View source for Penrose–Lucas argument

{{short description|Claim that human mathematicians are not describable as formal proof systems}}
The '''Penrose–Lucas argument''' is a [[Logic|logical]] [[argument]] partially based on a theory developed by mathematician and logician [[Kurt Gödel]]. In 1931, he proved that every [[effective procedure|effectively generated]] theory capable of proving basic arithmetic either fails to be  [[consistency|consistent]] or fails to be [[completeness (logic)|complete]]. Due to [[human]] ability to see the truth of formal systems' [[Gödel's incompleteness theorems|Gödel sentences]], it is argued that the human mind cannot be computed on a Turing Machine that works on [[Peano axioms|Peano arithmetic]] because the latter can't see the truth value of its Gödel sentence, while [[Mind|human minds]] can. Mathematician [[Roger Penrose]] modified the argument in his first book on [[consciousness]], ''[[The Emperor's New Mind]]'' (1989), where he used it to provide the basis of his theory of consciousness: [[orchestrated objective reduction]].

==Background==
Gödel showed that any such theory also including a statement of its own consistency is inconsistent. A key element of the proof is the use of [[Gödel numbering]] to construct a "Gödel sentence" for the theory, which encodes a statement of its own incompleteness: "This theory can't prove this statement"; or "I am not provable in this system". Either this statement and its negation are both unprovable (the theory is incomplete) or both provable (the theory is inconsistent). In the first eventuality the statement is intuitively true<ref>Gödel's theorem deals with a [[formal system]], in which a syntax is defined (i.e., one can talk of provability) but a semantic is not necessarily defined (there is no implicit notion of "truth"). However, Gödel's statement is actually true in the standard model of natural numbers. See {{cite book |author=Mendelson, Elliot |title=Introduction to Mathematical Logic
 |edition=5th
 |series=Discrete Mathematics and Its Applications
 |type=hardcover
 |year=2009 |publisher=Chapman and Hall/CRC |location=Boca Raton
 |isbn=978-1-58488-876-5
}}</ref> (since it is not provable); otherwise, the statement is intuitively false - though provable. 

An analogous statement has been used to show that humans are subject to the same limits as machines: “Lucas cannot consistently assert this formula”. In defense of [[J. R. Lucas|Lucas]], J. E. Martin and K. H. Engleman argued in ''The Mind's I Has Two Eyes''<ref>{{Cite journal |last1=Martin |first1=J. E. |last2=Engleman |first2=K. H. |date=1990 |title=The Mind's I Has Two Eyes |url=https://www.jstor.org/stable/3751287 |journal=Philosophy |volume=65 |issue=254 |pages=510–515 |doi=10.1017/S003181910006472X |jstor=3751287 |issn=0031-8191}}</ref> that Lucas can recognise that the sentence is true, as there's a point of view from which he can understand how the sentence tricks him.<ref>{{Harvnb|Hofstadter|1979|pp=476–477}}, {{Harvnb|Russell|Norvig|2003|p=950}}, {{Harvnb|Turing|1950}} under "The Argument from Mathematics" where he writes "although it is established that there are limitations to the powers of any particular machine, it has only been stated, without sort of proof, that no such limitations apply to the human intellect."</ref> From this point of view Lucas can appreciate that he can't assert the sentence-and consequently he can recognise its truth.<ref>{{Cite web |title=Details view: Lucas tricks machines into contradicting themselves |url=https://debategraph.org/Details.aspx?nid=1157 |access-date=2023-06-14 |website=debategraph.org}}</ref> Still, this criticism only works if we assume that we can replace Lucas' reasoning with a formal system whose theorems can be listed by an algorithm that has a Gödel sentence, but the Penrose-Lucas argument tries to prove otherwise: our ability to [[Understanding|understand]] this level of [[arithmetic]] is not an [[effective procedure]] that can be simulated in a [[Deterministic Turing machine|Turing machine]].

Penrose argued that while a formal proof system cannot prove its own consistency, Gödel-unprovable results are provable by human mathematicians.<ref name=Penrose1989/> He takes this disparity to mean that human mathematicians are not describable as formal proof systems (which [[Theorem|theorems]] can be proved using an [[abstract object]] such as a computer), and are therefore running a [[computable function|non-computable algorithm]]. Similar claims about the implications of Gödel's theorem were originally espoused by [[Alan Turing|Turing]] in the late 1940s, by [[Kurt Gödel|Gödel]] himself in his 1951 ''Gibbs lecture,'' by E. Nagel and J.R. Newman in 1958,<ref name="stanfordgoedel" /> and were subsequently popularized by the philosopher [[John Lucas (philosopher)|John Lucas]] of [[Merton College, Oxford|Merton College, Oxford in 1961]].<ref>{{Cite journal|last=Lucas|first=John R.|date=1961|title=Minds, Machines and Godel|url=https://philpapers.org/rec/LUCMMA|journal=Philosophy|volume=36|issue=April–July|pages=112–127|doi=10.1017/s0031819100057983|s2cid=55408480 }}</ref>
{{blockquote|The inescapable conclusion seems to be: Mathematicians are not using a knowably sound calculation procedure in order to ascertain mathematical truth. We deduce that mathematical understanding – the means whereby mathematicians arrive at their conclusions with respect to mathematical truth – cannot be reduced to blind calculation!|Roger Penrose<ref>Roger Penrose. Mathematical intelligence. In Jean Khalfa, editor, What is Intelligence?, chapter 5, pages 107–136. Cambridge University Press, Cambridge, United Kingdom, 1994.</ref>}}

==Consequences==
If correct, the Penrose–Lucas argument creates a need to understand the physical basis of non-computable behaviour in the brain.<ref>{{cite web | url=https://iep.utm.edu/lp-argue/ | title=Lucas-Penrose Argument about Gödel's Theorem &#124; Internet Encyclopedia of Philosophy }}</ref> Most physical laws are computable, and thus algorithmic. However, Penrose determined that [[wave function collapse]] was a prime candidate for a non-computable process.

In [[quantum mechanics]], particles are treated differently from the objects of [[classical mechanics]]. Particles are described by [[wave function]]s that evolve according to the [[Schrödinger equation]]. Non-stationary wave functions are [[linear combination]]s of the [[eigenstate]]s of the system, a phenomenon described by the [[superposition principle]]. When a quantum system interacts with a classical system—i.e. when an [[observable]] is measured—the system appears to [[wave function collapse|collapse]] to a random eigenstate of that [[observable]] from a classical vantage point.

If collapse is truly random, then no process or algorithm can deterministically predict its outcome. This provided Penrose with a candidate for the physical basis of the non-computable process that he hypothesized to exist in the brain. However, he disliked the random nature of environmentally induced collapse, as randomness was not a promising basis for mathematical understanding. Penrose proposed that isolated systems may still undergo a new form of [[wave function collapse]], which he called [[Orchestrated objective reduction|objective reduction]] (OR).<ref name="Penrose-Hameroff2014">{{cite journal|title=Consciousness in the universe: A review of the 'Orch OR' theory|last1=Hameroff|first1=Stuart|last2=Penrose|first2=Roger|date=March 2014|journal=Physics of Life Reviews|publisher=Elsevier|volume=11|issue=1|pages=39–78|doi=10.1016/j.plrev.2013.08.002|pmid=24070914|bibcode=2014PhLRv..11...39H|doi-access=free}}</ref>

Penrose sought to reconcile [[general relativity]] and quantum theory using his own ideas about the possible structure of [[spacetime]].<ref name=Penrose1989>{{Cite book |last=Penrose |first= Roger |author-link=Roger Penrose |title=The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics |publisher=Oxford University Press |year=1989 |page=480|isbn=978-0-19-851973-7|title-link= The Emperor's New Mind }}</ref><ref name= Penrose1994>{{Cite book |last=Penrose |first=Roger |author-link=Roger Penrose |title=Shadows of the Mind: A Search for the Missing Science of Consciousness |publisher=Oxford University Press |year=1989 |page=[https://archive.org/details/shadowsofmindsea00penr_0/page/457 457] |isbn=978-0-19-853978-0 |url-access=registration |url=https://archive.org/details/shadowsofmindsea00penr_0/page/457 }}</ref> He suggested that at the [[Planck scale]] curved spacetime is not continuous, but discrete. Penrose postulated that each separated [[quantum superposition]] has its own piece of [[spacetime curvature]], a blister in spacetime. Penrose suggests that gravity exerts a force on these spacetime blisters, which become unstable above the Planck scale of <math>10^{-35} \text{m}</math> and collapse to just one of the possible states. The rough threshold for OR is given by Penrose's indeterminacy principle:
:<math>
\tau \approx \hbar/E_G
</math>
where:
* <math>\tau</math> is the time until OR occurs, 
* <math>E_G</math> is the gravitational self-energy or the degree of spacetime separation given by the superpositioned mass, and
* <math>\hbar</math> is the [[reduced Planck constant]]. 
Thus, the greater the mass-energy of the object, the faster it will undergo OR and vice versa. Atomic-level superpositions would require 10 million years to reach OR threshold, while an isolated 1 [[kilogram]] object would reach OR threshold in 10<sup>−37</sup>s. Objects somewhere between these two scales could collapse on a timescale relevant to neural processing.<ref name="Penrose-Hameroff2014"/>{{citation needed|date=May 2013 |reason=Search for supporting evidence besides the papers from the main authors of the theory.}}<ref>{{cite web | url=https://physicsworld.com/a/physicists-place-fresh-limits-on-gravitys-role-in-wavefunction-collapse/ | title=Physicists place fresh limits on gravity's role in wavefunction collapse | date=10 October 2020 }}</ref>

An essential feature of Penrose's theory is that the choice of states when objective reduction occurs is selected neither randomly (as are choices following [[wave function collapse]]) nor algorithmically. Rather, states are selected by a "non-computable" influence embedded in the [[Planck]] scale of spacetime geometry. Penrose claimed that such information is [[Platonism|Platonic]], representing pure mathematical truth, aesthetic and ethical values at the Planck scale. This relates to Penrose's ideas concerning the three worlds: physical, mental, and the Platonic mathematical world. In his theory, the Platonic world corresponds to the geometry of fundamental spacetime that is claimed to support noncomputational thinking.<ref name="Penrose-Hameroff2014"/>{{citation needed|date=May 2013 |reason=Search for supporting evidence besides the papers from the main authors of the theory.}}<ref>{{cite book | chapter-url=https://plato.stanford.edu/entries/kant-spacetime/ | title=The Stanford Encyclopedia of Philosophy | chapter=Kant's Views on Space and Time | year=2022 | publisher=Metaphysics Research Lab, Stanford University }}</ref><ref>{{cite arXiv |eprint=gr-qc/9409007 |year=1994 |last1=Rosu |first1=H. C. |title=Essay on mesoscopic and quantum brain }}</ref>

==Criticism==
The Penrose–Lucas argument about the implications of [[Gödel's incompleteness theorems|Gödel's incompleteness theorem]] for computational theories of human intelligence was criticized by mathematicians,<ref name=laforte_1998>LaForte, Geoffrey, Patrick J. Hayes, and Kenneth M. Ford 1998.''[https://web.archive.org/web/20060901233518/http://www.cs.uwf.edu/~glaforte/papers/whyGodel.ps Why Gödel's Theorem Cannot Refute Computationalism]''. Artificial Intelligence, 104:265–286.</ref><ref name=solomon_1996>{{cite journal | author-link = Solomon Feferman | last = Feferman | first = Solomon | year = 1996 |citeseerx = 10.1.1.130.7027 | title = Penrose's Gödelian argument | journal = [[Psyche (consciousness journal)|Psyche]]| volume = 2 | pages = 21–32 }}</ref><ref name=krajewski_2007>Krajewski, Stanislaw 2007. ''On Gödel's Theorem and Mechanism: Inconsistency or Unsoundness is Unavoidable in any Attempt to 'Out-Gödel' the Mechanist.'' Fundamenta Informaticae 81, 173–181. Reprinted in [https://books.google.com/books?id=0jSS-3Bl06cC&pg=PA173 Topics in Logic, Philosophy and Foundations of Mathematics and Computer Science:In Recognition of Professor Andrzej Grzegorczyk (2008), p. 173]</ref><ref>P. Pudlak, A note on applicability of the incompleteness theorem to human mind, Annals of Pure and Applied Logic, 96 (1999), 335-342 doi://10.1016/S0168-0072(98)00044-X</ref> computer scientists,<ref name=putnam_1995>[[Hilary Putnam|Putnam, Hilary]] 1995. ''Review of Shadows of the Mind.'' In Bulletin of the American Mathematical Society 32, 370–373 (also see Putnam's less technical criticisms in his [https://www.nytimes.com/books/97/04/27/nnp/17540.html New York Times review])</ref> and philosophers,<ref name=mindpapers>{{cite web|url=http://consc.net/mindpapers/6.1b |title=MindPapers: 6.1b. Godelian arguments |publisher=Consc.net |date= |accessdate=2014-07-28}}</ref><ref name=lucas_criticisms>{{cite web|url=http://users.ox.ac.uk/~jrlucas/Godel/referenc.html |title=References for Criticisms of the Gödelian Argument |publisher=Users.ox.ac.uk |date=1999-07-10 |archive-url=https://web.archive.org/web/20200917122035/http://users.ox.ac.uk/~jrlucas/Godel/referenc.html |accessdate=2021-07-07|archive-date=2020-09-17 }}</ref><ref name=Boolos_1990>[[George Boolos|Boolos, George]], et al. 1990. ''An Open Peer Commentary on The Emperor's New Mind.'' Behavioral and Brain Sciences 13 (4) 655.</ref><ref name=martin_1993>[[Martin Davis (mathematician)|Davis, Martin]] 1993. ''How subtle is Gödel's theorem? More on Roger Penrose.'' Behavioral and Brain Sciences, 16, 611–612. Online version at Davis' faculty page at http://cs.nyu.edu/cs/faculty/davism/</ref><ref name=lewis_1969>[[David Kellogg Lewis|Lewis, David K.]] 1969.''[http://www2.units.it/etica/2003_1/7_monographica.doc Lucas against mechanism]''. Philosophy 44 231–233.</ref> and the consensus among experts{{which|date=February 2021}}<ref name="stanfordgoedel">{{cite book | chapter-url=https://plato.stanford.edu/entries/goedel-incompleteness/ | title=The Stanford Encyclopedia of Philosophy | chapter=Gödel's Incompleteness Theorems | year=2022 | publisher=Metaphysics Research Lab, Stanford University }}</ref> in these fields is that the argument fails,<ref>Bringsjord, S. and Xiao, H. 2000. ''[http://kryten.mm.rpi.edu/refute.penrose.pdf A Refutation of Penrose's Gödelian Case Against Artificial Intelligence].'' [[Journal of Experimental and Theoretical Artificial Intelligence]] 12: 307–329. The authors write that it is "generally agreed" that Penrose "failed to destroy the computational conception of mind."</ref><ref>In an article at {{cite web |url=http://www.mth.kcl.ac.uk/~llandau/Homepage/Math/penrose.html |title=Penrose's Philosophical Error |accessdate=2010-10-22 |url-status=dead |archiveurl=https://web.archive.org/web/20010125011300/http://www.mth.kcl.ac.uk/~llandau/Homepage/Math/penrose.html |archivedate=2001-01-25 }} L.J. Landau at the Mathematics Department of King's College London writes that "Penrose's argument, its basis and implications, is rejected by experts in the fields which it touches."</ref><ref name="Burgess">Princeton Philosophy professor John Burgess writes in ''[http://www.princeton.edu/~jburgess/Montreal.doc On the Outside Looking In: A Caution about Conservativeness]'' (published in Kurt Gödel: Essays for his Centennial, with the following comments found on [https://books.google.com/books?id=83Attf6BsJ4C&pg=PA131 pp. 131–132]) that "the consensus view of logicians today seems to be that the Lucas–Penrose argument is fallacious, though as I have said elsewhere, there is at least this much to be said for Lucas and Penrose, that logicians are not unanimously agreed as to where precisely the fallacy in their argument lies. There are at least three points at which the argument may be attacked."</ref> with different authors attacking different aspects of the argument.<ref name="Burgess" /><ref>[[Nachum Dershowitz|Dershowitz, Nachum]] 2005. ''[http://www.cs.tau.ac.il/~nachumd/papers/FourSonsOfPenrose.pdf The Four Sons of Penrose]'', in ''Proceedings of the Eleventh Conference on [[Logic for Programming, Artificial Intelligence, and Reasoning]] (LPAR; Jamaica)'', G. Sutcliffe and A. Voronkov, eds., Lecture Notes in Computer Science, vol. 3835, Springer-Verlag, Berlin, pp. 125–138.</ref>

[[Solomon Feferman|Feferman]] faulted detailed points in Penrose's second book, ''Shadows of the Mind''. He argued that mathematicians do not progress by mechanistic search through proofs, but by trial-and-error reasoning, insight and inspiration, and that machines do not share this approach with humans. He pointed out that everyday mathematics can be formalized. He also rejected Penrose's [[Platonism]].<ref name="solomon_1996" /> Still, this does not account for his core argument of the alleged ability of the [[Mind|human mind]] to prove [[Gödel sentence|Gödel-unprovable sentences]]. Also, [[Artificial intelligence|Artificial Intelligence]] based on [[Reinforcement learning|Reinforcement Learning]] can work by taking [[Action selection|actions]] in an environment in order to maximize the notion of [[Reward-based selection|cumulative reward]], acting like trial-and-error procedures.<ref>{{Cite web |date=19 April 2019 |title=Medium |url=https://towardsdatascience.com/reinforcement-learning-101-e24b50e1d292}}</ref><ref>{{Cite web |title=What is Machine Learning? |date=22 September 2021 |url=https://www.ibm.com/topics/machine-learning |access-date=2023-06-10 |publisher=IBM |language=en-us}}</ref><ref>{{Cite web |title=Building a Machine Learning Model through Trial and Error |url=https://www.kdnuggets.com/building-a-machine-learning-model-through-trial-and-error.html |access-date=2023-06-10 |website=KDnuggets |language=en-US}}</ref>

LaForte pointed out that in order to know the truth of an unprovable Gödel sentence, one must already know the formal system is consistent (although this was not the point Lucas tried to make); referencing [[Paul Benacerraf|Benacerraf]], he tried to demonstrate that humans cannot prove that they are consistent,<ref name="laforte_1998" /> and in all likelihood human brains are inconsistent algorithms that use some sort of [[paraconsistent logic]], pointing to alleged contradictions within Penrose's own writings as examples. Similarly, [[Marvin Minsky|Minsky]] argued that because humans can believe false ideas to be true, human mathematical understanding need not be consistent and consciousness may easily have a deterministic basis.<ref>Marvin Minsky. "Conscious Machines." Machinery of Consciousness, Proceedings, National Research Council of Canada, 75th Anniversary Symposium on Science in Society, June 1991.</ref> Penrose argued against [[Marvin Minsky|Minsky]] stating that mistakes human mathematicians make are irrelevant because they are correctable, while [[Logical truth|logical truths]] are “unassailable truths” to [[Person|persons]], wich are the outputs of a sound system and the only ones that matter.<ref>{{Cite encyclopedia |title=Lucas-Penrose Argument about Gödel's Theorem |encyclopedia=Internet Encyclopedia of Philosophy |url=https://iep.utm.edu/lp-argue/ |access-date=2023-06-11 |language=en-US}}</ref> Mistakes do not directly imply that the human mind is inconsistent per se: biological organisms are subject to cognitive turmoils, reduced [[long-term memory]] and [[attention]] shifts; these reduce our [[Reason|reasoning]] capabilities and make humans act [[Unconscious mind|unconsciously]] without taking into consideration all the possible variables of a system. 

Thus, a [[Disjunction (logical connective)|disjunction]] holds: either the [[Mind|human mind]] is not a [[computation]] of a [[Turing machine|Turing Machine]], and thus not a [[formal system]]; or it is a product of an [[Consistency|inconsistent]] Turing Machine that could be reasoning using some sort of [[paraconsistent logic]]. Gödel himself commented about this disjunction in 1953.<ref>{{Cite web |title=Are We Paraconsistent? On the Luca-Penrose Argument and the Computational Theory of Mind |url=https://journals.ku.edu/auslegung/article/download/12649/11945}}</ref><ref>{{Cite journal |last=Wrigley |first=Wesley |date=2022-10-01 |title=Gödel’s Disjunctive Argument† |url=https://academic.oup.com/philmat/article/30/3/306/6634878 |journal=Philosophia Mathematica |volume=30 |issue=3 |pages=306–342 |doi=10.1093/philmat/nkac013 |issn=1744-6406}}</ref>

==See also==
* [[Hypercomputation]]
* [[Orchestrated objective reduction]]
* [[Gödel's incompleteness theorems]]
* [[Logic]]
* [[Consistency]]
* [[Logical truth]]
* [[Quantum consciousness]]
* [[Mind]]
* [[Consciousness]]
* [[Hard problem of consciousness]]
* [[Qualia]]
* [[Reason]]

==References==
{{Reflist}}

==Sources==
* {{Citation | last=Hofstadter | first = Douglas | title = Gödel, Escher, Bach: an Eternal Golden Braid | year = 1979 | author-link = Douglas Hofstadter | title-link = Gödel, Escher, Bach }}
* {{Russell Norvig 2003}}
* {{Turing 1950}}

{{Roger Penrose|state=collapsed}}

{{DEFAULTSORT:Penrose-Lucas argument}}
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[[Category:Theorems]]
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