, IOS Press (1997) pp 202-210
Bringsjord, Selmer Is Gödelian model-based deductive reasoning computational? PDF Philosophica (1998) 61 pp 51-76
Bringsjord, Selmer & Xiao, H A refutation of Penrose’s new Gödelian case against the computational conception of mind PDF Journal of Experimental and Theoretical Artificial Intelligence (2000) 12 pp 307-329
Bruni, Riccardo Gödel, Turing, the Undecidability Results and the Nature of Human Mind PDF (2006)
Chalmers, David J [Return to link: P2] Minds, machines, and mathematics Psyche (1996) 2 pp 11-20
Coder, David Gödel’s theorem and mechanism (MS Word doc file) Philosophy (1969) 44 pp 234-237
Copeland, Jack The Mathematical Objection: Turing, Gödel, and Penrose on the Mind PDF Machinery (2008) July, pp 1-14
Davis, Martin How Subtle is Gödel’s Theorem Available online How Subtle is Gödel’s Theorem (PS file): here but only as a postscript file Behavioral and Brain Sciences (1993) 16 pp 611-612.
Davis, Martin Is Mathematical Insight Algorithmic? Available online Is Mathematical Insight Algorithmic? (PS file): here but only as a postscript file  
Dennett, Daniel C Murmurs in the cathedral: Review of R Penrose, The Emperor’s New Mind The Times Literary Supplement, September 29 (1989)  
Dershowitz, Nachum The Four Sons of Penrose PDF Lecture Notes in Computer Science (2005) vol 3835, pub Springer-Verlag, Berlin pp 125-138
Edis, Taner How Gödel’s theorem supports the possibility of machine intelligence PDF Minds and Machines (1998) 8 (2) pp 251-262
Feferman, Solomon Gödel’s incompleteness theorems, free will and mathematical thought PDF (2011)
Feferman, Solomon [Return to link: P3] Penrose’s Gödelian argument PDF Psyche (1996) 2 pp 21-32
Gaifman, H What Gödel’s incompleteness result does and does not show PDF Journal of Philosophy (2000) 97 (8) pp 462-471
George, A & Velleman, Daniel J Leveling the playing field between mind and machine: A reply to McCall PDF Journal of Philosophy (2000) 97 (8) pp 456-452
Good, I J Human and machine logic (MS Word doc file) British Journal for the Philosophy of Science (1967) 18 (August) pp 145-6
Grush, Rick & Churchland, Patricia Gaps in Penrose’s toiling PDF Book: Conscious Experience (1995) pub Schoningh
Hutton, Anthony This Gödel is killing me (MS Word doc file) Philosophia (1976) 6 pp 135-144
Jacquette, Dale Metamathematical criteria for minds and machines PDF Erkenntnis (1987) 27 (July) pp 1-16
Jongeneel, Christian & Koppelaar, H Gödel pro and contra AI: dismissal of the case Engineering Applications of Artificial Intelligence (1999) 12:5 pp 655-659
Ketland, Jeffrey Deflationism and Tarski’s paradise PDF Mind (1999) 108 pp 69-94
Ketland, Jeffrey & Raatikainen, Panu Truth and provability Again PDF a reply to a paper by Lucas and Redhead see below for details of that paper.
King, D Is the human mind a Turing machine? PDF Synthese (1996) 108 (3) pp 379-89
Kirk, Robert E Mental machinery and Gödel PDF Synthese (1968) 66 (March) pp 437-452
Klein, Stanley A [Return to link: P4] Is quantum mechanics relevant to understanding consciousness?  
Krajewski, Stanisław On Gödel’s Theorem and Mechanism: … Fundamenta Informaticae (2007) 81.1 pp 173-181
LaForte, G, Hayes, P J & Ford, K M Why Gödel’s Theorem Cannot Refute Computationalism: A reply to Penrose PDF Artificial Intelligence (1998) 104 pp 265-286
Lewis, David K Lucas against mechanism (MS Word doc file) Philosophy (1969) 44 231-233
Makey, Jeff Gödel’s Incompleteness Theorem is Not an Obstacle to Artificial Intelligence (1981)
Maudlin, Tim [Return to link: P5] Between the motion and the act Psyche (1996) 2 pp 40-51  
McCarthy, John [Return to link: P6] Awareness and understanding in computer programs  
McCullough, Daryl [Return to link: P7] Can humans escape Gödel?  
McDermott, Drew [Return to link: P8] Penrose is wrong  
Moravec, Hans [Return to link: P9] Roger Penrose’s gravitonic brains  
Putnam, Hilary Review of Shadows of the Mind PDF Bulletin of the American Mathematical Society (1995) 32.3 pp 370-373
Raatikainen, Panu McCall’s Gödelian argument is invalid PDF Facta Philosophica (2002) Vol 4, No 1 pp 167-169
Raatikainen, Panu On the philosophical relevance of Gödel’s incompleteness theorems (PDF) Revue Internationale de Philosophie (2005) 59 (4) pp 513-534
Raatikainen, Panu Truth and provability: A comment on Redhead PDF British Journal for the Philosophy of Science (2005) 56 (3) pp 611-613
Shapiro, Stewart Incompleteness, Mechanism, and Optimism PDF Bulletin of Symbolic Logic (1998) Vol 4-3 pp 273-302
Sloman, Aaron The Emperor’s Real Mind PDF Book: Artificial Intelligence and Its Applications (1986) pub John Wiley and Sons
Smart, J J C Gödel’s theorem, Church’s theorem, and mechanism PDF Synthese (1961) 13 pp 105-110
Tennant, Neil On Turing Machines Knowing Their Own Gödel-Sentence PDF Philosophia Mathematica 9, no. 1 (2001) pp 72-79.
Tassinari, R P & D’Ottaviano, I M. L Cogito ergo sum non machine PDF Anti-mechanist article
van de Putte, Xandra Is Lucas’ Gödelian argument valid? PDF (2005)
Yu, Qiuen Consistency, mechanicalness, and the logic of the mind PDF Synthese (1992) 90 (1) pp 145-79
Yu, Qiuen Further Explanations of the Gödel Scenario of the Mind - a Reply to Prof Graham Priest PDF Synthese (1992) 95-3, pp 461-465

 

 

Books and non-free online material

The articles are not available free online, except at sci hub, http://sci-hub.bz/ or http://sci-hub.cc/.

 

The principal anti-mechanist authors

Lucas, J R The Freedom of the Will Book: Oxford University Press (1970), ISBN‑13: 978‑0198243434 The Freedom of the Will: Details
Lucas, J R Metamathematics and the Philosophy of Mind: A Rejoinder Philosophy of Science (1971) 38 (2) pp 310-13 Metamathematics and the Philosophy of Mind: A Rejoinder: Details
Lucas, J R and Redhead, Michael Truth and Provability British Journal for the Philosophy of Science (2007) 58 (2) pp 331-332 Truth and Provability: Details
McCall, Storrs Can a Turing machine know that the Gödel sentence is true? Journal of Philosophy (1999) (10) pp 525-32 Can a Turing machine know that the Gödel sentence is true?: Details
Penrose, Roger The Emperor’s New Mind: Concerning Computers, Minds and The Laws of Physics Book: Oxford University Press, (1989) ISBN 0-198-51973-7 The Emperor’s New Mind: Details
Penrose, Roger Shadows of the Mind: A Search for the Missing Science of Consciousness Book: Oxford University Press, (1994) ISBN 0198539789 Shadows of the Mind: Details
Redhead, M Mathematics and the Mind British Journal for the Philosophy of Science (2004) 55 (4) pp 731-737 Mathematics and the Mind: Details
Seager, William E Yesterday’s algorithm: Penrose and the Gödel argument

Croatian Journal of Philosophy (2003) 3 (9) pp 265-273 Yesterday’s algorithm: Penrose and the Gödel argument: Details

 

 

Other authors

Bowie, G Lee Lucas’ number is finally up Journal of Philosophical Logic (1982) 11-3 pp 279-285 Lucas’ number is finally up: Details
Boyer, David L R Lucas, Kurt Gödel, and Fred Astaire The Philosophical Quarterly (1983) 33, No. 131 pp 147-159 Lucas, Kurt Gödel, and Fred Astaire: Details
Chari, C T K Further comments on minds, machines and Gödel Philosophy (1963) 38 (April) pp 175-8 Further comments on minds, machines and Gödel: Details
Chihara, C On alleged refutations of mechanism using Gödel’s incompleteness results Journal of Philosophy (1972) 69 pp 507-26 On alleged refutations of mechanism using Gödel: Details
Copeland, Jack Turing’s o-machines, Searle, Penrose, and the brain Analysis (1998) 58 (2) pp 128-138 Turing’s o-machines, Searle, Penrose, and the brain: Details
Hadley, Robert F Gödel, Lucas, and mechanical models of mind Computational Intelligence (1987) 3 pp 57-63 Gödel, Lucas, and mechanical models of mind: Details
Krajewski, Stanislaw On Gödel’s Theorem and Mechanism: Inconsistency or Unsoundness is Unavoidable in any Attempt to ‘Out-Gödel’ the Mechanist Fundamenta Informaticae (2007) 81, 173-181 On Gödel’s Theorem and Mechanism: Details
Landau, L J Penrose’s philosophical error Concepts for Neural Networks (1998) Springer pp 171-223 Details
Lindstrom, Per Remarks on Penrose’s new argument Journal of Philosophical Logic (2006) 35 (3) pp 231-237 Remarks on Penrose’s new argument: Details
Lyngzeidetson, A E Massively parallel distributed processing and a computationalist foundation for cognitive science British Journal for the Philosophy of Science (1990) 41 (March) pp 121-127 Massively parallel distributed processing: Details
Lyngzeidetson, A E & Solomon, Martin K Abstract complexity theory and the mind-machine problem British Journal for the Philosophy of Science (1994) 45 (2) pp 549-54 Abstract complexity theory and the mind-machine problem: Details
Martin, J & Engleman, K The mind’s I has two eyes Philosophy (1990) 65 (264) pp 510-515 The mind’s I has two eyes: Details
Priest, Graham Gödel’s theorem and the mind … again Kluwer Philosophical Studies Series, (1994) Volume 60 pp 191-201 Gödel’s theorem and the mind … again: Details
Putnam Hilary Reflexive reflections Erkenntnis (1985) 22, 1/3, Jan Reflexive reflections: Details
Robinson, William S Penrose and mathematical ability Analysis (1992) 52 (2) pp 80-88 Penrose and mathematical ability: Details
Serény, György How do We Know that the Gödel Sentence of a Consistent Theory Is True? Philosophia Mathematica (2011) Volume 19 (1) Oxford University Press How do We Know that the Gödel Sentence of a Consistent Theory Is True?: Details
Shapiro, Stewart Mechanism, Truth, and Penrose’s New Argument Journal of Philosophical Logic (2003) 32 (1) pp 19-42 Mechanism, Truth, and Penrose’s New Argument: Details
Slezak, Peter Artificial Intelligence, Gödelian Arguments Against Encyclopedia of Cognitive Science (2006) Artificial Intelligence, Gödelian Arguments Against: Details
Slezak, Peter Descartes’ diagonal deduction British Journal for the Philosophy of Science (1983) 34 pp 13-36 Descartes’ diagonal deduction: Details
Slezak, Peter Gödel’s theorem and the mind British Journal for the Philosophy of Science (1982) 33 pp 41-52 Gödel’s theorem and the mind: Details
Slezak, Peter Minds, machines and self-reference Dialectica (1984) 38 pp 17-34 Minds, machines and self-reference: Details
Webb, Judson Metamathematics and the philosophy of mind Philosophy of Science (1968) 35 pp 156-78 Metamathematics and the philosophy of mind: Details
Whitely, C Minds, machines and Gödel: A reply to Mr Lucas Philosophy (1962) 37 pp 61-62 Minds, machines and Gödel: A reply to Mr Lucas: Details

The Lighter Side

 

NEWS

There’s something about Gödel by Francesco Berto

There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book There’s something about Gödel comes under scrutiny.

 

 

Easy Footnotes

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).

 

 

O’Connor’s “computer checked” proof

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.

 

 

New page on Chaitin’s Constant

There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.

 

 

Updated: Diagonal Lemma

Flawed proofs of the Diagonal Lemma by Panu Raatikainen and Vann McGee have been added to the Diagonal Lemma web page.

Previous Blog Posts  

 

16th Mar 2015 Bishops Dancing with Pixies?

 

23rd Feb 2015 Artificial Intelligence

 

31 Mar 2015 Cranks and Crackpots

 

30 Apr 2015 The Chinese Room

 

Links  

 

For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

 

– a page with general links:

Gödel Links

 

– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines

 

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