European Mathematical Society - Roger Penrose
https://euro-math-soc.eu/author/roger-penrose
enFashion, Faith, and Fantasy in the New Physics of the Universe
https://euro-math-soc.eu/review/fashion-faith-and-fantasy-new-physics-universe
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In this book Roger Penrose gives his controversial opinion about some generally accepted theories in modern physics and cosmology. His ideas have been around for a while because the book is based on lectures that he gave in 2003 in Princeton. There were three main parts in those lectures which are reflected in the title of this book. <em>Fashion</em> is his critique to spin theory that he thinks to be relying on its reputation as a fashionable subject while there are some serious defects. <em>Faith</em> refers to the belief of physicists in the applicability of quantum theory at all scales, also at human scales and in cosmological applications where the rules of nuclear physics are not valid anymore. Finally he thinks that the cosmological vision of the Big Bang followed by cosmic inflation is basically wrong and is therefore to be categorized as <em>fantasy</em>. The book has a fourth part that in a sense covers recent evolution after his 2003 lectures in which he gives his own vision on the new physics. His old twistor theory can be integrated with string theory and it becomes fashionable again and his conformal cyclic cosmology (CCC), a periodic repetition of Big Bangs, may just be a fantasy like other proposals, but it is certainly rubbing up against general belief.</p>
<p>
This is not his first book on this topic. His book <a href="http://www.euro-math-soc.eu/review/emperors-new-mind"><em>The Emperor's New Mind</em></a> (1989, but recently reprinted by Oxford University Press in their <em>Landmark Science</em> series) and the sequels <em>Shadows of the Mind</em> (1994) and <em>The Road to Reality</em> (2004) were bestsellers. And so were several others, some of which he wrote jointly with Hawking. About his conformal cyclic cosmology he recently published a full length book <em>Cycles of Time</em> (2010). In the present book he obviously is riding all his pet horses once more and repeats several of his ideas, critiques, and beliefs from his previous books. The style and level of technicality is comparable to his other books too, but since this concerns an intense entanglement of physics and mathematical equations, the level of mathematics is somewhat higher. Therefore it has an extensive appendix on mathematics such as complex numbers, vector spaces, manifolds, complex geometry, and harmonic analysis.</p>
<p>
Whether it is the work of a scrupulous editor or Penrose's own doing I do not know, but the text is quite densely seeded with forward and backward references almost each time some technical concept is met. Thus besides the many references to papers and books (<em>The Road to Reality</em> is often cited) there are also many reading interrupts like ‘ ... alluded to in §2.6 — and also in §1.4 — and described in more detail below ’ or ‘ ... by the arguments of §§1.10 and 1.11, that at the end of §1.14, and...issues of §§A.2, A.4 and A.11... ’ and when it becomes more mathematical there are references to the appendix almost on every line. This looks like very helpful, and in a sense it is, but the book is not intended to be a textbook and then it is somewhat hindering a smooth story being told. It addresses a broader audience of somewhat informed readers who are not so deeply interested that they want to solve the actual equations, but they are not satisfied with hollow slogans not underpinned by some understandable technical justification either. Penrose gives as much mathematics to be credible but avoids the equations themselves, so that he has to do a bit of hand waving at some points too. The material is however very broad and complex, while Penrose needs enough of the physics and mathematics to enable him to formulate his criticism. His first part is about string theory, but to arrive there, one has to go through all the physics developed in the last 100 years. Penrose uses some stepping stones to build this up, but the approach is not systematic and for some topics the reader is submerged without warning and learning to swim is postponed to a later section or he is supposed to do it on his own. Thus the reader should either be well prepared or he has to reread the book a couple of times to really grasp all of the ideas properly. There are many illustrations in the text that make it easier to understand the ideas. Most of the time these are straightforward, but they also may represent ideas in higher dimensions that cannot be drawn or they are formal schemes (for which you need to learn the conventions and interpretation) or they are even just a tentative evocation of a concept (which may require some imaginations and fantasy from the reader).</p>
<p>
In the first part Penrose introduces general relativity theory, quantum theory, Kaluza-Klein theory (a unified field theory of gravitation and electromagnetism), Yang-Mills theory (a gauge theory trying to unify electromagnetic and weak forces and quantum chromodynamics, i.e., the theory of strong interactions), Feynman diagrams, and of course string theory and many other things along the way. An important argument for his critique is the excessive dimensions that spin theory requires beyond the four space-time dimensions of a Minkowski space. That dimension was 26 at some point most of them curled up at an unobservable Planck scale, but the 26 could later be reduced to 10 by using supersymmetry. The extra 6 spatial dimensions are often assumed to take the form of a Calabi-Yau manifold. The extra dimensions create extra functional freedom for which Penrose uses the Wheeler notation $\infty^{c\,\infty^d}$ where $c$ is the number of components of the field considered and $d$ is the dimension of the space. The $\infty$ refers to the fact that fields and functions are not a set of discrete points but they vary continuously. In this expression $d$ is the dominant parameter that defines the magnitude of this functional freedom. String theory claims that the Calabi-Yau manifold will not be perturbed because it would require too much energy, taking the Planck scale into account, but Penrose places this at a cosmological scale and then there would be enough energy for perturbations, which would result in very unstable nonlinear interactions on a macroscopic scale. String theory persists says Penrose because it is led by string theorists approving projects on string theory and guiding students in their own subject. Moreover it fails in giving a unique theory explaining physics. There has been a proliferation of string theories and then there is M-theory and the anti-de Sitter/conformal field theory (AdS/CFT) correspondence to which Penrose devotes the remainder of this first chapter. This is the source of many new ideas but it does not correspond to observable reality and moreover it gives Einstein's cosmological constant the wrong sign.</p>
<p>
The second chapter is devoted to quantum theory (QT). A description is given of Planck's work, of Schrödinger's equation and superposition. The quantum state evolves in a U (unitary) quantum world until there is an observation from the C (classical) world which causes a discontinuous collapse R (reduction) in the state. This mechanism gives quantum physicists a good computational framework, conforming the observations, strengthening a faith in the theory, but it does not explain what is really (i.e. in reality) going on. Quantum physicists consider the result of a quantum observation as real, but the wave function itself is not an observable reality. Penrose argues that there must be a reality associated with it. The reduction during measurement is a jump between two different worlds and superposition is difficult to explain. Many explanations beside the Copenhagen interpretation exist but the multiverse interpretation of Everett is according to Penrose absurd and so constitutes a proof by <em>reductio ad absurdum</em> that quantum theory can not be generally applicable. The problem is the linearity that is assumed in QT, but which is in real physics not very common. Gravitation itself is nonlinear and in this sense QT forms a clash with Einstein's general relativity theory. In QT a superposition of states should persist forever in stationary situations. Using Heisenberg's uncertainty principle Penrose shows that uncertainty in the energy would make the situation unstable and there would be a collapse after a finite time. This time can be long in quantum experiments performed today, but would be much shorter when applied to mass displacement between superposed states of human size objects and distances. Experiments are underway and Penrose hopes to see verification of this theory in the next decade.</p>
<p>
In the fantasy chapter cosmology is under fire. The cosmic microwave background (CMB) radiation and the cosmological redshift are seen as the confirmation of the Big Bang (BB) theory. Other experiments showed that the geometry of the universe is essentially flat which triggered the proposal of a cosmic inflation shortly after the BB. This gave rise to the standard model i.e. the cosmology of the FLRW model named after Friedmann, Lemaître, Robertson, and Walker. But what <em>is</em> actually the BB and what was before the BB? Since all equations fail and just give singularities, this is guesswork and may just be fantasy. A similar amount of imagination is needed to predict the long time future of our universe. Will the universe oscillate? If so how would the transition look like? Much of the answers depend on our understanding of black holes and other singularities and the 2nd law of thermodynamics (i.e. entropy). The latter tells us that entropy is constantly increasing with probability 1. Reversing time, this would mean that the BB would be a state of minimal entropy. However, the CMB indicates that the BB should be a state of an enormous entropy. Increase in entropy is traditionally illustrated by gas molecules spontaneously spreading uniformly over a box in which they are contained. However what happened in the early universe is the opposite: gravitational forces made the chaotic plasma clump into material stars, galaxies and black holes. Inflation was invented to explain the so called horizon problem (how can we see events so far apart that they cannot possibly have had some interaction in the past), the smoothness problem (how come matter is so equally distributed), and the flatness problem (why is our universe flat). Penrose has counter arguments against each of these. He has also arguments against the antropic principle (the parameters had to be exactly as they are so that entropy can evolve to make intelligent beings possible, which explains why we are here). And there are several other even more `fantastic' cosmologies which can be refuted.</p>
<p>
In the trailing part, Penrose recapitulates and emphasizes his own ideas some of which were mentioned before. After his lectures in 2003 at Princeton, Edward Witten, a notorious string theorist, surprised Penrose by proposing a marriage between string theory and Penrose's old twistor theory from 1967. The twistor space is like the Minkowski space, but where the metric of the latter has signature 3 positive (space) components and 1 negative (time) component, the former has 2 positive and 2 negative components. The theory is extensively explained. Penrose also returns to his previous critique on quantum theory, namely that the assumed linearity is just an approximation. Nonlinearity causes instabilities limiting the duration of a superposition of states. Reduction (R) should be replaced by objective reduction (OR) which takes place at `ordinary' time scales. For cosmology he describes his conformal cyclic cosmology theory where the history of the universe is a sequence of aeons that start with a BB but the inflationary phase is replaced by an ultimately exponential expanding phase of the previous aeon. The singularity of the BB can be transformed in a smooth transition by a conformal rescaling.</p>
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There are of course many more topics discussed than what I could mention in the review. I consider myself to belong to the targetted readership for this book but since I am not a theoretical physicist, I am not going to judge Penrose's views, and after I read this book, I doubt that even a theoretical physicist would have indisputable grounds to refute his viewpoints. Although new observations keep pouring in and different conclusions can be drawn from them, they can push fashion or faith into one direction or another, but it still requires a lot of creative fantasy to come up with reasonably possible explanations for the underlying physics of our universe that respect all the equations. And yet even these equations are just models that seem to approximate what is actually happening. Sooner or later they probably will be superseded by more general theories. Think of Copernican epicircles which were an approximation that may have been sufficient in the days of Copernicus, but that we now consider as hopelessly difficult explanations of what is actually happening. It requires good observations and people thinking outside the fashionable box, even fighting faith and religion, to come up with better ideas. If you are interested in a less mathematical (and more conventional) description of all these physics and cosmology, I could recommend Ian Stewart's recent book <a href="http://www.euro-math-soc.eu/review/calculating-cosmos-how-mathematics-unveils-universe">Calculating the Cosmos. How Mathematics Unveils the Universe</a>.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This book is inspired by the lectures that Penrose gave in 2003 in Princeton. He gives arguments for his criticism about trendy or generally accepted visions on physics and cosmology. Spin theory is fashionable but it suffers seriously from an excessive functional freedom caused by the extra dimensions that is required by the theory. The faith in quantum theory is not justified when one tries to apply the laws valid on a nuclear particle scale at a human or a cosmic scale. In the current standard model of cosmology, it is supposed that shortly after the Big Bang, a cosmic inflation took place. That is a possible explanation but there are several flaws so that Penrose classifies it as pure fantasy. He proposes a conformal cyclic cosmology instead which was more extensively discussed in his previous book <em>Cycles of Time</em> (2010). Like in his other books, there is a lot of technical (physics but mostly mathematical) material, but it is intended for a broad knowledgeable readership. Therefore an extensive appendix is added to introduce the necessary mathematics.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/roger-penrose" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Roger Penrose</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691119793 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">US$29.95 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">520</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://press.princeton.edu/titles/10664.html" title="Link to web page">http://press.princeton.edu/titles/10664.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/85-astronomy-and-astrophysics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">85 Astronomy and astrophysics</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/85-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">85-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/81-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-01</a></li></ul></span>Fri, 23 Sep 2016 07:19:58 +0000Adhemar Bultheel47178 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/fashion-faith-and-fantasy-new-physics-universe#commentsThe Emperor's New Mind
https://euro-math-soc.eu/review/emperors-new-mind
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In 2016 Oxford University Press started a <em>Landmark Science</em> series with cheap reprints of classic books, the 'must-reads', about topics that have shaped current science. Here I review two of the more mathematical/physical ones from the first release of seven books in April 2016: this book and <a href="/review/hyperspace"><em>Hyperspace</em></a> (M. Kaku).</p>
<p>
In the early days of artificial intelligence it was hoped that one day it would be possible that a computer would be built that could perfectly simulate a human mind. More than half a century later AI scientists are still trying to write software that can understand human language in rather restrictive written situations and even more restricive in voice control. The loose application of syntax and certainly the often implied semantics that humans use are still a serious obstacle for machines. Successes came only at the end of the previous century with very powerful machines and big data so that data mining and the interaction of computer science with other scientific domains and often with a strong mathematical component.</p>
<p>
With this book (the original is from 1989, but this reprint has the preface of the paperback edition of a decade later) Roger Penrose refutes the original ambitions of AI: Conscious thinking can not be modelled and programmed on a machine. To formulate precisely what he is claiming, it needs to be explained first what is understood by "a machine" (i.e., the software operating it) and how to decide who is right and who is wrong. In other words, what is truth, what is real, what is our world made of that we (our minds) experience? And this points to another thesis that Penrose wants to defend in this book: We still do not fully understand yet the link between microscopic quantum physics, the macroscopic physics that we experience, and the physics of the cosmos. Understanding the underlying physical truth and its mathematics is our only hope to understand the conscious mind. Also here has recently been some progress with the detection of the Higgs boson (2013), and observation of gravitational waves (2016). So there are still good reasons to read the book 25 years after its first publication. The original publication had raised some controversy and in the 1999 preface of this reprint, Penrose defends his viewpoints and refers to his follow-up book <em>Shadows of the Mind</em> (1994). Given the skeptic reactions, Penrose's viewpoint has never been very popular or generally accepted. If you read this book, you might also be interested in reading a more recent vision by another theoretical physicist M. Kaku who wrote <em>The Future of the Mind</em> (2014).</p>
<p>
To give meaning to all the components mentioned above, Penrose had to start this book with a prequel similar to the well known encyclopedic survey <em>A Short History of Nearly Everything</em> by B. Bryson (2003), although Penrose is somewhat more focused, and certainly more technical and with much more detail. Since he has to address scientists from several different fields, but also the sophisticated layman, Penrose has chosen to verbally explain most of the topics he wants to address, although there are still formulas (for which he almost apologizes). However reducing the number of formulas to a minimum does not mean that the reading is easy. A mathematical readership would perhaps have preferred a bit more formulas because once properly defined, a formula can exactly and compactly represent what is meant and replace many words.</p>
<p>
To begin with, Penrose introduces AI, and a discussion of Searle's ideas, who also is a protagonist of "strong AI". Next he introduces Turing Machines and Universal Turing Machines (the latter do not only accept data as input but also the instructions to process the data). He does so to a painfully detailed level of bitstrings and even gives examples of programs. I do not think it is necessary to go that deep to explain an algorithm and the Halting Problem that Turing did solve. Also the lambda calculus by Church as an alternative is described, although with a bit less details.</p>
<p>
The next topic is the Mandelbrot set, recursive sets, the real and complex number systems, and Cantor's infinite numbers, tilings of the plane, Platonism vs. intuitionism, complexity theory, etc. Of course also the Gödel theorems are discussed and the incompleteness of formal systems, touching on the problem of defining what is provable or what is truth. If something can not be proved, this does not mean that it is not true, and if something takes an infinite time to prove, we cannot even say that it is true or not until it is proved, which takes forever.</p>
<p>
In a second part Penrose gives an introduction to physics. He classifies theories as superb, useful, or tentative. To the superb category (basically the only ones he discusses) belong the classical views: Newtonian mechanics, Maxwell's theory, and Einstein's relativity theory, as opposed to the more recent quantum electrodynamics. The big-bang theory is useful, while supersymmetry, string theory, and GUT (Grand Unified Theory) are considered only tentative. Here he differs clearly from the views Kaku is defending in his <em>Hyperspace</em> book. So he introduces the billiard-ball Newtonian and Hamiltonian visions of the world, Maxwell's electromagnetics, and special and general relativity theory. For the latter he explains the Riemannian tensor as a sum of a Weyl (defining shape) and a Ricci (defining volume) tensor, but leaves out the technical details. Because of time-shifts experienced by an observer in motion, causality and determinism, hence computability, need revision, and the fuzzy boundary between matter and energy revives the question of what is real. The latter is even more problematic when quantum theory is introduced blurring the boundary between particle behavior and wave behavior. Here he goes to some detail with Hilbert spaces of orthogonal state vectors of probabilistic nature, Heisenberg's uncertainty principle, Schödinger's equation, and spins. Of course we meet Schrödinger's cat and other well known phenomena.</p>
<p>
To explain the 'arrow of time' (although the equations are symmetric in time, we experience it only in one direction), he refers to the second law of thermodynamics: entropy does not decrease. This means that in the beginning the entropy should have been extremely low. Where does that come from? An incentive to embark upon a chapter on cosmology. Penrose conjectured the <em>Weyl curvature hypothesis</em> already in 1979 It is an alternative for the cosmic inflation in the early stage of the cosmos. A vanishing Weyl tensor at the time of the big-bang that is constantly increasing should explain the homogeneity of mass and the increase of entropy. An explanation is hidden in gravitational radiation, which supposedly is time-asymmetric. A somewhat speculative idea.</p>
<p>
So after this long excursion, in his last two chapters Penrose returns to the original problem of modelling the human mind. He first gives a biophysical description of the brain and what is known about its centers and how it works. He does not give a precise definition of consciousness because it is seemingly impossible. For example, a brain seems to be able to register things, even when the person is 'unconscious' like during an operation. An indirect characterization is that consciousness is linked to for example common sense, judgement of truth, understanding and artistic appraisal while the opposite is automatic and algorithmic behaviour. What is there in the evolution that has made our brain the way it is and what advantage does it bring to the creatures able of conscious thinking? Then he applies what has been explained before and draws the conclusion that "...neither classical nor quantum mechanics [...] can ever explain the way in which we think" but "A plausible case can be made that there is a non-algorithmic ingredient to (conscious) thought processes" (p. 521). This idea is partly inspired by his experience as a mathematician and rests on Gödel's theorem. Mathematicians can know the truth of a proposition by 'insight' while the Gödel theorem will claim that there are propositions that can not be proved. For examples his nonperiodic tilings and quasi-crystals do exist and yet these are not algorithmic. He gives several other examples of scientists who, by a spark of inspiration came up with a superb result, while they were not `working' on the subject following algorithmic rules. A machine will never be able to achieve this. Thus classic computers in the sense of Turing machines will never be able to simulate this kind of consciousness. QED. It can only be hoped that a massive parallel quantum computer could ever simulate such complex interaction of atoms. His hope lies in grasping and understanding quantum radiation that will make such a monstrously complex objective possible. His statement has raised objection and is critisized by several of his colleagues.</p>
<p>
After the publication of the book, psychologist Stuart Hameroff suggested that there is some biological analog of quantum computing in the brain that involve microtubules within the neurons. This was taken up by Penrose and that ingredient formed the basis of a follow-up book <em>Shadows of the Mind</em> (1989), in which the current idea is further developed into the so called Orchestrated objective reduction (Orch-OR) theory. Still today arguments in favor and against the Penrose-Hameroff conjecture are published and the last word has not been said or written about this challenging and controversial hypothesis. All the more reasons to read or re-read this book. But besides the controversy, the larger part of the book is just an introduction to a Cathedral of Science brought to the educated layman in an extraordinary masterpiece brought by a brilliant mind. Even though the book has many pages already, you still get the feeling that Penrose is deliberately confining himself to the essence, which makes you hungry for more. This might explain that it won the Science Book Prize in 1990.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In this reprint of the well known book, Penrose defends his thesis that the human mind can not be simulated by a (Turing) machine because it does not operate algorithmically. To make this point clear, the larger part of the book is about Turing machines and AI and about the physical modeling of the "real world" from Newtonian mechanics over Einstein's relativity theory up to supersymmetry. Simulation of the mind will only be possible if we understand how the missing piece of gravity radiation can be consistently included in the standard model of physics.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/roger-penrose" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Roger Penrose</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780198784920 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£10.99 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">640</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://global.oup.com/academic/product/the-emperors-new-mind-9780198784920" title="Link to web page">http://global.oup.com/academic/product/the-emperors-new-mind-9780198784920</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a99</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/70-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a69" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a69</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/03d10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03D10</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/68q05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68Q05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/80-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">80-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/81-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/82-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">82-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/83-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/85-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">85-01</a></li></ul></span>Mon, 02 May 2016 10:47:25 +0000Adhemar Bultheel46913 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/emperors-new-mind#comments