Roger Penrose
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Sir Roger Penrose
British mathematician and theoretical physicist (1931–
Penrose, the son of the geneticist Lionel Penrose, was born at Colchester in Essex. He graduated from University College, London, and obtained his PhD in 1957 from Cambridge University. After holding various lecturing and research posts in London, Cambridge, and in America at Princeton, Syracuse, and Texas, Penrose was appointed professor of applied mathematics at Birkbeck College, London, in 1966. In 1973 he was elected Rouse Ball Professor of Mathematics at Oxford.
Penrose has done much to elucidate the fundamental properties of black holes. These result from the total gravitational collapse of large stars that shrink to such a small volume that not even a light signal can escape from them. There is thus a boundary around a black hole inside which all information about the black hole is trapped; this is known as its ‘event horizon’. With Stephen Hawking, Penrose proved a theorem of Einstein's general relativity asserting that at the center of a black hole there must evolve a ‘space–time singularity’ of zero volume and infinite density where the present laws of physics break down. He went on to propose his hypothesis of ‘cosmic censorship’, that such singularities cannot be ‘naked’; they must possess an event horizon. The effect of this would be to conceal and isolate the singularity with its indifference to the laws of physics.
Despite this Penrose went on in 1969 to describe a mechanism for the extraction of energy from a Kerr black hole, an uncharged rotating body first described by Roy Kerr in 1963. Such bodies are surrounded by an ergosphere within which it is impossible for an object to be at rest. If, Penrose demonstrated, a body fell into this area it would split into two particles; one would fall into the hole and the other would escape with more mass-energy than the initial particle. In this way rotational energy of the black hole is transferred to the particle outside the hole.
From the mid-1960s Penrose has been working on the development of a new cosmology based on a complex geometry. Penrose began with ‘twistors’ – massless objects with both linear and angular momentum in twistor space. From these he attempted to reconstruct the main outlines of modern physics. The matter is pursued not only by Penrose but through a number of ‘twistor groups’ who communicate through a Twistor Newsletter. The fullest account of twistor theory is to be found in Spinors and Space-Time (2 vols., 1984–86) by Penrose and W. Windler.
In 1974 Penrose introduced a novel tiling of the affine plane (Penrose tiling). Periodic tilings in which a unit figure is endlessly repeated can be constructed from triangles, squares, and hexagons – figures with three-, four-, or six-fold symmetry. The plane cannot be tiled by pentagons, which have a five-fold symmetry; three pentagons fitted together always leave a crack, known to crystallographers as a ‘frustration’. It was also known that crystal structures could have two-, three-, four-, or six-fold rotational symmetries only. No crystal, that is, could have a five-fold rotational symmetry.
Penrose's method of tiling the plane involved constructing two rhombuses by dividing the diagonal of a regular parallelogram by a golden section. These could be combined according to simple rules so as to cover the plane, even though there was no simple repeated unit cell. The rhombuses can be assembled in such a way as to have an almost five-fold symmetry. As such they were seen as an interesting oddity, usually discussed in columns devoted to recreational mathematics. However, things changed dramatically in 1984 when Dany Schectman of the National Bureau of Standards and his colleagues found that a rapidly cooled sample of an aluminum–manganese alloy formed crystals that displayed a five-fold symmetry. ‘Quasicrystals’, as they soon became known, developed rapidly into a major new research field and became the subject of hundreds of papers.
In addition to continuing his work on twistor theory Penrose also published a widely read book, The Emperor's New Mind (1989). The book is an attack on aspects of artificial intelligence. In it he argues that there are aspects of mathematics that cannot be tied to a set of rules. We cannot allow “one universally formal system…equivalent to all the mathematicians' algorithms for judging mathematical truth.” Such a system would violate Gödel's theorem. Nor can we accept that algorithms used are so complicated and obscure that their validity can never be known. We do not in fact ascertain mathematical truth solely through the use of algorithms. “We must see the truth of a mathematical argument to be convinced of its validity,” Penrose has insisted. Consequently when we see the validity of a theorem, in seeing it “we reveal the very nonalgorithmic nature of the ‘seeing’ process itself.”
He further developed his arguments in Shadows of the Mind (1994), in which he also answered many of the objections raised against the earlier work. Penrose has also published (in collaboration with Stephen Hawking) The Nature of Space and Time (1996), in which they develop their own cosmological viewpoints. Thus while Penrose presents his own twistor view of the universe, Hawking concentrates on problems connected with quantum cosmology.
Gale Encyclopedia of Biography:
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Roger Penrose
The British mathematician and physicist, Sir Roger Penrose (born 1931), made striking and original contributions to the study of geometry, relativity, quantum mechanics, and the human mind.
Roger Penrose was born in Colchester, England, on August 8, 1931. His father was the geneticist Lionel Penrose, an expert on mental defects, whose interest in geometry was communicated to his son. The Penrose family was illustrious in British intellectual life in the 20th century. Jonathan Penrose won the British chess championship ten times in the 1950s and 1960s. It is not surprising that the intellectual life of the Penrose household was lively.
Penrose received his undergraduate degree from University College, London, and then proceeded to Cambridge for his doctorate. While an undergraduate he discovered a theorem concerning conic sections from which some of the basic theorems of projective geometry follow as special cases. As part of his work for his doctorate he rediscovered some important results in the theory of matrices. From 1964 to 1966 he was a reader in applied mathematics at Birkbeck College at the University of London, advancing to full professor in 1966.
The study of mathematics in Britain has always included a large amount of applied mathematics and even physics, so it is not unexpected that much of Penrose's best-known work looks more like physics than pure mathematics. He and Stephen Hawking studied black holes in collaboration and the two of them identified the basic characteristics of black holes, which result from the collapse of large stars. The mass becomes so concentrated that even photons (light particles) are unable to escape. As a result, even if it is possible to recognize the existence of a black hole from its effects on nearby objects, it would be impossible to observe the interior of the black hole itself.
Starting from his interest in the question of whether space and time are smooth or divided into discrete units, Penrose investigated many aspects of quantum mechanics. While he was at Cambridge, Penrose tried to build mathematical models for quantum mechanics using the basic elements of real numbers. One of the long-standing problems of 20th-century physics has been to combine the apparently conflicting fields of relativity and quantum mechanics. Penrose attempted to find a resolution via twistor geometry, which is based on complex numbers. This ambitious project remains far from completion, but the study of twistors has become an industry within physics in its own right.
Penrose collaborated with his father on the creation of a visual illusion that was incorporated into lithographs by the Dutch artist M. C. Escher, whose work included many mathematical elements. Also within the area of geometry, Penrose made a striking contribution to the study of tilings. A tiling is a method of covering the entire plane with polygons, for example squares or equilateral triangles. Tilings using those figures are called periodic because the pattern repeats regularly in moving about the plane. The question was whether it would be possible to tile (cover) the plane with a nonrepeating pattern.
Before Penrose made his contribution, others had already shown that it was possible to tile the plane in a nonperiodic fashion. The first solution used an immense number of different tiles, and the best solution known in 1974 still used six tiles of different shapes. In that year Penrose found a nonperiodic tiling using only two different shapes. Although this geometric contribution seems far removed from his studies of astrophysics and quantum mechanics, it also reflects the width of his scientific background.
In 1966 Penrose received the Adams Prize from Cambridge University and in 1971 the Dannie Heineman Prize for Physics from the American Physical Society. The next year he was elected to the Royal Society and in 1973 he succeeded to the prestigious Rouse Ball Chair of Mathematics at Oxford University. He shared two awards with his collaborator Stephen Hawking; the 1975 Royal Astronomical Society's Eddington Medal and the 1988 Wolf Prize for physics. Penrose held visiting positions at many leading universities in the United States including Cornell, Texas, California, and Princeton.
Penrose became known to the general public thanks to the best-selling book The Emperor's New Mind, which appeared on both sides of the Atlantic in 1989. Hawking had written a book to similar acclaim a couple of years before but had not tried to include any equations other than Einstein's e = mc2. Penrose's book includes that equation and hundreds of others as it ranges over computers, minds, and the laws of physics, to mention just the subjects explicitly named in the subtitle. The Emperor's New Mind may have been the best book about modern science yet written. Within 18 months it had run through numerous printings.
During a historic lecture series at the Isaac Newton Institute for Mathematical Sciences at Cambridge University in 1994, Penrose and Hawking recreated the famous Bohr-Einstein debate. In public lectures Penrose and Hawking presented their distinctive views on the universe, its evolution and impact on quantum theory. The same year, Penrose was knighted for his numerous contributions to science. Shadows of the Mind (1994) once again demonstrated the ability of Penrose to communicate complex theoretical physics to a general audience.
What distinguished Roger Penrose among the physicists and mathematicians of his time was the breadth and depth in his work. Some of the essays that he wrote illustrate the attention that he gave to his intellectual ancestors, such as Sir Isaac Newton. His influence on his students was profound.
Further Reading
There is an article on Penrose in the McGraw-Hill set on Modern Scientists and Engineers (1980). A more personal glimpse is available in Martin Gardner's introduction to Penrose's The Emperor's New Mind (1989). A good discussion of tilings and Penrose's work is in B. Grunbaum and G. Shephard's book Tilings and Patterns (1986). Articles on Penrose can be found in the popular science journals Scientific American and Science. An account of the 1994 Penrose-Hawking debate is presented in The Nature of Space and Time (1996).
Wikipedia on Answers.com:
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Roger Penrose
Sir Roger Penrose OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. He has received a number of prizes and awards, including the 1988 Wolf Prize for physics which he shared with Stephen Hawking for their contribution to our understanding of the universe.[1] He is renowned for his work in mathematical physics, in particular his contributions to general relativity and cosmology. He is also a recreational mathematician and philosopher.
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Career
He was born in Colchester, Essex, England, Roger Penrose is a son of Lionel S. Penrose and Margaret Leathes.[2] Penrose is the brother of mathematician Oliver Penrose and of chess Grandmaster Jonathan Penrose. Penrose attended University College School and University College, London, where he graduated with a first class degree in mathematics. In 1955, while still a student, Penrose reintroduced the E. H. Moore generalized matrix inverse, also known as the Moore–Penrose inverse[3], after it had been reinvented by Arne Bjerhammar (1951). Penrose earned his Ph.D. at Cambridge (St John's College) in 1958, writing 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 tri-bar – 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.[4]
In 1965, at Cambridge, Penrose proved that singularities (such as black holes) could be formed from the gravitational collapse of immense, dying stars.[5] This work was extended by Hawking to prove the Penrose–Hawking singularity theorems.
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). In 1969, he conjectured the cosmic censorship hypothesis. This proposes (rather informally) that the universe protects us from the inherent unpredictability of singularities (such as the one in the centre of a black hole) by hiding them from our view behind an event horizon. This form is now known as the "weak censorship hypothesis"; in 1979, Penrose formulated a stronger version called the "strong censorship hypothesis". Together with the BKL 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.[6] Penrose and James Terrell independently realized 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.[7][8]
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[9] (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.[10] 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 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.[11] 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.[12]
Penrose is the Francis and Helen Pentz Distinguished (visiting) Professor of Physics and Mathematics at Pennsylvania State University.[13] Penrose is married to Vanessa Thomas, head of mathematics at Abingdon School,[14][15] with whom he has one son.[14] He has three sons from a previous marriage to American Joan Isabel Wedge, whom he married in 1959.
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. (E.g., imagine a system with only two states, ON and OFF. If the system's state is ON if a given Turing machine halts, and OFF if the Turing machine does not halt, then the system's state is completely determined by the Turing machine, however there is no algorithmic way to determine whether the Turing machine stops.) Penrose believes that such deterministic non-algorithmic processes may come in 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 criticized 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.[16] 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."[17]
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,[18] 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'".[19] 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[20][21] 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. It was 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, it is 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.[22] 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.[23]
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.[19]
Religious views
Penrose does not hold to any religious doctrine,[24] and refers to himself as an atheist.[25] In the film A Brief History of Time, he 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."[26] Penrose is a Distinguished Supporter of the British Humanist Association.
Awards and honours
Penrose has been awarded many prizes for his contributions to science. He was elected a Fellow of the Royal Society of London 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.[27] In the same year he was also awarded an Honorary Degree (Doctor of Science) by the University of Bath.[28] 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 (Honoris Causa) 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 ETHZ for his contributions to science and strengthening the connection between science and society.
Works
- 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)
- The Emperor's New Mind: Concerning Computers, Minds, and The Laws of Physics (1989, ISBN 0-14-014534-6 (paperback); it received the Rhone-Poulenc science book prize in 1990)
- Shadows of the Mind: A Search for the Missing Science of Consciousness (1994, ISBN 0-19-853978-9 (hardback))
- The Nature of Space and Time (with Stephen Hawking, 1996, ISBN 0-691-03791-4 (hardback), ISBN 0-691-05084-8 (paperback))
- The Large, the Small, and the Human Mind (with Abner Shimony, Nancy Cartwright, and Stephen Hawking, 1997, ISBN 0-521-56330-5 (hardback), ISBN 0-521-65538-2 (paperback), Canto edition: ISBN 0-521-78572-3)
- White Mars or, The Mind Set Free (with Brian W. Aldiss, 1999, ISBN 978-0-316-85243-2 (hardback))
- The Road to Reality: A Complete Guide to the Laws of the Universe (2004, ISBN 0-224-04447-8 (hardcover), ISBN 0-09-944068-7 (paperback))
- Cycles of Time: An Extraordinary New View of the Universe (Bodley Head (23 Sep 2010) ISBN 978-0-224-08036-1)
Penrose also wrote forewords to Quantum Aspects of Life and Zee's book Fearful Symmetry.
See also
- Conformal Cyclic Cosmology
- Orch-OR
- Quantum mind
- Multiverse
- Quantum Aspects of Life (book)
- Illumination problem
References
- ^ Penrose, R (2005). The Road to Reality: A Complete guide to the Laws of the Universe. Vintage Books. ISBN 0-09-944068-7
- ^ Penrose and his father shared mathematical concepts with Dutch graphic artist M. C. Escher which were incorporated into a lot of pieces, including Waterfall, which is based on the 'Penrose triangle', and Up and Down.
- ^ Penrose, R. "A Generalized Inverse for Matrices" Proc. Cambridge Phil. Soc. 51, 406–413, 1955)
- ^ Cycles of Time: An Extraordinary New View of the Universe by Roger Penrose – review | Books | The Guardian
- ^ Ferguson, 1991: 66
- ^ R. Penrose (1979). "Singularities and Time-Asymmetry". In S. W. Hawking and W. Israel. General Relativity: An Einstein Centenary Survey. Cambridge University Press. pp. 581–638.
- ^ Terrell, James (1959). "Invisibility of the Lorentz Contraction". Physical Review 116 (4): 1041–1045. Bibcode 1959PhRv..116.1041T. DOI:10.1103/PhysRev.116.1041.
- ^ Penrose, Roger (1959). "The Apparent Shape of a Relativistically Moving Sphere". Proceedings of the Cambridge Philosophical Society 55: 137–139. Bibcode 1959PCPS...55..137P. DOI:10.1017/S0305004100033776.
- ^ 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. http://www.pnas.org/content/93/25/14267.full.
- ^ "If an Electron Can Be in Two Places at once, Why Can't You?". http://discovermagazine.com/2005/jun/cover/article_view?b_start:int=0&-C=. Retrieved 2008-10-27.
- ^ 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].
- ^ "Dr. Roger Penrose at Penn State University". http://www.phys.psu.edu/people/display/index.html?person_id=233&mode=contact.. Retrieved 2007-07-09.
- ^ a b http://www.gruberprizes.org/SelectingRecipients/SelectionAdvisoryBoard_Bio.php?id=4
- ^ http://www.abingdon.org.uk/vanessa_penrose/
- ^ 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.
- 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."
- Dershowitz, Nachum 2005. The Four Sons of Penrose, in Proceedings of the Eleventh Conference on Logic Programming for 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.
- ^ 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. vol 61. pp. 4194–4206.
- ^ a b Tetlow, Philip (2007). The Web's Awake: An Introduction to the Field of Web Science and the Concept of Web Life. Hoboken, NJ: John Wiley & Sons. p. 166. ISBN 978-0-470-13794-9. http://books.google.com/?id=3mPI9rUuhJ8C&printsec=frontcover&dq=penrose+%22thought+process%22&q=penrose%20.
- ^ Hagan, S., Hameroff, S., and 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.
- ^ Hameroff, S. (2006). "Consciousness, Neurobiology and Quantum Mechanics". In Tuszynski, Jack. The Emerging Physics of Consciousness. Springer. pp. 193–253
- ^ http://www.youtube.com/watch?v=LXFFbxoHp3s
- ^ [1]
- ^ Harris, Sam. "Letter to A Christian Nation". SamHarrisOrg. http://www.samharris.org/site/book_letter_to_christian_nation/. Retrieved 5 June 2010. Quoting Penrose's blurb for Harris's book Letter to a Christian Nation.
- ^ "Big Bang follows Big Bang follows Big Bang". BBC News. 25 September 2010. http://news.bbc.co.uk/today/hi/today/newsid_9032000/9032626.stm. Retrieved 1 Dec 2010..
- ^ See A Brief History of Time, quote starts at about 1:12:43 in the video.
- ^ Official announcement knighthood. The London Gazette 11 June 1994.
- ^ "Honorary Graduates 1989 to present". bath.ac.uk. University of Bath. http://www.bath.ac.uk/ceremonies/hongrads/. Retrieved 18 February 2012.
Further reading
- 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.; see Box 34.2.
External links
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Wikiquote has a collection of quotations related to: Roger Penrose |
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Wikimedia Commons has media related to: Roger Penrose |
- Dangerous Knowledge – 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, Nov 11, 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
- Roger Penrose on The Forum
- Penrose on sidestepping reason
- O'Connor, John J.; Robertson, Edmund F., "Roger Penrose", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Penrose.html.
- Shaking Up Foundations of Math: Roger Penrose on Kurt Gödel's Groundbreaking Work
- Hilary Putnam's review of Penrose's 'Shadows of the Mind' claiming that Penrose's use of Godel's Incompleteness Theorem is fallacious
- 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. vol 61. pp. 4194–4206.
- "Biological feasibility of quantum states in the brain" – (a disputation of Tegmark's result by Hagan, Hameroff, and Tuszyński)
- "Toilet Paper Plagiarism" – 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
- Quantum-Mind
- Audio: Roger Penrose in conversation on the BBC World Service discussion show
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