European Mathematical Society - 85 Astronomy and astrophysics
https://euro-math-soc.eu/msc/85-astronomy-and-astrophysics
enCalculating the Cosmos. How Mathematics Unveils the Universe
https://euro-math-soc.eu/review/calculating-cosmos-how-mathematics-unveils-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>
Since ancient times humans have been fascinated with the sun, the moon, the other planets of our solar system and all the the stars out there. Meticulous observations were collected in almanacs and the celestial movements were listed. The <em>Almagest</em> is an astronomical and mathematical treatise written by the Greek Ptolemy in the 2nd century. It was originally called <em>Mathematical Syntaxis</em>, and later <em>He Megale Syntaxis</em>. These were lost, but fortunately, via an Arab translation the <em>al-majisti</em>, it came back to us as the <em>Almagest</em>. It illustrates that mathematics and astronomy were very close in those days, and they were going hand in hand in the days of Kepler and Galileo, and now that we can `see' so much further, mathematics is still a core instrument for astrophysicists and cosmologists. Mathematicians were forced to design models that explain the data, and the other way around, experimentalists are forced by predictions made by mathematical models to look for objects or phenomena that had not been observed before. There is nothing new under the sun and there is nothing new beyond it.</p>
<p>
Ian Stewart who has written so many books on popular mathematics, engages here in guiding us through the playground of the extraterrestrial universe and points us to where and how mathematics were instrumental in obtaining insights in the laws of physics that govern the whole cosmological pinball machine. The subtitle of this book <em>How mathematics unveils the universe</em> is of course correct, but if you would be curious of the mathematical formulas and details, you would be disappointed. There are practically no formulas! There are diagrams and illustrations though. The mathematics are present on nearly every page but they are not explicit.</p>
<p>
In a more or less chronological order, Stewart takes the reader through the development of several topics that are relevant in this context. Take for example gravity. That is directly connected with the switch from a geocentric to a heliocentric system in which gravitation is modelled by mathematical formulas such as the laws by Newton and Kepler. Later, to resolve some deficiencies that showed up in extreme circumstances, Einstein's relativistic approach brought the necessary corrections. In different geometries, a shortest path is no more a straight line, and bodies move differently. Stewart explains next how our solar system came about and he explores the different theories that may explain how we got our moon. Assumptions are made, checked with ever increasingly accurate simulations, which are verified against observations. What was originally considered a planetary system that ran like a clockwork turns out to be anything but consisting of fixed regular movements. As soon as three or more bodies are involved, chaotic behavior is the rule as Poincaré has shown. The geometric progression of distances of satellites from the core body predicted the existence of missing planets and of satellites around these planets. So one started looking for them and they were eventually found indeed. That's how Uranus and Neptune were detected. Pluto is a small trans-Neptunion object that only recently was degraded to the status of a dwarf-planet together with Ceres and Eris. Yet smaller objects are found in the astroide belt. For the origin of the rings of Saturn we only have several speculations available. The chaotic looking trajectory of the Rosetta space probe, launched in 2004, illustrates the slingshot manoeuvre, a very indirect trajectory to meet the comet 67P at a speed of 55,000 kph. It is obtained by using the gravity field of Mars and the Earth several times.</p>
<p>
Outside our solar system there is our galaxy consisting of a dazzling number of stars, and this is just one galaxy among so many others. Stars are mainly observable by a spectral analysis of the light they emit. Astrophysicists investigate the properties of all the different stars and how they are organized in galaxies. There are different forms of galaxies and different theories exist about how their shape evolves or just how they came about. Not all of them have the familiar spiral form. Again mathematical simulations are used to reject or back up some of the theories. A realistic discussion is given about how many habitable exoplanets one can reasonably expect, and how, if ever, one could possibly try to reach them. Science-fiction has many solutions like wormholes, traveling faster than light, etc. This may assume the existence of parallel universes and how they relate to black holes. Once more, with current technology. these can only be discussed in a mathematical context. The origin and the evolution of the universe as the Big Bang and inflation theory, is generally accepted, but it is not the only theoretical explanation. These require to assume the existence of dark matter and dark energy, whatever these may be. The possible futures of the universe is in the hands of topology and the geometry that can come out of the equations. The same holds for the different multiverse theories and the basic question about why we are so special. Do all these parameters have to be indeed exactly as they are, within very tight limits, to make life possible? Nothing is absolutely certain. Maverick scientists propose alternative theories outside the mainstream. For them it is difficult to get funding and hence these theories are perhaps too easily left behind. This is a viewpoint put forward by Penrose in his book <a href="http://www.euro-math-soc.eu/review/fashion-faith-and-fantasy-new-physics-universe">Fashion, Faith, and Fantasy in the New Physics of the Universe</a> (reading it requires however much more mathematics than this book).</p>
<p>
The nice thing is that Stewart not only gives the evolution of different historical trials, mistakes, additions, and generalizations, into what is now mainstream belief, he also discusses alternatives that are also possible, with a slight bias towards what he believes himself. One thing is for sure, after this guided tour on a dazzling large space-time scale, it becomes clear that we learned very much in the last hundred years, but all models designed so far needed sooner or later corrections, modifications, and generalizations so that many theories are still very shaky and unstable. There are still so many questions unanswered forcing us to be very humble and admit that we only understand just a tiny bit. But Stewart is a marvelous guide on this voyage to places where many have been before, showing us the myth and fallacies of the past and what is realistic enough to be generally accepted today. The text is remarkably up-to-date with the latest results up to 2015 and the first months of 2016. Even data from June 2016 are given 'at the moment of writing' while the book is in the bookshop only two months later. It illustrates how hot the topic is.</p>
<p>
The level is just right for a general public: just enough scientific details but not too technical that it would prevent smooth reading. No mathematics, but many pointers to it, and there are just enough funny phrases. Some examples of Stewart's sense of humour. Poincaré had to prove that he was wrong in his first opinion and Stewart remarks: <em>unlike politicians, mathematicians do that kind of thing</em>; after describing some cosmic event of galaxies floating on the surface of bubbles in a cosmic foam he has the laconic remark: <em>don't try this at home</em>; when explaining superposition in quantum theory and referring to the faith of Schödinger's cat: <em>this is not more mysterious than not knowing what's in your Christmas present from Auntie Vera until you unwrap it. Yes she always sends either socks or a scarf, but that doesn't mean her present is a superposition of the two</em>; and when discussing the precise values of some cosmic constants: <em>they could have been 2.7742, or 842,006,444.998, or 42 for different physics</em>. The latter is a reference to the Hitchhiker's Guide to the Galaxy, where 42 is the answer to the ultimate question of life, the universe and everything.</p>
<p>
I already mentioned there are many illustrations and a number of color plates. There are of course appropriate references to the literature for the interested reader. I also appreciated the few pages listing briefly the most common jargon that has been used and the carefully compiled index to find your way in this enormous pile of information. A marvelous breathtaking read this is. It brings you in one boost from the historical preliminary models to all the current up-to-date models and the beliefs and conjectures about our solar system and the universe.</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>
Ian Stewart tells us what is currently known about our solar system, our galaxy, and our universe, and what may possibly be beyond that. In thematic chapters, he sketches the historical models, and how these were modified and adapted as our knowledge improved. The periodic or otherwise regular observations were formulated as mathematical laws that governed the physical world, until deviating observations triggered mathematical modifications to the model. Or the other way around: mathematical models based on available observations predicted the presence of lacking data, perhaps a planet that had not yet been discovered or dark matter that has never been observed. As our instruments improved, we could see further in distance and in time, and a Big Bang was proposed, later modified by an inflation theory to fit the observations, and later the existence of dark matter and dark energy, not yet observed, which may imply that, even though the other data are matched almost perfectly, the model may still be wrong. Stewart tells this cosmic story pointing to all the mathematics and the calculations that are behind it, but avoids explaining the mathematics itself. There are almost no formulas in the book. Thus anybody can read about this adventurous voyage that brings you to places where so many have been before, no mathematics required.</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/ian-stewart" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ian stewart</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/profile-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">profile books</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">978-1781254318 (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">£20</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">352</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-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</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="https://profilebooks.com/calculating-the-cosmos.html" title="Link to web page">https://profilebooks.com/calculating-the-cosmos.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/97m50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97M50</a></li></ul></span>Fri, 23 Sep 2016 07:26:31 +0000Adhemar Bultheel47179 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/calculating-cosmos-how-mathematics-unveils-universe#commentsFashion, 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>
<p>
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#comments