{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T14:57:58Z","timestamp":1771513078110,"version":"3.50.1"},"reference-count":33,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":5615,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[1998,9]]},"abstract":"<jats:p><jats:bold>\u00a71. Overview<\/jats:bold>. Philosophers and mathematicians have drawn lots of conclusions from G\u00f6del's incompleteness theorems, and related results from mathematical logic. Languages, minds, and machines figure prominently in the discussion. G\u00f6del's theorems surely tell us something about these important matters. But what?<\/jats:p><jats:p>A descriptive title for this paper would be \u201cG\u00f6del, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility\u201d. Adding \u201cGod and the Devil\u201d would probably be redundant. Despite the breath-taking, whirlwind tour, I have the modest aim of forging connections between different parts of this literature and clearing up some confusions, together with the less modest aim of not introducing any more confusions.<\/jats:p><jats:p>I propose to focus on three spheres within the literature on incompleteness. The first, and primary, one concerns arguments that G\u00f6del's theorem refutes the mechanistic thesis that the human mind is, or can be accurately modeled as, a digital computer or a Turing machine. The most famous instance is the much reprinted J. R. Lucas [18]. To summarize, suppose that a mechanist provides plans for a machine,<jats:italic>M<\/jats:italic>, and claims that the output of<jats:italic>M<\/jats:italic>consists of all and only the arithmetic truths that a human (like Lucas), or the totality of human mathematicians, will ever or can ever know. We assume that the output of<jats:italic>M<\/jats:italic>is consistent.<\/jats:p>","DOI":"10.2307\/421032","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:11:46Z","timestamp":1146985906000},"page":"273-302","source":"Crossref","is-referenced-by-count":25,"title":["Incompleteness, Mechanism, and Optimism"],"prefix":"10.1017","volume":"4","author":[{"given":"Stewart","family":"Shapiro","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600007228_ref010","first-page":"113","volume-title":"The universal Turing machine","author":"Feferman","year":"1988"},{"key":"S1079898600007228_ref002","unstructured":"Boolos G. , Introductory note to [11], in [12], (1995), pp. 290\u2013304."},{"key":"S1079898600007228_ref027","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780195046724.001.0001","volume-title":"G\u00f6del's incompleteness theorems","author":"Smullyan","year":"1992"},{"key":"S1079898600007228_ref020","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093637647"},{"key":"S1079898600007228_ref031","doi-asserted-by":"publisher","DOI":"10.1093\/mind\/LIX.236.433"},{"key":"S1079898600007228_ref016","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139171472"},{"key":"S1079898600007228_ref022","volume-title":"Shadows of the mind: A search for the missing science of consciousness","author":"Penrose","year":"1994"},{"key":"S1079898600007228_ref011","unstructured":"G\u00f6del K. , Some basic theorems on the foundations of mathematics and their implications, in [12], (1951), pp. 304\u2013323."},{"key":"S1079898600007228_ref017","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511624926"},{"key":"S1079898600007228_ref024","first-page":"138","volume-title":"Dimensions of mind: A symposium","author":"Putnam","year":"1960"},{"key":"S1079898600007228_ref014","doi-asserted-by":"publisher","DOI":"10.2307\/2272975"},{"key":"S1079898600007228_ref032","volume-title":"From mathematics to philosophy","author":"Wang","year":"1974"},{"key":"S1079898600007228_ref003","doi-asserted-by":"publisher","DOI":"10.2307\/2218741"},{"key":"S1079898600007228_ref008","doi-asserted-by":"publisher","DOI":"10.4064\/fm-49-1-35-92"},{"key":"S1079898600007228_ref006","first-page":"140","article-title":"The philosophical significance of G\u00f6del's theorem","volume":"5","author":"Dummett","year":"1963","journal-title":"Ratio"},{"key":"S1079898600007228_ref015","volume-title":"Wittgenstein on rules and private language","author":"Kripke","year":"1982"},{"key":"S1079898600007228_ref004","article-title":"Minds, machines, and mathematics","volume":"2","author":"Chalmers","year":"1995","journal-title":"Psyche"},{"key":"S1079898600007228_ref023","article-title":"Beyond the doubting of a shadow: A reply to commentaries on \u2018Shadows of the mind\u2019","volume":"2","author":"Penrose","year":"1996","journal-title":"Psyche"},{"key":"S1079898600007228_ref019","volume-title":"Machines and thought: The legacy of Alan Turing, Volume 1","author":"Lucas","year":"1996"},{"key":"S1079898600007228_ref018","doi-asserted-by":"publisher","DOI":"10.1017\/S0031819100057983"},{"key":"S1079898600007228_ref028","volume-title":"Anti-realism and logic","author":"Tennant","year":"1987"},{"key":"S1079898600007228_ref025","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S1079898600007228_ref001","doi-asserted-by":"publisher","DOI":"10.5840\/monist196751112"},{"key":"S1079898600007228_ref033","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-7653-6"},{"key":"S1079898600007228_ref007","first-page":"329","volume-title":"The philosophy of Michael Dummett","author":"Dummett","year":"1994"},{"key":"S1079898600007228_ref021","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198519737.001.0001","volume-title":"The emperor's new mind: Concerning computers, minds, and the laws of physics","author":"Penrose","year":"1989"},{"key":"S1079898600007228_ref026","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70138-1"},{"key":"S1079898600007228_ref012","volume-title":"Collected works III","author":"G\u00f6del","year":"1995"},{"key":"S1079898600007228_ref029","volume-title":"The taming of the true","author":"Tennant","year":"1997"},{"key":"S1079898600007228_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-7731-1"},{"key":"S1079898600007228_ref009","doi-asserted-by":"publisher","DOI":"10.2307\/2964649"},{"key":"S1079898600007228_ref030","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s2-45.1.161"},{"key":"S1079898600007228_ref013","volume-title":"G\u00f6del, Escher, Bach","author":"Hofstadter","year":"1979"}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600007228","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T20:56:01Z","timestamp":1627332961000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600007228\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,9]]},"references-count":33,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1998,9]]}},"alternative-id":["S1079898600007228"],"URL":"https:\/\/doi.org\/10.2307\/421032","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,9]]}}}