Roger Penrose

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Roger Penrose

Roger Penrose at Festival della Scienza Oct 29 2011.jpg
Penrose in 2011
Born (1931-08-08) 8 August 1931 (age 90)
Known for
Joan Isabel Wedge
(m. 1959,divorced)

Vanessa Thomas
Scientific career
Fields Mathematical physics, tessellations
Thesis Tensor Methods in Algebraic Geometry  (1958)
Doctoral advisor John A. Todd
Other academic advisors W. V. D. Hodge
Doctoral students

Sir Roger Penrose OM FRS (born 8 August 1931) is a British mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. [1] 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. [2]


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, [3] 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". [4] [5] [lower-alpha 1]

Early life

Born in Colchester, Essex, Roger Penrose is a son of Margaret (Leathes) and psychiatrist and geneticist Lionel Penrose. [lower-alpha 2] His paternal grandparents were J. Doyle Penrose, an Irish-born artist, and The Hon. Elizabeth Josephine Peckover; and his maternal grandparents were physiologist John Beresford Leathes and his wife, Sonia Marie Natanson, [6] [7] a Jewish Russian who had left St. Petersburg in the late 1880s. [6] [8] His uncle was artist Roland Penrose, whose son with photographer Lee Miller is Antony Penrose. [9] [10] Penrose is the brother of physicist Oliver Penrose, of geneticist Shirley Hodgson, and of chess Grandmaster Jonathan Penrose. [11] [12]

Penrose spent World War II as a child in Canada where his father worked in London, Ontario. [13] Penrose attended University College School and University College London, where he graduated with a first class degree in mathematics. [11]


In 1955, whilst still a student, Penrose reintroduced the E. H. Moore generalised matrix inverse, also known as the Moore–Penrose inverse, [14] after it had been reinvented by Arne Bjerhammar in 1951. [15] 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. [16] 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. [17] [18] Escher's Waterfall, and Ascending and Descending were in turn inspired by Penrose. [19]

The Penrose triangle Penrose-dreieck.svg
The Penrose triangle

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. [20]

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. [21] 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) [11] in the words of Kip Thorne of Caltech, "Roger Penrose revolutionised the mathematical tools that we use to analyse the properties of spacetime". [22] [23] 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. [24] 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" [25] 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. [26] [27] [28] The Penrose–Hawking singularity theorems was inspired by Amal Kumar Raychaudhuri's Raychaudhuri equation.

Predicted view from outside the event horizon of a black hole lit by a thin accretion disc CNRSblackhole.jpg
Predicted view from outside the event horizon of a black hole lit by a thin accretion disc

It was in the local context of gravitational collapse that the contribution of Penrose was most decisive, starting with his 1969 cosmic censorship conjecture, [29] 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. [30] [31] [32]

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. [33] 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. [34] [35]

A Penrose tiling Penrose Tiling (Rhombi).svg
A Penrose tiling

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). [36] [37]

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 [38] (1938; an English translation Two Basic Types of Statistical Distribution) by Czech geographer, demographer and statistician Jaromír Korčák  [ cs ]. In 1984, such patterns were observed in the arrangement of atoms in quasicrystals. [39] Another noteworthy contribution is his 1971 invention of spin networks, which later came to form the geometry of spacetime in loop quantum gravity. [40] He was influential in popularizing what are commonly known as Penrose diagrams (causal diagrams). [41]

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. [42]

Later activity

In 2004, Penrose released The Road to Reality: A Complete Guide to the Laws of the Universe , a 1,099-page comprehensive guide to the Laws of Physics that includes an explanation of his own theory. The Penrose Interpretation predicts the relationship between quantum mechanics and general relativity, and proposes that a quantum state remains in superposition until the difference of space-time curvature attains a significant level. [43] [44]

Penrose is the Francis and Helen Pentz Distinguished Visiting Professor of Physics and Mathematics at Pennsylvania State University. [45]

An earlier universe

WMAP image of the (extremely tiny) anisotropies in the cosmic background radiation WMAP 2010.png
WMAP image of the (extremely tiny) anisotropies in the cosmic background radiation

In 2010, Penrose reported possible evidence, based on concentric circles found in Wilkinson Microwave Anisotropy Probe data of the cosmic microwave background sky, of an earlier universe existing before the Big Bang of our own present universe. [46] He mentions this evidence in the epilogue of his 2010 book Cycles of Time , [47] a book in which he presents his reasons, to do with Einstein's field equations, the Weyl curvature C, and the Weyl curvature hypothesis (WCH), that the transition at the Big Bang could have been smooth enough for a previous universe to survive it. [48] [49] He made several conjectures about C and the WCH, some of which were subsequently proved by others, and he also popularized his conformal cyclic cosmology (CCC) theory. [50] In this theory, Penrose postulates that at the end of the universe all matter is eventually contained within black holes which subsequently evaporate via Hawking radiation. At this point, everything contained within the universe consists of photons which "experience" neither time nor space. There is essentially no difference between an infinitely large universe consisting only of photons and an infinitely small universe consisting only of photons. Therefore, a singularity for a Big Bang and an infinitely expanded universe are equivalent. [51]

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. [30] 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. [52] 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. [53] [54] Alternatively, one can use the Einstein–Maxwell–Dirac equations. [55]

Physics and consciousness

Penrose at a conference Roger Penrose 9671.JPG
Penrose at a conference

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. [56] 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). [57] 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.) [58] [59]

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 computers today are unable to have intelligence because they are algorithmically deterministic systems. 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. [60] 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. [60] [61] These claims were originally espoused by the philosopher John Lucas of Merton College, Oxford. [62]

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. [63] 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 believed that humans are, in fact, machines, whose functioning, although complex, is fully explainable by current physics. Minsky maintained 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." [64]

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 those of anesthesiologist Stuart Hameroff. [65]

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, [66] 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'". [67] 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 Tuszyński and Hameroff [68] [69] 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. [68] They 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. [68] 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. [70]

Penrose in the University of Santiago de Compostela to receive the Fonseca Prize Penrose in University of Santiago de Compostela to picked up the Fonseca prize.jpg
Penrose in the University of Santiago de Compostela to receive the Fonseca Prize

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. [70]

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. [71] In addition to this, a 2011 paper by Roger Penrose and Stuart Hameroff published in the Journal of Cosmology gives an updated model of their Orch-OR theory, in light of criticisms, and discusses the place of consciousness within the universe. [72]

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. [67]

In January 2014, Hameroff and Penrose ventured that a discovery of quantum vibrations in microtubules by Anirban Bandyopadhyay of the National Institute for Materials Science in Japan [73] supports the hypothesis of Orch-OR theory. [74] 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 . [75]

Personal life

Family life

Penrose is married to Vanessa Thomas, director of Academic Development at Cokethorpe School and former head of mathematics at Abingdon School, [76] [77] with whom he has one son. [76] He has three sons from a previous marriage to American Joan Isabel Penrose (née Wedge), whom he married in 1959. [78] [79]

Religious views

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." [80] He regards himself as an agnostic. [81] 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." [82]

In an interview to The Jerusalem Post , Penrose stated that he could be considered a Jew according to religious laws but he doesn't identify himself as one. [83] Penrose is a patron of Humanists UK. [84]

Awards and honours

Penrose in a lecture RogerPenrose CapturingInfinity cropped.jpg
Penrose in a lecture

Penrose has been awarded many prizes for his contributions to science. In 2020, Penrose was awarded one half of the Nobel Prize in Physics for the discovery that black hole formation is a robust prediction of the general theory of relativity, a half-share also going to Reinhard Genzel and Andrea Ghez for the discovery of a supermassive compact object at the centre of our galaxy. [4]

In 2017, he was awarded the Commandino Medal at the Urbino University for his contributions to the history of science

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 2006, he also won the Dirac Medal given by the University of New South Wales. In 2008, Penrose was awarded the Copley Medal. He is also a Distinguished Supporter of Humanists UK 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). [85]

In 2004, he was awarded the De Morgan Medal for his wide and original contributions to mathematical physics. [86] 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. [87]

In 1971, he was awarded the Dannie Heineman Prize for Astrophysics. 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. [88] In the same year, he was also awarded an Honorary Degree (Doctor of Science) by the University of Bath, [89] and became a member of Polish Academy of Sciences. In 1998, he was elected Foreign Associate of the United States National Academy of Sciences. [90] In 2000, he was appointed to the Order of Merit. [91] He was elected to the American Philosophical Society in 2011. [92]



Academic books

Foreword to other books

See also


  1. The other half was awarded jointly to Reinhard Genzel and Andrea Ghez for their work on black holes.
  2. 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.

Related Research Articles

The weak and the strong cosmic censorship hypotheses are two mathematical conjectures about the structure of gravitational singularities arising in general relativity.

General relativity Theory of gravitation as curved spacetime

General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.

Gravitational singularity Spacetime location where density and gravitational field become infinite

A gravitational singularity, spacetime singularity or simply singularity is a location in spacetime where the density and gravitational field of a celestial body is predicted to become infinite by general relativity in a way that does not depend on the coordinate system. The quantities used to measure gravitational field strength are the scalar invariant curvatures of spacetime, which includes a measure of the density of matter. Since such quantities become infinite at the singularity point, the laws of normal spacetime break down.

<i>A Brief History of Time</i> 1988 book by Stephen Hawking

A Brief History of Time: From the Big Bang to Black Holes is a book on theoretical cosmology by English physicist Stephen Hawking. It was first published in 1988. Hawking wrote the book for readers who had no prior knowledge of physics and people who are interested in learning something new about interesting subjects.

Max Tegmark Swedish-American cosmologist

Max Erik Tegmark is a Swedish-American physicist, cosmologist and machine learning researcher. He is a professor at the Massachusetts Institute of Technology and the president of the Future of Life Institute. He is also a scientific director at the Foundational Questions Institute and a supporter of the effective altruism movement, and has received research grants from Elon Musk to investigate existential risk from advanced artificial intelligence.

Penrose–Hawking singularity theorems Key results in general relativity on gravitational singularities

The Penrose–Hawking singularity theorems are a set of results in general relativity that attempt to answer the question of when gravitation produces singularities. The Penrose singularity theorem is a theorem in semi-Riemannian geometry and its general relativistic interpretation predicts a gravitational singularity in black hole formation. The Hawking's singularity theorem is based on the Penrose's theorem and it is interpretated as a gravitational singularity in the Big Bang situation. Penrose was awarded the Nobel Prize in Physics in 2020 "for the discovery that black hole formation is a robust prediction of the general theory of relativity", which he shared with Reinhard Genzel and Andrea Ghez.

Stuart Hameroff

Stuart Hameroff is an American anesthesiologist and professor at the University of Arizona known for his studies of consciousness and his controversial contention that consciousness originates from quantum states in neural microtubules. He is the lead organizer of the Science of Consciousness conference.

<i>The Emperors New Mind</i> Book by SIr Roger Penrose

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Orchestrated objective reduction is a controversial hypothesis that postulates that consciousness originates at the quantum level inside neurons, rather than the conventional view that it is a product of connections between neurons. The mechanism is held to be a quantum process called objective reduction that is orchestrated by cellular structures called microtubules. It is proposed that the theory may answer the hard problem of consciousness and provide a mechanism for free will. The hypothesis was first put forward in the early 1990s by Nobel laureate for physics, Roger Penrose, and anaesthesiologist and psychologist Stuart Hameroff. The hypothesis combines approaches from molecular biology, neuroscience, pharmacology, philosophy, quantum information theory, and quantum gravity.

Dennis W. Sciama

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Amal Kumar Raychaudhuri

Amal Kumar Raychaudhuri was an Indian physicist, known for his research in general relativity and cosmology. His most significant contribution is the eponymous Raychaudhuri equation, which demonstrates that singularities arise inevitably in general relativity and is a key ingredient in the proofs of the Penrose–Hawking singularity theorems. Raychaudhuri was also revered as a teacher during his tenure at Presidency College, Kolkata.

Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories.

The Weyl curvature hypothesis, which arises in the application of Albert Einstein's general theory of relativity to physical cosmology, was introduced by the British mathematician and theoretical physicist Sir Roger Penrose in an article in 1979 in an attempt to provide explanations for two of the most fundamental issues in physics. On the one hand one would like to account for a universe which on its largest observational scales appears remarkably spatially homogeneous and isotropic in its physical properties, on the other hand there is the deep question on the origin of the second law of thermodynamics.

The Penrose interpretation is a speculation by Roger Penrose about the relationship between quantum mechanics and general relativity. Penrose proposes that a quantum state remains in superposition until the difference of space-time curvature attains a significant level.

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<i>Shadows of the Mind</i> Book by Roger Penrose

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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. 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 the theory of orchestrated objective reduction.

Models of consciousness are used to illustrate and aid in understanding and explaining distinctive aspects of consciousness. Sometimes the models are labeled theories of consciousness. Anil Seth defines such models as those that relate brain phenomena such as fast irregular electrical activity and widespread brain activation to properties of consciousness such as qualia. Seth allows for different types of models including mathematical, logical, verbal and conceptual models.


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Further reading