European Mathematical Society - 97A80
https://euro-math-soc.eu/msc-full/97a80
enPatently 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#commentsEuler's pioneering equation
https://euro-math-soc.eu/review/eulers-pioneering-equation
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Since The Mathematical Intelligencer conducted a poll in 1988 about which was the most beautiful among twenty-four theorems. Euler's equation $e^{i\pi}+1=0$ or $e^{iπ}=−1$ turned out to be the winner, and that is still today largely accepted among mathematicians. Even among physicists this is true. In a similar poll from 2004, it came out second after Maxwell's equations. The subtitle of this book is therefore <em>The most beautiful theorem in mathematics</em>.</p>
<p>This may immediately raise some controversy, not about the choice of the formula, but perhaps about what it should be called: a theorem, an identity, an equality, a formula, an equation,... A theorem or a formula applies but these are quite general terms. The others refer to formulas with an equal sign. The term identity assumes that there is a variable involved and that the formula holds whatever the value of that variable. That applies to Euler's identity, which is the related formula $e^{ix}=\cos(x)+i\sin(x)$. The previous formula appears as a special case of this identity. Wilson calls the former formula and "equation" but the reader with some affinity to the French language would probably prefer to call it an equality because the French équation means it has to be solved for an unknown variable. But all the previous names have been used interchangeably to indicate the formula. Calling it <em>Euler's identity</em> may not be the most correct but it is probably the most common terminology.</p>
<p>Whatever it is called, the description, if not <em>most beautiful</em>, then certainly the qualification <em>most important</em> or <em>most remarkable</em>, would be well deserved. It involves five fundamental mathematical constants: 1,0,π,e, and i in one simple relation. The 1 generates the counting numbers. The zero took a while to be accepted as a number but also negative numbers were initially considered to be exotic. Rational numbers were showing up naturally in computations, but so did numbers like √2 and π. These required an extension of the rationals with algebraic irrationals like √2 and the transcendentals like π which results in the reals that include all of them. The constant e (notation by Euler) relates to logarithms and its inverse the exponential function growing faster than any polynomial. Finally the imaginary constant i = √-1 (which is another notation introduced by Euler) was needed to solve any quadratic equation. This i allowed to introduce the complex numbers so that the fundamental theorem of algebra could be proved. The exponential and complex exponential are essential in applied mathematics. Euler's identity is most remarkable because it relates exponential growth or decay of the real exponential, and the oscillating behaviour of sines and cosines in the complex case.</p>
<p>All these links allow Wilson to tell many stories about mathematics that are usually discussed in books popularizing mathematics for the lay reader. There are indeed five chapters whose titles are the five previous constants and a sixth one is about Euler's equation. He does this in a concise way. The amount of information compressed in only 150 pages is amazing. This doesn't mean that it is so dense that it becomes unreadable. Quite the opposite. Because there are no long drawn-out detours, the story becomes straightforward and understandable. For example the first chapter (only 17 pages including illustrations) introduces children's counting rhymes, compares the names for numbers in seven different languages, and compares number systems: Roman, Egyptian, Mesopotamian, Greek, Chinese, Mayan, and the Hindu-Arabic. The latter was popularized in the West by Fibonacci and Pacioli. There are many illustrations not only of the notation of these different numerals in this chapter, but there are in fact many other illustrations throughout the book. This does not increase the number of pages needlessly because a picture sometimes says more than a thousand words. There are no colour illustrations but colour is not relevant for what they represent.</p>
<p>This is not the first book on Euler's equation. For example Paul Nahin. <a target="_blank" href="/review/dr-eulers-fabulous-formula"><em>Dr. Euler's Fabulous Formula</em></a>, Princeton University Press (2006), which is a bit more mathematically advanced, and a more recent one by David Stipp. <em>A Most Elegant Equation</em>, Basic Books (2017), which has more info about the person Euler. In the current book Euler's name appears frequently but as a person he is largely absent. For most of the five constants, separate popularizing books have been written or they are discussed in a chapter of more general popular books about mathematics, too many to list them here. Wilson refers to some of them in an appendix with a short list of additional literature, conveniently listed by subject.</p>
<p>There is of course mathematics in this book. It would be weird if there wasn't. But there is nothing that should shy away a reader with a slight affinity for mathematics. Some of it can be skipped, but the exponential and trigonometric functions, series, and an occasional integral do appear. The more advanced definitions or computations, are put in one of the eleven grey-shaded boxes distributed throughout the book, so that skipping is easy. Most of the topics are placed in their historical context. For example, the history of the computation of π is well represented, and also the history of the logarithms as they were derived by Napier and Briggs and how they relate is nicely explained. There are some notes to explain how complex numbers can be generalised to quaternions and even octonions, and several examples from applied mathematics illustrate the meaning and relevance of the exponential function.</p>
<p>A minor glitch: Albert Girard (1595-1632) who was the first to have formulated the fundamental theorem of algebra, is called on page 116 a Flemish mathematician, which is strange because the man was born in France, but, as a religious refugee, moved to Leiden in what was then the Dutch Republic of the Netherlands. So I do not think that the characterization Flemish does apply here.</p>
<p>The book does not go deep into the subjects discussed, but I liked it because it is quite broad, touching upon so many mathematical subjects, mainly in their historical context, while readability remains most enjoyable notwithstanding its conciseness.</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 in which Wilson gives a popularizing account about the historical development of mathematics. His guidance is Euler's equality that connects five fundamental constants of mathematics: 1, 0, π, e, and i = √-1. Each of these is an incentive to discuss respectively different number systems, how counting extends to negative numbers and eventually the real numbers, the approximation and calculation of π, different logarithms, and complex numbers.</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/robin-wilson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robin Wilson</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">9780198794929 (hbk); 9780198794936 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 14.99 (hbk); £ 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">176</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://global.oup.com/academic/product/eulers-pioneering-equation-9780198794936" title="Link to web page">https://global.oup.com/academic/product/eulers-pioneering-equation-9780198794936</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/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li></ul></span>Sun, 14 Apr 2019 07:22:45 +0000Adhemar Bultheel49288 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/eulers-pioneering-equation#comments99 Variations on a Proof
https://euro-math-soc.eu/review/99-variations-proof
<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>
From the title, you might think that the book is showing different proofs of a crucial theorem that has been reinvented many times and hence has been proved in many different ways. The Pythagoras theorem could be a good candidate with many published proofs, but still it would be hard to find 99 different ways. No, the author wants to illustrate that there are many different styles of communicating mathematics, in particular of giving a proof of some theorem. What the theorem is, is not essential here. The choice of the author is to prove some totally unimportant statement: If $x^3-6x^2+11x-6=2x-2$, then $x=1$ or $x=4$. The cubic equation has indeed a double root at 1 and a simple root at 4.</p>
<p>
The idea of the book is inspired by <em>Exercises de style</em> (1947) by Raymond Queneau, a member of the experimental writers group <em>Oulipo</em> (Ouvroir de littérature potentielle). What "styles" could one think of? There is certainly a tradition of doing mathematics that is inherited from historical mathematical centres in a country. There is definitely a difference between some papers written in a German or a French tradition. When looking at history, then proofs by Euclid or by Newton are certainly different and they are not at all like computer (assisted) proofs. Digging into history along this line of ideas will result in many proofs of the same statement looking very differently. One would come a long way, but still, 99 variations is a lot. Another source for differentiating is the way things are represented: graphical (geometric or other), colours, prefix or postfix notation, hand waving, oral discussion, e-mails, encrypted, sign language,... and there are some fun variations. In this way Ording arrives at 99 proofs. In fact, there is also a proof numbered 0, which has the statement of the problem with a proof omitted, which is indeed a way in which a lemma is often formulated in current mathematical papers. This is nicely represented by the cover of the book which has a ten by ten square grid of black squares, except for the first one which is white. The grid fills almost the whole cover which is possible since the shape of the book is almost square too.</p>
<p>
Most of the proofs are short and take only a few lines. On the back of the page, Ording gives some comments, for example where he got his (historical) inspiration or how to read the proof (for geometric constructions) or explaining the notation, the symbols, or the language, etc. For example the psychedelic proof is just a black-and-white fractal plot of the attraction basins of Newton's method. Such a "proof" obviously requires some explanation. These backside notes give also cross references to related proofs in the book. Exceptionally a proof takes more than one page. The comments by Ording turn the book into a fragmented survey of many historical mathematical factoids. Even if some of the proofs are abstract and incomprehensible for the layman, it can still be considered a popular science book about the peculiarities and trivia of mathematics. We meet proofs as dialogues formatted after one of the oldest Chinese texts on mathematics, but also a dialogue as it would develop at a tea party in a mathematics institute, or a proof can take the form of a screenplay featuring Cardano and Tartaglia, etc. There is a proof in the form of a parody of the collaborative discussion modelled after the Polymath projects by Yitang Zhang and Terrence Tao to work on the twin prime conjecture. Fermi was famous for his proofs scribbled "on the back of an envelope" and that one is represented too with a hand written proof on the back of the page. There are proofs as given during an exam, or during a seminar, or on a blackboard, proofs discussed in a referee report, in a patent application, in blogs, in a newspaper article, preprints on arXiv, a tweet by Cardano (if he would have been able to twitter), ... Among the more surprising ones are the ones using a music score, colour spectra, origami, a slide rule,... In fact most of them are surprising and/or amusing.</p>
<p>
The book can be safely considered as one produced by constrained writing as an oulipo author would produce it, and indeed as Queneau did in 1947, not for mathematical proofs, but for some surrealistic short story. The influence of constrained writing is for example obvious in a verbal proof using only monosyllabic words. There are many more hidden trouvailles like the proof called "Ancient" (in Babylonian cuneiform signs) as opposed to the one called "Modern" (with high level of abstraction). These are not placed next to each other but the first is numbered 16 and the second gets the opposite number 61. It must also have been great fun inventing parodies for styles like doggerel, paranoid, mystical, or social media. But it is more than just fooling around. There is always some rationale behind the way it is presented.</p>
<p>
I love this book. It is so much fun to read, and there are many double layers to be discovered. It is temping to invent some extra variations of your own. It is so much more than a book about mathematics. It is indeed creative writing under constraints. I will pick it up regularly and I am sure there will be more hidden gems to be unravelled that are easily missed in a first reading.</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 an exercise in using different mathematical styles to prove a theorem about the solution of a particular (otherwise not important) cubic equation. It is inspired by, and written in the style of, the constrained writing embraced by the French oulipo writers group, in particular the book <em>Exercises de style</em> (1947) by Raymond Queneau was a main source of inspiration.</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/philip-ording" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Philip Ording</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-0-691-15883-9 (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 USD</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">272</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://press.princeton.edu/titles/13308.html" title="Link to web page">https://press.princeton.edu/titles/13308.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/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li></ul></span>Wed, 20 Mar 2019 09:50:58 +0000Adhemar Bultheel49212 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/99-variations-proof#commentsThe Calculus Gallery
https://euro-math-soc.eu/review/calculus-gallery
<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 slightly corrected reprint of the book originally published in 2005. The fact that it is now made available in the <em>Princeton Science Library</em> series as a cheaper version is a confirmation of its quality.</p>
<p>
Dunham has chosen to tell the history of calculus from its origin, as conceived by Leibniz and Newton, till the moment that Lebesgue redefined Riemann's concept of an integral. Of course there exist several books on the history of mathematics, but Dunham has chosen to tell the story as if he is the intendant of a mathematical art exhibition. He chose a number of key results that he discusses in some detail, that means including the ideas of the original proofs (although translated in a for us readable form). These are the stepping stones that tell us about the evolution taking place. So Dunham walks with the reader through the historical museum and tells us why a particular result is important in the chain of ideas that brought us to our current understanding of the subject, and eventually how the current abstraction became a necessity. The museum where his exhibition is displayed has twelve period rooms corresponding to as many chapters in the book, named after the artist-mathematicians who, if not produced, at least published the result(s). So each chapter has a short introductory biographical sketch but the emphasis lies on the discussion of the mathematics and why these are important in an historical perspective. The museum has also two lounge rooms, two interludes, where there is time to summarize the history so far, looking at remaining problems and at what is ahead, and where a somewhat broader bird's-eye view is given because the twelve mathematicians selected are of course not the only ones that have shaped the history of mathematics.</p>
<p>
The names of the twelve chapters chosen to support the evolution are Newton, Leibniz, Jakob and Johann Bernoulli, Euler, Cauchy, Riemann, Liouville, Weierstrass, Cantor, Volterra, Baire, and Lebesgue. This includes obviously some of the usual suspects but a somewhat surprising name in the list is Baire and one may wonder why Liouville and Volterra are featuring while for example Gauss is not. So Dunham justifies his choice in the introduction. To answer the question which functions were continuous, differentiable, or integrable, one needs to know something about the continuum of the real numbers. Here Liouville was important for the discussion about irrational (algebraic, transcendental) numbers and how close these could be approximated by rationals, somewhat similar to what Weierstrass did for the approximation of continuous functions by polynomials. Volterra was instrumental in helping to answer the question of how irregular a function can be and still be (Riemann-)integrable. He was able to construct some pathological example that had everywhere a bounded derivative and yet was not integrable. Baire fits in this story because with his category theory, functions were finally classified with respect to their irregularity, which settled the discussion.</p>
<p>
Because Dunham digs into primary sources, we learn how also these brilliant pioneers who paved the way, had their struggles with concepts and approaches that for us seem clumsy. But we should realize that our calculus courses are the results of many years of filtration, polishing and reshaping of these original ideas. For example we know how to deal with infinitesimals as quantities that go to zero in the limit, but in the early days, without limits, serious resistance against the new ideas of calculus was raised because the infinitesimals were non-zero at some points and were replaced by zero at others. Manipulations that were considered by opponents to be all but sound mathematics. This issue was only solved with the introduction of the limit by d'Alembert.</p>
<p>
We also see that although Newton's fluxion stands for the derivative, both Newton's and Leibniz' approach was via integration, heavily relying on series expansions for small perturbations. The role of the integral for the origin of calculus can be seen in an historical context where geometry was dominant in solving mathematical problems and computing a surface area is a geometric problem. But calculus gradually moves away from geometry as we read on. Series however remained important issues in the early days. The Bernoulli's as well as Euler have analysed their convergence or divergence, but Cauchy was the one to formulate sound convergence criteria, while Riemann later showed the importance of differentiating between absolute and conditional convergence.</p>
<p>
With Riemann we are back to integration. Integrability was however related to the construction of pathological functions which were often of "ruler type" like being equal to 1 for <em>x</em> rational and 0 for <em>x</em> irrational. Weierstrass could construct a function continuous everywhere and yet nowhere differentiable. So this goes hand in hand with a discussion about algebraic and transcendental irrational numbers (hence the Liouville chapter). With this fundamental discussion of the number system, set theory enters the scene with Cantor's fundamental contributions and Dedekind's cuts. Topological aspects such as density of a subset of an interval has eventually triggered Lebesgue to redefine the concept of the integral to circumvent the problems raised when using Riemann's concept. With this evolution, for the finer details of calculus one has to leave not only geometry but also algebra to take off in a more abstract topological realm.</p>
<p>
Many generations of students are currently instructed in calculus courses, more or less advanced. Some may feel annoyed with the abstraction and may not see why it is needed. This book will reveal how and why their modern calculus course was shaped into its current form. This book is unique in its content because it is not a full history book, and it is not a calculus course. There are however many proofs that require some knowledge of (modern) calculus, and some of them are quite involved. But by restricting the discussion to functions of one real variable, the mathematics stay within the reach of students familiar with a basic calculus course at the level of a first year at the university. The nice thing about these proofs is that they follow the original ideas. Also Dunham's style is pleasant and much more entertaining than a formal course text. Princeton University Press has made a proper choice by promoting this book to their <em>Science Library</em> series and making it in this cheaper form available to a broader readership. My warm recommendation is only appropriate.</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 paperback reprint of the book originally published in 2005. It sketches the history of calculus from Newton and Leibniz till Lebesgue by a selection of key results during the evolution from a geometric/algebraic approach to a more abstract topological framework that was needed to cope with pathological cases when dealing with derivatives and integrals of functions. By restricting the discussion to functions of one real variable the book should be readable for students familiar with a basic calculus course.</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/william-dunham" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">William Dunham</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-0-691-18285-8 (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">£ 24.00</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/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/titles/14169.html" title="Link to web page">https://press.princeton.edu/titles/14169.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/26-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</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>Thu, 13 Dec 2018 14:40:33 +0000Adhemar Bultheel48936 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/calculus-gallery#commentsMillions, Billions, Zillions
https://euro-math-soc.eu/review/millions-billions-zillions
<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>
Journalists reporting on somebody else's results may easily be mistaken in citing the numbers that are not theirs, Perhaps authors change numbers on purpose to bias their arguments. In such cases unprepared and naive readers are easily deceived. In this book you can learn how to defend yourself from mistakes as an author and from deceptions as a reader.</p>
<p>
Kernighan is the guide on a tour where he shows all the number traps that people can easily fall into. There are of course the big numbers like millions and billions mentioned in the title. It is hard to get a mental idea of what they actually mean. As a consequence an interchange of millions and billions is a mistake easily made without being noticed. One should also be aware of what the really big numbers actually stand for when they are indicated by prefixes like mega, giga, tera, and peta. Even exa, zetta, and maybe yotta can come into the picture. In modern texts these terms regularly occur referring to the large amounts of digital data stored in for example the Deep Web. At the other end of the spectrum we should know something about micro, nano, pico, femto, atto etc. when reading about the tiny parts of the hardware on which these data are stored, or even further down the scale when reading about the Theory of Everything where the natural playground is at a subatomic level. Like in mathematics the super large and the super small often match to keep everything within finite boundaries.</p>
<p>
To avoid errors of course the units have to be right. Mistaking barrels for gallons, years for months, or hours for seconds may give quite unexpected and hard to believe interpretations of the numbers that are on display. Besides mistakes in scales, there is the extra complication that there are different systems of units like American miles, gallons, or degrees Fahrenheit that should somehow match with European kilometres, litres and degrees Celsius. Even a mile can have many different meanings in different contexts. All this requires carefully introducing the proper conversions. Just picking up a number from a website may easily lead to such mistakes if the proper conversion is not made. Another typical mistake is to confuse a square mile and a mile squared. This means that you should be aware that if you double the length of the sides of a square you get a surface four times as large. For a volume it is even more dramatic because that will give a volume or mass that is eight times as large.</p>
<p>
The previous examples are possible sources of mistakes. How should we detect them and how could we protect ourselves against malicious attempts to deceive us? We could compare different sources or different ways of computing. When the results are approximately the same we can probably trust the numbers. It is of course useful to check with some numbers from your own experience or numbers that you know like for example the population of your country. Approximate rounded values for these numbers can be used in crude and simple computations to get at least an idea about the magnitude of the number that should be expected. If the number presented to you deviates considerably, either it is fake or your computations are wrong. For example <em>Little's Law</em> is applicable for a simple check in many situations. Given the population of your country and an average human life span, Little's Law allows you to estimate for example the number of people that will turn 64 twenty years from now.</p>
<p>
Usually numbers appear in non-scientific texts with approximate rounded values. If one spots a specious number that is given with many digits, then it is probably the result of a computation or conversion and only the most significant digits or rounded values are actually meaningful. They are probably the result of some conversion, like a mountain over 4.000 meter high should not be referred to as over 13.123 feet. Statistics is another possible source of deception: the median is not the average, a correlation does not imply causation. It might also be interesting to know who did the statistics and the sampling or polling. Results may be consistently biassed towards the results of which some lobbyist or pressure group wants to convince you. Another well known trick is to fiddle with the scales used in the graphical representation of the numbers in pie or bar charts. Numbers representing a percent are again possible pitfalls that can put you on the wrong foot. A percent is definitely different from a percent point and you should also be aware that a percent increase is computed in terms of the lower number, while a percent decrease is referring to a percent of the larger number: a 50% decrease can only be compensated by a 100% increase.</p>
<p>
Kernighan gives many examples of all these issues, mostly from newspapers and websites. He also keeps his readers alert by continuously pushing them to do some mental calculation to estimate some results for themselves. As some kind of a test at the end of the book he gives many such problems that one should be able to approximately solve (he also gives his own estimates): How many miles did Google drive to get the pictures for Street View (for your country)? How long did that take? How much did it cost? Or, if you have a garden with some trees in it, how many leaves do you have to rake every autumn? And there are many of these so called <em>Fermi problems</em> throughout the book. Kernighan gives some tricks to solve them, hence the "test" at the end. However it certainly requires a lot of practising and training which the reader has to do for him or herself to acquire some routine in this,</p>
<p>
In this time of "fake news" and in a society that is more and more spammed by numbers, it seems like problems of numeracy among a general public is gaining interest and public awareness. More books devoted to different aspects of this issue seem to appear lately. Among the earlier examples are the books by John Allen Paulos <em>Innumeracy</em> (1988) and <em>A mathematician reads the newspaper</em> (1996). Kernighan mentions them in his survey of "books for further reading". However several more books appeared since 2010. Almost simultaneously with this one I received <em>Is This a Big Number?</em> (2018) by Andrew Elliott which is also reviewed <a href="/review/big-number" target="_blank">here</a>. Elliott has a more positive approach to the problem: how should I interpret the numbers presented to me (assuming they are correct) while Kernighan is more defensive: how to to stay away from mistakes or deceiving numbers.</p>
<p>
It should also be noted that Kernighan uses American units most of the time, and his examples are mostly related to the American situation, or from the American newspapers. His advise is of course generally applicable, but it might be an extra hurdle to take for European readers who are used to the metric system of metre-litre-gram. It is surprising that Kernighan does not discuss the difference between the short scale (a billion is $10^9$) and long scale (a billion is $10^{12}$) hence also missing the milliard ($10^9$) and billiard ($10^{15}$) and giving different meaning to trillions, and nomenclature higher up. These are also obvious ways to get the wrong numbers cited.</p>
<p>
This book does not need mathematics to read and it is actually not about mathematics at all. There is not a formula made explicit, even though the rule of 72 is explained (it takes 72/x units of time to double your capital when it has a compounding interest of x percent) and an idea is given about what it means to grow exponentially or by powers of 10 or powers of 2. The style of Kernighan is fluent and casual, but not particularly funny. The charm sits in his continuous teasing to make you think of these Fermi problems, and of course in his ample illustrations of how often authors are mistaken in citing numbers and how easily a reader can be deceived.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This book about numeracy learns you how to defend yourself against making mistakes with numbers or to recognize incorrectly cited numbers by explaining what the possible sources of these errors are. On the positive side, one learns how to solve Fermi problems, that is to make rough numerical estimates of certain quantities, using little available data and where the few computations can me made "on the back of an envelope".</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/brian-w-kerninghan" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Brian W. Kerninghan</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-0-691-18277-3 (hbk), 978-0-691-19013-6 (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">£ 17.99 (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">176</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://press.princeton.edu/titles/14171.html" title="Link to web page">https://press.princeton.edu/titles/14171.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/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A80</a></li></ul></span>Wed, 24 Oct 2018 12:08:48 +0000Adhemar Bultheel48773 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/millions-billions-zillions#commentsThe Joy of Mathematics
https://euro-math-soc.eu/review/joy-mathematics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The authors directly address the secondary school student pointing them to mathematical issues that are not covered by traditional curricula. They are of course addressing students in the USA, but most of what they mention applies to the European system as well. I doubt it that most of these young adults will spontaneously read this book for fun, but there are always exceptions of course. Clearly, through these students, the authors are indirectly reaching the teachers, or it may well be the other way around.</p>
<p>
The subtitle of the book: <em>Marvels, Novelties, and Neglected Gems That Are Rarely Taught in Math Class</em> catch the spirit. What are all these tricks, techniques, and theorems which are not usually covered in a regular curriculum because of a lack of time? The authors have organised them in five chapters collecting many of them around a central theme. The first chapter is called <em>Arithmetic Novelties</em>. I hesitate to call these "novelties", unless they are novelties for the student who may read about them here for he first time. The "novelties" are classic arithmetic tools but that may have been forgotten because many computations are performed on computing machines nowadays and not so much in the heads of students anymore. Examples are shortcuts for divisibility checks, formulas to sum numbers or squares of numbers, the Euclidean algorithm, and fun things to know about numbers like palindromic, triangular or square numbers, perfect numbers and the likes, and more material of that style.</p>
<p>
The second chapter collects some algebraic items. Here are some classics like the irrationality of the square root of 2, why a division by zero allows to prove anything true or false, and there are again useful computational methods: the bisection method to find a zero, the Horner scheme for polynomial evaluation, and problems like solving Diophantine equations, generating Pythagorean triples, Descartes's sign rule for zeros of polynomials, and more.</p>
<p>
The geometry topics of chapter 3 take more pages, but that is mainly because these require many graphical illustrations. As you might expect, we find here several less conventional proofs of the Pythagorean theorem and several of its possible generalisations. Also many theorems involve circles (not surprising since the authors published a year earlier in 2016 <a href="/review/circle-mathematical-exploration-beyond-line" target="_blank">The Circle. A Mathematical Exploration Beyond the Line</a>, a book completely devoted to such circle theorems). But there are many other properties as well that involve triangles, spirals, polygons, Platonic solids and star polyhedra, and much more.</p>
<p>
The chapter on probability is relatively short. Here the surprise effect of unexpected results are a central theme. Benford's law, coinciding birthdays, the Monty Hall problem and the related paradox of Bertrand's box, the false positive paradox, and the poker wild-card paradox. Other topics are surprising properties of Pascal's triangle and random walks.</p>
<p>
The last chapter is a collection of miscellaneous problems. About the origin of some of the familiar mathematical symbols, compound interest and the rule of 72 to double your investment, the Goldbach conjecture, countability and the different levels of infinity, properties and constructions of the parabola, the speed of a bicycle as a function of the sprocket wheel used, and several others.</p>
<p>
Anyone who is a bit familiar with the literature on popular and recreational mathematics will find that most items collected in this book are not really novelties, and as a gem, they are not really neglected, but they certainly are rarely taught in math class. However, if you know some teenager who loves mathematics, then this will be a fantastic gift. All the content is up to the level of her mathematics and it are marvels and gems, which are most probably novelties to her. The good thing is that everything is not just raising wonder and surprise, but it is explained why it works and proved when appropriate. It is not a regular textbook tough with formal theorems, proofs and exercises. It is kept at an entertaining level. If, as a teacher, you have some spare time within the strict framework of the curriculum, you can use the book as an inspiration for examples that are stimulating the interest of your pupils.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of popular mathematical topics that are brought at the level of secondary school students but that is usually not included in the regular curriculum because of time constraints. Things are explained and proved at an appropriate level, but it is recreational in the sense that it is not a textbook with formal theorems, proofs and exercises.</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/alfred-s-posamentier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">alfred s. posamentier</a></li><li class="vocabulary-links field-item odd"><a href="/author/robert-geretschl%C3%A4ger" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robert Geretschläger</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781633882973 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 18.00 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">300</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.edelweiss.plus/#sku=1633882977&amp;amp;page=1" title="Link to web page">https://www.edelweiss.plus/#sku=1633882977&page=1</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/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</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/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</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/97a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A20</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li></ul></span>Mon, 08 Jan 2018 20:58:08 +0000Adhemar Bultheel48155 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/joy-mathematics#commentsPower-Up: Unlocking the Hidden Mathematics in Video Games
https://euro-math-soc.eu/review/power-unlocking-hidden-mathematics-video-games
<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>
Matthew Lane is a mathematician who maintains an interesting blog <a href="http://www.mathgoespop.com/" target="_blank">Mathematics Goes Pop!</a> where he links mathematics to popular culture, and this book is perfectly in line with that. Six years before, Keith Devlin in his book <em>Mathematics Education for a New Era: Video Games as a Medium for Learning</em> (A K Peters, 2011), already argued that video games could be a tool for (mathematical) education. Devlin, as well as Lane have some sensible ideas about how to make use of the fact that gaming is so popular among youngsters into a tool or an incentive to learn some mathematics. That this can be obtained by especially designed or pimped versions of existing games is rather obvious, but Lane claims here that any game is suitable. Just analysing the winning strategies or the way in which adventurous problems have to be solved in different games can be concrete examples of a mathematical abstraction. Even, just the creativity that goes into exploring the possibilities within the rules of the game and the endurance with which it is played may promote an attitude of trial and error within the rules of mathematics and a culture of perseverance in solving mathematical problems. Anyway, it would be a waste if the popularity of gaming would not be exploited to serve a higher purpose.</p>
<p>
In different chapters, Lane gives examples of how these ideas can be brought into practice by just relying on some popular video games that were <em>not</em> especially designed with an educational purpose in mind.</p>
<p>
The first chapter introduces several games, in which physical reality is overly simplified. Gravitation and inertia are missing and worlds may be even just two-dimensional. However in the game <em>A Slower Speed of Light</em>, as you may have guessed, the speed of light is lowered so that one moves through the landscape and it will be observed just as relativity theory predicts when you are travelling close to the speed of light. In <em>Miegakure</em> the environment is the familiar three-dimensional setting, but one can move into a fourth dimension to avoid obstacles. Moving to a fourth space dimension is not possible in reality, but there is no problem to experience it in a video game. The gamer can experience a mathematical abstraction or a physical observation that is impossible in real life.</p>
<p>
Chapter two is about guessing games like <em>Family Feud</em> where two teams have to guess the five most popular answers to some question. The popularity of these games dropped drastically after a short time because the number of questions was finite, and hence the questions keep repeating after a while. This can be the hook on which to attach some statistics and combinatorics and to design a procedure to avoid repetition as much as possible by attaching weights to questions that have already been asked. This is like interpolating between drawing balls from an urn with and without replacement, something that simply studying combinatorics mathematically does not offer.</p>
<p>
The pitfalls of voting systems is another popular, yet tricky business to analyse mathematically. This applies not only to politics but also to games in which the user has to grade some components and also to the scores and the ranking of the users themselves. The way in which the player collects his points can be very complicated, and it may not always be clear what will be the score, positive or negative, that can be earned by their actions. Inverse engineering of your final score is not at all a simple problem. But if you succeed, then it should be possible to detect impossible scores, or perhaps screen configurations revealing partial information that is not possible, given the rules of the game. Of course the latter remotely refers to the consistency of a logical system. There are two chapters devoted to this kind of problems.</p>
<p>
Chapter five is all about chasing and shooting. This is the chapter that is the most mathematical or at least the one with most formulas. As far as shooting is concerned, one may consider two kinds of missiles: those that go in a straight line and bounce off walls or the heat seeking missiles that lock in on the target and adapts its trajectory continuously. In the first case, the mathematics involves some simple trigonometry, but still the moving target complicates things, and it becomes really tricky when there are multiple reflections on walls. This is the part that has most of the formulas. The trajectories of heat seeking missiles are not piecewise linear anymore. They can in principle still hit a target that disappears behind a corner. This is a more involved issue and it is worked out to some extent in an addendum. But even a simple interception problem of an enemy missile moving on a straight line towards a target that has to be neutralised by your own missile, also moving in a straight line, is interesting to investigate. A blast with a certain radius can help you still destroying the enemy missile when the interception point is slightly missed. There are some quite interesting mathematics involved here.</p>
<p>
As we progress in the book, the mathematics and the abstraction is cranked up a bit. The next chapter is about computational complexity and the P vs NP problem. These complexity concepts are introduced by explaining Kevin Beacon numbers. This is the distance of an actor to Kevin Beacon measured in coactor-of-coactorship. It is the analogue of the Erdős number which is the co-authorship distance from Paul Erdős, which is quite popular among mathematicians (I wonder why the Erdős number is not even mentioned). Finding these numbers is a shortest path problem in a graph and that is a problem from class P, but finding the longest path or the path of a certain length between two nodes are known to be NP-complete, i.e. easy to check but difficult to solve. So are some problems related to <em>Tetris</em>. Another well known example is the travelling salesman problem. This is a problem a gamer has to solve when he has to pick up some potions, treasures or weapons at fixed places in a maze. Finding a fast algorithm for solving them will earn you instant fame and a 1 million dollar prize from the Clay Mathematical Institute. Games in the class NP are usually the more challenging and perhaps therefore the more attractive ones. There is however little hope that you will crack the P vs. NP problem by playing video games.</p>
<p>
There is a game called <em>Sims</em> which is all about getting (and keeping) friendship relations. Chapter 7 is about modelling such relations between two persons. Several models are proposed in discrete and in continuous time. The latter involves differential equations. It is not explained how to solve systems of differential equations, but solutions are plotted graphically, so that interpretations can be given. This moves seamlessly to the next chapter where nonlinear elements cause chaotic behaviour. For example when a third person competes with the second for the friendship of the first: a three-body problem. Chaotic trajectories may also result when a shell is fired that behaves like a ball on a billiard table. Even when these tables have simple geometries like squares or ovals or when there are a few obstacles inside.</p>
<p>
In a final chapter Lane reflects on how video games can help in solving pedagogical issues. He explicitly refers to Devlin's book mentioned above and to other publications and reports on experiments that have been conducted at several places.</p>
<p>
From this summary, it is clear that this is not about the mathematics of video games which would be much more involved with modelling the physics of the scenes, and the involved mathematics of computer graphics needed for rendering realistic characters. On the contrary, this is all about relatively simple mathematics and logical questions that the gamer could ask spontaneously or with a little help from his teacher. It's the mathematics hidden behind the game, the one not really explicitly visible. The game or its modes of operation can be the hook on which to hang the meaning of some abstraction or it can justify why a certain mathematical concept is useful. The mathematics itself is not really the focus of the book. Differential equations are mentioned but not their solution method for example. Lane just gives some examples of where a game can be an incentive to engage in a mathematical problem, and these problems go well beyond the cuddling mathematics of kindergarten. Lane is certainly convinced of the idea and he has a broad knowledge of the many different games, probably earned with a lot of experience. He does a good job in making his point and the ideas are not naive and they do make sense. If, as a teacher, you are game-phobic and feel like an alien in this virtual world of your students, don't be afraid of this book. Lane does a marvellous job in explaining what all these games do, or at least you are informed about what you need to know, and the book is amply illustrated. We shall not be teaching all our mathematics using games in the near future, but who knows what will happen when the ideas are elaborated further in games especially designed with an educational purpose. It is not unthinkable that they become standard ingredients in our educational toolboxes.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The point made in this book is that with little effort video games can be used as a hook on which to hang a mathematical problem and hence they can be used for educational purposes. By directing the interest of the pupil for gaming towards questions about the rules of the game, or the winning strategies, or the models that were used in the design of the game, this can serve as an incentive to study its abstracter version or to analyse the sequence of events or to generalize the problem, hence to illustrate the usefulness and the meaning of mathematics</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/matthew-lane" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Matthew Lane</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">9780691161518 (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 29.95 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">264</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://press.princeton.edu/titles/10954.html" title="Link to web page">http://press.princeton.edu/titles/10954.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/00a35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A35</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/97a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A20</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97c70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97C70</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97m70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97M70</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97u80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97U80</a></li></ul></span>Mon, 19 Jun 2017 06:54:14 +0000Adhemar Bultheel47725 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/power-unlocking-hidden-mathematics-video-games#commentsThe Magic of Math. Solving for x and Figuring Out Why
https://euro-math-soc.eu/review/magic-math-solving-x-and-figuring-out-why
<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>
Arthur Benjamin is a mathematics professor at Harvey Mudd College in Claremont, CA. He has made it part of his mission to bring mathematics to the general public. He performs before audiences and gave TED talks. He also wrote a book on how to perform better in mental calculation, and he recorded the video course <em>The Joy of Mathematics</em> produced by <em>The Great Courses</em>. Elements of this course are used to write this book. Like so many others, he is an admirer of Martin Gardner, and likes puzzles, magic, and mathemagic. Hence probably the title of this book.</p>
<p>
The title <em>The Magic of Math</em> and the cover picture referring to a magician will make you expect magic tricks with numbers and cards, but do not be mistaken. The message of the book is obviously 'mathematics is fun' with fun-elements omnipresent and there are indeed some suggestions to use mathematical properties (like the casting-out nines check) to surprise your audience, but it becomes in the second half of the book also very course-like in the sense that there are theorems and proofs, all at an elementary level, but still.</p>
<p>
Because everything is kept at this introductory mathematical level, there is much in the realm of numbers (meaning natural numbers) to start with, but we also get some algebra, and later geometry and calculus. To some extent, Benjamin follows the historical evolution of mathematics. He starts with numbers and geometry and deals with the properties of numbers much like the ancient Greek did using essentially geometric elements to 'prove' these properties.</p>
<p>
The first four chapters are generally dealing with numbers and a bit of algebra. There are number patterns (e.g., triangular and rectangular numbers, i.e., numbers that can be arranged in this geometric form). But there are also many patterns to be discovered in Pascal's triangle. Furthermore the reader is instructed about modulo calculus, Fibonacci numbers, and combinatorics to do all the counting. The algebra is essentially restricted to first and second order equations. There are almost no formal proofs here, but evidence is sometimes given on a geometric basis with graphical arguments.</p>
<p>
The sixth chapter introduces 'the magic of proofs' with some elementary examples like a formal proof of the property that "the product of two integers is odd if and only if both numbers are odd". The proof that the square root of 2 is irrational, and the proof that there are infinitely many prime numbers are classics.</p>
<p>
Once the reader is familiar with the rigor of a formal proof, it is time to switch to a more axiomatic environment. The most classic rigorous system is provided by Euclid's <em>The Elements</em>. So the next chapter introduces some elements of geometry and it has more formal proofs, ending in several variants for the proof of the Pythagoras theorem.</p>
<p>
Chapter eight is semi-geometry semi-calculus. It is a discussion of the number pi and how it relates to circular area and circumference but also with mnemonics to memorize its digits and a mock tribute to pi, in the form of a parody of a popular song. The number pi is the best known number that everybody knows about, so it deserves a separate chapter in a popular book on mathematics.</p>
<p>
Somewhat less popular, but mathematically equally important are the numbers e and the imaginary unit i. These are however more 'mathematical', meaning that they are further away from people's daily common experience. Therefore the subsequent chapters are more serious lecture-like dealing with trigonometry, the numbers e and i with logarithms and complex numbers, some elements of calculus such as differentiation, and finally some infinite series. The 'fun-element' returns at the end with the proof that the sum of all natural numbers equals −1/12 (an amazing paradox that you may find on the Web in several versions) and magic squares.</p>
<p>
The book is richly illustrated and it has many grey boxes, called 'asides', that give some more information, or a proof, or something extra that will appeal to the more advanced readers. These can be skipped without any harm.</p>
<p>
From the previous, it will be clear that this is a minimal introduction to the mathematics that one would get at a secondary school level. Is it the mathematics book that I would have loved to I have had then? I doubt it. I did not need that much of show element to be interested. But it might help for others who find more traditional textbooks terribly boring. If, as a teacher, you need to 'force' the math upon some unwilling student, this might be a very helpful alternative.</p>
<p>
On the other hand the market for popular science books, and that includes popular mathematics, has never been as big as it is today. So there is great interest for this kind of books. I doubt that the buyers of this kind of books are the secondary school pupils. Perhaps the main target readers are the adults who lost interest during adolescence and regret that later. So they want to catch up, but in a less scholarly way. For this kind of readers this is a marvelous read.</p>
<p>
And then there are the ones who already became mathematicians or math teachers. They will not find new mathematical elements and they do not need the motivation anymore, but they can always pick up some of the fun elements. I'm sure some of these will be new even for them. So also for them, there is a reason to enjoy the book.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a popularized introduction to elementary secondary school mathematics. There are many fun-elements, but also theorems and proofs. The text is readable for anyone after primary school.</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/arthur-benjamin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Arthur Benjamin</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">2015</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">978-0-465-05472-5</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$26.99</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">336</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.perseusacademic.com/book/hardcover/the-magic-of-math/9780465054725" title="Link to web page">http://www.perseusacademic.com/book/hardcover/the-magic-of-math/9780465054725</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/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</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/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</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/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li></ul></span>Fri, 02 Oct 2015 13:55:27 +0000Adhemar Bultheel46425 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/magic-math-solving-x-and-figuring-out-why#commentsWho killed professor X?
https://euro-math-soc.eu/review/who-killed-professor-x
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The author has won several prizes for his education project <em>Who killed professor X?</em>. Now we have this graphical novel with beautiful graphics by Thanasis Gkiokas also available in English (the Greek original is from 2010).</p>
<p>
It is a detective story in which several of the greatest historic mathematicians become all suspects for a murder on a colleague. A French inspector Gérard, assisted by a mathematician Kurt, has to find the murderer of a mathematician X. The problem is to find out which suspect was at a short distance from the place of the delict. The readers can find out for themselves by going through the files and computing the distances on a map of the hotel. A short sketch characterizing the achievements of each suspect and some reason why they could have wanted to murder X is explained by Kurt to the inspector as their possible guilt is investigated. The file of each suspect is concluded by a statement made under oath about where they were at the time of the murder and some geometric information about the hotel. All the suspects are indicated by their first name but at the end, after the mystery has been solved, some more historical information is provided about René (Descartes), Constantin (Carathéodory), Pierre (Fermat), Isaac (Newton), Blaise (Pascal), Leonhard (Euler), Carl Friedrich (Gauss), Bernhard (Riemann), (Marie-)Sophie (Germain), Évariste (Galois), and Kurt (Gödel). There is one unfortunate play of words that is lost in translation. The X in Greek is pronounced as Chi, while it refers to (David) Hilbert.</p>
<p>
Meanwhile we learn about Hilbert's <em>Wir müssen wissen — wir werden wissen! (``We must know — we will know!'')</em>, about the Leibniz-Newton controversy, the Köninggsberg bridges problem, the golden ratio (explained by Pheidias — a Greek sculptor of 5th century BC), that Marie-Sophie Germain was inspired by Archimedes and wanted to become a mathematician but she had to pretend to be a man (<em>Monsieur Le Blanc</em>) because she was not accepted as a female among mathematicians, that Voltaire described Émily du Châtelet as <em>``She was a great man whose only fault was to be born a woman''</em>, that the revolutionary Galois had problems in getting his results accepted by Cauchy and Poisson, and many other anecdotes and stories.</p>
<p>
The conscientious and studious reader who has been solving the distance problems while reading along will find solutions to check his or her computations at the end of the book. Only mathematics of secondary school is needed. However, the finale is a marvelous <em>coup de théâtre</em>. The surprising outcome is not illogical though since it corresponds to the historical development that was triggered by Hilbert who formulated his great unsolved mathemaical problems at the International Congress of Mathematicians in Paris in 1900.</p>
<p>
This is a wonderful booklet of fiction, but based on historical incidents. In the foreword, Andriopoulos states that the book <em>... is aimed at two kinds of readers: those who have some knowledge of mathematics, and those who have no knowledge of mathematics</em>. Because "some knowledge" is a flexible concept ranging from zero to infinity, it is safe to say that anybody will love the book. It is a fantastic present that you can give to anybody between 9 and 99.</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>
A marvelous graphical novel with beautiful decors of Paris in which the greatest mathematicians become suspects for the murder of a colleague. But don't worry inspector Gérard, assisted by the brilliant mathematician Kurt is on the trail.</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/thodoris-andriopoulos" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Thodoris Andriopoulos</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">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-0348-0883-5 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">21,19 €</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/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</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="http://www.springer.com/gp/book/9783034808835" title="Link to web page">http://www.springer.com/gp/book/9783034808835</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/97-01-97a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01 97A30</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li></ul></span>Tue, 23 Jun 2015 15:13:56 +0000Adhemar Bultheel46275 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/who-killed-professor-x#commentsCakes, Custard + Category Theory
https://euro-math-soc.eu/review/cakes-custard-category-theory
<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>
Eugenia Cheng is a senior mathematics lecturer at Sheffield University (UK) whose domain is higher-dimensional category theory. She has gained some popularity from her YouTube videos where she mixes her love for cooking and for mathematics to show the analogy between both and to show that knowledge of one can help understanding the other. This is exactly what she also wants to achieve in this book as the subtitle promises: <em>Easy recipes for understanding complex maths</em>. She is also active as a pianist, but that is not used much in this context. This illustrates that she is a very enthusiast communicative and talkative ambassador for the popularization of mathematics. She definitely wants to convince people that it is not mathematics that is difficult, but that it is life that is complicated and mathematics is just there to simplify it and make it much easier to solve problems.</p>
<p>
In this book she engages in the task to explain what category theory is to mathematical lay people, which is certainly not an easy or obvious choice. I doubt that a mathematical illiterate after reading the book will be able to tell you what category theory really is. But they will have gotten at least a vague idea. Fortunately, Cheng starts from scratch and is meandering along many other topics along the way. In fact, there are two parts: the first explains what mathematics is about, and the second part explains what category theory is: the mathematics of mathematics. The two parts are not much different. Cheng is following the mathematical river of concepts flowing to its estuary of understanding. She also tells about the many brooks, streamlets, and bourns that feed it. The recipes that she starts each chapter with, are not really essential in my opinion. Of course cookery programs are currently very popular and it is a kind of a opening sentence to start a discussion about something that really matters. The recipes sum up the ingredients and give a brief description of the method, and you will get some ideas of how to deal with certain allergies in your cooking, but I believe you should know something about cooking if you want to really use them since not many details are given. More or less the same holds for the mathematics. The most elementary topics of mathematics are explained, but it is advisable that you know a bit of mathematics to keep apace with Cheng. You do learn that the concept of a number is not that obvious, you learn about logic, what a proof is, how one arrives at an axiomatic system by repeatedly asking `why?', you learn about complex numbers, and a group, about the unsuccessful attempts to prove the fifth axiom of Euclidean geometry, you are convinced that distance is not always the same as a Euclidean distance, and you are introduced to topology. That's a whole lot if you only have secondary school mathematics in you backpack, and certainly if it has been a while since you needed it. All this is wrapped up in much story telling featuring Fermat, Poincaré, and Riemann, and a lot of foody and cookery stuff. And this is just the mathematics part.</p>
<p>
In the category part, relations (morphisms) represented as arrows connecting objects become important. The example of genetic and mathematical family ties (e.g. the Erdős number), are examples. It is all about structures and removing as much as possible to keep the simplest skeleton. Some of the properties of the mappings are explained and simple examples are given, but a clear and strict axiomatic definition is not really given. However you learn about what it can mean to say that structures are `the same', what a monoid or a universal property is, and even what a colimit is. And again the wrapping consists of many stories e.g. about Nelson's last message to his fleet before the battle of Trafalgar, the three domes of St. Paul's Cathedral and Battenberg cakes. I find the discussion in the concluding chapter about truth most interesting. It is about different gradations or meanings of `truth' depending on (1) what we know, (2) what we understand, and (3) what we believe. The most `secure' truth is what is in the intersection of the three.</p>
<p>
The enthusiasm of Cheng is contagious, and she knows how to take the reader along on her hiking tour (not really a stroll in the park). Do not expect that after reading the book you will be ready to start reading current research in category theory. Even the reader that is a mathematician may be somewhat confused because it is too different from the top-down axiomatic and much less verbose books that he probably is more used to. But I do not think professional mathematicians are the first targets that Cheng had in mind when writing this book. Nevertheless, it is entertaining reading stuff that the professional and the non-professional will appreciate.</p>
<p>
To avoid some confusion, let me finally point out that this book is published in UK by Profile Books, but that the same book is available in the US under a different title <em>How to bake pi: an edible exploration of the mathematics of mathematics</em> published by Basic Books.</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 brave attempt to show us that mathematics is there to make our lives easy and not the other way around. Cheng does not use applied mathematics to convince the reader, but instead explains the layman what category theory, her own research field, is about, and how it simplifies structures to their bare minimum, so that the proof of a certain property still holds.</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">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-781-25287-1 (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">£ 12.99</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">302</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/algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebra</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="http://www.profilebooks.com/isbn/9781781252871/" title="Link to web page">http://www.profilebooks.com/isbn/9781781252871/</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/18-category-theory-homological-algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18 Category theory, homological algebra</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/18-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">18-01</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/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li></ul></span>Sat, 20 Jun 2015 07:33:51 +0000Adhemar Bultheel46268 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/cakes-custard-category-theory#comments