Book reviews
http://euro-math-soc.eu/book-reviews
Book reviews published on the European Mathematical Society websiteenThe Outer Limits of Reason
http://euro-math-soc.eu/review/outer-limits-reason
<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>
This is a paperback edition of a book that was previously published in 2013. It is generally believed that reason distinguishes humans from other life forms. <em>The Outer Limits of Reason</em> is exploring the boundaries of what can be known by proper logical reasoning without meeting a contradiction. It is only after 345 pages of exploring what is <em>reason</em>-ably possible and what is not that Yanofsky comes to the following definition: the set of processes or methodologies that do not lead to contradiction or falsehood. This means that reason defines what we will ever know and understand.</p>
<p>
Exploring the boundaries of our knowledge immediately reminds one of a more recent book by Marcus Du Sautoy <em>What We Cannot Know</em> (2016). The subjects discussed in both books are about the same and both explore the outer limits, or edges as Du Sautoy calls them, of what we can ever know and understand, but the books do reflect the background of their authors: Du Sautoy is a the mathematician and Yanofsky a professor of Computer and Information Science. Both discuss logic, the foundations of mathematics, quantum relativity, and cosmology. Du Sautoy pays some special attention to whether all this may point to a Creator or a God, while Yanofsky's discussions are more philosophical, and he uses an approach that is indeed more AI and computer science-like. The recurrent motto of his argument is that if an assumption implies a contradiction, then we have crossed a boundary.</p>
<p>
Paradoxes are the tools par excellence that lead to contradictions. So that is where this book starts with. You find all the classics like the liar who says he is lying and Russell's barber paradox where a barber claims to shave everyone in the village who is not shaving himself, and all the other paradoxes that result from self-referencing, such as the first uninteresting number that by definition negates its own existence. More contradictions can be concluded in connection with the ship of Theseus, the Zeno paradoxes, time travel, while the Monty Hall problem is just counter-intuitive. Yanofsky analyses why we arrive at these contradictions and how they can be solved. Most of them are thought experiments that cannot happen or exist in a real world.</p>
<p>
When infinity is involved, then again reasoning can easily lead to a contradiction just like a division by zero is the trick that allows to prove anything. To understand infinity, Yanofsky includes a discussion of countability of the natural and rational numbers and the infinity of the reals. This leads to the development of set theory and the Zermelo-Fraenkel system. To answer the question whether this says something about what exists outside or only inside our brain requires a discussion about the Platonist view of reality.</p>
<p>
Computational complexity is a subject that is somewhat closer to the heart of Yanofsky. Searching and sorting, graphs, and hard problems like the travelling salesman problem, and the Hamiltonian cycling problem, are examples to be classified in the P vs NP classification. In the higher regions of the complexity scale we find the non-computable problems we find computer algorithms analysing algorithms. For example using a Cantor-like diagonal process, Yanofsky proves that the halting problem is undecidable, unless an oracle can be consulted. But then we may wonder whether an algorithm will stop after a finite time when it is verifying whether another algorithm will halt. So there can be iterations of halt-computable problems leading to an hierarchy of uncomputable problems like there is an hierarchy of infinities. The philosophical question can then be raised whether the human brain can outperform a machine, or is the brain just a very complex machine. For the moment our brain can beat computers for certain tasks but the gap is closing.</p>
<p>
Next follows a discussion of traditional boundaries of what we do not know (yet). Chaotic systems and of course quantum mechanics with Heisenberg's uncertainty principle, Schrödinger's cat, Bell's theorem, and the quantum eraser experiment. These quantum theoretical aspects raise again philosophical questions about free will and all the different interpretations of quantum mechanics. More familiar and a bit less weird is relativity theory that mixes space and time and mass and energy, but still, there can be strange consequences like time travelling leading to the twins paradox and the grandfather paradox. We currently also hit a theoretical boundary since relativistic quantum mechanics can join quantum mechanics and special relativity theory but when we want to weave general relativity into this theoretical fabric to also include gravity, it leads to a contradiction. The attempts of scientists are focused on a more general theory of everything to include all the four fundamental interactions (electromagnetics, weak and strong forces and gravity). Current theoretical derivations resulted in several kinds of string theory.</p>
<p>
On a further philosophical level we may question the foundations of how science increases knowledge. For example the problem of induction: Can we be absolutely sure that something holds because we never observed the contrary so far? Instead of proving that a certain scientific method is correct, we can turn the argument around as Popper proposed and accept a theory as long as it is not empirically falsified. Thomas Kuhn's vision of how science progresses was controversial in the 1960s. He claims that science doesn't develop linearly. Competing incommensurable paradigms coexist for a while until progress happens with discontinuous paradigm shifts. Since mathematics seems to be a driving factor for progress, this leads to an investigation of Wigner's "Unreasonable Effectiveness of Mathematics". Yanofsky seems to support a Platonic view that this effectiveness comes from the fact that mathematics is derived from physics. Einstein's relativity and Emma Noether's conservation laws are based on symmetry and symmetry is a driving force in today's fundamental physics. Thus the structure is there for us to detect. But if that is true, then it shifts only the question to the next level: why is all this structure in the universe, and why is intelligent life possible to discover the mathematics in that universe? This starts an extensive discussion of the (weak) anthropic principle but (of course) without a definitive answer.</p>
<p>
Then Yanofsky comes to analyse mathematics itself. There are definitely limitations to what is possible in mathematics. There are the classical Greek problems of squaring the circle, trisecting an angle, and doubling a cube using only a straight edge and compass. Galois theory puts an end to the search for formulas for the zeros of higher order polynomials, and Robert Berger proved that the tiling problem is undecidable, and so are Diophantine equations, and the word problem for groups. On a fundamental level, Peano arithmetic was proved to be consistent in the Zermelo-Fraenkel system with the axiom of choice, (but is ZFC itself consistent?). Furthermore Tarski's theorem and the incompleteness theorems of Gödel are discussed in some detail. This also relates to Parikh's theorem from computer science and Löb's paradox of logic.</p>
<p>
This brief enumeration sketching the contents shows that in this book many topics are discussed on a broad scale. The approach is rather fundamental, but the most technical aspects are skipped, so that the arguments remain accessible for a broad audience, but it is not leisure reading. You need to stay focussed to follow all the arguments. Yanofsky is not just hopping on the surface, but penetrates the epidermic of the subject and actually proves things, especially those that are somewhat related to computer science. Also the mathematics part is not be easy for an outsider. Fortunately, he takes his time and is rather verbose, gently taking the reader along on a slow pace not avoiding repetitions if appropriate. Chapters have an introduction announcing what will be discussed in the next sections, and they end with notes and suggestions for further reading, and within the chapters there are ample pointers to specific notes and references that are gathered at the end of the book. And if you need to recall or reread something from previously chapters there is an extensive index. Yanokofsky keeps asking why at every stage and he places the many, often opposing, philosophical opinions next each other. In most cases, he leaves it to the reader to decide what side to choose.</p>
<p>
The year 2018 has been announced as the year of artificial intelligence because of the progress made in recent years. The announcement of the singularity where human intelligence will be surpassed by computers has raised serious concern. Several tycoons of science and computer industry started thinking more concretely about how to deal with this evolution. Since the hard cover version of this book was out of print, this paperback edition is most welcome these days. Although the contents dates back from 2013, it is still an important read for its deep, yet accessible approach to human and computer intelligence and a thorough discussion of its philosophical aspects. </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 discusses the limitations of mathematics and logic and where science in general meets its current and future boundaries. This is the fuel for many classical philosophical controversies and our vision on what science can and cannot explain. And indeed it raises the very question of the reason of the existence of life and of human intelligence in this universe that we observe and try to understand. </p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/noson-s-yanofsky" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Noson S. Yanofsky</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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">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">978-0-262-01935-4 (hbk); 978-0-262-52984-6 (pbk); 978-0-262-31676-7 (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">out of print (hbk); £14.95 (pbk); $13.95 (ebk) (net)</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">428</div></div></div><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/outer-limits-reason" title="Link to web page">https://mitpress.mit.edu/books/outer-limits-reason</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/mathematical-aspects-computer-science" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Aspects of Computer Science</a></li>
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A30</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/03a10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03A10</a></li>
<li class="field-item odd"><a href="/msc-full/81p99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81P99</a></li>
</ul>
</span>
Wed, 13 Dec 2017 07:11:57 +0000adhemar48096 at http://euro-math-soc.euThe Dialogues. Conversations about the Nature of the Universe
http://euro-math-soc.eu/review/dialogues-conversations-about-nature-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>
Clifford V. Johnson is a physics professor at the University of Southern California who is often asked as advisor for science programs on television. He often appears himself in the media, he is active on the web, and he gives popularising science lectures. You may for example look up his 2012 TED-Ed Lecture about <a href="https://ed.ted.com/lessons/string-theory-and-the-hidden-structures-of-the-universe-clifford-johnson" target="_blank">String theory and the hidden structures of the universe</a> or find out about his blog called <a href="http://asymptotia.com/" target="_blank">Asymptotia</a>.</p>
<p>
Many theoretical physicists and mathematicians have written popularising books on elementary particles, cosmology, and the theory of everything like <em>Big Bang: The Origin of the Universe</em> (Simon Singh). Some other examples have been reviewed here like <a href="/review/our-mathematical-universe-my-quest-ultimate-nature-reality" target="_blank"><em>Our Mathematical Universe</em></a> (Max Tegmark); <a href="/review/emperors-new-mind" target="_blank"><em>The Emperor's New Mind</em></a> (Roger Penrose); <a href="/review/calculating-cosmos-how-mathematics-unveils-universe" target="_blank"><em>Calculating the Cosmos</em></a> (Ian Stewart); <a href="/review/theories-everything-ideas-profile" target="_blank"><em>Theories of Everything</em></a> (Frank Close); <a href="/review/beautiful-question" target="_blank"><em>A Beautiful Question</em></a> (Frank Wilczek); <a href="/review/fashion-faith-and-fantasy-new-physics-universe" target="_blank"><em>Fashion, Faith, and Fantasy in the New Physics of the Universe</em></a> (Roger Penrose) to name just a few. You might consider this book to be another contribution in this style, except that the concept is totally different. In all the previously mentioned examples, the author acts as an expert in the field who is explaining the material in a more or less accessible way to the reader. Johnson may have been inspired by classical examples of Plato or Galileo who have instructed the reader by having some characters discussing the subject and report this in a book in the form of dialogues. Also Paul Ribenbaum's <a href="/review/prime-numbers-friends-who-give-problems-trialogue-papa-paulo" target="_blank">Prime Numbers, Friends Who Give Problems</a> contains a "trialogue" on number theory following the same idea. But Johnson's concept is different yet. There is not an all-knowing person who is directing the conversation. In this case, it are conversations between ordinary people (some of them are scientists, but I believe they can also be considered to be ordinary people) who have a conversation which comes at some point down to particle physics or cosmology, but they are also discussing many other things as well. One thing leads to another like conversations usually run in practice. It is just like the reader happens to be in the same place which may be a costume party, a restaurant, a train, a museum, a coffee shop, or a sunny terrace, and he or she accidentally overhears what the two are discussing about. Thus there is no "instructor" who is "teaching" the reader but the discussion is between people looking for an answer themselves. The most surprising, and certainly the most unusual, is that everything is presented to the reader as a graphic novel. Sydney Padua did something similar in his (bio)graphic novel <a href="/review/thrilling-adventures-lovelace-and-babbage">The Thrilling Adventures of Lovelace and Babbage</a>. However in the latter, the pictures are in black and white and the characters are true caricatures. The drawing style here is much more realistic and in colour. The background attributes and the decors are remarkably detailed and realistic and we are observing the conversation from all possible physical perspectives literally like a fly on the wall. Some samples of the pictures used in the book can be seen on the <a href="https://thedialoguesbook.com/" target="_blank">book's website</a> where also the link to a YouTube video with samples can be found.</p>
<p>
So what is the point of producing such a graphic novel if the reader is not properly instructed about anything? In my opinion, the true message can be found in the one or two pages of notes that follow each of the eleven conversations. If you are not into the subject that you were eavesdropping, you probably have heard words, concepts, theories, etc. that you did not understand, or you might just be curious about what exactly the two were discussing. Then you should look that up on the web, or if you want to do it properly, you should consult some literature. This is what these notes are providing: they are pointing to the proper books to consult. You find something like "On page x, in panels y-z subject S is discussed" and then the note is telling you where you can read more about whatever came up there, or there is some note about a formula (yes there are formulas! – for example the Maxwell equations and their modification to include relativity theory). Several of the books mentioned above are referred to. In fact Wilczek who authored one of them, also wrote the preface for this book.</p>
<p>
As a consequence, the reader has to do some homework if he/she wants to learn something here. The material is not simple and it is not much easier just because it is in the form of a graphic novel. All the difficult concepts and buzz words of the topic pop up sooner of later in the conversations. You overhear discussions about multiverses, the cosmological constant, gluons, the Higgs particle, string theory, D-branes, relativity theory, a dispute about mathematics being invented or discovered, or the existence of God, and there are black holes, quantum field theory, gauge theory, quantum gravity, gravitational waves and the LIGO, and so much more. Is there a happy ending? Perhaps not in the usual sense, but it gives a message of hope at least for the layperson who is still completely lost. The last conversation happens in a bus. The mother tells her daughter that string theorists have an extra brain that can think in all those extra dimensions, whereupon a third person joins in and claims that it is not true. Some twelve pages later she concludes her arguments saying that scientists do not need special brains different from ours. When she is asked whether she is a scientist, she answers "No, but I love learning about science like you". And she gives some good advise for the daughter: Thus if you want to become a scientist, you do not need special brains. It's all about being interested and hard work to develop the skills and tools to answer the questions your curiosity will come up with.</p>
<p>
This is a most unusual book for this subject and the way this is approached is most surprising. Not only the contents is heavy stuff, it is also physically heavy to read. Some 250 pages on thick glossy paper makes it a quite heavy book to hold. You probably do not want to read this in bed or take it on a train, unless you have a table in front of you to put it on. Many subjects are mentioned, but not all are explained in detail. The reader should definitely be prepared to do some extra reading to understand things better. Since most references concern other popularising books on the subject, it may require quite a lot of extra reading. But all this hard science is happening in conversations by young enthusiastic people in casual locations and it is all wrapped up in beautiful graphics showing marvellous realistic decors.</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 graphic novel, the reader is witnessing conversations between people discussing cosmology and particle physics and the possible theory of everything. Each of the eleven conversations are followed by notes pointing to other books where details about the topic that was discussed can be found.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/clifford-v-johnson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Clifford V. Johnson</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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">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">978-0-262-03723-5 (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">£ 24.95 (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">246</div></div></div><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/dialogues-0" title="Link to web page">https://mitpress.mit.edu/books/dialogues-0</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a69" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a69</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</a></li>
</ul>
</span>
Fri, 08 Dec 2017 07:02:19 +0000adhemar48089 at http://euro-math-soc.euHow to Count to Infinity
http://euro-math-soc.eu/review/how-count-infinity
<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 2017 Quercus launched a new series <em>Little Ways to Live a Big Life</em> which consists of small sized booklets of approximately 60 pages of the "how to" type. In 2017 five titles were made available: <em>How to Play the Piano, How to Draw Anything, How to Land a Plane</em> and in the more technical-scientific sphere: <em>How to Understand $E=mc^2$</em> and the current text.</p>
<p>
Marcus Du Sautoy starts with an introduction formulating the following problem. If you want to count to infinity by enumeration: 1,2,3,..., you will never be able to reach infinity, no matter how fast you will count. So is it possible to count to infinity? To start with the beginning: counting is one of the earliest human "mathematical" activities. However, a sum of infinitely many numbers can still be finite. Suppose you count the first ten numbers at a slow pace, but with every subsequent 10 numbers you count twice as fast, then he proves that you will reach infinity in a finite time. But that requires you to eventually count infinitely fast. Some primitive languages have words for one, two and three, but everything beyond is "many". However these people can still work out whether a set with more than three elements is bigger or smaller than another set. The method is pairing the elements one by one and the bigger set will have elements that cannot be paired with elements of the smaller set. This pairing idea is used in the metaphor of the Hilbert hotel to illustrate that there are as many rational numbers as natural numbers. Then Du Sautoy illustrates that people needed irrational numbers like for example the square root of 2 and pi. With Cantor's diagonal principle he can illustrate that there are more irrational numbers than rationals. And there we are: we reached infinity and even went beyond to a next level. Du Sautoy concludes: "The trick was not to start counting, '1,2,3,' and then to hope to reach infinity. Instead, a change of perspective allowed us to think of infinity in one go and, by doing so, to show that infinity is a many-headed beast. Amazingly it took just 48 pages for us to get to infinity. That's the power of mathematical thought. Using our finite equipment in our head we can transcend our finite surroundings and touch the infinite", a poetical ode to mathematics.</p>
<p>
If you want to know what mathematicians mean when they talk about infinity. Why is infinity plus one or even two times infinity not bigger than infinity? How to compare two sets that both have infinitely many elements? Is it then still possible that one of them is bigger than the other? If you are confronted with this kind of questions and you ignore the answers, then you have no more excuse. This little booklet has all the answers, and the great news is that you don't need to know any mathematics for that, and it takes not more than a jiffy to finish. So, what are you waiting for? </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 little booklet of only 50 pages, Du Sautoy explains to the layperson that counting the infinitely many natural numbers and the infinitely many rational numbers is the same. Using the metaphor of Hilbert's hotel and Cantor's diagonal principle he can show that there are definitely more irrationals than rationals. No mathematics required. The most mathematical part is when he shows that an infinite sum can still result in a finite value.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/marcus-du-sautoy" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Marcus Du Sautoy</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/quercus" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Quercus</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">978-1-78648-497-0 (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">£ 9.99 (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">64</div></div></div><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.quercusbooks.co.uk/books/detail.page?isbn=9781786484970" title="Link to web page">https://www.quercusbooks.co.uk/books/detail.page?isbn=9781786484970</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
Tue, 05 Dec 2017 09:12:34 +0000adhemar48082 at http://euro-math-soc.euHypatia. Mathematician, Philosopher, Myth
http://euro-math-soc.eu/review/hypatia-mathematician-philosopher-myth
<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>
Hypatia was a female mathematician and philosopher who lived in Alexandria from around 350 to 415 CE. Alexandria had become a centre of Hellenistic culture with its famous Great Library which suffered several fires and became iconic for the loss of knowledge collected over centuries. Egypt was at that time part of the Byzantine Empire, and in this period there coexisted Christian communities with competing interpretations of the concept of the Trinity while there were still other religious groups like Jews and Greek polytheists, who were indicated as Pagans. Religious intolerance by Christians and a struggle for power between the local Roman prefect Orestes and the bishop Cyril resulted in violent rioting during which Hypatia, being Pagan, was brutally murdered.</p>
<p>
Hypatia's father was a mathematician and philosopher too. It was however unusual for women to be educated beyond the basics and to become teachers themselves. Standing out as a female influential teacher in an otherwise exclusively male profession and being brutally murdered, made her a mythical figure. She became that fair, intelligent, strong, independent, woman senselessly killed by a male fanatic gang. This is an ideal entry point to be fictionalized in numerous novels, paintings, sculptures, stage plays, and films. Among the best known today are Charles Kingsley's 1853 novel <em>Hypatia, Or, New Foes with an Old Face</em>, her iconic portrait sketched by Gasparo which appeared in Hubbard's <em>Little journeys to the homes of the great teachers</em> (1908) and more recently the film <em>Agora</em> (2009) by Alejandro Amenábar.</p>
<p>
All these evocations of Hypatia were based on many historical secondary sources to which the authors added their own fantasy to improve the dramatic elements of a story well told. But what did happen in reality and who was the historical Hypatia really? Charlotte Rooth who is currently working towards a PhD in Egyptology took up the challenge to check all the sources and make well thought conclusions about the historical truth. Not an easy task because there are only five written texts left relating to Hypatia's life and none of them contemporary. The oldest is a collection of Christian records written 25 years after her death. It contains her biography written by Socrates Scholasticus (who might have known the living Hypatia) and some letters from Synesius (who was one of Hypatia's followers) addressed to her and some others that mention her. The <em>Suda Lexicon</em> is an encyclopedia written in the tenth century, but it reproduces <em>The Life of Isidorus</em> which was written some 50 years after Hypatia's death (which is not only about Isidorus but it describes many other lives too among which Hypatia's) and it also quotes from another account in the <em>Onomatologion</em>, an earlier encyclopedia from the sixth century (which had an entry on Hypatia too).</p>
<p>
So it was not an easy task, especially with all the juicy stories that had been spread in recent times. These stories and their inconsistencies and impossibilities is what Rooth goes through in the first chapter. In the second chapter she sketches the city of Alexandria and the political and religious climate of Hypatia's time. There were Pagan temples such as the Serapium (dedicated to Serapis, the Greek version of Osiris), which was later destroyed as commanded by the bishop Thelonius. It may be that in this Pagan temple accommodated part of the Great Library, so that after previous disastrous fires, this is sometimes considered to be the definitive end of the Library. On page 47, Rooth includes a plan of the temple, copied from a paper by McKenzie et al., but for some strange reason the wrong mirrored image is inserted.<br />
This bishop Thelonius was a friend of Synesius who was an admirer of Hypatia. This might explain why Hypatia was spared at the time the temple was destructed. However with Cyril, the successor of Thelonius, the tide turned against her. Cyril, representing the ecclesiastic power in Alexandria struggled for supremacy against the prefect Orestes. Most scholars, as does Rooth, claim that this conflict directly or indirectly caused the dramatic consequences for Hypatia.</p>
<p>
The date of Hypatia's birth can only be guessed by deduction, and nothing is known about her mother. Her father, Theon, was a known mathematician who studied Euclid's <em>Elements</em> and the <em>Almagest</em> a book of astronomical data by Ptolemy and he wrote comments about them. We do not know how, where, and how much education Hypatia got. Perhaps she has helped editing her father's texts. We know that later she wrote some comments of her own. Her teaching was Neoplatonist, but we do not know how and where she was teaching. Her father was member of the Mauseon, an established institution (we would probably call it a university) but it might be that she was teaching elsewhere, perhaps at the Separium, or at the Kol el Dikka, a religiously neutral educational institute, or she may have been a wandering teacher.</p>
<p>
Not so much is known with certainty about her mathematical legacy. There are the books with comments on the <em>Almagest</em> but it is not clear how much of it is original and how much is her contribution, and how much is her father's work. Some difference in style may point to her work. Other work that is attributed to her are comments on Diophantus. When the originals were copied, some comments were added by the copyist, so that it is was not always clear what the original was and what the comments. Moreover in later copies, the added comments may have been deleted again or replaced by others. It is also believed that she wrote a comment on Apollonius's <em>Conics</em> but no copy has survived. Thus there is a strong belief that some mathematical work is indeed by Hypatia, but irrevocable proof is actually missing. From a letter by Synesius we know he asked her how to construct a silver astrolabe which he later claims to have built himself with her helping. In another letter he asks her to construct an hydrometer but it is not sure that she knew how to construct it. Rooth concludes that "The modern day reputation held by Hypatia as a philosopher, mathematician, astronomer, and mechanical inventor, is disproportional to the amount of surviving evidence of her life's work."</p>
<p>
In another chapter, Hypatia's marriage and friendship is discussed. Again, there is no evidence that she was or was not married. Some of Synesius's letters to her were rather devotional, but that could be because in those days philosophy, mathematics, astronomy and education all had some religious flavour and teachers had some semi-divine status. Knowledge and abstraction was for the Neoplatonists a way to come closer to "the One". So Synesius may just have expressed his deep respect for Hypatia. Later historians and novelists have attributed Hypatia whatever status they thought was most suitable for their story. Of course the dramatic event that has boosted Hypatia's reputation is how she was killed. So many contradictory versions of that event exist, which are often tendentious because she became the victim of a religious conflict. She was obviously not Christian, but there is no evidence of an explicit religion, except the spiritualism of her Neoplatonist philosophy and her pursuit of intellect (she is considered to have been the head of the Neoplatonist school in Aexandria). Rooth suggests that the murder was indirectly a consequence of a conflict between Orestes and Cyril. The latter was fiercely against Jews and Pagans. Orestes, who wasn't always keen on following his suggestions, at some point had one of Cyril's men arrested. The <em>Parabolani</em>, a group of violent aggressive monks from a nearby monastery were summoned by Cyril and they assaulted Orestes. One of the monks was captured and tortured to death whereupon Cyril declared him a martyr. Hypatia must have taken the side of Orestes in this conflict, which made her an easier target than Orestes. So it is generally believed that Hypatia's murder was indirectly caused by Cyril who controlled the rioting <em>Parabolani</em>.</p>
<p>
The caution taken by Rooth to interpret all the contradictory sources of this "cold case" give you confidence in what she concludes, which unfortunately is that not much is certain. She may break down the mythical hype that has grown around Hypatia as a role model for feminism and a case against religious fanaticism, but it doesn't diminish the fact that Hypatia has played an important role for philosophy, mathematics, and astronomy. The whole text is very well documented and all the prolific versions are scrutinised carefully. Yet the story she tells remains lively and is never dull or boring. It may well be that some of her colleagues do not agree with her conclusions, or they may have preferred to have the many citations in the original language (i.e. not translated in English) but I think it remains an interesting read for everybody. </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 book Rooth is analysing the historical sources which are often contradictory and which have feeded the hype created in the many fictional stories about Hypatia, a female Greek mathematician and philosopher in Alexandria around 400 CE. Rooth comes to the conclusion that Hypatia's modern reputation is not in correspondence with what is known with certainty about the historical Hypatia.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/charlotte-rooth" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Charlotte Rooth</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/fonthill-media" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Fonthill Media</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">978-1-781-55546-0 (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">£ 18.00 (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">192</div></div></div><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://fonthillmedia.com/Hypatia" title="Link to web page">http://fonthillmedia.com/Hypatia</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/01a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A20</a></li>
</ul>
</span>
Tue, 05 Dec 2017 08:51:05 +0000adhemar48081 at http://euro-math-soc.eu The Beauty of Numbers in Nature
http://euro-math-soc.eu/review/beauty-numbers-nature
<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>
The copyright notice says 2001, 2017, from which I conclude that the book was published before, but there is no information about what this previous publication might have been. I do not have a copy, but from the reviews of Stewart's book <em>What Shape Has a Snowflake?</em> from 2001, it looks like that was the previous version of the present one. By checking the references and recent developments of some of the topics discussed, I also assume that this is still the text from 2001, that has not been updated.</p>
<p>
In his introduction and in the first chapter of the book Stewart starts by wondering about the shape of a snowflake (this is also a pointer to the previous title of the book). Then follow some 200 richly illustrated pages, discussing all kinds of patterns that can be found in nature, until chapter 16 (the last one, entitled <em>The Answer</em>) which includes a discussion of what we know so far about the shape of a snowflake. Since the snowflake is basically only at the start and at the finish, the new current title is in fact a better description of the bulk of the contents, and the subtitle <em>Mathematical patterns and principals from the natural world</em> is the most accurate. Nature makes ample use of patterns and humans are fascinated by them. Patterns are the toys and the feedstock of mathematicians. Finding out what the rules and the (physical) laws are that generate the patterns is what mathematicians, and scientists in general, are looking for. Since most patterns can be visualised, this is a rewarding subject to illustrate mathematical principles, while avoiding all formulas. All what is required is to detect a pattern and then ask the question "why?". This is exactly what Stewart does in this book.</p>
<p>
There are three parts. Part 1 is an overview of what patterns we can look for, part 2 relates this to some mathematics, and finally in part 3, some more advanced issues like fractals, chaos, and cosmology, are tackled, a journey that ends back on earth with a discussion of the shape of the snowflake, since indeed all the rules that govern all these complex phenomenons can all be found in the many different appearances of a tiny snowflake.</p>
<p>
The first problem is to find an answer to the question: what exactly is a pattern?. In the first part, Stewart presents a sample book of what a pattern could be. Patterns are everywhere. We are surrounding by patterns. Think of the stripes of the zebra, the spots of a leopard, the ripples in the sand on the beach, the hexagonal honeycomb, a five-pointed starfish, a fern, the spiral of the nautilus shell, the arrangement of the seeds of a sunflower, the trajectories of the planets,... The list is almost endless. There is obviously some mathematics that can catch the regularity of a pattern like for example Fibonacci, triangular, or square number sequences, or laws for the planetary trajectories that Kepler detected, or the number of symmetries we can detect in an object, or some self-similarity rule like in the fern. All these just describe the phenomenon, but they do not explain why the pattern occurs. They are the handles with which we get a hold on the pattern and by which we can manipulate it, trying to find an answer to the why-question.</p>
<p>
Finding an answer to the latter requires more mathematics. These elements are introduced in part two. the pattern can be one dimensional like the gait of an animal, which also has a cyclic component. So we also need a description of periodic movement like in a pendulum. Two dimensional patterns are for example the rotational symmetry of a flower or the bilateral symmetry of the human body. Stewart gives many inspiring examples of symmetry, often corresponding to the symmetry in physical laws. However the mathematical perfection is only rarely followed by nature. The two halves of the human brain look symmetric, but they perform asymmetric tasks, the bowels of the body are not symmetric, the DNA helix is only right-handed, etc.<br />
This raises another question: if the physical laws are symmetric, and one starts from a symmetric situation, why is symmetry lost so often? If life starts from a spherically symmetric egg, why does it evolve into something that is definitely not sphere-like? If our universe started from an homogeneous point, why is it now clustered in planets, stars, and galaxies? All these components are maybe similar, but certainly not identical. How come?<br />
More patterns are detected in tilings. They form the basis of crystal structures (an occasion to mention the sphere packing problem and the hypothesis formulated by Kepler in his treatise on the six-pointed snowflake, and proved by Hales in 1998). We are used to the symmetry and regularity in patterns so much that when in 1970 Roger Penrose found a non-periodic tiling of the plane it came as a surprise, and when in 1982 Shechtman and coworkers detected quasi crystals they were not taken seriously: nature would never be able to do this. However his finding was confirmed and in 2011 he finally got the Nobel Prize in chemistry for his discovery. (The Nobel Prize is not mentioned in the text, which might be an illustration that the text was not updated in 2017. On the other hand, by 2000 Shechtman had received many other Prizes already which were not mentioned either. A side remark: Stewart got Shechtman's name misspelled with an 'Sch'.)<br />
Stripes can be explained as wave phenomena, and when two wave fronts interact, they may form spots. The mathematician Turing designed a complex mathematical theory of morphogenesis, but the modern view is that the patterns must develop according to an interaction between genetic switching instructions and chemical dynamics.<br />
In three dimensions the sphere is obviously the most symmetrical object satisfying some physical optimality condition. Soap bubbles can join in much more complex structures, some of which are definitely mathematical challenges. It is remarkable that soap bubbles (minimal surfaces) meet at either 109 or 120 degrees and no other. The double bubble conjecture describes the structure of two soap bubbles meeting as a minimal surface. The conjecture was proved assuming that the joining surfaces and their interface were all parts of spheres. The theorem was only fully proved under general assumptions in 2001. (That is not included in the text either, although it is mentioned that they were closing in on a solution. This is another indication that the text has not been updated after 2001.) Besides the sphere there are of course crystal structures with only a finite number of symmetries like the popular Buckminster ball which is the basic building block in some fullerenes.<br />
This second part ends with a discussion of spirals, whirlpools, music, colour patterns on sea shells, and the gaits of men and animals and more generally the way that animals move.</p>
<p>
In the third part, we meet complexity and nonlinear dynamics that can result in chaotic behaviour. Bifurcations, catastrophes and symmetry breaking can and will occur under certain conditions. In Darwinian evolution theory, several bifurcations took place. Iterative systems can create whimsical shapes that can be analysed with fractal geomrtry, but even chaotic systems can sometimes be described by very simple mathematical rules, like for example the complex structure of the Mandelbrot set. Our planetary system, turbulence, weather forecasting, and population dynamics, are all examples of chaotic systems depending on the time scale they are discussed. At a cosmic scale one has to reconcile the symmetry of its origin and its current constellation and derive estimates about its possible shape and its ultimate future. After this cosmic excursion Stewart returns to the original question since even the symmetry in the shape of a snowflake is the result of chaotic dynamics and it has a fractal structure. He discusses this in a rather philosophical way. Snowflakes are born in a cloud with complex weather conditions and what shape will result depends on complex rules, too complicate to fully analyse what exactly will be the shape of the flake. Many, but not all of them, are six-pointed, and even when six-pointed, they can have many different shapes. We are only able to classify them in general groups and find some configuration of temperature and saturation to predict what type of shapes will be produced (needles, dendrites, plates,...). A carbon atoms in a living being or in a rock, are in principle the same atoms subject to the same physical laws. What are exactly the rules that define how and why the atoms evolve and stick together in so very different forms of matter? These rules are still far too complex to be fully understood by our limited set of brain cells.</p>
<p>
This is a marvellous book, not only because of its abundance of colour pictures, but also because of the knowledgeable text written in Stewart's pleasing style that is never pedantic, always informative and smooth. Text and illustrations are perfectly in balance. I hesitate to call it a coffee table book because of its soft cover and the normal size for a book printed in a two-column format. It is however printed on glossy paper which gives it a certain cachet lifting it above an average book.</p>
<p>
Tvy Press is the European publisher if this book. In the US and Canada, the book is published by MIT Press under the same title with a different cover and with isbn 978-0-26-253-428-4.</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 richly illustrated book in which Stewart explores patterns that appear in nature. Detecting the parameters describing the pattern, and trying to answer why these patterns occur is the task of mathematicians and scientists. In some fractal or chaotic systems the pattern is not visible but hidden in the simplicity of the rules that are able to generate very complex phenomena. As a guideline, Stewart selects the problem of explaining the shape of a snowflake. For the six pointed snowflake this problem was already considered by Kepler. It is only at the very end of the book that Stewart is able to give a partial solution to the problem.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="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="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/ivy-pressquarto" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ivy Press/Quarto</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">978-1-78240-471-2 (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">£ 14.99 (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">224</div></div></div><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.quartoknows.com/books/9781782404712/The-Beauty-of-Numbers-in-Nature.html" title="Link to web page">https://www.quartoknows.com/books/9781782404712/The-Beauty-of-Numbers-in-Nature.html</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
Tue, 05 Dec 2017 08:05:54 +0000adhemar48080 at http://euro-math-soc.euEssentials of Mathematical Thinking
http://euro-math-soc.eu/review/essentials-mathematical-thinking
<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>
With a title like "Essentials of Mathematical Thinking" one might expect a philosophical treatise, or possibly a research exposition about cognitive processes and math education. But at the top of the cover, you can see that it is announced as a "Textbook in Mathematics". Since that is what it is: a textbook in mathematics, but a rather unconventional one. Several writers of popular science or recreational mathematics have written books in which they collect mathematical topics that are accessible for a general public and that should illustrate that mathematics can be fun and that there are many practical applications in everyday life involving mathematics. The items discussed in these books can involve integers, prime numbers, geometry, probability, counting problems, logic and paradoxes, games, puzzles, etc. But they are mostly "recreational" or at most they can serve as a source of inspiration for math teachers to embellish their courses and candy-coat the theorems and proofs of the actual textbook.</p>
<p>
Here however, Steven Krantz uses all these entertaining subjects to use them as an actual textbook to teach mathematical awareness and some skills to students who have not the slightest ambition of using mathematics in their further career. For example if undergraduate students are required to broaden their curriculum with some math course. There is no point in imposing mathematical abstraction on them or to force them to memorize proofs of theorems they will never need in life. So the idea is to use all these entertaining subjects to develop their ability to use logic arguments, to solve problems, and to convince them that mathematics is indeed everywhere, but that it is nothing to be afraid of. They will not become better mathematicians in the narrow sense of the word, but at the end of the journey they should have acquired some skills one could call mathematical and they should be more open minded towards mathematics and mathematicians.</p>
<p>
Obviously the book should not have the usual definition-theorem-proof structure and, although formulas have not been abolished completely, there are fewer than in a classical textbook. There are some exercises, but the usual long lists of drilling exercises are absent. Some exercises are meant to drill, some can be more challenging, and chapters are concluded with an open-ended problem. It is like Krantz is telling his story in a stream of consciousness which results in a surprising meandering succession of ideas that will hold the attention of his public or his readers.</p>
<p>
In Chapter 2 the breadth of the field is explored covering many different problems. Some examples: the Monty Hall problem, the four colour problem, minimal surfaces, P vs. NP, Bertrand paradox, etc. This sounds impressive as a starter, but these are actually pretexts for introducing the reader to probability, logarithms,... and to modern tools such as proofs by computer, algebraic computer systems, etc.</p>
<p>
This is followed by seven relatively short chapters. Now problems are solved. Some are classic (When will be the first time after midnight that the hands of an analogue clock will coincide?) others are less classic (Will new years day fall more frequently on a Saturday than on a Sunday? How many trailing zeros will 100! have?...).<br />
To illustrate how ideas are linked, let us consider a section of Chapter 5 as an example. It starts by telling that Kepler derived his laws for the motion of the planets not by solving equations but by analysing observed data (Newton came later), which leads to the meaning of average and standard deviation, which in turn leads to big data and their analysis such as DNA used in forensics, social studies based on Street View and other big sets of data collected by companies such as Google. A remarkable arc that connects Kepler to Google.<br />
Furthermore many of the classics are passing by on the catwalk: the pigeonhole principle, conditional probability, Benford's law, lottery and roulette problems, Conway's Game of Life, Towers of Hanoi, Buffon's needle problem, Euler's characteristic, sphere packing, Platonic solids, voting systems, interpretation of medical tests, facial recognition, wavelets, prisoner's dilemma, Hilbert's hotel and others. Some of these are worked out and actually solved, others are only mentioned as illustration of what is possible, or what they have been used for.</p>
<p>
Up to this point, the text is easily accessible with minimal mathematical background. In the remaining chapters, somewhat more is needed. Chapter 10 is about cryptography (explaining the basics of RSA encryption), the next one gathers some diverse discrete problems (a.o. divergence of the harmonic series, surreal numbers, graphs and the bridges of Königsberg, scheduling problems), and finally a chapter with more advanced problems (Google's Pagerank, needle problem of Kakeya, non-Euclidean geometry, the area of a circle as the limit of the area regular polygons).</p>
<p>
Besides mathematical monographs, Steven Krantz has written books on how to write mathematics, and some books that may be considered as introductions to mathematics for a general public and he won several prizes for his writing. He wrote also one on mathematical education before: <em>How to Teach Mathematics</em> (3rd ed., AMS, 2015) which is about "how" one has to teach. This one is about "what" to teach to a particular type of students. Whether he has experience teaching the "essentials of mathematical thinking" using the material presented in this textbook, I do not know. It might not be a bad idea for students that are somehow obliged to take a math course but that have no the intention to take subsequent courses. I have no information about experiments with this type of course. It would certainly be interesting to know the results.<br />
The text is typeset in LaTeX with the quality of lecture notes. There are many illustrations, but pictures do not have the resolution of high professional quality, and some are not really necessary (a picture of 3 arbitrary dice is not really helpful in solving a probability problem). There are many line drawings too which are usually quite helpful, but by resizing them to fit properly on the page, sometimes circles are distorted and become ellipses or the text in the figure is stretched and resized out of proportion. </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 mathematical anecdotes and applications of mathematics that are presented in the form of a textbook. It is the intention that this can be used as a broadening course for (undergraduate) students who do not have an ambition to take further mathematics courses. The topics mainly deal with numbers, geometry, probability, and logic. Analysis is less represented. </p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/steven-g-krantz" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Steven G. Krantz</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/chapman-and-hallcrc-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Chapman and Hall/CRC 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">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">978-1-138-19770-1 (pbk); 978-1-138-04257-5 (hbk); 978-1-315-11682-2 (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">£ 44.99 (pbk); £ 115.00 (hbk); £ 40.49 (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">336</div></div></div><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.crcpress.com/Essentials-of-Mathematical-Thinking/Krantz/p/book/9781138197701" title="Link to web page">https://www.crcpress.com/Essentials-of-Mathematical-Thinking/Krantz/p/book/9781138197701</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li>
</ul>
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<span class="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
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<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
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</span>
Sat, 25 Nov 2017 07:02:54 +0000adhemar48043 at http://euro-math-soc.euThe Mathematics Lover’s Companion
http://euro-math-soc.eu/review/mathematics-lover%E2%80%99s-companion
<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>
This book consists of a collection of mathematical vignettes that illustrate the fun, the beauty, and the applicability of mathematics to neophytes. This is not the first, and certainly not the last book of this type that will be published. There is necessarily some overlap between these books, and yet every author has his or her specific preferences for selecting the topics, the level of the mathematics, and the style adopted in telling their story.</p>
<p>
The overall level of the book is really basic mathematics. For example the concept of a proof is illustrated in the introduction by proving that the sum of two odd numbers is even. Later the principle of a proof by contradiction (there are infinitely many primes) and a proof by induction (for the Fibonacci numbers it holds that $F_0+F_1+\cdots+F_n=F_{n+2}−1$) is illustrated. For the latter identity also an alternative combinatorial proof is given. That is by showing that the left-hand side and the right-hand side correspond to two different ways of counting the same number of events. Some chapters are slightly more advanced, but only after new concepts are introduced like the exponential or the logarithmic function, complex numbers,... but everything stays at a really modest level. The reader is supposed to work a bit to assimilate the material. Therefore in every chapter on one or two occasions it happens that after some example is explained, you, as a reader are asked to work out the next step or do an analogous example for yourself. Solutions are then provided at the end of the chapter. There are thus no long lists of "exercises" to be solved, and actually they do not feel like exercises, but more like an interesting puzzle you are challenged to solve. It remains entertainment all the way.</p>
<p>
The subjects discussed involve the usual suspects at this level of introductory mathematics: numbers, geometry, and probability. The 23 relatively short chapters on these respective subjects are brought together in three separate parts. The first part, simply called <em>Number</em>, is the largest. It fills about the first half of the book. The second half is approximately equally divided over the other two topics. Although the subjects are entertaining, the book is building up the material in steps like in a textbook, proving the results (at least the simple ones) and sequentially introducing new concepts as the reader progresses. To illustrate this concept, we shall give a quick (and incomplete) survey of the topics that are discussed.</p>
<p>
In the first part we obviously meet the natural numbers, prime numbers, and binary representation, but in subsequent chapters the integers are followed by rationals, irrationals (even transcendentals) and complex numbers. So we find a proof that 0.9999999... is equal to 1 and that $\sqrt{2}$ is irrational. We are introduced to $i=\sqrt{-1}$ and complex numbers, the transcendent number $\pi$, Euler's number e and the exponential function, infinity (and the transfinite numbers). And then there are some applications: Fibonacci numbers, factorials, Benford's law, and eventually algorithms for sorting, and for computing the GCD and LCM. Some famous problems and applications are only mentioned briefly (sometimes in one of the other parts): the Goldbach conjecture, Fermat's last theorem, the twin prime conjecture, the Collatz or 3n+1 conjecture, Kepler's sphere packing problem, cryptography, music theory,...</p>
<p>
The part about geometrical topics is called <em>Shape</em> and proves theorems about triangles (of course several proofs of the Pythagoras theorems, but other properties too like formulas for area and perimeter, and the four possible centres that can be associated with a triangle), circles (kissing circles, Kepler's problem, Pascal's hexagon theorem), platonic solids (with Euler's characteristic linking the number of vertices, edges and faces), fractals (defining fractal dimension by box counting), and hyperbolic geometry (including several ways of tiling the hyperbolic plane).</p>
<p>
The title of the third part is <em>Uncertainty</em>. Probability is employed to show that calculating the chances that some event will occur can give rather counter intuitive results. For example how in a game involving rolling dice it is possible that player A beats player B and player B beats player C, and yet player C beats player A. Another chapter investigates how you should interpret the result of a medical test predicting that you have (or have not) a disease if the test in only correct 95% of the time. There is an illustration that for the slightest change in the initial conditions of some dynamical system that originally converged to a stable steady state, can result in a chaotic, hence unpredictable, behaviour. Arrow's theorem shows that designing a fair democratic voting system is far from trivial. Finally the classic Newcomb paradox is discussed. It refers to a thought experiment where some player can either choose to take both boxes presented to him or only box 2. Box 1 contains a small amount, but the content of box 2 is unknown and depends on an oracle. The oracle is almost always correct and it has predicted what choice the player will make but does not reveal this. If the oracle predicted both boxes are chosen, then box 2 contains nothing, otherwise, box 2 contains a large amount. What choice should the player make? If you think this problem through, this touches upon the philosophical problem of free will.</p>
<p>
So there is really a great diversity of entertaining topics that can catch the interest of any reader that has the mathematical level of a regular young secondary school (junior high) student from the age of about 12-14. Not only the content, sprinkled with some sparks of Scheinerman's dry sense of humour, also the layout of the page helps to make reading agreeable. The main text is in one column that takes about two thirds of the width of a page and to the right of it a wide margin is left mostly blank, but it is used to add in smaller font some extra explanation or a side remark or a graph. It is truly a mathematics lover's companion, containing masterpieces for everyone. </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 brings 23 chapters on mathematical subjects at the level of early secondary school mathematics. The subjects are similar to what is often found in popular mathematics books. There is a part about numbers and the number systems, one about geometry and one on uncertainty. It is written in a style that remotely reminds of a textbook, but it is definitely much more easy going on the reader. Math lovers will enjoy it, math neutrals will be entertained, and it may even cure some of the math phobics too.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/edward-r-scheinerman" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Edward R. Scheinerman</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/yale-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Yale 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">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">978-0-300-22300-2 (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">USD 20.55 (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">296</div></div></div><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://yalebooks.yale.edu/book/9780300223002/mathematics-lovers-companion" title="Link to web page">https://yalebooks.yale.edu/book/9780300223002/mathematics-lovers-companion</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
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<li class="field-item even"><a href="/msc-full/97f60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F60</a></li>
<li class="field-item odd"><a href="/msc-full/97g30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97G30</a></li>
<li class="field-item even"><a href="/msc-full/97k50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97K50</a></li>
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Thu, 16 Nov 2017 15:00:35 +0000adhemar48019 at http://euro-math-soc.euThe Geometrical Beauty of Plants
http://euro-math-soc.eu/review/geometrical-beauty-plants
<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>
About 12 years elapsed since Johan Gielis wrote his book <em>Inventing the circle: The geometry of nature</em> in which he describes his superformula. This formula is a generalisation of the equation for the ellipse. In its simplest form it is $(x/a)^n+(y/b)^n=R^n$ with the Euclidean circle for $n=2$ and $a=b$ as a special case and more generally it represents an ellipse in the $\ell^n$-metric of $\mathbb{R}^2$. These graphs are also known as Lamé curves. The Danish artist and mathematician Piet Hein declared that $n=2.5$ resulted in the most aesthetic shape and made the super ellipse and its 3D analog, the super egg, an icon of the 1970's Swedish design. Martin Gardner picked it up in his Scientific American column popularising it worldwide. In the superellipse, the four points $(\pm a,0)$ and $(0,\pm b)$ are fixed for all $n$. For $n\gt2$ the ellipses are convex like a rectangle with rounded corners in between the ellipse and the circumscribed rectangle to which they converge as $n\to\infty$. For $1\lt n\lt 2$ they are in between the circle and the inscribed rhombus and for $n\lt 1$ they become concave like 4 pointed stars and converge to the coordinate axes as $n\to0$.</p>
<p>
Gielis was inspired by the shape of bamboo stems whose cross section had this rounded square shape. As a bio-engineer he got interested in describing also other shapes like flowers and leaves and 3D botanical components. He transformed the equation to polar coordinates and introduced more parameters to arrive at $r(\theta)=[|x(\theta)|^{n_2}+|y(\theta)|^{n_3}]^{−1/n_1}$ with $x(\theta)=\frac{\cos(m_1\theta/4)}{a}R$ and $y(\theta)=\frac{\sin(m_2\theta/4)}{b}R$. In a second stage also $R$ can depend on $\theta$ which allows for example to generate spirals. The result is that almost any appealing shape from nature can be generated. The <a href="https://en.wikipedia.org/wiki/Superformula">wikipedia page on the superformula</a> shows several examples, and provides matlab and octave scripts to generate them as well as links to interactive websites where one may experiment with the parameters to generate different shapes.</p>
<p>
In the present book the adjective "super" is replaced by "Gielis". So there are Gielis transforms, Gielis curves, Gielis surfaces, etc. Gielis is announced as an acronym on page 5, but the explanation comes only in chapter 7 on page 118: '<strong>G</strong>eneralised (or <strong>G</strong>eometric) <strong>I</strong>ntrinsic and <strong>E</strong>xtrinsic <strong>L</strong>engths <strong>I</strong>n <strong>S</strong>ubmanifolds'. The idea is still the same as in the previous book: namely to illustrate that the Gielis transform (of the basic circle and spiral) can describe almost any 2D or 3D form appearing spontaneously in nature (in particular in botany). Some slightly further generalised variants are considered which for example allow Möbius band-like objects, which combine symmetry in certain coordinate directions and twists in others. But Gielis has explored the material much further and he has collaborated with mathematicians which resulted in several published papers relating the transform to several other, not always plant-related, applications. Besides giving more historical, mathematical and even philosophical background on the formula, this book is also a summary of some of these papers.</p>
<p>
The book has 6 parts as described in the comments by Proclus (412-485 CE) on Euclid's Elements. These are the 6 steps that have to be followed to prove a theorem in the Platonic tradition. In this way a bridge is spanned between the classical Greek geometric approach to mathematics (including the Pythagorean theorem and the commensurability problem) on the one hand and on the other hand the proof that when we use an elastic notion of length, that is if the unit used to measure the length $r$ of the radius depends on the angle $\theta$, then the Gielis transform just describes a circle or a spiral in this flexible metric, and it can be applied in particular to describe many natural shapes in 2D, or, with obvious adaptations, curves and surfaces in 3D.</p>
<p>
The first part, the <em>Propositio</em>, Gielis explores the idea and provides some elements that will be used later. In this case it concerns the algebraic, geometric, and harmonic means of two numbers; some historical background on the problem; and it gives a discussion about the relation between mathematical shapes and how these shapes appear in nature.</p>
<p>
In the <em>Expositio</em> he then generalises how these different notions of means can be generalised giving different weights to each of the two elements. But there are also reflections on Newton's fluxions, derivatives and multiplications to conclude that polynomials are in fact transforms of monomials. These prepare the reader to accept flexibility (via parameterisation) in how we should look at mathematical definitions. This fundamental idea is applied when finally the 2D and 3D Gielis transforms are introduced to generate far reaching variants of the circle in 2D and the ball in 3D.</p>
<p>
The more advanced mathematical elements are introduced in the <em>Determinatio</em>. Some slight generalisations for the Gielis transform can be generated or the transform can consist of combinations of Gielis transforms for different values of the parameters. Other excursions into the mathematics are dealing with Pythagorean trees, Lindemayer-systems, fractals, and R-functions. The latter are functions named after Rvachev who introduced them in 2D and that later were generalised by Fichera for 3D. These are multivariate functions whose sign only changes when one of its arguments changes sign. When the sign is interpreted as true or false, these functions can be applied in logic and define cobordisms. An object can be defined as for example the set of points $x\in\mathbb{R}^d$ for which $R(x)>0$. Highly complex objects can be described in this way.<br />
The Gielis formula can transform the circle or the spiral into almost anything. That may lead to a complexity theory for topology: the oligomials (oligo is Greek for few). The complexity of a curve can be expressed by the degree of its polynomial equation. If it is rational, the polynomial degree is infinite, but the degree of the rational expression is a finite alternative. Similarly the (topological) complexity of an object can be expressed in terms of the Gielis transform (or transforms) required to represent it. In that sense, the circle is the simplest among all Lamé curves (its radius does not depend on the angle) but all other curves in that class have essentially the same complexity.<br />
Furthermore the concepts of intrinsic and extrinsic measure are introduced. These notions point us to the idea of an elastic, anisotropic measure of length that can be applied on manifolds. This gives rise to all kinds of geometries (Minkowski, semirigid, Riemann, ...), curves on manifolds (with applications in phyllotaxis), etc. The Gielis transform which so far is only described in polar coordinates can also be described in Euclidean coordinates thanks to the connection with Chebyshev polynomials: $T_n(x)=\cos(n\theta)$ if $x=\cos\theta$. That completes the picture showing that Gielis is the natural extension of the historical line that connects Pythagoras and Lamé.</p>
<p>
The <em>Constructio</em> lifts the machinery developed from botanical observations to a higher level, showing that it can me employed in a much broader (mathematical) context. While the rest of the text is more descriptive, here we find definitions and proper mathematical theorems. All the elements of the previous step are set to work in studying solutions of differential equations (boundary value problems) which can be applied to explain for example colour patterns in flowers. Differential geometry and definitions of curvature are also employed to explain natural shapes of manifolds i.e., surfaces with constant mean <em>anisotropic</em> curvature, where anisotropic refers to this notion of elastic length.</p>
<p>
In the <em>Demonstratio</em> the actual demonstration is given that plant and other shapes in nature can be described with a Gielis transform of the circle and the spiral and they can be explained by satisfying some natural optimality condition. The keyword is that all this is possible thanks to an elastic anisotropic concept of what a unit (of length) is.</p>
<p>
In the last step, the <em>Conclusio</em>, some reflections are attached to what has eventually been obtained. Is everything just based on a generalization of the Pythagorean theorem? Should plants be at the core of our world view, rather than physics or humans?</p>
<p>
The book has many beautiful colour pictures of natural forms that illustrate the mathematical shapes generated by the relatively elementary superformula. In general the book is rather accessible, at least for those who can understand and appreciate the Gielis transform. Not much more than elementary mathematics is needed. Some chapters (in particular those of the <em>Expositio</em> and the <em>Constructio</em>) where text reaches out to the more mathematical aspects of some connections and applications, are more demanding. For the interested readers, a more in-depth analysis of the references is probably needed since not everything is fully explained. But since it often concerns summaries of published papers, it should be easy for the interested reader to look up the originals. The Gielis transform is clearly the North Star that shows the way through the book, but it is not always very clear where the excursions taken along the winding road will lead the reader to. For example the discussion of arithmetic, geometric, harmonic, and Gaussian means show up at different places but it is not very clear how they contribute to the understanding of the Gielis transform, except perhaps in the chapter on curvature. Finally I could spot some typos like on page 114 where $M(\sqrt{2)},1)$ has an extra bracket and $n'''=\sqrt{m'''n'}$ is missing a prime in the second $n$, and on page 37, it is written "Sine and cosine are examples of simple polynomials...", but otherwise it is a well-groomed text.</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 beautifully illustrated sequel to Gielis' previous book <em>Inventing the circle</em> from 2002. Fifteen years ago he developed his superformula which increases the number of parameters in a polar description of Lamé curves. With this formula he is able to describe many 2D and 3D shapes from nature, especially in botany. This sequel brings more historical and mathematical background and reports on the new insights and applications of his formula (here renamed as Gielis transform) that were obtained since the previous book.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/johan-gielis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Johan Gielis</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="field-item even"><a href="/publisher/atlantis-pressspringer-nature" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Atlantis Press/Springer Nature</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">978-94-6239-150-5 (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">60,41 € (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">254</div></div></div><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/gp/book/9789462391505" title="Link to web page">http://www.springer.com/gp/book/9789462391505</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li>
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<li class="field-item even"><a href="/msc/51-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51 Geometry</a></li>
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<ul class="field-items">
<li class="field-item even"><a href="/msc-full/51n20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51N20</a></li>
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<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/92b05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92B05</a></li>
<li class="field-item odd"><a href="/msc-full/92c80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92c80</a></li>
<li class="field-item even"><a href="/msc-full/53-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53-01</a></li>
<li class="field-item odd"><a href="/msc-full/53c201" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53C201</a></li>
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Thu, 16 Nov 2017 14:52:08 +0000adhemar48018 at http://euro-math-soc.euThe Calculus Story
http://euro-math-soc.eu/review/calculus-story
<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>
David Acheson's popular math book <em>1089 and all that</em> (Oxford U. Press, 2002) was rather successful, and it got with this one a worthy successor. Small size, short chapters, amply illustrated, large font, airy layout, all properties that turn it into a storybook indeed. By storybook, I mean the kind of books you want to read to your children or grandchildren, that keeps their attention and makes them impatient and eager for the adventure to come in the next episode. In this sense, the title is very well chosen. It is indeed a story book, the story of calculus and how it came about, once upon a time...</p>
<p>
Since it is still about mathematics, you shouldn't try this on a toddler, but you might catch the attention of any novice to calculus from the age of about twelve or thirteen on. What exactly should be familiar before you can start telling the calculus story? You need some algebra to be able to manipulate equations. Also the concept of a proof, and curves as representations of functions are assumed, but that's about all. The mathematical content of the story is then the same as what you find in a classic textbook: limits, differentiation, integration, and infinite series. In the early chapters of the book Acheson sketches the preliminaries and sheds some light on what one wants to achieve with the next steps about to be taken. And later in the second half of the book, while he got the attention of the reader, Acheson moves on to differential equations, optimisation, complex numbers and chaos.</p>
<p>
For reasons of efficiency, a classic textbook will introduce a new concept by following a certain paradigm that consists of giving some motivating examples, followed by a precise and polished definition, or it is done the other way around: first the definition, and then some examples. However, this is not how the concept caught by the definition was developed historically. It is a much more natural approach to follow the historical path. In retrospect, the side tracks that turned out to be dead ends, can be pruned away, but still, letting the concept grow organically usually is a better choice. Decoupling mathematics from its history makes it abstract and dull. Including the history makes it less of a top-down rigid dogmatic doctrine that is forced upon the pupil, but something developed by real people, hence more "human".</p>
<p>
These historical elements make Acheson's book a mathematical (his)story(book). For example it is interesting to learn how people struggled with $\infty\times0=?$. Summing infinitely many infinitely small elements was used, and sometimes misused, by the founding fathers such as Kepler, Cavalieri, Wallis, Torricelli and others to compute and area or a volume, hence indirectly summing infinite series. Many of these computations were inspired by physics. The speed and acceleration of falling objects subject to gravity had been investigated but it was Newton who formulated the more general fundamental laws of motion. When applied to the force of gravity, these eventually explained the orbits in our planetary system. Kepler had described the "how" of the orbits, and Newton provided the "why". This has influenced Newton strongly in the way he developed his calculus. He made use of fluxions as it was based on the dynamics of coordinates like an object exposed to some action will move along a path describing its position as a function of time. Independently also Leibniz developed calculus. He used these infinitely small increments. The ratio $\delta y/\delta x$ of small changes gave rise to the notation $dy/dx$ for the derivative that mathematicians still use today, while Newton used $\dot{x}$ for the derivative of $x$, which is more commonly used in physics. With this climax in the calculus story, the birth of calculus, Acheson is about half way in his book.</p>
<p>
In the second half, the sine and cosine functions are used to connect an angle to periodic motion and for example to show the Leibniz formula $1−1/3+1/5−1/\cdots=\pi/4$ and other ways to compute <em>π</em>. The Leibniz formula had been hinted to in previous chapters, building up some tension. So, it feels like yet another success of calculus that it can explain why this formula holds. But periodic motion also means differential equations describing the pendulum or a vibrating string. The towering historical mathematician is now Euler. With calculus at our disposal, now topics such as calculus of variations, optimization, logarithms (including e and $i=\sqrt{-1}$), Taylor series expansions, Fourier series and other topics now come in rapid succession. It also requires to reconsider the definition of a limit to the more rigorous $(\epsilon,\delta)$ definition as developed by Weierstraß. Thus it is illustrated how this precise definition of the limit is the endpoint of a whole evolution and perhaps should not be the first definition to which a novice should be exposed. Acheson ends his story with a glimpse on Maxwell's equation of electromagnetism, Schrödinger's equation, and chaos theory.</p>
<p>
Thus Acheson introduces the reader with elementary steps to the concepts that matter, creating insight, and answering the why's and how's by calling the historical mathematicians to the stage and by citing from their papers. Thus it is not a bedtime story after all, but it should awaken the interest of the youngsters for the fascinating mathematics that in the end is describing the physics of the world we live in. Even for those students who had to assimilate calculus from a dull textbook, this story underlying all these definitions, theorems and computational rules, may soften their aversive attitude towards the subject. To pull youngsters away from the dark side of mathphobia, this booklet acts as a medicine to be applied to your wishes: preventively, remedially, or supplementary. This calculus story can be applied at all times to create a mathematical success story. </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>
Acheson uses an historical approach to introduce calculus to beginners. Starting from the most elementary concepts, the reader is exposed to a world of limits, differentiation, integration, and series and their entanglement with physics. Describing our planetary system was an incentive for the development of calculus. A flavour is given of differential equations describing a pendulum or a vibrating string, and there is a preview of Maxwell's and Schrödinger's equations and chaos theory. While previously physics pushed the boundaries of mathematics, it is now mathematics that pushes the limitations of physics.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/david-acheson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Acheson</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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">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">9780198804543 (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">£ 11.99 (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">208</div></div></div><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://global.oup.com/academic/product/the-calculus-story-9780198804543" title="Link to web page">https://global.oup.com/academic/product/the-calculus-story-9780198804543</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li>
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<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
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<li class="field-item even"><a href="/msc-full/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li>
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<li class="field-item even"><a href="/msc-full/97i99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97I99</a></li>
</ul>
</span>
Tue, 31 Oct 2017 06:17:37 +0000adhemar47981 at http://euro-math-soc.euFoolproof, and Other Mathematical Meditations
http://euro-math-soc.eu/review/foolproof-and-other-mathematical-meditations
<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>
Brian Hayes started his career as a member of the editorial staff of <em>Scientific American</em> in 1973. From 1983 on, his <em>Computer Recreations</em> continued the columns previously written by Martin Gardner and Douglas Hofstadter in <em>Scientific American</em>. After he left <em>Scientific American</em> he was mainly active as a contributor to (and for two years also as the editor of) <em>American Scientist</em>, a bimonthly magazine devoted to science and technology. Two books with selections of his articles appeared before. The present book is a collection of updated and extended versions of 13 of his contributions that appeared previously in <em>American Scientist</em>. The texts as they appeared originally were exposed to a broad readership and so many the reactions and additions could be implemented in the current version.</p>
<p>
The topics covered by these 13 essays, as the author calls them, are diverse. Several of the topics are familiar subjects in popular science writing, but what I appreciate most is how Hayes transfers his interest in the subject to the reader. He is not just transferring knowledge of other authors, collecting the ideas from the literature, but he takes the reader along, showing how he explored the topic himself in his quest to understand the underlying truth or the mathematical law.</p>
<p>
Anyone with an interest in puzzles and/or science and mathematics will love this book. No specific mathematical knowledge is required. To give a flavour of the contents, here is a quick survey.</p>
<p>
The first text investigates the well known story about Gauss, who as a schoolboy had to add the numbers from 1 to 100, and how he surprised the school teacher putting the right answer on his desk with a "ligget se". He had inventing the sum formula for an arithmetic sequence by adding symmetric pairs arriving at 50 pairs whose sum is 101. Just telling this story (adding a personal pinch of drama), is where most authors stop, but it is where Hayes starts. Where does this story come from? How much is authentic? Did Gauss indeed count symmetric pairs? How much time does it take to add the 100 numbers sequentially? And many more similar questions. Hayes concludes after dragging the historical literature that the origin seems to be the biography of Gauss by Sartorius written in 1856 as an obituary for Gauss who died the year before. However the story has been moulded and modified a lot since then. Hayes has compiled and investigated the most complete collection of the different versions of how the story has been told since then.<br />
Another historical detective work is the chapter where Hayes investigates what went wrong when in 1873 William Shanks published 707 digits of pi that he had calculated by hand. However, only the first 527 places were correct as D.L. Ferguson detected in 1944. Like a forensic investigator, Hayes analyses the computations of this cold case to find out which exactly were the errors that Shanks made and succeeds in identifying several omissions in transcribing the numbers.</p>
<p>
Statistics is used in several of the chapters. We all know that to compute an average, we can increase the quality of the estimate by increasing the sample size, unless this process does not converge. A counter example investigated here is the factorial-like function <em>n</em>? To find the function value, one has to keep multiplying randomly selected integers between 1 and <em>n</em> (possibly selecting the same number repeatedly) until one hits the number 1. The function value is then the product obtained so far. Playing with this function, Hayes shows that the average outcome increases with the sample size.<br />
More statistics are used in the chapter on Zeno's game. You and another players start with the same amount of coins. In step <em>n</em> you bet 1/<em>n</em> for the outcome of a fair coin toss. If you win you get 1/<em>n</em> of the other one's budget, it not, you loose 1/<em>n</em> of yours. Suppose you loose the first tosses, can you ever recover from your loss and win in the end? This is of course related to the divergence of the harmonic series, but it can also be connected with Cantor sets and with random walks on binary trees.</p>
<p>
Random walks, space filling curves, and self-avoiding walks have practical applications but they are also fun to play with. For example, when walking on a rectangular grid, how many <em>n</em>-step self-avoiding walks do there exist. No exact formula is known, but an heuristic one has been proposed. The asymptotic behaviour as <em>n</em> approaches infinity seems to give rise to a so called connectivity constant in this formula. Experimental values were obtained, but so far no exact solution has been found. It is a tantalizing challenge to find out whether it is a (simple) expression in terms of known transcendental numbers.<br />
In another chapter space filling curves are defined recursively leading to fractals, Cantor's calculus of the infinite, and to approximate solutions of the travelling salesman problem.<br />
Another counting problem is to figure out the number of different (uniquely defined) solutions for a sudoku of order <em>n</em>. The history of sudokus is briefly discussed and some heuristics are given for solving them. The number of givens does not seem to be a good criterion to classify a sudoku as easy or hard. More mathematical approaches to the problem are found in the references. It is strange that the book by J. Rosenhouse and L. Taalman <em>Taking sudoku seriously</em> Oxford U. Press, 20012, is missing from the list. Anyway the sudoku problem is hard and assumed to be NP complete. The proof that 17 is the minimum number of givens for the usual order 3 sudoku was only proved in 2012 by exhaustively checking all the approximately 5 billion solutions.<br />
Markov chains is another statistical subject. Finding the probability of a letter following a group of letters or words following a group of words, can be characteristic for the text produced by a particular author. With these probabilities, one may write a computer program that will generate (gibberish) text in the style of that author.</p>
<p>
The distribution of discrete random variables are caught in a spectrum, like the eigenvalues of a random matrix, or the zeros of the Riemann zeta function or seismic activity, etc. Hilbert and Polya conjectured that there is some universal operator whose spectrum corresponds to the distribution of the Riemann zeta zeros. This can be interpreted as corresponding to the jumps in the energy levels of the nucleus of some imaginary chemical element that could be named Riemannium.<br />
The distinction between pure random numbers and the more advantageous quasi- and pseudo-random numbers in computer applications is clearly explained in a chapter where it is also explained how Monte Carlo-type methods can solve high dimensional integration problems.</p>
<p>
It is a known paradox that volume of the largest ball that can be inscribed in the <em>n</em> dimensional unit cube when <em>n</em> is large is surprisingly small. Hayes gives arguments that convince the reader that this is not so surprising, but that it should actually be an obvious fact.<br />
Sometimes Hayes includes short computer code snippets that he used in his experiments. One chapter is devoted to the representation of floating point numbers on computers. Assigning a fixed finite number of bits to represent a floating point number results in finite precision and rounding errors that accumulate during the computations. This may have catastrophic consequences if not kept under control. Besides the standard IEEE representation, alternatives exist that are more flexible in distributing the available bits for representing a number between the significant and the exponent.</p>
<p>
The final chapter is called <em>Foolproof</em> which is also the title of the book. It sketches how the concept of a mathematical proof has evolved since Socrates. Nowadays computer proofs are more common, but the first computer assisted proof (of the four colour problem in 1976) had a hard time to get accepted. But also proofs provided by humans can be very long or perhaps very cryptic so that it takes years to check them. The incentive for this chapter is the proof by Wanzel in 1837 that the trisection of an angle using only ruler and compass is impossible. This was known since ancient Greece, but no proof was given until then, and yet it remained in obscurity for quite a while.</p>
<p>
I have tried to give an idea of what the different subjects are without including too many spoilers. There is plenty left to discover and savour for the connoisseur. Hayes has a pleasant style and he is taking you along his personal exploration of the subject. There are ample references provided for those who are hungry for more details. Hayes also refers at several instances to his web site at <a href="http://bit-player.org/" target="_blank">bit-player.org</a> for additional material. These web pages claim to be <em>An amateur's outlook on computation and mathematics</em>. You can find many animations and texts that will also be of interest for the readers of this 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>
Hayes has contributed many articles to the bimonthly magazine <em>American Scientist</em> discussing some mathematical topics for a broad audience. This book is a collection of 13 of these texts that have been updated and polished. Recommended for lovers of popular science and recreational mathematics. Hayes' own explorations are gentle invitations to do similar computer experiments. In this way, skin diving just below the surface of an intriguing phenomenon, you will have the personal satisfaction of discovering some of the hidden mathematical rules, the "why" of what is observed.</p>
</div></div></div></div>
<span class="field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Author: </span>
<ul class="field-items">
<li class="field-item even"><a href="/author/brian-hayes" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Brian Hayes</a></li>
</ul>
</span>
<span class="field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix">
<span class="field-label">Publisher: </span>
<ul class="field-items">
<li class="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">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">9780262036863 (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">248</div></div></div><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/foolproof-and-other-mathematical-meditations" title="Link to web page">https://mitpress.mit.edu/books/foolproof-and-other-mathematical-meditations</a></div></div></div>
<span class="field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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>
<span class="field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="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="field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li>
</ul>
</span>
<span class="field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden">
<ul class="field-items">
<li class="field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li>
</ul>
</span>
Fri, 27 Oct 2017 05:41:06 +0000adhemar47967 at http://euro-math-soc.eu