Sir Roger Penrose OM FRS (born 8 August 1931) is a British mathematical physicist, mathematician, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics at the University of Oxford, an emeritus fellow of Wadham College, Oxford, and an honorary fellow of St John's College, Cambridge, and University College London.
Penrose has made contributions to the mathematical physics of general relativity and cosmology. He has received several prizes and awards, including the 1988 Wolf Prize in Physics, which he shared with Stephen Hawking for the Penrose–Hawking singularity theorems, and one half of the 2020 Nobel Prize in Physics "for the discovery that black hole formation is a robust prediction of the general theory of relativity".[lower-alpha 1]
- 1 Research
- 2 Later activity
- 3 An earlier universe
- 4 Physics and consciousness
- 5 Personal life
- 6 Awards and honours
- 7 Depictions in popular culture
- 8 Works
- 9 See also
- 10 Notes
- 11 References
- 12 Further reading
- 13 External links
In 1955, whilst still a student, Penrose reintroduced the E. H. Moore generalised matrix inverse, also known as the Moore–Penrose inverse, after it had been reinvented by Arne Bjerhammar in 1951. Having started research under the professor of geometry and astronomy, Sir W. V. D. Hodge, Penrose finished his PhD at St John's College, Cambridge, in 1958, with a thesis on "tensor methods in algebraic geometry" under algebraist and geometer John A. Todd. He devised and popularised the Penrose triangle in the 1950s, describing it as "impossibility in its purest form", and exchanged material with the artist M. C. Escher, whose earlier depictions of impossible objects partly inspired it. Escher's Waterfall, and Ascending and Descending were in turn inspired by Penrose.
As reviewer Manjit Kumar puts it:
As a student in 1954, Penrose was attending a conference in Amsterdam when by chance he came across an exhibition of Escher's work. Soon he was trying to conjure up impossible figures of his own and discovered the tribar – a triangle that looks like a real, solid three-dimensional object, but isn't. Together with his father, a physicist and mathematician, Penrose went on to design a staircase that simultaneously loops up and down. An article followed and a copy was sent to Escher. Completing a cyclical flow of creativity, the Dutch master of geometrical illusions was inspired to produce his two masterpieces.
Penrose spent the academic year 1956–57 as an Assistant Lecturer at Bedford College, London and was then a Research Fellow at St John's College, Cambridge. During that three year post, he married Joan Isabel Wedge, in 1959. Before the fellowship ended Penrose won a NATO Research Fellowship for 1959–61, first at Princeton and then at Syracuse University. Returning to the University of London, Penrose spent two years, 1961–63, as a researcher at King's College, London, before returning to the United States to spend the year 1963–64 as a Visiting Associate Professor at the University of Texas at Austin. He later held visiting positions at Yeshiva, Princeton and Cornell during 1966-67 and 1969.
In 1964, while a reader at Birkbeck College, London, (and having had his attention drawn from pure mathematics to astrophysics by the cosmologist Dennis Sciama, then at Cambridge) in the words of Kip Thorne of Caltech, "Roger Penrose revolutionised the mathematical tools that we use to analyse the properties of spacetime". Until then, work on the curved geometry of general relativity had been confined to configurations with sufficiently high symmetry for Einstein's equations to be soluble explicitly, and there was doubt about whether such cases were typical. One approach to this issue was by the use of perturbation theory, as developed under the leadership of John Archibald Wheeler at Princeton. The other, and more radically innovative, approach initiated by Penrose was to overlook the detailed geometrical structure of spacetime and instead concentrate attention just on the topology of the space, or at most its conformal structure, since it is the latter – as determined by the lay of the lightcones – that determines the trajectories of lightlike geodesics, and hence their causal relationships. The importance of Penrose's epoch-making paper "Gravitational Collapse and Space-Time Singularities" was not its only result, summarised roughly as that if an object such as a dying star implodes beyond a certain point, then nothing can prevent the gravitational field getting so strong as to form some kind of singularity. It also showed a way to obtain similarly general conclusions in other contexts, notably that of the cosmological Big Bang, which he dealt with in collaboration with Dennis Sciama's most famous student, Stephen Hawking. The Penrose–Hawking singularity theorems was inspired by Amal Kumar Raychaudhuri's Raychaudhuri equation.
It was in the local context of gravitational collapse that the contribution of Penrose was most decisive, starting with his 1969 cosmic censorship conjecture, to the effect that any ensuing singularities would be confined within a well-behaved event horizon surrounding a hidden space-time region for which Wheeler coined the term black hole, leaving a visible exterior region with strong but finite curvature, from which some of the gravitational energy may be extractable by what is known as the Penrose process, while accretion of surrounding matter may release further energy that can account for astrophysical phenomena such as quasars.
Following up his "weak cosmic censorship hypothesis", Penrose went on, in 1979, to formulate a stronger version called the "strong censorship hypothesis". Together with the Belinski–Khalatnikov–Lifshitz conjecture and issues of nonlinear stability, settling the censorship conjectures is one of the most important outstanding problems in general relativity. Also from 1979, dates Penrose's influential Weyl curvature hypothesis on the initial conditions of the observable part of the universe and the origin of the second law of thermodynamics. Penrose and James Terrell independently realised that objects travelling near the speed of light will appear to undergo a peculiar skewing or rotation. This effect has come to be called the Terrell rotation or Penrose–Terrell rotation.
In 1967, Penrose invented the twistor theory which maps geometric objects in Minkowski space into the 4-dimensional complex space with the metric signature (2,2).
Penrose is well known for his 1974 discovery of Penrose tilings, which are formed from two tiles that can only tile the plane nonperiodically, and are the first tilings to exhibit fivefold rotational symmetry. Penrose developed these ideas based on the article Deux types fondamentaux de distribution statistique (1938; an English translation Two Basic Types of Statistical Distribution) by Czech geographer, demographer and statistician Jaromír Korčák. In 1984, such patterns were observed in the arrangement of atoms in quasicrystals. Another noteworthy contribution is his 1971 invention of spin networks, which later came to form the geometry of spacetime in loop quantum gravity. He was influential in popularizing what are commonly known as Penrose diagrams (causal diagrams).
In 1983, Penrose was invited to teach at Rice University in Houston, by the then provost Bill Gordon. He worked there from 1983 to 1987.
In 2004 Penrose released The Road to Reality: A Complete Guide to the Laws of the Universe, a 1,099-page book aimed at giving a comprehensive guide to the laws of physics. He has proposed a novel interpretation of quantum mechanics.
Penrose is the Francis and Helen Pentz Distinguished (visiting) Professor of Physics and Mathematics at Pennsylvania State University. He is also a member of the editorial board of The Astronomical Review.
An earlier universe
In 2010, Penrose reported possible evidence, based on concentric circles found in WMAP data of the CMB sky, of an earlier universe existing before the Big Bang of our own present universe. He mentions this evidence in the epilogue of his 2010 book Cycles of Time, a book in which he presents his reasons, to do with Einstein's field equations, the Weyl curvature C, and the Weyl curvature hypothesis, that the transition at the Big Bang could have been smooth enough for a previous universe to survive it. He made several conjectures about C and the WCH, some of which were subsequently proved by others, and the smoothness is real.
In simple terms, he believes that the singularity in Einstein's field equation at the Big Bang is only an apparent singularity, similar to the well-known apparent singularity at the event horizon of a black hole. The latter singularity can be removed by a change of coordinate system, and Penrose proposes a different change of coordinate system that will remove the singularity at the big bang. One implication of this is that the major events at the Big Bang can be understood without unifying general relativity and quantum mechanics, and therefore we are not necessarily constrained by the Wheeler–DeWitt equation, which disrupts time. Alternatively, one can use the Einstein–Maxwell–Dirac equations.
Physics and consciousness
Penrose has written books on the connection between fundamental physics and human (or animal) consciousness. In The Emperor's New Mind (1989), he argues that known laws of physics are inadequate to explain the phenomenon of consciousness. Penrose proposes the characteristics this new physics may have and specifies the requirements for a bridge between classical and quantum mechanics (what he calls correct quantum gravity). Penrose uses a variant of Turing's halting theorem to demonstrate that a system can be deterministic without being algorithmic. (For example, imagine a system with only two states, ON and OFF. If the system's state is ON when a given Turing machine halts and OFF when the Turing machine does not halt, then the system's state is completely determined by the machine; nevertheless, there is no algorithmic way to determine whether the Turing machine stops.)
Penrose believes that such deterministic yet non-algorithmic processes may come into play in the quantum mechanical wave function reduction, and may be harnessed by the brain. He argues that the present computer is unable to have intelligence because it is an algorithmically deterministic system. He argues against the viewpoint that the rational processes of the mind are completely algorithmic and can thus be duplicated by a sufficiently complex computer. This contrasts with supporters of strong artificial intelligence, who contend that thought can be simulated algorithmically. He bases this on claims that consciousness transcends formal logic because things such as the insolubility of the halting problem and Gödel's incompleteness theorem prevent an algorithmically based system of logic from reproducing such traits of human intelligence as mathematical insight. These claims were originally espoused by the philosopher John Lucas of Merton College, Oxford.
The Penrose/Lucas argument about the implications of Gödel's incompleteness theorem for computational theories of human intelligence has been widely criticised by mathematicians, computer scientists and philosophers, and the consensus among experts in these fields seems to be that the argument fails, though different authors may choose different aspects of the argument to attack. Marvin Minsky, a leading proponent of artificial intelligence, was particularly critical, stating that Penrose "tries to show, in chapter after chapter, that human thought cannot be based on any known scientific principle." Minsky's position is exactly the opposite – he believes that humans are, in fact, machines, whose functioning, although complex, is fully explainable by current physics. Minsky maintains that "one can carry that quest [for scientific explanation] too far by only seeking new basic principles instead of attacking the real detail. This is what I see in Penrose's quest for a new basic principle of physics that will account for consciousness."
Penrose responded to criticism of The Emperor's New Mind with his follow up 1994 book Shadows of the Mind, and in 1997 with The Large, the Small and the Human Mind. In those works, he also combined his observations with that of anesthesiologist Stuart Hameroff.
Penrose and Hameroff have argued that consciousness is the result of quantum gravity effects in microtubules, which they dubbed Orch-OR (orchestrated objective reduction). Max Tegmark, in a paper in Physical Review E, calculated that the time scale of neuron firing and excitations in microtubules is slower than the decoherence time by a factor of at least 10,000,000,000. The reception of the paper is summed up by this statement in Tegmark's support: "Physicists outside the fray, such as IBM's John A. Smolin, say the calculations confirm what they had suspected all along. 'We're not working with a brain that's near absolute zero. It's reasonably unlikely that the brain evolved quantum behavior'". Tegmark's paper has been widely cited by critics of the Penrose–Hameroff position.
In their reply to Tegmark's paper, also published in Physical Review E, the physicists Scott Hagan, Jack Tuszynski and Hameroff claimed that Tegmark did not address the Orch-OR model, but instead a model of his own construction. This involved superpositions of quanta separated by 24 nm rather than the much smaller separations stipulated for Orch-OR. As a result, Hameroff's group claimed a decoherence time seven orders of magnitude greater than Tegmark's, but still well short of the 25 ms required if the quantum processing in the theory was to be linked to the 40 Hz gamma synchrony, as Orch-OR suggested. To bridge this gap, the group made a series of proposals.
He supposed that the interiors of neurons could alternate between liquid and gel states. In the gel state, it was further hypothesized that the water electrical dipoles are oriented in the same direction, along the outer edge of the microtubule tubulin subunits. Hameroff et al. proposed that this ordered water could screen any quantum coherence within the tubulin of the microtubules from the environment of the rest of the brain. Each tubulin also has a tail extending out from the microtubules, which is negatively charged, and therefore attracts positively charged ions. It is suggested that this could provide further screening. Further to this, there was a suggestion that the microtubules could be pumped into a coherent state by biochemical energy.
Finally, he suggested that the configuration of the microtubule lattice might be suitable for quantum error correction, a means of holding together quantum coherence in the face of environmental interaction. In the last decade, some researchers who are sympathetic to Penrose's ideas have proposed an alternative scheme for quantum processing in microtubules based on the interaction of tubulin tails with microtubule-associated proteins, motor proteins and presynaptic scaffold proteins. These proposed alternative processes have the advantage of taking place within Tegmark's time to decoherence.
Hameroff, in a lecture in part of a Google Tech talks series exploring quantum biology, gave an overview of current research in the area, and responded to subsequent criticisms of the Orch-OR model. In addition to this, a recent 2011 paper by Roger Penrose and Stuart Hameroff gives an updated model of their Orch-OR theory, in light of criticisms, and discusses the place of consciousness within the universe.
Phillip Tetlow, although himself supportive of Penrose's views, acknowledges that Penrose's ideas about the human thought process are at present a minority view in scientific circles, citing Minsky's criticisms and quoting science journalist Charles Seife's description of Penrose as "one of a handful of scientists" who believe that the nature of consciousness suggests a quantum process.
In January 2014 Hameroff and Penrose announced that a discovery of quantum vibrations in microtubules by Anirban Bandyopadhyay of the National Institute for Materials Science in Japan confirms the hypothesis of Orch-OR theory. A reviewed and updated version of the theory was published along with critical commentary and debate in the March 2014 issue of Physics of Life Reviews.
Penrose is married to Vanessa Thomas, director of Academic Development at Cokethorpe School and former head of mathematics at Abingdon School, with whom he has one son. He has three sons from a previous marriage to American Joan Isabel Penrose (née Wedge), whom he married in 1959.
During an interview with BBC Radio 4 on 25 September 2010, Penrose stated, "I'm not a believer myself. I don't believe in established religions of any kind." He regards himself as an agnostic. However, in the 1991 film A Brief History of Time, he also said, "I think I would say that the universe has a purpose, it's not somehow just there by chance … some people, I think, take the view that the universe is just there and it runs along—it's a bit like it just sort of computes, and we happen somehow by accident to find ourselves in this thing. But I don't think that's a very fruitful or helpful way of looking at the universe, I think that there is something much deeper about it."
Awards and honours
Penrose has been awarded many prizes for his contributions to science. He was elected a Fellow of the Royal Society (FRS) in 1972. In 1975, Stephen Hawking and Penrose were jointly awarded the Eddington Medal of the Royal Astronomical Society. In 1985, he was awarded the Royal Society Royal Medal. Along with Stephen Hawking, he was awarded the prestigious Wolf Foundation Prize for Physics in 1988. In 1989 he was awarded the Dirac Medal and Prize of the British Institute of Physics. In 1990 Penrose was awarded the Albert Einstein Medal for outstanding work related to the work of Albert Einstein by the Albert Einstein Society. In 1991, he was awarded the Naylor Prize of the London Mathematical Society. From 1992 to 1995 he served as President of the International Society on General Relativity and Gravitation. In 1994, Penrose was knighted for services to science. In the same year he was also awarded an Honorary Degree (Doctor of Science) by the University of Bath. In 1998, he was elected Foreign Associate of the United States National Academy of Sciences. In 2000 he was appointed to the Order of Merit. In 2004 he was awarded the De Morgan Medal for his wide and original contributions to mathematical physics. To quote the citation from the London Mathematical Society:
His deep work on General Relativity has been a major factor in our understanding of black holes. His development of Twistor Theory has produced a beautiful and productive approach to the classical equations of mathematical physics. His tilings of the plane underlie the newly discovered quasi-crystals.
In 2005 Penrose was awarded an honorary doctorate by Warsaw University and Katholieke Universiteit Leuven (Belgium), and in 2006 by the University of York. In 2008 Penrose was awarded the Copley Medal. He is also a Distinguished Supporter of the British Humanist Association and one of the patrons of the Oxford University Scientific Society. In 2011, Penrose was awarded the Fonseca Prize by the University of Santiago de Compostela. In 2012, Penrose was awarded the Richard R. Ernst Medal by ETH Zürich for his contributions to science and strengthening the connection between science and society. In 2015 Penrose was awarded an honorary doctorate by CINVESTAV-IPN (Mexico).
Depictions in popular culture
- In the 2004 BBC TV movie Hawking, Penrose is played by Tom Ward
- In the 2014 movie The Theory of Everything, Penrose is played by Christian McKay
- The Emperor's New Mind: Concerning Computers, Minds, and The Laws of Physics (1989)
- Shadows of the Mind: A Search for the Missing Science of Consciousness (1994)
- The Road to Reality: A Complete Guide to the Laws of the Universe (2004)
- Cycles of Time: An Extraordinary New View of the Universe (2010)
- The Nature of Space and Time (with Stephen Hawking) (1996)
- The Large, the Small and the Human Mind (with Abner Shimony, Nancy Cartwright, and Stephen Hawking) (1997)
- Techniques of Differential Topology in Relativity (1972, ISBN 0-89871-005-7)
- Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields (with Wolfgang Rindler, 1987) ISBN 0-521-33707-0 (paperback)
- Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry (with Wolfgang Rindler, 1988) (reprint), ISBN 0-521-34786-6 (paperback)
Foreword to other books
- Foreword to Beating the Odds: The Life and Times of E. A. Milne, written by Meg Weston Smith. Published by World Scientific Publishing Co in June 2013.
- Foreword to Quantum Aspects of Life by Derek Abbott, Paul C. W. Davies, and Arun K. Pati. Published by Imperial College Press in 2008.
- Foreword to Fearful Symmetry' by Anthony Zee's. Published by Princeton University Press in 2007.
- Foreword to "A Computable Universe" by Hector Zenil. Published by World Scientific Publishing Co in December 2012.
- Conformal cyclic cosmology
- Illumination problem
- Orchestrated objective reduction
- Penrose graphical notation
- Quantum Aspects of Life
- Quantum mind
- ↑ The other half was awarded jointly to Reinhard Genzel and Andrea Ghez for their work on black holes.
- ↑ https://www.britannica.com/biography/Roger-Penrose
- ↑ "Oxford Mathematician Roger Penrose jointly wins the Nobel Prize in Physics | University of Oxford". www.ox.ac.uk. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Siegel, Matthew (8 January 2008). "Wolf Foundation Honors Hawking and Penrose for Work in Relativity". Physics Today. 42 (1): 97–98. doi:10.1063/1.2810893. ISSN 0031-9228.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "The Nobel Prize in Physics 2020". NobelPrize.org. Retrieved 6 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Overbye, Dennis; Taylor, Derrick Bryson (6 October 2020). "Nobel Prize in Physics Awarded to 3 Scientists for Work on Black Holes – The prize was awarded half to Roger Penrose for showing how black holes could form and half to Reinhard Genzel and Andrea Ghez for discovering a supermassive object at the Milky Way's center". The New York Times. Retrieved 6 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Penrose, R. (1955). "A generalized inverse for matrices". Mathematical Proceedings of the Cambridge Philosophical Society. 51 (3): 406–413. Bibcode:1955PCPS...51..406P. doi:10.1017/S0305004100030401.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Zheng, Wenjie. "The 100th anniversary of Moore-Penrose inverse and its role in statistics and machine learning". www.zhengwenjie.net. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Roger Penrose wins 2020 Nobel Prize in Physics for discovery about black holes". University of Cambridge. 6 October 2020. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Welch, Chris (23 March 2012). "'Frustro' typeface applies the Penrose impossible triangle concept to words". The Verge. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Baggini, Julian (2012). Philosophy: All That Matters. John Murray Press. ISBN 978-1-4441-5585-3.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Ascending and Descending by M.C. Escher – Facts about the Painting". Totally History. 21 May 2013. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Kumar, Manjit (15 October 2010). "Cycles of Time: An Extraordinary New View of the Universe by Roger Penrose – review". The Guardian.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Professor Sir Roger Penrose awarded the 2020 Nobel Prize in Physics". King's College London. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Cite error: Invalid
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- ↑ "The second Cambridge Cutting Edge Lecture: Professor Sir Roger Penrose". Cambridge Society of Paris. 12 March 2019. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Thorne, Kip; Thorne, Kip S.; Hawking, Stephen (1994). Black Holes and Time Warps: Einstein's Outrageous Legacy. W. W. Norton & Company. ISBN 978-0-393-31276-8.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Ellis, George F. R.; Penrose, Sir Roger (1 January 2010). "Dennis William Sciama. 18 November 1926 – 19 December 1999". Biographical Memoirs of Fellows of the Royal Society. 56: 401–422. doi:10.1098/rsbm.2009.0023. ISSN 0080-4606. S2CID 73035217.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Penrose, Roger (January 1965). "Gravitational Collapse and Space-Time Singularities". Physical Review Letters. 14 (3): 57–59. Bibcode:1965PhRvL..14...57P. doi:10.1103/PhysRevLett.14.57.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Clark, Stuart. "A brief history of Stephen Hawking: A legacy of paradox". New Scientist. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Roger Penrose". New Scientist. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Wolchover, Natalie. "Physicists Debate Hawking's Idea That the Universe Had No Beginning". Quanta Magazine. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Fernandez, Rodrigo L. (21 July 2020). "Cosmic Censorship Conjecture violation: A semiclassical approach". arXiv:2007.10601 [gr-qc].<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Curiel, Erik (2020), "Singularities and Black Holes", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Summer 2020 ed.), Metaphysics Research Lab, Stanford University, retrieved 7 October 2020<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Kafatos, M.; Leiter, D. (1979). "1979ApJ...229...46K Page 46". The Astrophysical Journal. 229: 46. Bibcode:1979ApJ...229...46K. doi:10.1086/156928. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Penrose process". Oxford Reference. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ R. Penrose (1979). "Singularities and Time-Asymmetry". In S. W. Hawking; W. Israel (eds.). General Relativity: An Einstein Centenary Survey. Cambridge University Press. pp. 581–638.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Terrell, James (1959). "Invisibility of the Lorentz Contraction". Physical Review. 116 (4): 1041–1045. Bibcode:1959PhRv..116.1041T. doi:10.1103/PhysRev.116.1041.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Penrose, Roger (1959). "The Apparent Shape of a Relativistically Moving Sphere". Proceedings of the Cambridge Philosophical Society. 55 (1): 137–139. Bibcode:1959PCPS...55..137P. doi:10.1017/S0305004100033776.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "New Horizons in Twistor Theory | Mathematical Institute". www.maths.ox.ac.uk. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "twistor space in nLab". ncatlab.org. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Jaromír Korčák (1938): Deux types fondamentaux de distribution statistique. Prague, Comité d'organisation, Bull. de l'Institute Int'l de Statistique, vol. 3, pp. 295–299.
- ↑ Steinhardt, Paul (1996). "New perspectives on forbidden symmetries, quasicrystals, and Penrose tilings". PNAS. 93 (25): 14267–14270. Bibcode:1996PNAS...9314267S. doi:10.1073/pnas.93.25.14267. PMC 34472. PMID 8962037.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Penrose on Spin Networks". math.ucr.edu. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Penrose diagrams". jila.colorado.edu. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Roger Penrose at Rice, 1983–87". Rice History Corner. 22 May 2013.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "If an Electron Can Be in Two Places at once, Why Can't You?". Archived from the original on 1 November 2012. Retrieved 27 October 2008.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Dr. Roger Penrose at Penn State University". Retrieved 9 July 2007.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Gurzadyan, V.G.; Penrose, R. (2010). "Concentric circles in WMAP data may provide evidence of violent pre-Big-Bang activity". arXiv:1011.3706 [astro-ph.CO]. Unknown parameter
|volume=ignored (help)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Roger Penrose, Cycles of Time, Vintage; Reprint edition (1 May 2012)
- ↑ Criticism of the Lucas/Penrose argument that intelligence can not be entirely algorithmic:
- MindPapers: 6.1b. Godelian arguments
- References for Criticisms of the Gödelian Argument*
- Boolos, George, et al. 1990. An Open Peer Commentary on The Emperor's New Mind. Behavioral and Brain Sciences 13 (4) 655.
- 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/
- Feferman, Solomon (1996). "Penrose's Gödelian argument". PSYCHE. 2: 21–32. CiteSeerX: 10.1.1.130.7027.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- 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
- LaForte, Geoffrey, Patrick J. Hayes, and Kenneth M. Ford 1998. Why Gödel's Theorem Cannot Refute Computationalism. Artificial Intelligence, 104:265–286.
- Lewis, David K. 1969. Lucas against mechanism. Philosophy 44 231–233.
- 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)
- Bringsford, 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."
- In an article at http://www.mth.kcl.ac.uk/~llandau/Homepage/Math/penrose.html 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."
- 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."
- Nachum Dershowitz 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 Andrei Voronkov, eds., Lecture Notes in Computer Science, vol. 3835, Springer-Verlag, Berlin, pp. 125–138.
- ↑ Marvin Minsky. "Conscious Machines." Machinery of Consciousness, Proceedings, National Research Council of Canada, 75th Anniversary Symposium on Science in Society, June 1991.
- ↑ Tegmark, Max (2000). "The importance of quantum decoherence in brain processes". Physical Review E. 61: 4194–4206. arXiv:quant-ph/9907009. Bibcode:2000PhRvE..61.4194T. doi:10.1103/physreve.61.4194.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ 43.0 43.1 Tetlow, Philip (2007). The Web's Awake: An Introduction to the Field of Web Science and the Concept of Web Life. Hoboken, New Jersey: John Wiley & Sons. p. 166. ISBN 978-0-470-13794-9.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Hagan, S.; Hameroff, S. & Tuszyński, J. (2002). "Quantum Computation in Brain Microtubules? Decoherence and Biological Feasibility". Physical Review E. 65: 061901. arXiv:quant-ph/0005025. Bibcode:2002PhRvE..65f1901H. doi:10.1103/PhysRevE.65.061901.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Hameroff, S. (2006). "Consciousness, Neurobiology and Quantum Mechanics". In Tuszynski, Jack (ed.). The Emerging Physics of Consciousness. Springer. pp. 193–253<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Clarifying the Tubulin bit/qubit – Defending the Penrose-Hameroff Orch OR Model (Quantum Biology)". YouTube. 22 October 2010. Retrieved 13 August 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Roger Penrose & Stuart Hameroff (4 July 1992). "Consciousness in the Universe: Neuroscience, Quantum Space-Time Geometry and Orch OR Theory". Journal of Cosmology. Quantumconsciousness.org. Retrieved 13 August 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Anirban Bandyopadhyay". ResearchGate. Retrieved 22 February 2014.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Discovery of quantum vibrations in 'microtubules' inside brain neurons supports controversial theory of consciousness". ScienceDaily. Retrieved 22 February 2014. Italic or bold markup not allowed in:
- ↑ S. Hameroff; R. Penrose (2014). "Consciousness in the universe: A review of the 'Orch OR' theory". Physics of Life Reviews. 11 (1): 39–78. Bibcode:2014PhLRv..11...39H. doi:10.1016/j.plrev.2013.08.002. PMID 24070914. Retrieved 29 March 2014.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ 51.0 51.1 "The Peter & Patricia Gruber Foundation, St. Thomas US Virgin Islands – Grants and International Awards". Gruberprizes.org. 8 August 1931. Retrieved 13 August 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Vanessa Penrose". Abingdon School. 6 July 2012. Archived from the original on 27 March 2012. Retrieved 13 August 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "7+ Out of This World Facts About Physicist Sir Roger Penrose". interestingengineering.com. 27 October 2019. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Roger Penrose". The Gifford Lectures. 18 August 2014. Retrieved 7 October 2020.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Big Bang follows Big Bang follows Big Bang". BBC News. 25 September 2010. Retrieved 1 December 2010.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Thomas Fink (19 December 2020). "A singular mind: Roger Penrose on his Nobel Prize". The Spectator. The Spectator. Retrieved 18 May 2021.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ See A Brief History of Time (1991) film script – springfieldspringfield.co.uk Archived 24 September 2015 at the Wayback Machine
- ↑ "Supplement 53696,10 June 1994,London Gazette". The Gazette. Retrieved 16 August 2015.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ "Honorary Graduates 1989 to present". University of Bath. Retrieved 18 February 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ London Mathematical Society
- Ferguson, Kitty (1991). Stephen Hawking: Quest For A Theory of Everything. Franklin Watts. ISBN 0-553-29895-X.
- Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> (See Box 34.2.)
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- Roger Penrose at the Internet Movie Database
- Awake in the Universe - Penrose debates how creativity, the most elusive of faculties, has helped us unlock the country of the mind and the mysteries of the cosmos with Bonnie Greer.
- Dangerous Knowledge on YouTube – Penrose was one of the principal interviewees in a BBC documentary about the mathematics of infinity directed by David Malone
- Penrose's new theory "Aeons Before the Big Bang?":
- Original 2005 lecture: "Before the Big Bang? A new perspective on the Weyl curvature hypothesis" (Isaac Newton Institute for Mathematical Sciences, Cambridge, 11 Nov 2005).
- Original publication: "Before the Big Bang: an outrageous new perspective and its implications for particle physics". Proceedings of EPAC 2006. Edinburgh. 2759–2762 (cf. also Hill, C.D. & Nurowski, P. (2007) "On Penrose's 'Before the Big Bang' ideas". Ithaca)
- Revised 2009 lecture: "Aeons Before the Big Bang?" (Georgia Institute of Technology, Center for Relativistic Astrophysics)
- BBC interview on the new theory on YouTube
- Roger Penrose on The Forum
- Penrose on sidestepping reason on YouTube
- O'Connor, John J.; Robertson, Edmund F., "Roger Penrose", MacTutor History of Mathematics archive, University of St Andrews<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
- Hilary Putnam's review of Penrose's 'Shadows of the Mind' claiming that Penrose's use of Godel's Incompleteness Theorem is fallacious
- Beyond the Doubting of a Shadow: A Reply to Commentaries on Shadows of the Mind at the Wayback Machine (archived June 18, 2008)
- Penrose Tiling found in Islamic Architecture
- Two theories for the formation of quasicrystals resembling Penrose tilings
- Tegmark, Max (2000). "The importance of quantum decoherence in brain processes". Physical Review E. 61: 4194–4206. arXiv:quant-ph/9907009. Bibcode:2000PhRvE..61.4194T. doi:10.1103/physreve.61.4194.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- "Biological feasibility of quantum states in the brain" – (a disputation of Tegmark's result by Hagan, Hameroff, and Tuszyński)
- "Toilet Paper Plagiarism" at the Wayback Machine (archived March 12, 2005) – D. Trull about Penrose's lawsuit concerning the use of his Penrose tilings on toilet paper
- Roger Penrose: A Knight on the tiles (Plus Magazine)
- Penrose's Gifford Lecture biography
- Audio: Roger Penrose in conversation on the BBC World Service discussion show
- Roger Penrose speaking about Hawking's new book on Premier Christian Radio
- "The Cyclic Universe – A conversation with Roger Penrose", Ideas Roadshow, 2013
- Forbidden crystal symmetry in mathematics and architecture, filmed event at the Royal Institution, October 2013
- Oxford Mathematics Interviews: "Extra Time: Professor Sir Roger Penrose in conversation with Andrew Hodges." These two films explore the development of Sir Roger Penrose’s thought over more than 60 years, ending with his most recent theories and predictions. 51 min and 42 min. (Mathematical Institute)
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