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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.
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.
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 | Minds, Machines and Gödel: a Retrospect (PDF) | Ethics & Politics, V (2003) 1 |
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). |
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 |
The articles are not available free online, except at sci hub, http://sci-hub.bz/ or http://sci-hub.cc/.
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 |
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 |
It has come to my notice that, when asked about the demonstration of the flaw in his proof (see A Fundamental Flaw in an Incompleteness Proof by Peter Smith PDF), Smith refuses to engage in any logical discussion, and instead attempts to deflect attention away from any such discussion. If any other reader has tried to engage with Smith regarding my demonstration of the flaw, I would be interested to know what the outcome was.
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.
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).
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.
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.
16th Mar 2015 Bishops Dancing with Pixies?
23rd Feb 2015 Artificial Intelligence
31 Mar 2015 Cranks and Crackpots
For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem
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