Penrose–Lucas argument

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Short description: Claim that human mathematicians are not describable as formal proof systems

The Penrose–Lucas argument is a logical argument partially based on a theory developed by mathematician and logician Kurt Gödel. In 1931, he proved that every effectively generated theory capable of proving basic arithmetic either fails to be consistent or fails to be complete. Due to human ability to see the truth of formal system's Gödel sentences, it is argued that the human mind cannot be computed on a Turing machine that works on Peano arithmetic because the latter cannot see the truth value of its Gödel sentence, while 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

Kurt 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[1] (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 philosopher John Lucas, J. E. Martin and K. H. Engleman argued in The Mind's I Has Two Eyes[2] 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.[3] From this point of view Lucas can appreciate that he can't assert the sentence-and consequently he can recognise its truth.[4] Still, this criticism only works if we assume that we can replace Lucas' reasoning with a formal system that has a Gödel sentence, but the Penrose-Lucas argument tries to prove otherwise: our ability to understand this level of arithmetic is not an effective procedure that can be simulated in a Turing machine.

Penrose argued that while a formal proof system cannot prove its own consistency, Gödel-unprovable results are provable by human mathematicians.[5] He takes this disparity to mean that human mathematicians are not describable as formal proof systems (which theorems can be proved using an abstract object such as a computer), and are therefore running a non-computable algorithm. Similar claims about the implications of Gödel's theorem were originally espoused by Alan Turing in the late 1940s, by Gödel himself in his 1951 Gibbs lecture, by E. Nagel and J.R. Newman in 1958,[6] and were subsequently popularized by Lucas in 1961.[7]

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[8]

Consequences

If correct, the Penrose–Lucas argument creates a need to understand the physical basis of non-computable behaviour in the brain.[9] 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 functions that evolve according to the Schrödinger equation. Non-stationary wave functions are linear combinations of the eigenstates 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 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 objective reduction (OR).[10]

Penrose sought to reconcile general relativity and quantum theory using his own ideas about the possible structure of spacetime.[5][11] 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]\displaystyle{ 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]\displaystyle{ \tau \approx \hbar/E_G }[/math]

where:

  • [math]\displaystyle{ \tau }[/math] is the time until OR occurs,
  • [math]\displaystyle{ E_G }[/math] is the gravitational self-energy or the degree of spacetime separation given by the superpositioned mass, and
  • [math]\displaystyle{ \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 10−37s. Objects somewhere between these two scales could collapse on a timescale relevant to neural processing.[10][citation needed][12]

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 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.[10][citation needed][13][14]

Criticism

The Penrose–Lucas argument about the implications of Gödel's incompleteness theorem for computational theories of human intelligence was criticized by mathematicians,[15][16][17][18] computer scientists,[19] and philosophers,[20][21][22][23][24] and the consensus among experts[which?][6] in these fields is that the argument fails,[25][26][27] with different authors attacking different aspects of the argument.[27][28]

Philosopher and mathematician Solomon 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.[16] Still, this does not account for his core argument of the alleged ability of the human mind to prove Gödel-unprovable sentences. Also, artificial Intelligence based on reinforcement learning can work by taking actions in an environment in order to maximize the notion of cumulative reward, acting like trial-and-error procedures.[29][30][31]

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 Paul Benacerraf, he tried to demonstrate that humans cannot prove that they are consistent,[15] 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 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.[32] Penrose argued against Minsky stating that mistakes human mathematicians make are irrelevant because they are correctable, while logical truths are “unassailable truths” to persons, which are the outputs of a sound system and the only ones that matter.[33] 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 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 Turing Machine that could be reasoning using some sort of paraconsistent logic.

See also

References

  1. ↑ 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 Mendelson, Elliot (2009). Introduction to Mathematical Logic (hardcover). Discrete Mathematics and Its Applications (5th ed.). Boca Raton: Chapman and Hall/CRC. ISBN 978-1-58488-876-5. 
  2. ↑ Martin, J. E.; Engleman, K. H. (1990). "The Mind's I Has Two Eyes". Philosophy 65 (254): 510–515. ISSN 0031-8191. https://www.jstor.org/stable/3751287. 
  3. ↑ Hofstadter 1979, pp. 476–477, Russell & Norvig 2003, p. 950, 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."
  4. ↑ "Details view: Lucas tricks machines into contradicting themselves". https://debategraph.org/Details.aspx?nid=1157. 
  5. ↑ 5.0 5.1 Penrose, Roger (1989). The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics. Oxford University Press. p. 480. ISBN 978-0-19-851973-7. 
  6. ↑ 6.0 6.1 "Gödel's Incompleteness Theorems". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. 2022. https://plato.stanford.edu/entries/goedel-incompleteness/. 
  7. ↑ Lucas, John R. (1961). "Minds, Machines and Godel". Philosophy 36 (April–July): 112–127. doi:10.1017/s0031819100057983. https://philpapers.org/rec/LUCMMA. 
  8. ↑ Roger Penrose. Mathematical intelligence. In Jean Khalfa, editor, What is Intelligence?, chapter 5, pages 107–136. Cambridge University Press, Cambridge, United Kingdom, 1994.
  9. ↑ "Lucas-Penrose Argument about Gödel's Theorem | Internet Encyclopedia of Philosophy". https://iep.utm.edu/lp-argue/. 
  10. ↑ 10.0 10.1 10.2 Hameroff, Stuart; Penrose, Roger (March 2014). "Consciousness in the universe: A review of the 'Orch OR' theory". Physics of Life Reviews (Elsevier) 11 (1): 39–78. doi:10.1016/j.plrev.2013.08.002. PMID 24070914. Bibcode: 2014PhLRv..11...39H. 
  11. ↑ Penrose, Roger (1989). Shadows of the Mind: A Search for the Missing Science of Consciousness. Oxford University Press. p. 457. ISBN 978-0-19-853978-0. https://archive.org/details/shadowsofmindsea00penr_0/page/457. 
  12. ↑ "Physicists place fresh limits on gravity's role in wavefunction collapse". 10 October 2020. https://physicsworld.com/a/physicists-place-fresh-limits-on-gravitys-role-in-wavefunction-collapse/. 
  13. ↑ "Kant's Views on Space and Time". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. 2022. https://plato.stanford.edu/entries/kant-spacetime/. 
  14. ↑ Rosu, H. C. (1994). "Essay on mesoscopic and quantum brain". arXiv:gr-qc/9409007.
  15. ↑ 15.0 15.1 LaForte, Geoffrey, Patrick J. Hayes, and Kenneth M. Ford 1998.Why Gödel's Theorem Cannot Refute Computationalism. Artificial Intelligence, 104:265–286.
  16. ↑ 16.0 16.1 Feferman, Solomon (1996). "Penrose's Gödelian argument". Psyche 2: 21–32. 
  17. ↑ 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 Topics in Logic, Philosophy and Foundations of Mathematics and Computer Science:In Recognition of Professor Andrzej Grzegorczyk (2008), p. 173
  18. ↑ 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
  19. ↑ 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 New York Times review)
  20. ↑ "MindPapers: 6.1b. Godelian arguments". Consc.net. http://consc.net/mindpapers/6.1b. Retrieved 2014-07-28. 
  21. ↑ "References for Criticisms of the Gödelian Argument". Users.ox.ac.uk. 1999-07-10. http://users.ox.ac.uk/~jrlucas/Godel/referenc.html. Retrieved 2021-07-07. 
  22. ↑ Boolos, George, et al. 1990. An Open Peer Commentary on The Emperor's New Mind. Behavioral and Brain Sciences 13 (4) 655.
  23. ↑ 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/
  24. ↑ Lewis, David K. 1969.Lucas against mechanism. Philosophy 44 231–233.
  25. ↑ Bringsjord, S. and Xiao, H. 2000. 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."
  26. ↑ In an article at "Penrose's Philosophical Error". Archived from the original on 2001-01-25. https://web.archive.org/web/20010125011300/http://www.mth.kcl.ac.uk/~llandau/Homepage/Math/penrose.html. Retrieved 2010-10-22.  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."
  27. ↑ 27.0 27.1 Princeton Philosophy professor John Burgess writes in On the Outside Looking In: A Caution about Conservativeness (published in Kurt Gödel: Essays for his Centennial, with the following comments found on 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."
  28. ↑ Dershowitz, Nachum 2005. 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.
  29. ↑ "Medium". 19 April 2019. https://towardsdatascience.com/reinforcement-learning-101-e24b50e1d292. 
  30. ↑ "What is Machine Learning?" (in en-us). IBM. https://www.ibm.com/topics/machine-learning. 
  31. ↑ "Building a Machine Learning Model through Trial and Error" (in en-US). https://www.kdnuggets.com/building-a-machine-learning-model-through-trial-and-error.html. 
  32. ↑ Marvin Minsky. "Conscious Machines." Machinery of Consciousness, Proceedings, National Research Council of Canada, 75th Anniversary Symposium on Science in Society, June 1991.
  33. ↑ "Lucas-Penrose Argument about Gödel's Theorem" (in en-US). Internet Encyclopedia of Philosophy. https://iep.utm.edu/lp-argue/. Retrieved 2023-06-11. 



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