417:. 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. Still, this does not account for his core argument of the alleged ability of the human mind to prove
1169:) 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."
391:, 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.
121:
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
88:
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. From this point of view Lucas can appreciate that he cannot assert the sentence-and consequently he can recognise its truth. Still, this criticism only works if we
76:
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
454:
and attention shifts; these reduce our reasoning capabilities and make humans act unconsciously without taking into consideration all the possible variables of a system. Thus, a disjunction holds: either the human mind is not a computation of a Turing
Machine; or it is a product of an inconsistent
449:
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. Penrose argued against Minsky stating that mistakes human mathematicians make are irrelevant because they are correctable,
89:
assume that Lucas is attaining the truth of the sentence using a logical proof, but the
Penrose-Lucas argument tries to prove otherwise: our ability to understand this level of arithmetic is not based on the classical idea of provability nor is it an effective procedure which can be simulated in a
188:
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
450:
while logical truths are “unassailable truths” to persons, which are the outputs of a sound system and the only ones that matter. Mistakes do not directly imply that the human mind is inconsistent per se: biological organisms are subject to cognitive turmoils, reduced
403:
for computational theories of human intelligence was criticized by mathematicians, computer scientists, and philosophers, and the consensus among experts in these fields is that the argument fails, with different authors attacking different aspects of the argument.
616:
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
501:, 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
96:
Penrose argued that while a formal proof system cannot prove its own consistency, Gödel-unprovable results are provable by human mathematicians. He takes this disparity to mean that human mathematicians are not describable as
137:
If correct, the
Penrose–Lucas argument creates a need to understand the physical basis of non-computable behaviour in the brain. Most physical laws are computable, and thus algorithmic. However, Penrose determined that
436:
Geoffrey 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
1117:
1357:
1190:
376:
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
289:
1147:
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."
1544:
245:
387:) 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
80:
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 philosopher
1249:
Marvin Minsky. "Conscious
Machines." Machinery of Consciousness, Proceedings, National Research Council of Canada, 75th Anniversary Symposium on Science in Society, June 1991.
745:
Roger
Penrose. Mathematical intelligence. In Jean Khalfa, editor, What is Intelligence?, chapter 5, pages 107–136. Cambridge University Press, Cambridge, United Kingdom, 1994.
982:
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
77:
inconsistent). In the first eventuality the statement is intuitively true (since it is not provable); otherwise, the statement is intuitively false - though provable.
366:
344:
315:
441:, he tried to demonstrate that humans cannot prove that they are consistent, and in all likelihood human brains are inconsistent algorithms that use some sort of
189:
induced collapse, as randomness was not a promising basis for mathematical understanding. Penrose proposed that isolated systems may still undergo a new form of
383:
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
1490:
212:, a blister in spacetime. Penrose suggests that gravity exerts a force on these spacetime blisters, which become unstable above the Planck scale of
1391:
1301:
380:
object would reach OR threshold in 10s. Objects somewhere between these two scales could collapse on a timescale relevant to neural processing.
972:
Topics in Logic, Philosophy and Foundations of Mathematics and Computer Science:In Recognition of Professor Andrzej Grzegorczyk (2008), p. 173
843:
668:
611:
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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
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859:
208:
curved spacetime is not continuous, but discrete. Penrose postulated that each separated quantum superposition has its own piece of
1120:
12: 307–329. The authors write that it is "generally agreed" that Penrose "failed to destroy the computational conception of mind."
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468:
194:
58:
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916:
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and collapse to just one of the possible states. The rough threshold for OR is given by Penrose's indeterminacy principle:
1495:
968:
On Gödel's Theorem and Mechanism: Inconsistency or Unsoundness is Unavoidable in any Attempt to 'Out-Gödel' the Mechanist.
1480:
1195:, G. Sutcliffe and A. Voronkov, eds., Lecture Notes in Computer Science, vol. 3835, Springer-Verlag, Berlin, pp. 125–138.
1455:
1377:
1073:
90:
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1325:
998:
In Bulletin of the American Mathematical Society 32, 370–373 (also see Putnam's less technical criticisms in his
658:
53:
253:
346:
is the gravitational self-energy or the degree of spacetime separation given by the superpositioned mass, and
1440:
422:
369:
170:
158:
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and quantum theory using his own ideas about the possible structure of spacetime. He suggested that at the
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1287:
947:
430:
426:
384:
190:
178:
139:
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because the latter cannot see the truth value of its Gödel sentence, while human minds can. Mathematician
507:(hardcover). Discrete Mathematics and Its Applications (5th ed.). Boca Raton: Chapman and Hall/CRC.
215:
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787:
456:
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32:
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28:
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445:, pointing to alleged contradictions within Penrose's own writings as examples. Similarly,
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1165:(published in Kurt Gödel: Essays for his Centennial, with the following comments found on
438:
23:
is a logical argument partially based on a theory developed by mathematician and logician
756:"Lucas-Penrose Argument about Gödel's Theorem | Internet Encyclopedia of Philosophy"
117:
by E. Nagel and J.R. Newman in 1958, and were subsequently popularized by Lucas in 1961.
31:
theory capable of proving basic arithmetic either fails to be consistent or fails to be
999:
791:
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Behavioral and Brain Sciences, 16, 611–612. Online version at Davis' faculty page at
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can work by taking actions in an environment in order to maximize the notion of
110:
81:
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723:
545:
182:
174:
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57:(1989), where he used it to provide the basis of his theory of consciousness:
553:
1450:
388:
809:
660:
The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics
16:
Claim that human mathematicians are not describable as formal proof systems
860:"Physicists place fresh limits on gravity's role in wavefunction collapse"
377:
1113:
A Refutation of Penrose's Gödelian Case Against Artificial Intelligence
1081:
561:
529:
173:. When a quantum system interacts with a classical system—i.e. when an
897:
830:
Shadows of the Mind: A Search for the Missing Science of Consciousness
706:
1012:
628:"Details view: Lucas tricks machines into contradicting themselves"
776:"Consciousness in the universe: A review of the 'Orch OR' theory"
1279:
1259:
755:
1283:
1118:
Journal of Experimental and Theoretical Artificial Intelligence
1358:
Fashion, Faith, and Fantasy in the New Physics of the Universe
914:
LaForte, Geoffrey, Patrick J. Hayes, and Kenneth M. Ford 1998.
1191:
Logic for Programming, Artificial Intelligence, and Reasoning
891:
Rosu, H. C. (1994). "Essay on mesoscopic and quantum brain".
1227:"Building a Machine Learning Model through Trial and Error"
1162:
On the Outside Looking In: A Caution about Conservativeness
39:, it is argued that the human mind cannot be computed on a
35:. Due to human ability to see the truth of formal system's
455:
Turing Machine that could be reasoning using some sort of
51:
modified the argument in his first book on consciousness,
149:, particles are treated differently from the objects of
769:
767:
765:
1159:
Princeton Philosophy professor John Burgess writes in
880:. Metaphysics Research Lab, Stanford University. 2022.
694:. Metaphysics Research Lab, Stanford University. 2022.
1078:
How subtle is Gödel's theorem? More on Roger Penrose.
399:
The Penrose–Lucas argument about the implications of
354:
325:
303:
256:
218:
1034:"References for Criticisms of the Gödelian Argument"
142:
was a prime candidate for a non-computable process.
1568:
1423:
1368:
1317:
599:
harvnb error: no target: CITEREFRussellNorvig2003 (
1062:An Open Peer Commentary on The Emperor's New Mind.
970:Fundamenta Informaticae 81, 173–181. Reprinted in
917:Why Gödel's Theorem Cannot Refute Computationalism
827:
411:faulted detailed points in Penrose's second book,
360:
338:
309:
283:
239:
682:
680:
587:harvnb error: no target: CITEREFHofstadter1979 (
113:in the late 1940s, by Gödel himself in his 1951
928:
926:
910:
908:
774:Hameroff, Stuart; Penrose, Roger (March 2014).
119:
1260:"Lucas-Penrose Argument about Gödel's Theorem"
1295:
649:
647:
594:
169:of the system, a phenomenon described by the
8:
1036:. Users.ox.ac.uk. 1999-07-10. Archived from
610:harvnb error: no target: CITEREFTuring1950 (
84:, J. E. Martin and K. H. Engleman argued in
1471:Penrose interpretation of quantum mechanics
105:. Similar claims about the implications of
1302:
1288:
1280:
1189:Proceedings of the Eleventh Conference on
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1153:
582:
433:, acting like trial-and-error procedures.
1064:Behavioral and Brain Sciences 13 (4) 655.
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353:
330:
324:
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284:{\displaystyle \tau \approx \hbar /E_{G}}
275:
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232:
223:
217:
663:. Oxford University Press. p. 480.
1392:The Large, the Small and the Human Mind
936:(1996). "Penrose's Gödelian argument".
920:. Artificial Intelligence, 104:265–286.
878:The Stanford Encyclopedia of Philosophy
692:The Stanford Encyclopedia of Philosophy
528:Martin, J. E.; Engleman, K. H. (1990).
490:
355:
263:
1013:"MindPapers: 6.1b. Godelian arguments"
606:
7:
1491:Penrose–Hawking singularity theorems
1082:http://cs.nyu.edu/cs/faculty/davism/
161:. Non-stationary wave functions are
1264:Internet Encyclopedia of Philosophy
834:. Oxford University Press. p.
1110:Bringsjord, S. and Xiao, H. 2000.
505:Introduction to Mathematical Logic
240:{\displaystyle 10^{-35}{\text{m}}}
177:is measured—the system appears to
14:
688:"Gödel's Incompleteness Theorems"
1536:Orchestrated objective reduction
1409:White Mars or, The Mind Set Free
874:"Kant's Views on Space and Time"
469:Orchestrated objective reduction
185:from a classical vantage point.
59:orchestrated objective reduction
27:. In 1931, he proved that every
1132:"Penrose's Philosophical Error"
474:Gödel's incompleteness theorems
181:to a random eigenstate of that
996:Review of Shadows of the Mind.
407:Philosopher and mathematician
401:Gödel's incompleteness theorem
1:
1496:Riemannian Penrose inequality
497:Gödel's theorem deals with a
157:that evolve according to the
153:. Particles are described by
122:reduced to blind calculation!
1403:and Stephen Hawking) (1997)
1378:The Nature of Space and Time
317:is the time until OR occurs,
200:Penrose sought to reconcile
109:were originally espoused by
101:and are therefore running a
1206:"What is Machine Learning?"
966:Krajewski, Stanislaw 2007.
801:10.1016/j.plrev.2013.08.002
707:"Minds, Machines and Godel"
530:"The Mind's I Has Two Eyes"
1658:
1560:Conformal cyclic cosmology
1521:Penrose graphical notation
503:Mendelson, Elliot (2009).
419:Gödel-unprovable sentences
1461:Weyl curvature hypothesis
724:10.1017/s0031819100057983
595:Russell & Norvig 2003
546:10.1017/S003181910006472X
86:The Mind's I Has Two Eyes
1481:Newman–Penrose formalism
1184:The Four Sons of Penrose
1101:. Philosophy 44 231–233.
103:non-computable algorithm
1441:Abstract index notation
1098:Lucas against mechanism
780:Physics of Life Reviews
717:(April–July): 112–127.
705:Lucas, John R. (1961).
423:artificial Intelligence
370:reduced Planck constant
171:superposition principle
1600:John Beresford Leathes
1540:Penrose–Lucas argument
1531:Penrose–Terrell effect
1326:The Emperor's New Mind
786:(1). Elsevier: 39–78.
427:reinforcement learning
385:wave function collapse
362:
361:{\displaystyle \hbar }
340:
311:
285:
241:
191:wave function collapse
140:wave function collapse
130:
54:The Emperor's New Mind
21:Penrose–Lucas argument
1476:Moore–Penrose inverse
1451:Geometry of spacetime
1000:New York Times review
363:
341:
339:{\displaystyle E_{G}}
312:
310:{\displaystyle \tau }
286:
242:
29:effectively generated
1606:Illumination problem
1466:Penrose inequalities
457:paraconsistent logic
443:paraconsistent logic
352:
323:
301:
254:
216:
159:Schrödinger equation
99:effective procedures
1342:The Road to Reality
1334:Shadows of the Mind
792:2014PhLRv..11...39H
414:Shadows of the Mind
210:spacetime curvature
195:objective reduction
163:linear combinations
151:classical mechanics
1179:Dershowitz, Nachum
862:. 10 October 2020.
585:, pp. 476–477
358:
336:
307:
281:
237:
202:general relativity
193:, which he called
1619:
1618:
1555:Andromeda paradox
1526:Penrose transform
1456:Cosmic censorship
1129:In an article at
934:Feferman, Solomon
845:978-0-19-853978-0
670:978-0-19-851973-7
514:978-1-58488-876-5
431:cumulative reward
235:
147:quantum mechanics
1649:
1588:Jonathan Penrose
1545:FELIX experiment
1511:Penrose triangle
1416:
1404:
1401:Nancy Cartwright
1386:
1369:Coauthored books
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1134:. Archived from
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540:(254): 510–515.
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452:long-term memory
409:Solomon Feferman
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45:Peano arithmetic
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1594:Shirley Hodgson
1564:
1550:Trapped surface
1501:Penrose process
1486:Penrose diagram
1446:Black hole bomb
1419:
1413:Brian W. Aldiss
1407:
1389:
1383:Stephen Hawking
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1093:Lewis, David K.
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1060:, et al. 1990.
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583:Hofstadter 1979
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37:Gödel sentences
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1576:Lionel Penrose
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1350:Cycles of Time
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992:Putnam, Hilary
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91:Turing machine
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43:that works on
41:Turing machine
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1611:Quantum mind
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1234:. Retrieved
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1200:
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1136:the original
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1038:the original
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133:Consequences
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114:
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18:
1167:pp. 131–132
1015:. Consc.net
617:intellect."
607:Turing 1950
167:eigenstates
111:Alan Turing
1626:Categories
1269:2023-06-11
1236:2023-06-10
1212:2023-06-10
1142:2010-10-22
1044:2021-07-07
1019:2014-07-28
711:Philosophy
637:2023-06-14
534:Philosophy
485:References
183:observable
175:observable
82:John Lucas
70:Kurt Gödel
65:Background
25:Kurt Gödel
1637:Arguments
1590:(brother)
1584:(brother)
1415:) (1999)
1385:) (1996)
1231:KDnuggets
948:CiteSeerX
946:: 21–32.
570:170576086
554:0031-8191
425:based on
395:Criticism
356:ℏ
305:τ
264:ℏ
261:≈
258:τ
225:−
1642:Theorems
1596:(sister)
1578:(father)
1424:Concepts
826:(1989).
810:24070914
733:55408480
657:(1989).
463:See also
421:. Also,
389:Platonic
378:kilogram
179:collapse
125:—
33:complete
1569:Related
788:Bibcode
562:3751287
368:is the
294:where:
165:of the
1411:(with
1395:(with
1381:(with
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197:(OR).
1632:Logic
1318:Books
1208:. IBM
1187:, in
1095:1969.
893:arXiv
729:S2CID
566:S2CID
558:JSTOR
840:ISBN
806:PMID
665:ISBN
612:help
601:help
589:help
550:ISSN
509:ISBN
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