European Mathematical Society - 00a06
https://euro-math-soc.eu/msc-full/00a06
enSleight of Mind
https://euro-math-soc.eu/review/sleight-mind
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Matt Cook is an economist, a composer, a storyteller (as the author of thrillers), and he performs as a magician. Several magical tricks rely on creating an intuitive expectation and then come up with a totally different result. This creates amazement and unbelief in the audience. This is also very much the effect of a paradox. Given that Cook is not a professional mathematician himself, it comes as a surprise to find rather much abstraction and mathematics in this book.</p>
<p>Logical paradoxes are often found in popular science books discussing mathematics, games, and puzzles. Many of these "popular" paradoxes you can also find in this book but there are many more. Although this book is written for a general public, it is not leisure reading, since the discussion of the paradoxes goes in depth and that requires precise definitions and sometimes it touches upon the foundations of logic, mathematics, probability, or whatever topic the paradox is about.</p>
<p>The different topics are arranged in different chapters and the format is always similar. There is a general introduction to the subject, and that involves the definition of the concepts that are required for the discussion of the paradoxes to follow. These are precise but the selected terms and the associated technicalities are restricted to a minimum. Only what is essential is defined and only as precise as needed. For example in the chapter on probability it is defined what a probability space is, and that involves a sample space, a sigma-algebra, and a probability function, which are described by words, rather than formulas. Of course it is also explained how a random variable and its density function are defined and the Bayes theorem is introduced (this inevitably results in a formula). So, there are some formulas, but they are suppressed as much as possible, describing the definitions mostly in words and by using examples. I guess this is intended not to shy away the non-mathematician, but if you are a mathematician, then, given the intended rigour, it feels a bit awkward and verbose. Of course some formulas cannot be avoided, for example to illustrate what is in the Principia Mathematica of Whitehead and Russell a formula here and there is unavoidable.</p>
<p>When Cook comes to the many examples of paradoxes, it assumes an attentive reader because the lack of formulas requires sometimes complicated sentences that are often almost philosophical. Also here, a returning format is used. First the paradox is formulated, wherever possible, mentioning its origin. Cook usually tells a story to make the paradox concrete for the reader, rather than formulating it in its mathematical or abstract form. Then the opposing explanations (often there are only two) are formulated. The main discussion then explains why one is wrong and the other is correct. Sometimes there are more possibilities and more than one explanation is possible depending on how some components are defined or interpreted, which happens when the problem is ill-posed or under-defined.</p>
<p>Let me give some examples that illustrate the types of paradoxes and the depth of the discussion. A first chapter is dealing with infinity, which is not the simplest one to start with, but it is also the underlying concept in some subsequent chapters. It is clearly a concept that has caused a lot of confusion throughout the history of mathematics and logic. First we are instructed about bijections and countable sets, Cantor's diagonalization process, the cardinals $\aleph_k$, and the continuity hypothesis. Then the paradoxes can be explained: Hilbert's Hotel, Stewart's HyperWebster Dictionary, and many more. After introducing some additional group theory also the Banach-Tarsky theorem is explained in some detail. Not really a proof, but still the reader is given some idea of why this seemingly impossible result holds. Zeno's paradoxes of motion are of course somewhat related to the concept infinity, and so these are discussed making use of what was obtained in the previous chapter. Thomson's lamp is also related. If a lamp is alternately switched on and off at time instances $1−2^{−n}$, then deciding whether at time $t=1$ the lamp will be on or off is impossible.</p>
<p>With chapter four, probability is introduced. The Simpson paradox and the Monty Hall problem are probably the best known but there are others that allow much more variations and require much more discussion. In the chapter on voting systems we are introduced to social choice theory and Arrow's impossibility theorem. This is not completely unrelated to the topic of game theory which plays a role in, for example, price setting in a economic system. The Braess paradox is the unexpected result that by adding an extra road to a traffic system, the traffic may be slowed down.</p>
<p>With self-reference we are back to the foundations of mathematics with axiomatic set theory, and, among others, the paradoxes of Russell (the set of al sets that are not a member of themselves) and the liar (I am always lying). Inevitably this leads to Gödel's incompleteness theorems, a theory of types, the ZFC axiomatic system, etc. Also the unexpected hanging is a tough paradox discussed here. Somewhat in the same style is the chapter on induction, where some elements of formal logic are introduced.</p>
<p>A chapter involving geometry has curves, areas, and volumes with fractal dimension. There is not really a paradox here, but the fact that a dimension can be a fraction and need not be integer is considered to be paradoxical. But there are other simpler geometric examples. In many calculus books, we find the hard-to-believe fact that we can create an infinitely large overhang by stacking bricks if brick $k$ (numbered from top to bottom) overhangs the underlying one by $1/(k+1)$. This is an example where the mathematical fact that $\sum_{k=}^\infty 1/k$ diverges is replaced by a "story" of stacking overhanging bricks. Some typical mathematical beginners errors can also give some unexpected results, dividing by zero for example, or summing divergent series.</p>
<p>Finally Matt Cook has invited some colleagues to discuss paradoxes from physics. With statistical mechanics, the reader learns about entropy, Maxwell's Demon, and other classics such as the Brownian Ratchet driven by Brownian motion, and the Feynman's sprinkler problem. The unexpected results of special relativity are well known, and quantum physics is still difficult to understand in all its consequences and different interpretations are still discussed today.</p>
<p>In the final chapter the age-old question whether mathematics is discovered or invented is tackled. As one might expect, the answer is not exclusive for one or the other.</p>
<p>Mind, the paradoxes that are mentioned in this survey, are only few and exemplary for the many examples that can be found in this book (there are over 75). I can imagine that for readers who are totally mathematically illiterate, some steps may be hard, if these use terminology or arguments that are taken for granted. Nevertheless also those are considered potential readers because there is a short addendum introducing some very elementary mathematical notation. Cook also added a rather extensive bibliography, but many of the references are papers where the paradoxes were originally formulated, or papers discussing the solution. Thus not really the popularizing kind of literature for further reading. The index though is well stuffed and useful, since there is sometimes cross referencing across the chapters.</p>
<p>I could spot a typo in the discussion of the Banach-Tarsky theorem. When discussing successions of irrational spherical rotations left, right, up, down, denoted as L,R,U,D, strings of these letters are formed to denote points on a sphere. Uniqueness requires eliminating the succession of opposite rotations (free group). Thus UD, DU, LR, or RL are not allowed in a string. However in the table page 25 appears the string DUL which is not allowed.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Several paradoxes are analysed in depth. Some are well known others are less familiar: Zeno's paradoxes, Monty Hall problem, Banach-Tarsky theorem, paradoxes related to voting systems, self reference, but also statistical mechanics, special relativity and quantum physics and many more pass the review. The finale is a discussion of the ultimate question: Is mathematics invented or discovered?</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/matt-cook" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Matt Cook</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/mit-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MIT Press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2020</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780262043465 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 34.95 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">368</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Logic and Foundations</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://mitpress.mit.edu/books/sleight-mind" title="Link to web page">https://mitpress.mit.edu/books/sleight-mind</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01a15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A15</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/81p05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81P05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/63a10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">63A10</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/70-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/83-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83-01</a></li></ul></span>Wed, 01 Apr 2020 11:54:56 +0000Adhemar Bultheel50644 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/sleight-mind#commentsPatently Mathematical
https://euro-math-soc.eu/review/patently-mathematical
<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>Jeff Suzuki is a mathematician teaching at Brooklyn College who has written two books about mathematics in an historical context, but in his previous one he shifted gear and he wrote about the mathematics as used in the US Constitution. That is quite revealing since it is well known that politicians and lawyers are usually not the most skilled mathematicians. He is an active blogger and vlogger with a <a target="_blank" href="https://www.youtube.com/user/jeffsuzuki1">YouTube channel</a> about mathematics. He has the knack for explaining mathematical concepts in a remarkably simple way. This skill is again one of the major characteristics of the present book which is exploring the mathematics that is underlying approved patents in the US.</p>
<p>Mathematics is characterized by abstraction and implementing a formula or a method in some specific application is often a straightforward thing to do. Applying the theory in the context of a diversity of applications is pretty obvious to anyone "skilled in the art". When the abstract result is obtained, the mathematician is satisfied and then often looses interest in the implementation or the application. Patents are approved to encourage innovation, but it should not prevent the exploration of a broad range of related research. So, mathematics or an algorithm are often considered as general abstract ideas that cannot be patented, and only when it is implemented on a device for a specific application, a patent can be approved for that particular case. Mathematicians and researchers in general have a culture of publishing and sharing their results with the idea of advancing science. Companies on the other hand want to hide away their innovative results from competing companies by claiming their ownership in patents and preventing others to build on the same idea. But what if that idea is basically just mathematics? Unfortunately patent agents are not mathematicians and patents have been approved whose core element is basically a simple implementation of a mathematical idea or formula. As we are living in a world that is becoming more and more digitized, mathematics has penetrated the smallest pores of society, and therefore these issues become more and more relevant. Can mathematical innovations be the subject of a patent, hopefully not, but where is the boundary and under what conditions can a patent essentially based on a mathematical idea be approved? Suzuki gives many examples of patents based on a mathematical idea and gives in his epilogue some concluding recommendations. First the mathematical core of any patent should be clearly defined and it should be proved that it does what it claims to do. This should prevent claims that are too broad and prevent any other development in the area. Secondly, since in the US patent agents have to prove their expertise, Suzuki suggests that developing mathematics coursework should be allowed as a proof of mathematical expertise. This is kind of a strange conclusion but it might refer to his own position. Finally, also mathematicians should be allowed as patent agents, which, in the US, is currently restricted to engineers or scientists.</p>
<p>The book describes in a very accessible way all the mathematics that are behind many patents. It starts with several indexing systems in the early days of the Internet. These indices or keywords should allow to detect similar or related documents. Then of course along came Google linking the queries to the appropriate pages with ranking. That was their reason for success putting the most probable sites sought for on top of the (long) list. This was based on Pagerank, which is basically just computing an eigenvector of a large network matrix. Patents were approved to competing search engines and for methods to prevent link farms, spamming or other fraudulent practices or techniques that abuse or disrupt the system. What is done for text documents can also be done for images, which poses additional issues of the way in which pictures are represented, compressed, and transformed. The same person or scene can be represented by images that correspond to possibly different views or the image has been edited and manipulated. Facial recognition is certainly a well developed area. Copyright issues for images that are spreading over the Internet is another issue to be resolved.</p>
<p>In the very different area of match making companies and dating websites, remarkably similar problems arise. How to characterize a candidate, how to characterize his expectations, and how to find possible matches? This is almost like matching websites to a search query. An additional problem may be that the requirements for a match put forward by a person may not be exactly what he or she is really desiring. Suzuki investigates even whether these patented algorithms really work. No hard proof is available so far. The problem of formulating the proper questions in order to evaluate what is really intended is a subtle problem that teachers are faced with when they have to evaluate their students. That is an important problem for all kinds of rating systems. That can be educational platforms but it is similarly important for e-commerce and advertisement. For example in e-learning, the evaluation by multiple choice exams can be deceptive, or the kind of question asked may not really test the skill of a student or her understanding of a topic that one intended to test. Oral interviews can mitigate that, but computers and automation through algorithms is so much faster and objective. But don't forget that these are also very stupid and just follow the prescribed rules, and these may not always be the rules that were intended. The math underlying all these companies may not do what is claimed in their patent applications.</p>
<p>From this point on, the applications described by Suzuki become a bit more technical, but the mathematics are still explained in the same easily understandable way. How can we measure the strength of a password, and how to defend against eavesdropping? Here cryptography is an important tool, but that may not completely solve the problem of authentication or the related problem of how to prevent giving away our identity. We can be identified by our way of touching the keyboard, or by our surfing behaviour traced with cookies, all highly desired data for advertising, spamming, or phishing. Other data are collected about how we are digitally connected. This can be used to propagate an idea or a product in a network just like a virus spreading in an epidemy. This requires an analysis of a network graph. Optimization problems with constraints in large networks raises combinatorial problems that can only be solved with heuristics like simulated annealing.</p>
<p>Compression techniques of images (jpeg, DCT, wavelets), encoding of bit sequences (Huffman coding), fractals (e.g. fractal antennas) and space filling curves, cellular automata are all explained with simple examples. But also RSA and other crypto systems are illustrated for simple cases. These require more advanced mathematical techniques like modulo calculus, prime number factorization, discrete logarithm, Chinese remainder theorem, elliptic curves,..., but traditional techniques are challenged by quantum computing. It will not be a surprise that all these essentially mathematical techniques have been encapsulated in some patent.</p>
<p>This book illustrates why Suzuki has mixed feelings towards patents. There are a lot of mathematical ideas that can potentially be turned into a commercial patent, but at the same time there is the fear that a patent may kill the development and use of mathematics in a mathematically similar, although seemingly a quite different application.This is an important issue to be considered in an increasingly automated society. This is an important message and basically a political problem. What impressed me most in this book is the painless simplicity used by Suzuki to explain all these mathematics. Some illustrations and very few simple formulas suffice to communicate the mathematical ideas to inexperienced readers. This simplicity is of course an essential requirement if he wants to bring his message across to the politicians.</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>Mathematical formulas or pure algorithms cannot be patented. They need to be implemented in the framework of some application. But there are often simple mathematical ideas that form the heart and soul of an application or implementation that has been patented and that patent was the start of some very successful billion dollar companies. This book is a very readable introduction to the mathematics that are implemented in many approved US patents.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/jeff-suzui" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jeff Suzui</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/john-hopkins-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John Hopkins 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">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781421427058 (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">296</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematical-aspects-computer-science" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Aspects of Computer Science</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://jhupbooks.press.jhu.edu/title/patently-mathematical" title="Link to web page">https://jhupbooks.press.jhu.edu/title/patently-mathematical</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a69" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a69</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li></ul></span>Mon, 25 Nov 2019 08:46:14 +0000Adhemar Bultheel49944 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/patently-mathematical#commentsMath Art
https://euro-math-soc.eu/review/math-art
<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>
Mathematicians consider some mathematics to be beautiful, and there has indeed been scientific research measuring that mathematicians showed increasing brain activity in the frontal cortex when seeing mathematical formulas. This brain activity is similar to what is observed when people see a beautiful painting or listen to music. So there must be some link between mathematics and art. Several mathematicians are known to be also great musicians of produce visual art and most mathematicians have a soft spot for a certain kind of visual works of art that have some mathematical flavour. The <a href="http://bridgesmathart.org/" target="_blank">Bridges Organization</a> promotes the interaction between mathematics and visual art, music, architecture, and it organizes an annual conference around these ideas.</p>
<p>
Several books are available in which mathematics is linked to art. Some are coffee table books, mainly consisting of pictures, others are philosophical essays with some illustrations. This beautifully illustrated book by Ornes is about the "mathematical art" of nineteen contemporary visual artists. Its format is a careful balance between a sketch of the artist, a short discussion of his or her work, and an easily accessible introduction to the "mathematics behind the art". Some of the artists were once represented at these Bridges conferences and some are at the MoMath museum. For obvious reasons Ornes is mainly interested in visually pleasing work, with a strong mathematical background. Most of the artists are still living and Ornes is quoting some of them, which shows that he has interviewed or at least spoken with these artists.</p>
<p>
Divided into four parts, Ornes discusses 19 artists and their work. A telegraphic survey of the contents and the list of artists is given at the end of this review. Some of the artists are professional mathematicians, but others are just inspired by mathematics. The selection of the art is quite diverse. It can be monumental sculptures, weaving, computer generated curves, quilts, 3D-printed objects, wood carving, or crocheting. The work presented is selected to serve the purpose of this book. The artists can have other work of a quite different nature, or it can be early work and they may have moved more recently to a different kind of work. The appendices about the mathematics that served as an inspiration is diverse as well. There is pi and phi (the golden ratio), the Fibonacci numbers and primes, the Pythagoras theorem, set theory and infinity, geometry with classical Platonic and Archimedian solids, fractals and non-Euclidean geometry, topology and the Moebius band, space filling curves and tilings, computer science with complexity theory, algebra with symmetries and groups, and more. An appendix is linked to one artist, but there are cross references to other artists as well. Clearly the selection of topics and artists is very diverse, but this is only a very small section from a vast domain showing a growing interest for this kind of interaction between mathematics and art.</p>
<p>
The size of the book is nearly square (9 x 9.5 inches) and it is printed on glossy paper. So it can serve as a coffee table book but it has more to read than it has to see. The cover is black with a white design by Bathsheba Grossman. I could not find the reference in the book for the cover picture (although all other pictures are properly credited). The picture is actually a dodecahedron based design for a <a href="https://www.materialise.com/en/mgx/collection/quin-mgx">lamp</a> that is 3D printed by Materialise. It is also an illustration on Grossman's <a href="https://en.wikipedia.org/wiki/Bathsheba_Grossman" target="_blank">Wikipedia page</a> (3 Sept 2019). Grossman has also a Klein bottle opener, i.e., an operational bottle opener in the shape of a Klein bottle.</p>
<p>
I like the book very much. Unlike some other popularizing math books, it literally illustrates the beauty of mathematics, and makes this beauty accessible to non-mathematicians. Hopefully they will be triggered by the beauty of the pictures, to also read the mathematical appendices, which are written at a level that can be read and understood by anyone.</p>
<p>
To conclude, a quick summary of the 19 cases that are collected in four parts.</p>
<ul>
<li>
Part 1: <em>Making sense of the universe</em>.
<ul>
<li>
The art of pi - <em>John Sims</em>, who among other work, produces quilts like coloured QR codes where colours are defined by the digits of pi.</li>
<li>
Geometry in motion - <em>John Edmark</em> designs objects that require dynamics, and here one should consult <a href="https://www.johnedmark.com" target="_blank">his website</a> to understand and appreciate his work. The mathematics here deals with the Fibonacci numbers and the golden section.</li>
<li>
The proof is in the painting - <em>Crockett Johnson</em> has paintings that are inspired by graphical proofs of the Pythagoras theorem.</li>
<li>
One to one to infinity - <em>Dorothea Rockburne</em> produces abstract art, sculptures and installations, that draw inspiration from set theory. The mathematical appendix discusses set theory and different orders of infinity and gives a proof that there are infinitely many primes.</li>
<li>
The many faces of geometry - <em>George Hart</em> makes sculptures by weaving several identical components together that shapes Platonic solids. The mathematics is about regular and classical polyhedra and their stellations.</li>
</ul>
</li>
<li>
Part 2: <em>Stranger shapes</em>
<ul>
<li>
Space and beyond - <em>Bathsheba Grossman</em> makes sculptures that are periodic minimal surfaces or projections of the 120-cell in 3D space.</li>
<li>
The consequences of never choosing - <em>Helaman Ferguson</em> has monumental sculptures like an umbilic torus decorated with a Peano space filling curve. This and other space filling curves are discussed in the appendix.</li>
<li>
The tangled, torturous universe of fractals - <em>Robert Fathauer</em> produces fractal organic sculptures. Fractals are introduced in the appendix but is continued in the next case.</li>
<li>
The mystical and the mathematical - <em>Melina Green</em> focusses on the Mandelbrot set and generates an image of the set that suggests the shape of a Buddha.</li>
<li>
The equations of nature - <em>David Bachman</em> is a topologist and his art was originally the result of describing nature by equations and then generate artificial objects that look very natural. More recently his work visualizes more abstract ideas. The appendix is discussing topology.</li>
</ul>
</li>
<li>
Part 3: <em>Journeys</em>
<ul>
<li>
The wandering mathematician - <em>Robert Bosch</em> produces a piecewise linear Jordan curve that is denser at some places which, from a distance, gives the impression of a grey-scale reproduction of for example the Mona Lisa of whatever other image one cares to choose. The construction of the curve is based on a traveling salesman algorithm which is discussed in the appendix together with the P versus NP problem.</li>
<li>
The curves in the machine - <em>Anita Chowdry</em> is inspired by the Lissajous curves and produces some steampunk instrument to generate such complex curves.</li>
<li>
The algorithms of art - <em>Roman Verostko</em> (born in 1929) has embraced the first computers and designed algorithms to produce graphical art. The appendix discusses some elements from complexity theory and quantum computing.</li>
<li>
Projections - <em>Henry Segerman</em> has work inspired by stereographic projection, producing a Riemann sphere that is the projection on the sphere of for example a regular grid in the plane.</li>
</ul>
</li>
<li>
Part 4: <em>(near) Impossibilities</em>
<ul>
<li>
Following yarn beyond Euclid - <em>Daina Taimina</em> is known for her crochet work representing hyperbolic geometry. The appendix explains and illustrates rather well hyperbolic geometry with the Poincaré disk or half plane models.</li>
<li>
Bounding infinities - <em>Frank Farris</em> produces symmetric images and transitions in wall paper groups using deformed photographs as a stamp. Some of his work is discussed in <a href="/review/creating-symmetry-artful-mathematics-wallpaper-patterns" target="_blank"> <em>Creating symmetry</em></a>.</li>
<li>
Connections - <em>Carlo Séquin</em> is a computer scientist who produces complex large mathematically inspired sculptures. The appendix discusses symmetry and group theory.</li>
<li>
Math and the woodcarver's magic - <em>Bjarne Jespersen</em> is a wood carver who produces a wooden object where a sphere is capture inside a polyhedral structure that, unable to take it out or put it in. The appendix is about tessellations that cover the plane.</li>
<li>
The possibilities - <em>Eva Knoll</em> uses many different media to express herself among which weaving where some relative prime repetition of patterns creates some extra pattern on top of the underlying one. The appendix gives a discussion of algebra and all its different meanings in mathematics.</li>
</ul>
</li>
</ul>
</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 discussing the relation between mathematics and art by describing the work of 19 contemporary visual artists and explaining the mathematics that is behind their artwork.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/stephen-ornes" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Stephen Ornes</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/sterling-publishing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Sterling Publishing</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1454930440 (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">208</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.sterlingpublishing.com/9781454930440/" title="Link to web page">http://www.sterlingpublishing.com/9781454930440/</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Mon, 09 Sep 2019 14:07:48 +0000Adhemar Bultheel49707 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/math-art#commentsWeirder maths
https://euro-math-soc.eu/review/weirder-maths
<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 continuation of the previous book <a href="/review/weird-maths-edge-infinity-and-beyond" target="_blank"><em>Weird Math</em></a> by the same authors. We find more stories from the realm of popular mathematics. Some of the elements were already discussed in their previous book, but appear here in a slightly different context. Most of the stories told can also be found in other popular science or popularizing math books. The two books together form some kind of a round-up of all these curious or interesting (I would not call them weird) tales related to mathematics and mathematicians that can be found elsewhere, but perhaps not all collected in one (or two) books like it is done here. There are so many different items that an index would have been welcome, but unfortunately that has not been included.</p>
<p>
The subtitle of this volume is `at the edge of the possible', which is in my opinion a bit of an exaggeration, but it can be ascribed to the enthusiasm of the authors (an astronomer-science writer Darling and a young math prodigy Banerjee) who just discovered these stories often known for a while already. But it is a good thing they want to transfer their enthusiasm to generations of newcomers.</p>
<p>
What is in this volume that was not in the first one? Here is a telegraphic survey per chapter:<br />
"How to get out of that" deals with mazes: mythological (king Minos, Ariadne, and the Minotaur), historical (Chartres cathedral a.o.), imaginary (Borges), natural (different caves), computer games (Adventure), amusement parks, none of which is really related to mathematics, but there is also Euler and the bridges of Köningsberg, with the development of graph theory, the Internet graph, and escape algorithms from a maze and more.<br />
Something similar holds also for the other chapters: they are collections of diverse topical stories that are sometimes loosely connected to mathematics that is lingering in the background, but in some cases mathematics comes prominent to the foreground but always without being technical.<br />
"The vanishing point" deals with the number zero and more generally number systems, but also with perspective, infinitesimals, the limit, and the vacuum in physics.<br />
"Seven numbers that rule the universe" starts with physics, but then discusses the top-seven mathematical constants: 0 (zero), 1, π, e, i = √-1, γ (Euler constant), and ℵ₀,<br />
"Through the looking glass" is about symmetry. It includes the definition of a group, and the solution of algebraic equations which gave rise to Galois theory, and symmetry breaking (quasi crystals, and cosmic microwave background radiation).<br />
"Maths for art's sake" is obviously relating mathematics and art. There are many historical mutual influences in sculpture, paintings, architecture. Some more recent examples are Escher and Dalí. That the golden section is characterizing the ideal ratio that defines beauty, is only approximately true.<br />
"Beyond the imaginary" describes how numbers developed from integers to rationals, reals, complex numbers, quaternions, octonions and surreals.<br />
"Tilings: plain, fancy and downright" is obviously about tilings: the Alhambra, the 17 wallpaper groups, irregular tilings, and quasi crystals.<br />
"Weird mathematicians" do exist of course: Erdős, Pythagoras, Galois, Ramanujan, Hamilton, Turing, Boltzmann, Cantor, Gödel, Bourbaki (although the latter is not really a person).<br />
"In the realm of the quantum" sketches the development of quantum physics and hints to the necessity to develop quantum mathematics. Determining whether or not a spectral gap exists turns out to be an undecidable problem, which touches the foundations of mathematics with very important consequences for physics.<br />
"Bubbles, double bubbles, and bubble troubles" is a classic topic of soap bubbles, but now also turns out to be important in nanofoams as well as in cosmic bubbles.<br />
"Just for the fun of it" is about puzzles (tangram, soma cube, towers of Hanoi, Chinese rings,...) and great puzzlers (Gardner, Dudeney, Lucas,...).<br />
"Shapes weird and beautiful" is an interesting chapter of peculiar 3D objects: Gabriel's horn, minimal surfaces, Piet Hein's super egg, Reuleaux triangle, Gömböc, rattleback, sphericon, the horned sphere, and the amplituhedron.<br />
"The great unknowns" is about unsolved problems or at least problems that were only solved after a long period like the classical Greek problems (trisecting an angle, squaring the circle, doubling the cube), Fermat's last theorem, the continuum hypothesis, the Poincaré conjecture and the Riemann hypothesis.<br />
"Could math have been different" might relate to our use of a decimal system, but that is inspired by humans having accidentally 10 fingers. A more fundamental question is about mathematics being invented or discovered. Is π a universal constant? Would aliens have essentially the same mathematics as we have? It all depends on the axiomatic system one starts from.</p>
<p>
With their two books the authors bring a pretty general survey of what is found scattered in many other popularizing books on mathematics. The true mathematics behind all these stories is barely touched upon and the narration is kept at the story-telling level, so that it is really written for a general public. It is brought in a playful and entertaining way: no deep philosophical contemplations, no complicated technical discussions. These are all stories in the margin of mathematics (which is fortunately much broader than Fermat's margin), but not really including the mathematics itself, and it is intended to be that way because there is no reference list to look up further details or recover the source of their 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>
This is a continuation of the previous book by these authors, extending their collection of stories about mathematics and mathematicians at a level that is accessible for a general public.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-darling" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Darling</a></li><li class="vocabulary-links field-item odd"><a href="/author/agnijo-banerjee" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Agnijo Banerjee</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/one-world" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">One World</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781786075086 (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">£ 9.99</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">256</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://oneworld-publications.com/weirder-maths-pb.html" title="Link to web page">https://oneworld-publications.com/weirder-maths-pb.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span>Mon, 05 Aug 2019 10:04:33 +0000Adhemar Bultheel49604 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/weirder-maths#commentsWeird maths. At the edge of infinity and beyond
https://euro-math-soc.eu/review/weird-maths-edge-infinity-and-beyond
<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 is a joint venture of an experienced science writer (Darling) and an exceptionally bright young math student (Banerjee). The result is this book popularising mathematics by presenting a set of curious/interesting/surprising (I would not call them weird) mathematical facts in such a way that they are easily accessible for the layman. The topics they cover are close to the topics that are also discussed in other books written with the same objective. There is of course always a new approach and there is always something to learn from a surprising fact or an unexpected link that is made. The authors have adopted the rule that also Hawking used: avoiding mathematical formulas in a popular science book. Each formula would presumably halve the number of readers. Besides symbols like ℵ and ω (when discussing infinities) and notations like 3↑↑3↑↑3 (when discussing large numbers) there are only very few equations or formulas. The authors state in the preface: If we can not explain it in plain language, then we don't properly understand it. This does not mean that hard topics are avoided since there is quantum theory, cosmology, and physics, as well as the foundations of mathematics with for example Gödel's theorems. According to the preface, Darling is responsible for the philosophical and anecdotal aspects, the relation to music, and he polished everything into the final text. Banerjee was more involved with the technical aspects including large numbers, computation, and prime numbers. The result is a pleasant read that anyone with only a remote interested in mathematics will enjoy.</p>
<p>
The breadth of the topics covered is too wide to enumerate them all, but to give a rough idea, what follows is a fistful of topics that are discussed.</p>
<p>
Historically mathematics is of course inspired by the necessity to count and by our surrounding physical world and the stars up above. But how would a four-dimensional being see our three-dimensional world? For us hard to imagine, but mathematics has no problem to function in higher dimensions.</p>
<p>
With probability one can simply explain the birthday paradox, but it is also essential in quantum theory which is hard to understand, certainly when it eventually leads to vibrating strings in an attempt to construct a theory of everything. In chaotic systems such as the weather, the smallest perturbations, in spite of all the laws of probability, may prevent any valid prediction. On the other hand probabilistic systems can obey simple rules like in Brownian motion, or it may generate complex structures such as fractals. Think of automata like the Turing machine or Conway's Game of Life. If we can model a system, this does not mean that it is practically computable because one can hit the boundaries of complexity like problems of class NP.</p>
<p>
Music and prime numbers are classical topics for books like this one. With the title "music of the spheres", there is a reference to Kepler of course but the story of that chapter also meanders by mentioning the music disk sent into space in the SEFI project as well as singing whales. Obviously there is an obligatory extensive discussion of the mathematics of music. The next chapter is discussing the unavoidable prime numbers. We meet for example the 17 year cycle of cicadas, the Ulam spiral, the Riemann hypothesis, and the twin prime gap.</p>
<p>
Two more classical recreational topics are game theory and logical paradoxes. Game theory is discussed in connection with computers playing chess against humans and later also the more complex game go, but game theory can of course be applied to other games as well. One may for example investigate whether winning strategies exist when the human player can start? Game theory may have been developed to help people win an entertaining game, but when it was applied to very real economics, politics and other modelling and optimization problems it became a serious mathematical subject and John Nash won even the Nobel Prize in economics and the Abel Prize with his results. Paradoxes, the foundations of mathematics, and logic get their separate chapter. They may be applied to entertaining logical puzzles, but when digging a bit deeper, one bumps into much harder problems and the chapter ends by mentioning surprising mathematical results such as for example the Banach-Tarski theorem (a solid ball can be cut up into 5 pieces such that these can be reassembled into two solid balls of the same size as the original).</p>
<p>
With large numbers and transfinite numbers we are back in the realm of numbers and mathematics. Infinity and the orders of infinity are discussed including the continuity hypothesis and the existence or not of the absolute infinity Ω. Big numbers (the really really big ones) like googol, googolplex, power towers (as introduced by Knuth), Graham's number, TREE(3), and several other numbers are featuring in a chapter that is less common in popular math books. Yet there is a subculture of googologists challenging each other in competitions to define ever larger (finite) numbers.</p>
<p>
The last two chapters dive somewhat deeper into the less elementary mathematical topics. Of a more geometric nature is a chapter on topology with objects like the Moebius band, the Klein bottle, and different kinds of geometry and how this is applied to our universe. Finally we arrive at fundamentals discussing the completeness of the mathematical system, Gödel's theorems, proof theory, the axiom of choice, the Peano calculus, ZFC axioms, etc. These are clearly among the more advanced topics discussed in this last chapter.</p>
<p>
As can be seen from this (largely incomplete) enumeration of topics, the discussion is rather broad and sometimes also touching upon deeper theoretical problems. The text remains however very readable, even when more advanced problems are carefully dissected. This is a very nice addition to the popular math literature deserving a warm recommendation.</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 book to popularize mathematics written by a science writer (Darling) and a bright young math student (Banerjee). The authors cover many of the topics that are traditionally covered by this kind of books, but some surprising connections are made. For example the topic of transfinite numbers is a classic but the googology chapter on large (but finite) numbers is not so common.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-darling" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Darling</a></li><li class="vocabulary-links field-item odd"><a href="/author/agnijo-banerjee" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Agnijo Banerjee</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/basic-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">basic books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781541644786 (hbk), 9781541644793 (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">USD 27.00 (hbk), USD 16.99 (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">320</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.basicbooks.com/titles/david-darling/weird-math/9781541644786/" title="Link to web page">https://www.basicbooks.com/titles/david-darling/weird-math/9781541644786/</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span>Tue, 05 Feb 2019 08:21:18 +0000Adhemar Bultheel49080 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/weird-maths-edge-infinity-and-beyond#commentsThe Art of Logic
https://euro-math-soc.eu/review/art-logic
<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 book about applied logic. Books that popularize mathematics (and logic) often have chapters about paradoxes, or logic puzzles. If you were expecting something in that style, then it will soon become clear that you are mistaken. Eugenia Cheng who combined in her previous book <em>Cakes, Custard + Category Theory</em> her profession as a mathematician specialised in category theory and her hobby of cooking. In <em>Beyond Infinity</em> she discussed the infinitely small and the infinitely large, which is also rather mathematical, but it can also lead to paradoxes such as the Hilbert Hotel.</p>
<p>
In the current books she discusses the roots that govern all mathematics: the rules of logic and axioms that lay at the origin. Close as this may be to the heart of a mathematician, she considers here however how logic also applies to our daily life, although in a much fuzzier way and lacking the mathematical formalism. As a consequence misunderstandings will occur. These will result in endless and unsolvable discussions, because the opponents apply different rules or different axioms and so both claim to be correct in coming to opposing conclusions.</p>
<p>
Cheng applies this to unravel some of the currently hot topics that roam the media like (political) discussions about health care, or racism and sexism. She clearly explains why the different parties can not come to an agreement. People come to their own version of "The Truth" by making logical mistakes. For example a negation is mistaken for a contraposition, or they use a false premise which logically allows to imply anything, or people apply the rules to different classes of subjects, etc.</p>
<p>
To be able to point out where things go wrong in many practical situations, Cheng of course needs to explain some rules and terms form logic that are much more clearly defined in a mathematical situation. Mathematicians will agree on what is true or correct because they are arguing within a much more abstract and unambiguous universe, using generally accepted rules, even though they need not make all the details of their logical deduction steps explicit. As long as their peers will be able to see how the gaps need to be filled, they will accept the result. Only if the gaps are too large, a referee will require more details.</p>
<p>
So the first part of this book is explaining what logic actually is and how it is experienced every day by anyone. Using many examples from social discussions, political disagreement, or just parent-kid discussions, Chen introduces the different terms, using some necessary abstraction, to talk properly about the terms that are used in more formal proposition logic, including quantifiers, Venn diagrams, truth tables, negation of implications, equivalence, but also fuzzy logic (the world has many "grey zones"),... When there is a discussion about who is to blame for some unfortunate event, then one should first see how it is possible that the event came about, and those previous events are caused in turn by some other events, etc. So there can be a very complex network of causal coincidences that have eventually led to the event that is the subject of the dispute (Chen uses all these relations as a pretext to smuggle in some of her beloved category theory).</p>
<p>
In a second part she explains the limits of logic. In practice there is no peer reviewing process of some person's argument like in a mathematical environment of publishing a paper. Which mechanisms (correct or inappropriate) are used to convince people? Perhaps (Internet) memes are assumed correct while they are not. When one comes to paradoxes, some alarm should go off to revise the system applied. In other situations, logic will not be useful like in emergencies, or when we do not have all the information to act logically, in which cases we may perhaps just follow a reflex, a gut feeling, or trust the judgement of others.</p>
<p>
The third and last part is called beyond logic. This is where one should agree on axioms, the things that are accepted without a (logic) proof. Then there are of course the many grey zones where binary logic is not the proper tool to use. What universe is one talking about (all humans, all men, all women, all white women, all rich and white women,...?) Things may be considered equivalent (the same) for somebody, but not for the adversary. And then there are of coarse emotions that are important factors in everything we do or say.</p>
<p>
In this book Chen is strongly engaged in social justice, minority groups, gurus, religion, climate issues, the role of science, etc. So in her last chapter she somehow summarizes how logic can help you to be a reasonable and intelligent person. There should be some framework that one believes in, and one should be sceptical towards charismatic "superstars". You should realize that there are a lot of grey zones and that you are not alone so that reaching a joint objective can be more rewarding than reaching your own. Correct and reasonable logical arguments should be used, even in a world that is not always logical.</p>
<p>
This is an engaging book, that should be read by everyone. It will help solving disagreements, or direct discussions away from and "it is - it isn't" arguments, and help you focus on the underlying cause of the dispute. Of course real life is not mathematics, boundaries are fuzzy, and obviously, it can not prevent that people disagree, and they should if for the proper reasons and when using the correct arguments, and this is the main message of the book.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In this book, Cheng illustrates how by using logic, one can become a better, reasonable, and intelligent human. She describes the possibilities, the limitations, and the pitfalls of logic when it is applied beyond the abstract context of mathematics. Can it define what is right or wrong or help to resolve a deadlock in political or social discussions about subjects such as solidarity principles, climate issues, racism, or sexism?</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/eugenia-cheng" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Eugenia Cheng</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/profile-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">profile books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-78816038-4 (hbk); 978-1-78283442-7 (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">£14.99 (hbk); £12.99 (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">320</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://profilebooks.com/the-art-of-logic-hb.html" title="Link to web page">https://profilebooks.com/the-art-of-logic-hb.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/03-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97a40</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Thu, 03 Jan 2019 08:09:55 +0000Adhemar Bultheel48975 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/art-logic#commentsApplied Mathematics: A Very Short Introduction
https://euro-math-soc.eu/review/applied-mathematics-very-short-introduction
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
If you are a mathematician, try to define what exactly is applied mathematics and in what sense is it different form (pure) mathematics, and you will realize that it is not that easy. Existing definitions are not univocal. Although in most cases you recognize it when you see it. To that conclusion comes also Alain Goriely, who is an applied mathematician himself, in the introduction of this booklet. Yet he isolates three key elements characterizing the topic. First there is the modelling: some phenomenon is (approximately) described by choosing variables and parameters brought together in equations. Then there is of course a whole mathematical machinery to support and analyse the model theoretically, and finally there are the theoretical as well as the algorithmic and computational methods that solve the equations. The digital computers that emerged after WW II have certainly contributed to the development of applied mathematics bifurcating from pure mathematics. These three elements (model, theory, methods) form the framework for the rest of this (very short) introduction to applied mathematics which is intended for a mathematically interested outsider. Like the other booklets in this series it is a compact (17 x 11 x 0.6 cm) pocket book that is entertaining to read, even on a commuting train or during some idle moments.</p>
<p>
The data that an applied mathematician has to deal with are numbers, but these numbers have a certain dimension (length, weight, time,...) and they need to be expressed in proper units (like mks) and at a proper scale. Only when all this has been taken care of in a proper way, one can start building a model to, for example, predict the cooking time of poultry as a function of their weight or try to solve the inverse problem: how fast mammals can loose heat. With the answer to the latter problem, one may deduce something about their metabolism as a function of their volume. Keeping track of the proper dimensions throughout the modelling and the computations is called <em>dimensional analysis</em>.</p>
<p>
Choosing a model is a matter of deciding which are the most influential variables. The finer the model, the more computing time it will require while its predictive power or insight will not increase correspondingly. A simple mechanism to arrive at a model is illustrated with the model describing our solar system. First there was the geocentric system, but anomalies in the observations made Copernicus propose his heliocentric alternative. The more precise observations provided when telescopes were being used (a lot of data were provided by Tycho Brahe), allowed Kepler to derive his laws which fit the data, but it was only Newton's gravitational theory that gave the eventual explanation, not only for Kepler's laws, but for gravitation in general. Nowadays, models are constructed in a similar although a more interactive and more complex way. Observations lead to simple models, that are checked by experiments, which require subsequently refinements of the simple model, which is then checked against new observations, etc.</p>
<p>
Once the model is shaped in the form of equations, it requires theory to analyse under what conditions there exist solutions and what properties these solutions will have. For example one may analyse when they have an explicit solution (in terms of simple functions). If not, the equations can be considered as defining equations for new (less elementary) functions. The celestial gravitational problem of two mutually attracting bodies was generalized to the three-body problem, which was only solved partially by Henri Poincaré who, by doing so, created chaos theory. A deterministic world view had to be left behind and a qualitative analysis of (nonlinear) differential equations was born. The Lotka-Volterra equations describes a prey-predator model has periodic solutions, but with three species involved they will have chaotic solutions. Also the Lorenz equations, a set of three simple differential equations, originally describing an atmospheric convection problem is a famous model generating chaotic solutions.</p>
<p>
When it comes to periodicity, then the wave equation is the example that pops to mind. However when non-linearities are involves, like with seismic P-waves that travel trough the earth mantle, or phenomena like rogue waves, then solitons are involved, which are bump-like shapes that travel along without changing shape. They have a particle-like behaviour, and thus they have potential as carriers of digital information in optical communication, which is an exciting recent research field.</p>
<p>
The applications mentioned in the remaining chapters are computer tomography, the discovery of DNA, and the use of wavelets in JPEG2000 for image compression. Other examples are illustrating that what originally were theoretical developments, eventually turned out to be of the highest importance for applications like complex numbers, quaternions, and octonions (this line of complication was eventually replaced by the concept of a vector space), and knot theory (which found application in DNA modification). Finally large networks and big data are fairly recent topics that are used for describing global phenomena. Even with the complexity and magnitude of these networks, they are still inferior to what a human brain is capable of. Accurate modelling of our brain is momentarily still a (distant) target exceeding our current computational capacity but we are closing the gap.</p>
<p>
The previous enumeration is just a selection of some of the topics discussed that should illustrate what applied mathematics is about. Of course this limited booklet cannot be exhaustive. The approach is partially historical and still manages to refer to topics of current research. While examples are rather elementary in the beginning, towards the end, the topics tend to be more advanced. But even when discussing these more advanced subjects, Goriely tries to convince the reader that even if math is not always simple, still it is fun to do. The many quotations from the Marx brothers (most of them from Groucho) sprinkled throughout the text are funny of course. Goriely even provides a play-list of pop music that you could play in the background while reading (at least some of these he used while writing). This makes it clear that he has enjoyed writing the book and some of this joy radiates from the text when you read it.</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 short survey, Goriely gives examples (rather than a precise definition) of how applied mathematics relates to and interacts with pure mathematics. Applied mathematics fills the gap between the abstraction of pure mathematics and the world we live in. He describes historical models as well as more recent applications and even reaches out to future targets.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alain-goriely" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alain Goriely</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-1987-5404-6 (pbk), 978-0-1910-6888-1 (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">9.99 € (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">168</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/applied-mathematics-a-very-short-introduction-9780198754046" title="Link to web page">https://global.oup.com/academic/product/applied-mathematics-a-very-short-introduction-9780198754046</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a69" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a69</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span>Mon, 02 Jul 2018 09:27:22 +0000Adhemar Bultheel48570 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/applied-mathematics-very-short-introduction#commentsHow to Count to Infinity
https://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="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/marcus-du-sautoy" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Marcus Du Sautoy</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/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><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.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="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Tue, 05 Dec 2017 09:12:34 +0000Adhemar Bultheel48082 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/how-count-infinity#commentsA Mathematical Odyssey. Journey from the Real to the Complex
https://euro-math-soc.eu/review/mathematical-odyssey-journey-real-complex
<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>
It may happen that mathematicians are confronted with the question what the heck they are doing all day except for computing pi since the Greeks had everything written down and there is nothing to be invented. One and one is two and that's all there is to it. With this book Krantz and Parks try to give a nontrivial answer to counter this opinion. Just like Homer's Odyssey describes the adventurous return of the hero Odysseus, presumed dead after the Trojan war, yet eventually slaying Penelope's suitors, this Krantz and Parks book reports on some astonishing successes of mathematicians in the past century, showing that mathematics is far from being dead. With these fourteen examples, the authors illustrate that mathematics is still very much alive and that it has great influence on our lives. Here is the list of the topics.</p>
<p>
<em>The four-color problem</em>. Since its original formulation in 1875 many well known mathematicians had tried to find a proof that 4 colors were sufficient to color any map so that no two neighboring countries would have the same color. It was with a computer-assisted proof by Appel and Haken in 1976 and after many corrections and an acrimonious debate about this being a mathematical proof or not that in 1981 the Math Department of the University of Illinois acclaimed that <em>Four Colors Suffice</em>. By now this kind of proof has been accepted and it was applied in other situations, but long after 1981 alternative proofs were obstinately looked for.</p>
<p>
<em>Mathematics of finance</em> is approached with a long history starting with the origin of interest, compound interest, before arriving at stock markets with the Dutch East India Company. Next, the reader is introduced to financial terms like derivative (a future contract), forward, option, arbitrage, call option, and most importantly how to determine the price of, for example, a call option. Several stochastic models existed already but the grand entrance was for the Black-Scholes equation for which Scholes and Merton got the Nobel prize in 1997 (Black died in 1995). However as the 1987 crash illustrated, models can and need to be improved, which is an ongoing topic of investigation for many mathematicians currently employed in the financial sector.</p>
<p>
<em>Ramsey theory</em> named after F. Ramsey (1903-1930) may be a bit less known. Ramsey was convinced that absolute chaos did not exist. A typical question to answer is "How many elements of some structure must there be to guarantee that a particular property will hold?" For example P. Erdős formulated the following variant "How large should <em>N</em> be so that in a room with <em>N</em> people, at least <em>k</em> people are mutually acquainted or at least <em>k</em> are mutually unacquainted?" Of course the idea is to find the smallest number <em>N</em>. The inconceivably large Graham's number (powers of powers of powers...) is an upper bound for an equivalent problem in graph theory: If <em>N</em> points in the plane are connected with either a red or a blue line (nodes are acquainted or not), how large should <em>N</em> be so that either <em>m</em> of them are red or <em>n</em> of them are blue? The Ramsey number <em>R</em>(<em>m</em>,<em>n</em>) is the smallest value of <em>N</em>, and is, except for some simple cases, only known to be in some interval.</p>
<p>
<em>Dynamical systems</em> are better known in a wider audience, via the popular Mandelbrot and Cantor sets, fractals, the Lorentz attractor, and the butterfly effect. Visual effects and chaotic effects that can astonish and fascinate the observer. A three dimensional Mandelbrot set, the Mandelbulb, is used an an illustration on the cover of the book.</p>
<p>
<em>The Plateau problem</em> of minimal surfaces spanning a closed curve was originally posed by Lagrange in 1760, but it became popular and was named after the Belgian J. Plateau (1801-1883) who observed that soap films were good approximations to such surfaces. Plateau made several observations on curvature and the angle between soap bubbles where they meet. A formula was derived by Enneper and Weierstrass and Costa, Hoffman and Meeks designed a minimal surface that did not intersect itself. A beautiful shape that has inspired several artists. One of the first Fields medals were awarded to J. Douglas for his new approach for the solution of the problem, although an obscure but ill understood paper by Garnier preceded his. Proofs of Plateau's observations were given by J.E. Taylor in 1976 and generalizations of the problem are still under investigation.</p>
<p>
<em>Non Euclidean geometry</em> arises when we leave the 5 Euclidean postulates. Exploring this possibility was a consequence of the efforts to show that the last postulate (only one parallel line is possible through a point outside a given line) was independent of the others. It stimulated J. Bolyai and N. Lobachevsky to come up with hyperbolic geometry where it is possible to draw at least two distinct parallel lines through a point. The consistency was only proved by Beltrami in 1868. It was B. Riemann who revolutionized the way we now think of geometry and Riemannian geometry is what was used by Einstein. Calabi's conjecture of the existence of some nice Riemannian metric on a complex manifold was proved by Yau (1977) which earned him a Fields Medal, and nowadays Calabi-Yau manifolds are an essential tool for theoretical physicists working on string theory.</p>
<p>
<em>Special relativity</em> is another topic that is not too difficult to explain. Einstein got a Nobel Prize in 1921 especially for his discovery of the photoelectric effect. A major campaign was set up to award this Prize also to H. Poincaré (who, among many other things, developed relativity theory parallel to Einstein) but because, as Mittag-Leffer states, the Nobel committee "fears mathematics because they don't have the slightest possibility of understanding anything about it", Poincaré never got it.</p>
<p>
<em>Wavelets</em> revolutionized Fourier analysis and they have a remarkable track in mathematics. Morlet, a geophysicist, came up with an alternative for the windowed Fourier transform, and with the help of Grossmann, a theoretical physicist, developed a theory of frames. Y. Meyer, a mathematician specialized in harmonic analysis, got hold of their papers and this was the start of a new approach to the domain. I. Daubechies later developed an orthogonal set of wavelets, which can only be described by an algorithm. The time between the mathematical formulation and the extensive applications in engineering (e.g. jpeg compression code) is remarkably short as compared to many other mathematical ideas. This is a relatively short chapter, discussing more the history of Fourier analysis while it is rather short on the wavelet stuff itself.</p>
<p>
<em>RSA encryption</em>, named after Rivest, Ahamir and Adleman, is another widely used application. Most probably your browser uses RSA whenever you open a https website, where personal or private information is exchanged. The RSA code is freely available and has never been compromised in the 30 years that it has been around. It is remarkable that it completely rests on an old and simple idea of prime numbers. In particular the difficulty to find large prime factors is the fact that makes it work.</p>
<p>
<em>The P/NP problem</em> follows immediately from the previous topic. P indicates the class of problems that can be solved in polynomial time, i.e., the execution time is a polynomial in the size n of the problem. NP however stands for nondeterministic polynomial, which means that it can be checked in polynomial time that a given solution is indeed solving the problem. It is generally believed that the class NP is strictly larger than P, but that has not been proved as yet. The chapter elaborates extensively on automata, Turing machines, and formal languages.</p>
<p>
<em>Primality testing</em> is again related to the two previous problems. It was not until the AKS algorithm of 2002 by Agarwal, Kayal, and Saxena who generalized some ideas of Fermat (his little theorem) to get a polynomial time algorithm for primality testing, which places this problem in the class P. In the foundations of mathematics it is explained that a mathematical proof traditionally consists of a sequence of statements, in principle derived from some axiomatic system following some rules of (formal) logic. As seen in the first chapter, nowadays the computer can play an essential and active role in the concept of a proof, by algebraic computation or verifying an extensive set of possibilities. The whole chapter is building up via an introduction to formal logic to Gödel's incompleteness theorem.</p>
<p>
Wiles' proof of <em>Fermat's last theorem</em> of 1995 is one of the last triumphs of mathematics in the 20th century. An introduction is given to polynomials over finite fields, elliptic curves, the Taniyama-Shimura-Weil conjecture, and Frey curves, to explain how Wiles, by proving Serre's form of the Frey conjecture actually proved Fermat's last theorem.</p>
<p>
<em>Ricci flow and Poincaré's conjecture</em> are connected in the proof by G. Perelman that he published in three papers in the years 2002 and 2003 on arXiv. The conjecture dates from 1904 and says that every 3-dimensional surface on which a closed curve can be continuously deformed to a point is homeomorphic to a 3-sphere. It was one of the <em>Clay Millennium Problems</em>. It is of great importance because it relates to relativity theory and the shape of our universe. Hamilton introduces differential equations generating Ricci flows that can be considered as geometric evolution equations that result eventually in the geometric objects predicted by Thurston in his geometrization program to describe a possible classification of all n-dimensional manifolds. The mathematical community did not accept Perelman's proof immediately. Certainly too long to his taste and, deceived in the mathematical establishment, Perelman retired from the Steklov Institute and from mathematics. He was awarded the Clay Millennium Prize and the Fields Medal which he both declined.</p>
<p>
Although there is some relation between some of the 14 subjects, each chapter can be read independently. The text is intended for the layman, but some knowledge and affinity with mathematical concepts is advised to help assimilate the material. Some average undergraduate mathematics from secondary school should suffice, and to really enjoy the texts, the reader should not have an aversion of mathematics. The chapters differ a bit in style and extent. Sometimes it starts with an extensive account of the very early history to introduce the topic, some are more mathematical than others, but they obviously serve the same purpose. The treatment is usually not very deep. It's just enough to enlighten the readers and give them an idea of what the topic is about and by whom and how the problem was eventually solved, and what impact it may have on society. It is also made clear that as mathematics advances, the problems can only be solved by applying interdisciplinary techniques. That obviously requires much more collaboration and hence ease of communication between mathematicians to finally crack the problem. Every chapter ends with a short list of references, but it is noteworthy that these are every time preceded by a section called "A Look Back". Why this is done is not very clear, but it seems to collect the crumbles of what has not been mentioned before. Most often that section contains some historical remarks or side stories about some of the main players in the theory or some future perspectives or generalizations. All in all, a book that is in the tradition a Krantz's previous books, which gives the non-mathematician an idea of what mathematicians get so passionately involved with and how that has resulted recently in successes.</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>
By describing 14 recent success stories of mathematics, the authors give the non-mathematician an idea of what mathematicians do for a living, what kind of problems they are looking at, and how their joint efforts eventually resulted in successes.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/steven-krantz" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Steven Krantz</a></li><li class="vocabulary-links field-item odd"><a href="/author/harold-r-parks" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Harold R. Parks</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-verlag-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-4614-8938-2 (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">35,69 € (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">397</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/mathematics/history+of+mathematics/book/978-1-4614-8938-2" title="Link to web page">http://www.springer.com/mathematics/history+of+mathematics/book/978-1-4614-8938-2</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/68p25" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">68P25</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/42c40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42C40</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/91g70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">91G70</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/05d10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05D10</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/37f45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">37F45</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/53a10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53A10</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/11y16" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11Y16</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/83c15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83C15</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/03f03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03F03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/53c44" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">53C44</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/57r60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">57R60</a></li></ul></span>Wed, 13 Aug 2014 06:06:30 +0000Adhemar Bultheel45671 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/mathematical-odyssey-journey-real-complex#commentsWhat's Happening in the Mathematical Sciences - Volume 8
https://euro-math-soc.eu/review/whats-happening-mathematical-sciences-volume-8
<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 the eighth volume of the series "What's Happening in the Mathematical Sciences". The series, published by the American Mathematical Society, started in 1993 and its goal is to shed light on some of the outstanding recent progress in both pure and applied mathematics.</p>
<p>The book is divided into nine chapters which present some remarkable mathematical achievements.</p>
<p>The first chapter "Accounting for Taste" describes how Netflix, a movie rental company, offered a million-dollar prize for a computer algorithm to recommend videos to customers. The first year of competition identified matrix factorization as the best single approach. However to factor matrices with unknown elements the winner team had to devise their own strategy combining matrix factorization with regularization and gradient descent. After three years of competition the award was given to the team called BellKor’s Pragmatic Chaos. This is an example of the use of mathematics behind the scenes in everyday life.</p>
<p>The second chapter "A Brave New Symplectic World" is devoted to the conjecture of Weinstein saying that certain kinds of dynamical systems with two degrees of freedom always have periodic solutions. The conjecture was proposed in the late 1970s as a problem in symplectic topology and solved thirty years later by Cliff Taubes. The remarkable thing is that Taube's solution does not stay within the original discipline and borrows some ideas from string theory, developed by physicist Edward Witten.</p>
<p>"Mathematics and the Financial Crisis" described the collapse of the world's financial markets in 2008. The Black-Scholes formula to estimate the value of call options is explained in detail. For some time this formula was almost perfect but a mathematical model is only as good as its assumptions.</p>
<p>"The Ultimate Billiard Shot" deals with the game of outer billiards proposed in 1959 by Bernhard Neumann. The outer billiard table is infinitely large and it has a hole in the center. The question is: Does the table need to be infinitely large? In other words, is there any way a ball that starts near the central region can spiral out to infinity? The answer depends on the shape of the hole. In 2007, Schwartz proved that for certain shapes, an outer billiards shot cannot be contained in any bounded region. The game of outer billiards may seem a bit restricted but is of interest to mathematicians as a toy model of planetary motion.</p>
<p>The fifth chapter, "Simpatient", deals with the controversial recommendation in 2009 by the U.S. Preventive Services Task Force that women aged 40-49 should no longer be advised to have an annual mammogram. A public health panel used six breast cancer model to take this decision. This is an example of the growing acceptance of mathematical models for medical decision-making, at least behind the scenes.</p>
<p>"Instant Randomness" addresses questions of the following type: How long does it take to mix milk in a coffee cup, neutrons in an atomic reactor, atoms in a gas, or electron spins in a magnet? In many systems the onset of randomness is quite sudden. This abrupt mixing behavior is the "cutoff phenomenon", and the time when it occurs is called the mixing time.</p>
<p>Quantum chaos is the topic of the seventh chapter "In Search of Quantum Chaos". In the 1970s and 1980s chaos theory revolutionized the study of classical dynamical systems. In the atomic and subatomic realm chaos seems to be absent. However, there is a gray zone, the semiclassical limit, between he quantum world and the macroscopic world. Mathematicians have recently confirmed the occurrence of quantum chaos in this zone.</p>
<p>Even in the twenty-first century mathematics reveal new phenomena in the ordinary three-dimensional space. This is the topic of the chapter "3-D Surprises". In 2008 and 2009, some new ways to pack tetrahedra extremely densely were discovered. In 2005, two engineers in Hungary discovered a new three-dimensional object similar to a tetrahedron but with curvy sides. It is the first homogeneous, self-righting (and self-wronging!) object.</p>
<p>Last chapter is "As One Heroic Age Ends, a New One Begins". In the 1950s John Milnor constructed 7-dimensional "exotic spheres" which are identical to normal spheres from the viewpoint of continuous topology, but different from the viewpoint of smooth topology. This was the starting point of a new era of high-dimensional topology. But one question, the Kervaire Invariant One problem was open for more than forty years. In 2009 three mathematicians, Mike Hill, Michael Hopkins and Doug Ravenel, answered this question. But this may be just the beginning of what topologists will learn from the new machinery used to solved this problem.</p>
<p>The book is well written and can be of interest to both mathematicians and general public with some background in mathematics. Many pictures and illustrative diagrams are included in the book.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Antonio Díaz-Cano Ocaña</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">Universidad Complutense de Madrid, Spain</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 the eighth volume of the series "What's Happening in the Mathematical Sciences". The goal of this book, and of the whole series, is to give account for some recent progress in mathematics. The topics covered in the nine chapters of this book range from the high-dimensional topology to quantum chaos and include applications in computer science, medicine, financial markets, ...</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/dana-mackenzie" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">dana mackenzie</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/american-mathematical-society" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">american mathematical society</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">2011</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">ISBN-10: 0-8218-4999-9, ISBN-13: 978-0-8218-4999-6</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">US$ 23</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="/www.ams.org/bookpages/happening-8" title="Link to web page">www.ams.org/bookpages/happening-8</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a06</a></li></ul></span>Mon, 29 Jul 2013 11:29:44 +0000Anonymous45517 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/whats-happening-mathematical-sciences-volume-8#comments