European Mathematical Society - 70-01
https://euro-math-soc.eu/msc-full/70-01
enSleight of Mind
https://euro-math-soc.eu/review/sleight-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>Matt Cook is an economist, a composer, a storyteller (as the author of thrillers), and he performs as a magician. Several magical tricks rely on creating an intuitive expectation and then come up with a totally different result. This creates amazement and unbelief in the audience. This is also very much the effect of a paradox. Given that Cook is not a professional mathematician himself, it comes as a surprise to find rather much abstraction and mathematics in this book.</p>
<p>Logical paradoxes are often found in popular science books discussing mathematics, games, and puzzles. Many of these "popular" paradoxes you can also find in this book but there are many more. Although this book is written for a general public, it is not leisure reading, since the discussion of the paradoxes goes in depth and that requires precise definitions and sometimes it touches upon the foundations of logic, mathematics, probability, or whatever topic the paradox is about.</p>
<p>The different topics are arranged in different chapters and the format is always similar. There is a general introduction to the subject, and that involves the definition of the concepts that are required for the discussion of the paradoxes to follow. These are precise but the selected terms and the associated technicalities are restricted to a minimum. Only what is essential is defined and only as precise as needed. For example in the chapter on probability it is defined what a probability space is, and that involves a sample space, a sigma-algebra, and a probability function, which are described by words, rather than formulas. Of course it is also explained how a random variable and its density function are defined and the Bayes theorem is introduced (this inevitably results in a formula). So, there are some formulas, but they are suppressed as much as possible, describing the definitions mostly in words and by using examples. I guess this is intended not to shy away the non-mathematician, but if you are a mathematician, then, given the intended rigour, it feels a bit awkward and verbose. Of course some formulas cannot be avoided, for example to illustrate what is in the Principia Mathematica of Whitehead and Russell a formula here and there is unavoidable.</p>
<p>When Cook comes to the many examples of paradoxes, it assumes an attentive reader because the lack of formulas requires sometimes complicated sentences that are often almost philosophical. Also here, a returning format is used. First the paradox is formulated, wherever possible, mentioning its origin. Cook usually tells a story to make the paradox concrete for the reader, rather than formulating it in its mathematical or abstract form. Then the opposing explanations (often there are only two) are formulated. The main discussion then explains why one is wrong and the other is correct. Sometimes there are more possibilities and more than one explanation is possible depending on how some components are defined or interpreted, which happens when the problem is ill-posed or under-defined.</p>
<p>Let me give some examples that illustrate the types of paradoxes and the depth of the discussion. A first chapter is dealing with infinity, which is not the simplest one to start with, but it is also the underlying concept in some subsequent chapters. It is clearly a concept that has caused a lot of confusion throughout the history of mathematics and logic. First we are instructed about bijections and countable sets, Cantor's diagonalization process, the cardinals $\aleph_k$, and the continuity hypothesis. Then the paradoxes can be explained: Hilbert's Hotel, Stewart's HyperWebster Dictionary, and many more. After introducing some additional group theory also the Banach-Tarsky theorem is explained in some detail. Not really a proof, but still the reader is given some idea of why this seemingly impossible result holds. Zeno's paradoxes of motion are of course somewhat related to the concept infinity, and so these are discussed making use of what was obtained in the previous chapter. Thomson's lamp is also related. If a lamp is alternately switched on and off at time instances $1−2^{−n}$, then deciding whether at time $t=1$ the lamp will be on or off is impossible.</p>
<p>With chapter four, probability is introduced. The Simpson paradox and the Monty Hall problem are probably the best known but there are others that allow much more variations and require much more discussion. In the chapter on voting systems we are introduced to social choice theory and Arrow's impossibility theorem. This is not completely unrelated to the topic of game theory which plays a role in, for example, price setting in a economic system. The Braess paradox is the unexpected result that by adding an extra road to a traffic system, the traffic may be slowed down.</p>
<p>With self-reference we are back to the foundations of mathematics with axiomatic set theory, and, among others, the paradoxes of Russell (the set of al sets that are not a member of themselves) and the liar (I am always lying). Inevitably this leads to Gödel's incompleteness theorems, a theory of types, the ZFC axiomatic system, etc. Also the unexpected hanging is a tough paradox discussed here. Somewhat in the same style is the chapter on induction, where some elements of formal logic are introduced.</p>
<p>A chapter involving geometry has curves, areas, and volumes with fractal dimension. There is not really a paradox here, but the fact that a dimension can be a fraction and need not be integer is considered to be paradoxical. But there are other simpler geometric examples. In many calculus books, we find the hard-to-believe fact that we can create an infinitely large overhang by stacking bricks if brick $k$ (numbered from top to bottom) overhangs the underlying one by $1/(k+1)$. This is an example where the mathematical fact that $\sum_{k=}^\infty 1/k$ diverges is replaced by a "story" of stacking overhanging bricks. Some typical mathematical beginners errors can also give some unexpected results, dividing by zero for example, or summing divergent series.</p>
<p>Finally Matt Cook has invited some colleagues to discuss paradoxes from physics. With statistical mechanics, the reader learns about entropy, Maxwell's Demon, and other classics such as the Brownian Ratchet driven by Brownian motion, and the Feynman's sprinkler problem. The unexpected results of special relativity are well known, and quantum physics is still difficult to understand in all its consequences and different interpretations are still discussed today.</p>
<p>In the final chapter the age-old question whether mathematics is discovered or invented is tackled. As one might expect, the answer is not exclusive for one or the other.</p>
<p>Mind, the paradoxes that are mentioned in this survey, are only few and exemplary for the many examples that can be found in this book (there are over 75). I can imagine that for readers who are totally mathematically illiterate, some steps may be hard, if these use terminology or arguments that are taken for granted. Nevertheless also those are considered potential readers because there is a short addendum introducing some very elementary mathematical notation. Cook also added a rather extensive bibliography, but many of the references are papers where the paradoxes were originally formulated, or papers discussing the solution. Thus not really the popularizing kind of literature for further reading. The index though is well stuffed and useful, since there is sometimes cross referencing across the chapters.</p>
<p>I could spot a typo in the discussion of the Banach-Tarsky theorem. When discussing successions of irrational spherical rotations left, right, up, down, denoted as L,R,U,D, strings of these letters are formed to denote points on a sphere. Uniqueness requires eliminating the succession of opposite rotations (free group). Thus UD, DU, LR, or RL are not allowed in a string. However in the table page 25 appears the string DUL which is not allowed.</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>Several paradoxes are analysed in depth. Some are well known others are less familiar: Zeno's paradoxes, Monty Hall problem, Banach-Tarsky theorem, paradoxes related to voting systems, self reference, but also statistical mechanics, special relativity and quantum physics and many more pass the review. The finale is a discussion of the ultimate question: Is mathematics invented or discovered?</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/matt-cook" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Matt Cook</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/mit-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MIT 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">2020</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">9780262043465 (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">USD 34.95 (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">368</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/logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Logic and Foundations</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://mitpress.mit.edu/books/sleight-mind" title="Link to web page">https://mitpress.mit.edu/books/sleight-mind</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/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</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/01a15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A15</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/81p05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81P05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/63a10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">63A10</a></li><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/83-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83-01</a></li></ul></span>Wed, 01 Apr 2020 11:54:56 +0000Adhemar Bultheel50644 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/sleight-mind#commentsBreakfast with Einstein
https://euro-math-soc.eu/review/breakfast-einstein
<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>Orzel is a physicist who authored several popularizing books already. How to teach quantum mechanics to your dog is probably the best known because it was a bestseller. In this book, Orzel does something similar and explains all the quantum effects that we are constantly (and probably unknowingly) experience as one gets out of bed and have breakfast at sunrise.</p>
<p>For example the sun is a gigantic nuclear reactor essential for our existence. What is happening in the sun? Why it exists the way it does? And more generally how to explain the whole existence of the universe? These are the first topics to be explained. It is all a matter of particles being attracted, and repulsed as they are subject to several forces. This causes the collapsing of stars, evolving into a white dwarf, a neutron star, or a black hole depending on its total mass. It is all an interplay of gravity, electromagnetism, and strong and weak nuclear interaction. This introduces some model of atoms consisting of a kernel and electrons, forming a system that is kept in a stable state by all these forces. In fact the growing insight into the structure and the physics that take place at an atomic scale is used to lead us throughout the historical evolution sketched in this book.</p>
<p>The next trigger is the observation that heated material shows a red glow (like in the toaster). This is a reason to discuss light, wavelength and photons. It was Einstein's suggestion in his analysis of the photoelectric effect that light might be a particle. Recall that it was this, and not relativity theory, that eventually resulted in his Nobel Prize. So now that we got Einstein involved, we have a complete explanation of the title of the book. The alarm clock is triggering reflections on time keeping, which in modern times depends on the oscillations of a cesium atom. This requires a more detailed analysis of the atomic structure and the physics that happen there.</p>
<p>The Internet is the next trigger. This chapter is an exploration of glass fibers and deals again with the interaction of light and atoms. We are introduced to the physics of lasers. The smell of a cup of coffee is a reason to explain why and how we can smell something. This is related to how atoms bond and form molecules. The planetary model of an atom with kernel and electrons needs to be replaced by atoms that are sharing electrons. The Schrödinger equation characterizes the probability distribution of the position of the electron and we are confronted with the Heisenberg uncertainty principle. Next is the question why solid objects are solid. Why do they not collapse like stars do and cause a nuclear explosion? To answer this, one needs to look at the global behaviour of many atoms. And of course this can be applied to how we experience a loaf of bread as well as to how we can explain some astrophysical phenomena.</p>
<p>Computer chips, require semiconductors and diodes. These are explained starting from a single molecule with a discrete energy spectrum emitting particular wavelengths of light, but many molecules together emit a certain continuous frequency band. In this context Orzel explains a surprising fact about some parrot feathers that seem to be blue, while consisting of filaments of keratin which is actually grey and slightly translucent. It is all a matter of matching wavelengths. Ordinary magnets are even more mysterious to explain. It depends on the spin of the electrons and how they are paired in different states of excitation, taking into account Pauli's exclusion principle and how they interact when in bulk.</p>
<p>A smoke detector depends on light being reflected by tiny smoke particles. More advanced ones can detect particles that do not reflect light well and these use a small ionization chamber and detect disturbances. The latter depends on alpha particles generated during the decay of an artificial radioactive element. Radioactivity and quantum tunnelling are explained and these effects are also important for X-ray radiography and other medical applications. The final application explains entanglement, the EPR (Einstein-Podolsky-Rosen) paradox and sketches the related Bell theorem and the Aspect experiment. It explains how this can break down our whole encryption system when quantum computing becomes a common reality.</p>
<p>Orzel uses physics, not mathematics in his explanations of the quantum effects. These are mostly related to the structure of the atom and the interaction of electrons in molecules. All the atomic models are explained with the increasing complexity that grew as physicists got more and more insight. Historical context is given of all the physical models proposed and the related experiments but also about the application that is being considered like time keeping, the Internet, the use of the magnetic compass by sailors, etc. It is interesting to note that in this context some puzzling observations could only be explained by breaking with traditional views. The new model that was proposed depended partly on intuition and it was often only later confirmed by mathematical computations and experiments. Original ideas are what makes the quantum leap advances in science just like an original angle of approach from a different area can solve a long standing problem in mathematics. It is also noteworthy that in the course of history, several breakthroughs were proposed by scientists who spent some time away from their usual research environment. Newton is a well known example who proposed calculus when he returned from his stay in the countryside during the plague, and Orzel gives other examples.</p>
<p>Orzel often refers to some scientist "who did the mathematics" and could confirm what he or somebody else had proposed, or, that in a more complicated situation of more than just one atom "the mathematics become much more involved". Thus it is clear that Orzel suggests that there is mathematics underlying all these models, but he does not go anywhere into the mathematics itself. So it may be a bit disappointing that with Einstein in the title, there is not more mathematics. Even relativity theory is absent. It is not relevant at this level of detail anyway. Neither is Orzel visiting the zoo of subatomic particles that have been proposed more recently. He is not even hinting at the Theory of Everything, a theory where theoretical physicists are nowadays wandering is an almost purely mathematical maze. Some have criticized this modern evolution as pure mathematical speculation which is not even scientific because it is not verifiable by experiment any more.</p>
<p>Thus if you expect to find mathematics in this book you may be disappointed, but you will find a readable introduction of the quantum physics that are connected to atomic models and you will know that you walk on solid ground since it is all supported by mathematics as well as by experimental verification. There are some effective illustrations and exceptionally a formula, but most of the book is just text. Since there is a lot of material presented, even in a popular science book, (and perhaps especially in such a book), one might have expected a subject index which would facilitate to look up something from a previous chapter that you need to recall later. Orzel gives cross references in the text, but sometimes a reader will like to recall some concept, and then it is difficult to find it in previous chapters.</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 a popular introduction to quantum physics, somewhat following the historical development. It illustrates that quantum physics is not an exotic theory but that we constantly experience its effects in our everyday life.</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/chad-orzel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Chad Orzel</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/oneworld-bloomsburry-publishing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Oneworld / Bloomsburry Publishing</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">2019</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">9781786076403 (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">USD 26.99 (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">288</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="https://www.bloomsbury.com/au/breakfast-with-einstein-9781786076403/" title="Link to web page">https://www.bloomsbury.com/au/breakfast-with-einstein-9781786076403/</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/81-quantum-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81 Quantum theory</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/81-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-01</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/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</a></li><li class="vocabulary-links field-item odd"><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 even"><a href="/msc-full/78-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">78-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/82-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">82-01</a></li></ul></span>Mon, 25 Nov 2019 09:16:52 +0000Adhemar Bultheel49947 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/breakfast-einstein#commentsCelestial Calculations
https://euro-math-soc.eu/review/celestial-calculations
<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>There are several popular science books available about our solar system, astronomy, and even cosmology. Most of them are descriptive. Obviously, whatever we know about astronomy and our solar system depends upon observations, and these have clearly increased drastically since we have satellites and other probes that do the observations from space. That requires however a thorough knowledge of how celestial objects move with respect to each other, while eventually, all the observations are collected on Earth at some particular time and place. It is not difficult to describe with a formula an elliptic trajectory of a planet in a two-body system, but to know at what time it will show above the horizon for an observer on Earth at some particular place and date, requires a careful transformation between different coordinate systems. If you are an amateur astronomer, and you are interested in doing these computations for yourself, this is the book that will teach you how to do that. It is not high precision rocket science, but you will get reasonably accurate results using the computational methods described in this book. Not that it requires highly advanced mathematics. The subtitle "A gentle introduction to computational astronomy" is spot on. All you need is some analytic geometry and trigonometry. The rest is conversion of units and transformations between coordinate systems.</p>
<p>First there is unit conversion, for example AU (astronomical unit which is the average distance Earth-Sun) versus kilometres and miles, but a time is even more disturbing. Conversion of a fractional number of hours into hours, minutes and seconds (HMS) is relatively simple but 24 hours also correspond to a rotation of 360 degrees, so time can also be measured in degrees, arcminutes, and arcseconds, (DMS) and degrees can be expressed as radians. Moreover the time of the day depends on the longitude position (defining the local mean time (LMT)) of the observer and there is daylight saving time (DST) for some countries. The sidereal time refers to our position with respect to the stars while the Earth rotates, which is of course important in astronomical calculations. On a larger time scale there is a calender problem defining the year (Julian vs. Gregorian calendar).</p>
<p>Next problem is the choice of a coordinate system. To define a location on Earth we are used to spherical coordinates with the origin at the centre, and the z-axis though the North Pole, (in the current epoch defined as the direction towards Polaris) and choosing a main meridian (Greenwich). The Earth's equator lies in the ecliptic plane. This system is similar to the celestial sphere (with its own North Pole —which is close to but different from Polaris because of precession— and its own meridian). For trajectories, we know since Kepler that we need elliptic coordinates depending on the anomaly of the ellipse. A planet circles the Sun on an elliptic trajectory increasing speed as it approaches the Sun in its perihelion and it is slowest when it is farthest away in the aphelion. Then one has to realize that the orbital plane of the planet (or any other celestial object) need not be the same as our ecliptic plane The galactic coordinate system refers to the larger scale where the equator plane corresponds to the average plane of our galaxy (the Milky Way). Minor corrections are required for parallax (like observing the Moon from different directions on Earth) and precession (the rotation axis of the Earth circling the celestial North Pole). All this requires careful transformations between the different space-time coordinates.</p>
<p>Equipped with all these formulas and computer algorithms, one can finally start to put them to good use to predict the time and the position of a phenomenon we want to observe. It should be possible to find the position of the Sun, Moon, stars and planets at a certain day and time for a particular place on Earth. For example will Venus rise above my horizon today, and if it does where and when will I see it? In fact the next chapters discuss star rising and star settings, and for our solar system, there is a more descriptive part discussing the Sun, the Moon, and the planets and other objects in our solar system. Man made satellites are like all other celestial objects, but differ in the sense that they are closer and relatively small with respect to Earth.</p>
<p>So one can compute for example your own time for Sunrise and Sunset, solstices and equinoxes, and the angular diameter of the Sun. For the Moon there are similarly formulas to define instances of Moonrise and Moonset, to compute the phases of the Moon, its distance from the Earth, and moments of solar eclipses. Similar computations can be done for all the planets of our solar system. A distinction has to be made for the ones closer to the Sun (interior planets) and those beyond the Earth (exterior planets). Although the basic laws are the same, our satellites need a special discussion. Because they are closer to us, a higher precision is needed, it makes really a difference whether the origin is at the center of the Earth or at the observer's position, their orbit changes much faster, and they are subject to gravity and therefore are regularly repositioned.</p>
<p>All this illustrates that astronomical calculations are not at all trivial. Fortunately the author has made the Java, Python and Visual Basic code available via github at <a target="_blank" href="https://celestialcalculations.github.io/">celestialcalculations.github.io</a>. There is also a chapter with references to books, websites, almanacs and star catalogs with some explanation on how to use them. The extensive glossary with short descriptions of terms and the detailed index is very useful if one is lost in the terminology.</p>
<p>It is amazing to realize how complicate computations are for relatively simple observations like for example the daily Sunrise and Sunset. Fortunately the computer code makes this quite easy. One can only be in awe for the calendars and almanacs produced by ancient civilisations without any of our modern insights or equipment. Now this book brings this within the reach of anyone who can deal with simple computer programs easily downloadable and ready to be installed and executed. This book is quite an achievement bringing all this within the reach of a general public. Not only the clear explanation of the technical mathematical background with formulas and graphs of the coordinate systems, but also for the very informative descriptions, illustrated with pictures, of the astronomical objects and phenomena.</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>The book describes all the mathematics needed to compute positions at a certain time of the familiar celestial objects. In this sense, it is also a users manual for the computer code that is made available via github. On the other hand, it is also a description of the phenomena and objects in our solar system.</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/jackie-l-laurence" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jackie L. Laurence</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/mit-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MIT 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">2019</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">9780262536639 (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">&pound; 30.00 (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">392</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/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</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="https://mitpress.mit.edu/books/celestial-calculations" title="Link to web page">https://mitpress.mit.edu/books/celestial-calculations</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/70-mechanics-particles-and-systems" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70 Mechanics of particles and systems</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/70f15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70F15</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></ul></span>Sun, 14 Apr 2019 07:00:31 +0000Adhemar Bultheel49287 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/celestial-calculations#commentsDo Colors Exist?
https://euro-math-soc.eu/review/do-colors-exist
<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>
Many a mathematics or physics student will know the popular site <a href="http://www.askamathematician.com/" target="_blank">Ask a Mathematician / Ask a Physicist</a>. The main contributor of that blog is Seth Cottrell, "the physicist", who has however a mathematics degree in quantum information from NYU. In 2008 at the <em>Burning Man</em> festival (an annual experimental festival in the Nevada desert) he, together with a friend Spencer Greenberg "the mathematician", set up a little tent with a sign "Ask a Mathematician / Ask a Physicist", an experiment that was later repeated in public parks around New York City. The idea is that strangers can just ask any question about the physics of our universe, which the physicist and/or the mathematician try to answer as well as possible. Later (2009) this took the more convenient form of the previously mentioned blog on the Internet where "the physicist" is definitely more active than "the mathematician", or perhaps the physics questions are more popular. This book is a collection of some of the Q&A from that blog. Thus also here most of them are more physics-related than directly mathematics-related. It is however interesting to note that on the FAQ of the blog it is written:</p>
<blockquote><p>
<em>It cannot be overemphasized how important math is. If you’re bad at math, then doing more math is the only way to get better. If you can’t get past something (looking at you, fractions), then admit it to your teachers (or anyone else who can help), ask lots of questions, and then: math, math, math. Math.</em></p></blockquote>
<p>
Cottrell admits that he started mathematics studies because of his interest in the physics and he needed mathematics to understand the physics. This book is a selection of the more extensive blog entries (there are now hundreds Q&A in the searchable blog archive).</p>
<p>
The reader is warned by the author that some of the questions (and their answers) are controversial and may be subject to critique and neither "the mathematician" nor "the physicist" are infallible. The questions are however most interesting, and I can safely assume that most of you sooner or later in life have asked some of these and answering them is sometimes surprisingly nontrivial. Since inquirers are often students or certainly not specialists, the answer tries to balance between a proper (but deep and technical) answer and a superficial (with some hand-waving) reply that remains readable (at least to some extent) for the person who asked the question. As popularizing science texts usually are, the style is colloquial, entertaining, and even funny. A special warning is given when things become more technical. This more technical or more advanced material is placed at the end and gets a section-title "gravy".</p>
<p>
The book has four parts called "Big Things" (about cosmology and the universe), "Small Things" (about atoms, particles and quantum physics), "In-Between-Things" (mostly about classical physics), and "Not Things" (about mathematical topics). The title of the book "Do Colors Exist?" is for example a question discussed in the "In-Between" part. Although we can define color by wavelength and we can take pictures beyond the human visual boundaries, what our eyes register are basically only three components from which our brain makes up a color. Some other questions here discuss why wet stones look different from dry stones, but also carbon dating, entropy, energy, plasma, etc, The cosmological questions are related to the obligatory big bang, relativity theory, dark energy, and expansion of the universe, but also: 'What if the Earth were a cube?' and 'What if we drill a tunnel though the Earth and jump in it?'. The description of what we would experience just before the Earth were hit by another celestial object of a similar size is mind-bogglingly frightening. The "Small Things" section answers questions about true randomness, or whether an atom is besides a few particles mostly empty space, furthermore quantum decryption, anti-matter, particle-spin, etc.</p>
<p>
These is of course some mathematics involved already in answering some of the previous questions but the more "purely" mathematical section contains 11 questions, which form a curious collection. Some of them are classical topics in popularizing math books like why 0.999... = 1, and the problem of 1/0: stumble stones in undergrad mathematics. Others involve modern cryptography and the Enigma machine, transfinite numbers, the number pi, prime numbers, and chaos theory. Somewhat less obvious are a discussion of Fourier analysis, fractional derivatives, and a topological problem of knots in higher dimensions, and what the "Theory of Everything" (ToE) stands for.</p>
<p>
All in all, an entertaining collection with some interesting physics questions. A skilled mathematician, may not be thrilled by the mathematical subjects, but I can imagine that many people are pleased with the mathematics answers as much as they are by the physics explanations. The whole book has some nice illustrations (sometimes more intended to be fun or just to be `illustrating' than they are explaining). Of course the "Ask a Mathematician/Ask a Physicist" site is not the only one of its kind. There are many similar initiatives, which is a blessing of the World Wide Web, but entails also the danger of innocent students being spammed by fake and incorrect information. Science in general and mathematics in particular is certainly happy with people such as Cottrell who take such initiatives to their heart and serve the interested and the curious only driven by their enthusiasm, with little or no financial support.</p>
<p>
It is true that Cottrell is not really avoiding formulas, since there are quite a lot, perhaps more than what some people are prepared to swallow. On the other hand, if the readers had a phobia for formulas, they would probably not be asking the question. Most people will be more than satisfied with the answers provided. But be warned that to <em>really</em> understand the physics (or the mathematics), it will require a handbook to look up de details, although I must admit that for some explanations the answer will not directly be found there, and it will require to work up your way to a well founded answer starting from first principles. In that case Cottrell is your guide, pointing the way to follow.</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 is a collection of Q&A from the popular blog <em>Ask a Mathematician / Ask a Physicist</em>. The majority of the items discussed is physics-related but is has also a part that is more directly mathematics. Since questions are usually asked by non-specialists or students, the answers are as accurate as possible, but remain sometimes a bit on the surface to be understandable. The style of the answers is friendly, colloquial, sometimes funny, like popularizing texts usually are.</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/seth-cottrell" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Seth Cottrell</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/birkh%C3%A4user-basel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser basel</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">2018</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">978-3-319-64360-1 (pbk); 978-3-319-64361-8 (ebk)</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">42,39 € (pbk); 32,12 € (ebk)</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">291</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/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://www.springer.com/gp/book/9783319643601" title="Link to web page">https://www.springer.com/gp/book/9783319643601</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/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</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/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/70-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70-01</a></li></ul></span>Mon, 02 Jul 2018 08:50:24 +0000Adhemar Bultheel48568 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/do-colors-exist#commentsBlockbuster Science: The Real Science in Science Fiction
https://euro-math-soc.eu/review/blockbuster-science-real-science-science-fiction
<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>
Science and science fiction (SF) are not too far apart and the boundary can become fuzzy in some cases. Terms and ideas now generally accepted were used for the first time in fiction novels. The word "robot" is an example of such an SF product and Jules Verne's space gun is a well known prediction of Apollo 11 used to realize the moon landing realized much later. Where science reaches its current boundaries, fiction can extrapolate and take these results to the next level. Exploring these boundaries is the purpose of this book. There is virtually no limit to the fantasy of fiction writers, but somehow the adventures of their heroes and villains take place in worlds that are inspired by our familiar society and by the environment and technology that we know today. Hence the science in SF involves almost anything we know or struggle with today: from quantum mechanics to biology, to artificial intelligence, to cosmology, to relativity theory, and of course, some ethical and philosophical problems can arise when science is pushed a bit further. All this science can be discussed on a very technical level and that can involves deep mathematics and complicated physics. So it is a challenge to discuss advanced science topics and yet avoiding all the difficult technical details.</p>
<p>
Bernstein has written some science fiction himself and he is professionally involved in economics, statistics, and mathematics as a managing consultant. So he is well earthed to real science and has the fantasy to fictionalise it beyond its boundaries. Testimony of his scientific knowledge is the astonishing amount of well documented technical knowledge that he summarizes in this book. Most of this science is based on mathematics, but as he writes in the introduction:</p>
<blockquote><p>
Also, don't worry about the math that occurs here and there, because these references are very limited. Never fear math. It is the language of science. In fact, as with spoken languages, it is fraught with tongue twisters the scientists sometimes take too seriously.</p></blockquote>
<p>
Faithful to his promise, there is indeed not much mathematics explicitly present, and neither does he become technical about the physics. Nevertheless he starts with a discussion of the two pillars of twentieth century physics: quantum mechanics and relativity theory. These two chapters are characteristic in approach and format for the other chapters (there are 21 chapters, which is just half of 42; it might be a coincidence but 42 is, according to Douglas Adams, the answer to the ultimate question of life). Most of the chapters are short chunks, hashing up the complex themes in digestible bites. They place the real science results against the fictional extrapolations. For example spacetime has black holes which may be used for time travelling, or they may create wormholes which would allow just travelling from on point in spacetime to another without violating the speed limit of light. While these are theoretical constructs in real science, they are mostly presupposed trivialities in science fiction. Both chapters have so-called bonus material for example discussing the twin or the grandfather paradox of time travelling or using Einstein's formula to compute the amount of energy that is packed in a human body, or to explain what quantum suicide and quantum immortality means in a multiverse context. Other chapters have "parting comments" which are just take-along summaries of what has been discussed in that chapter. The book has also three "interludes" which are chapters that deviate from the standard format. These discuss only science, there is no fiction. They briefly introduce some basics: the first one is about atomic theory, the second about transhumanism, that is when the human body and brain are biologically and technically upgraded until the result can hardly be called human anymore, and the third is about mass, dark matter and dark energy.</p>
<p>
</p>
<p>
The broad spectrum of themes discussed include string theory and extra dimensions, the vastness of our universe, parallel worlds, energy resources, the origin of life, genetic modification and cyborgs, global warming and other catastrophes, colonisation of the galaxy, computers, robots and AI, extraterrestrial life, materials engineering, virtual reality, and possible ends of the universe as we know it. All of these have shown up in science fiction media (comics, stories, books, films) in some form and Bernstein gives several concrete examples. Everything he claims is well documented with references (to both the scientific and the SF literature). These references are collected chapter by chapter at the end of the book. There you can also find a useful glossary explaining many technical terms from "abiogenesis" and "absolute zero" to "wobble method" and "zombies" but also the useful index and extensive lists of SF literature, movies and songs. Extensive as the latter lists may be, it is obviously reflecting a selective choice made by the author, because the amount SF literature, movies, and TV-series is too vast to aspire any degree of completeness.</p>
<p>
So there is no explicit mathematics and there are no formula in the book, but many mathematicians work in applied areas of AI, computer science, theoretical physics, materials science, nanotechnology, and many other engineering applications. So I am sure there are enough geeks among them that love science fiction and thus will probably love this book. The author did an amazing lot of fact checking and he has ample illustrations from SF. His style is really crisp, up tempo, and to the point, but most of all I love the humour he uses. An example: where he explains the spaghettification effect when entering a black hole he writes:</p>
<blockquote><p>
Think of an event horizon as a fence with a big Keep Away sign hammered into it. Personally, I would do what it says. [...] If you do decide to ignore the warning and trespass on the event horizon, I hope you like pasta. This is not an adventure I would recommend. However, if you insist, the first thing you do is put on the latest spacesuit ad disembark from your starship. There is no reason to endanger the rest of the crew.</p></blockquote>
<p>
But don't be mistaken, the technical or philosophical material is serious, and he is not joking much when he discusses global warming, a hot topic these days, that unfortunately is no fiction. Even the geeks among the readers may learn a thing or two that is new to them, and it is also an interesting way to detect some new SF literature or movies you didn't know about. For potential authors who want to start writing their SF novel and they want it to be hard SF, they better check out all the facts that are provided here. If you can't get enough of this, I can refer to a similar book by Charles L. Adler <a href="/review/wizards-aliens-and-starships-physics-and-math-fantasy-and-science-fiction" target="_blank">Wizards, Aliens, and Starships</a> who also includes the fantasy literature. Also Paul Nahin wrote about the topic in <a href="/review/holy-sci-fi-where-science-fiction-and-religion-intersect" target="_blank">Holy Sci-Fi!</a> but he is more focussing on the religious aspects and less on the science.</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>
Bernstein explores in this book the boundaries between what is scientifically known today and what science may be capable of in the future and what will always be science fiction. Using many references from the scientific literature and the SF media (books and film) he succeeds in separating the science from the fiction. He has a scientific background and has written some SF himself, but he succeeds in discussing scientific topics avoiding the mathematical and technical material.</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/david-siegel-bernstein" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Siegel Bernstein</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/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus 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">2017</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">9781633883697 (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">USD 24.00 (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">304</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/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://www.penguinrandomhouse.com/books/557922/blockbuster-science-by-david-siegel-bernstein/9781633883697/" title="Link to web page">https://www.penguinrandomhouse.com/books/557922/blockbuster-science-by-david-siegel-bernstein/9781633883697/</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/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</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/00a69" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a69</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</a></li><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/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/83-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/85-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">85-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/92-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92-01</a></li></ul></span>Mon, 08 Jan 2018 20:29:58 +0000Adhemar Bultheel48154 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/blockbuster-science-real-science-science-fiction#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#commentsThe Art of Science
https://euro-math-soc.eu/review/art-science
<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>
If you were asked to think of a scientist, and next to imagine an artist, then probably two quite different persons will appear in your mind's eye. However, when you think of it more, it is clear that art and science are actually quite closely related. There is a mathematical theory of music, architects have to deal with the reality of physical laws, painters apply geometric properties while the laws of optics rule their observation, etc. In particular in the Renaissance the science of art and the art of science went hand in hand.</p>
<p>
Martin Kemp in his book <em>The Science of Art</em> (Yale University Press, 1990) emphasized how (optical) science has influenced art since Brunelleschi's (1377-1446) invention of perspective. This book wants to invert this idea and argues that science developed by using artistic `inventions'. The artist does not always paint what is real. For example parallel lines may not be parallel in a painting. The artist gives form to a symbol that represents an idea. This is in line with Ernst Cassirer's theory of symbolism. This suggests that here a conceptual vision of the world changed and modern science arose by moving from the concrete to the abstract. The visual is an essential element in the development of science. A picture is a (sometimes simplified) observable form of what can be imagined. Like Leonardo Da Vinci has an `pictorial style', Hilbert's `mathematical style' is his `general theory of forms' and Hermann Weyl's <em>Symmetry</em> (Princeton University Press, 1952) can be seen as an `art guide' to science. The Renaissance did not only change the way artists represented reality, but it also triggered new visions that were decisive for the way in which mathematics developed.</p>
<p>
The argumentation for this point of view is given in eight essays by different authors. Some contributions are more mathematical, others are more philosophical. The subtitle of this book <em>From Perspective Drawing to Quantum Randomness</em> reflects both the period and the span of subjects that are covered.<br />
This is the table of contents.</p>
<p>
<strong>I. Ways of Perspective</strong></p>
<ul>
<li>
1. From perspective drawing to the eight dimension (John Stillwell)</li>
<li>
2. Seeing reality in perspective: the art of optics and the science of painting (Nader El-Birzi)</li>
<li>
3. The role of perspective in the transformation of European culture (Dalibor Vesely)</li>
<li>
4. Visual differential geometry and Beltrami's hyperbolic plane (Tristan Needham)</li>
<li>
5. All done by mirrors: symmetries, quaternions, spinors, and Clifford algebras (Simon Altmann)</li>
</ul>
<p>
<strong>II. The Complex Route</strong></p>
<ul>
<li>
6. Artists and gambles on the way to quantum physics (Annarita Angelini & Rossella Lupacchini)</li>
<li>
7. <em>Radices sophisticae, racines imaginaires</em>: the origins of complex numbers in the late Renaissance (Veronica Gavagna)</li>
<li>
8. Random, complex, and quantum (Artur Ekert)</li>
</ul>
<p>
The first part concentrates on the impact of the discovery of perspective on the way mathematics has evolved. Before Brunelleschi, the natural perspective was a matter of optics. It is the way how light travels in `the real world', i.e., not on the canvas. But when the rules of perspective were discovered, the rules became internal to the painting. The painting got its own internal mathematical rules. That became a `science of painting' (El-Birzi). Although it also introduced a new approach to the science of optics and to science in general (Vesely). On a canvas the classical geometric rules do not always work. For example parallel lines meet in the vanishing point. This stimulated the discovery of projective geometry. The latter allowed for a geometric study of algebra and hence the analysis of what kind of algebra is possible in higher dimensions, which leads to complex numbers, quaternions, octonions,... (Stillwell). On the geometric trail, Lobachevsky and Bolyai gave an axiomatic definition of non-Euclidean geometry, but Eugenio Beltrami's role is under-estimated. Differential geometry illustrates his important contribution: hyperbolic geometry can be represented on a surface with negative curvature (Needham). In this context it is only a side remark, but only few people are aware that Beltrami was also the first to introduce the singular value decomposition. Although symmetry has always been an artistic element, there is yet a distinctive lack of symmetry in paintings. Portraits show the right cheek predominantly, motion is better appreciated if it is suggested to move from left to right, and there is a preference for an upward diagonal composition. All this left-right and upside-down symmetries also show up in mathematical objects like quaternions, spinors, etc. So there is some art in science as well (Altmann).</p>
<p>
</p>
<p>
Renaissance is not only the period where perspective became available, but it is also the birth period of complex numbers and probability. While the first part sketched the role of perspective, the second part focusses on the impact of complex numbers and probability on modern quantum physics. Cardano (1501-1576) was the originator of both probability (he was a notorious gambler) and the complex numbers. The latter showed up when applying formulas for the roots of a cubic. Sometimes `imaginary' roots resulted when the square roots had to be taken from negative numbers, however, when applying certain `sign rules' in the computations, arithmetic still worked (Gavagna). Randomness and complex numbers meet in quantum theory where a complex amplitude for probability of an event is needed, where the term making the difference with the classical theory reflects the quantum interference (Ekert). In their own contribution, the editors go though all the aspects of the second part, based on Cardano's work.</p>
<p>
There are many books that link art and mathematics, but this one is unique in that it brings a distinct message: the ideas that revolutionized painting in the Renaissance can be applied to the evolution of mathematics. Artists produce paintings and scientists develop new theories on a similar basis. What perspective was for painters, is what complex numbers and probability are for quantum physics. One should not read the book for the rules of perspective, for mathematics, and perhaps not even for the history of mathematics. Neither is it a treatise on aesthetics. Of course all this is present, and some of the contributions are rather mathematical and historical, but the main objective is the philosophical-epistomologocal analysis of the interplay, sometimes touching on the metaphysics. It's a philosophical book written for mathematicians. So, if you are a mathematician, be prepared to interpolate some time for reflection while reading the book.</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 is a collection of eight philosophical-mathematical essays that reflect on the influence that art had on the development of mathematics and the intimate relation between both disciplines since the Brunelleschi introduced perspective in painting, and Cardano initiated complex numbers and probability during the Renaissance.</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/rossella-lupacchini" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Rossella Lupacchini</a></li><li class="vocabulary-links field-item odd"><a href="/author/annarita-angelini" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Annarita Angelini</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</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/springer-verlag-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer verlag</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">2014</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">978-3-319-02110-2 (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">63,59 € (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">223</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/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History 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="http://www.springer.com/978-3-319-02110-2" title="Link to web page">http://www.springer.com/978-3-319-02110-2</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/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</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/20g20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">20G20</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/51p05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51P05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/58b32" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">58B32</a></li><li class="vocabulary-links field-item odd"><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 even"><a href="/msc-full/70-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70-03</a></li></ul></span>Mon, 20 Oct 2014 07:14:10 +0000Adhemar Bultheel45781 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/art-science#comments