European Mathematical Society - Chapman and Hall/CRC Press
https://euro-math-soc.eu/publisher/chapman-and-hallcrc-press
enEssentials of Mathematical Thinking
https://euro-math-soc.eu/review/essentials-mathematical-thinking
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
With a title like "Essentials of Mathematical Thinking" one might expect a philosophical treatise, or possibly a research exposition about cognitive processes and math education. But at the top of the cover, you can see that it is announced as a "Textbook in Mathematics". Since that is what it is: a textbook in mathematics, but a rather unconventional one. Several writers of popular science or recreational mathematics have written books in which they collect mathematical topics that are accessible for a general public and that should illustrate that mathematics can be fun and that there are many practical applications in everyday life involving mathematics. The items discussed in these books can involve integers, prime numbers, geometry, probability, counting problems, logic and paradoxes, games, puzzles, etc. But they are mostly "recreational" or at most they can serve as a source of inspiration for math teachers to embellish their courses and candy-coat the theorems and proofs of the actual textbook.</p>
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Here however, Steven Krantz uses all these entertaining subjects to use them as an actual textbook to teach mathematical awareness and some skills to students who have not the slightest ambition of using mathematics in their further career. For example if undergraduate students are required to broaden their curriculum with some math course. There is no point in imposing mathematical abstraction on them or to force them to memorize proofs of theorems they will never need in life. So the idea is to use all these entertaining subjects to develop their ability to use logic arguments, to solve problems, and to convince them that mathematics is indeed everywhere, but that it is nothing to be afraid of. They will not become better mathematicians in the narrow sense of the word, but at the end of the journey they should have acquired some skills one could call mathematical and they should be more open minded towards mathematics and mathematicians.</p>
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Obviously the book should not have the usual definition-theorem-proof structure and, although formulas have not been abolished completely, there are fewer than in a classical textbook. There are some exercises, but the usual long lists of drilling exercises are absent. Some exercises are meant to drill, some can be more challenging, and chapters are concluded with an open-ended problem. It is like Krantz is telling his story in a stream of consciousness which results in a surprising meandering succession of ideas that will hold the attention of his public or his readers.</p>
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In Chapter 2 the breadth of the field is explored covering many different problems. Some examples: the Monty Hall problem, the four colour problem, minimal surfaces, P vs. NP, Bertrand paradox, etc. This sounds impressive as a starter, but these are actually pretexts for introducing the reader to probability, logarithms,... and to modern tools such as proofs by computer, algebraic computer systems, etc.</p>
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This is followed by seven relatively short chapters. Now problems are solved. Some are classic (When will be the first time after midnight that the hands of an analogue clock will coincide?) others are less classic (Will new years day fall more frequently on a Saturday than on a Sunday? How many trailing zeros will 100! have?...).<br />
To illustrate how ideas are linked, let us consider a section of Chapter 5 as an example. It starts by telling that Kepler derived his laws for the motion of the planets not by solving equations but by analysing observed data (Newton came later), which leads to the meaning of average and standard deviation, which in turn leads to big data and their analysis such as DNA used in forensics, social studies based on Street View and other big sets of data collected by companies such as Google. A remarkable arc that connects Kepler to Google.<br />
Furthermore many of the classics are passing by on the catwalk: the pigeonhole principle, conditional probability, Benford's law, lottery and roulette problems, Conway's Game of Life, Towers of Hanoi, Buffon's needle problem, Euler's characteristic, sphere packing, Platonic solids, voting systems, interpretation of medical tests, facial recognition, wavelets, prisoner's dilemma, Hilbert's hotel and others. Some of these are worked out and actually solved, others are only mentioned as illustration of what is possible, or what they have been used for.</p>
<p>
Up to this point, the text is easily accessible with minimal mathematical background. In the remaining chapters, somewhat more is needed. Chapter 10 is about cryptography (explaining the basics of RSA encryption), the next one gathers some diverse discrete problems (a.o. divergence of the harmonic series, surreal numbers, graphs and the bridges of Königsberg, scheduling problems), and finally a chapter with more advanced problems (Google's Pagerank, needle problem of Kakeya, non-Euclidean geometry, the area of a circle as the limit of the area regular polygons).</p>
<p>
Besides mathematical monographs, Steven Krantz has written books on how to write mathematics, and some books that may be considered as introductions to mathematics for a general public and he won several prizes for his writing. He wrote also one on mathematical education before: <em>How to Teach Mathematics</em> (3rd ed., AMS, 2015) which is about "how" one has to teach. This one is about "what" to teach to a particular type of students. Whether he has experience teaching the "essentials of mathematical thinking" using the material presented in this textbook, I do not know. It might not be a bad idea for students that are somehow obliged to take a math course but that have no the intention to take subsequent courses. I have no information about experiments with this type of course. It would certainly be interesting to know the results.<br />
The text is typeset in LaTeX with the quality of lecture notes. There are many illustrations, but pictures do not have the resolution of high professional quality, and some are not really necessary (a picture of 3 arbitrary dice is not really helpful in solving a probability problem). There are many line drawings too which are usually quite helpful, but by resizing them to fit properly on the page, sometimes circles are distorted and become ellipses or the text in the figure is stretched and resized out of proportion. </p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of mathematical anecdotes and applications of mathematics that are presented in the form of a textbook. It is the intention that this can be used as a broadening course for (undergraduate) students who do not have an ambition to take further mathematics courses. The topics mainly deal with numbers, geometry, probability, and logic. Analysis is less represented. </p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/steven-g-krantz" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Steven G. Krantz</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/chapman-and-hallcrc-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Chapman and Hall/CRC Press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-138-19770-1 (pbk); 978-1-138-04257-5 (hbk); 978-1-315-11682-2 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 44.99 (pbk); £ 115.00 (hbk); £ 40.49 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">336</div></div></div><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.crcpress.com/Essentials-of-Mathematical-Thinking/Krantz/p/book/9781138197701" title="Link to web page">https://www.crcpress.com/Essentials-of-Mathematical-Thinking/Krantz/p/book/9781138197701</a></div></div></div><span class="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></ul></span>Sat, 25 Nov 2017 07:02:54 +0000Adhemar Bultheel48043 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/essentials-mathematical-thinking#commentsComputing: A Historical and Technical Perspective
https://euro-math-soc.eu/review/computing-historical-and-technical-perspective
<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 remarkable book. Written by four authors it consists of a collection of 31 self-contained papers that explain many different concepts related to computing and place them in an historical context. The papers are generally accessible for the layman and relatively short: only 10-15 pages or less. The whole collection is roughly ordered from the early notions of counting to quantum computing. It thus forms a compact encyclopedia of computing involving all aspects such as mathematics, software and hardware. The mathematical techniques and algorithms are obviously braided, and certainly with the early computing machines there was still a mutual influence in the sense that problems needed faster techniques and hardware, which in turn influenced the development of algorithms and the urge to tackle larger and more difficult problems. The hardware in this book is however restricted to describing the first computing machines. A discussion of the spectacular electrotechnical evolution of the hardware, was wisely left out. Computer science has been divorced from electrotechnical engineering some time ago and are co-parenting the computing topic.</p>
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The book starts off with numbers and number systems and the Greek mathematicians (Euclid, Diophantus) and their predecessors. Sometimes theorems are proved or exercises are solved. There is also a chapter on steganography and cryptography. True computing required tools such as the abacus but also the techniques such as described in Fibonacci's <em>Liber Abaci</em> and much later the logarithms. Soon computer science starts to arise with the introduction of more advanced mechanical devices (by Pascal, Jacquard, Babbage, Ada Lovelace, Hollerith). Some were originally made to mechanise certain programmable processes and others were true calculating machines. More general purpose machines were first doing numerical computation on which one could implement some of the methods designed to solve algebraic equations or Gauss elimination for linear systems (the method is explained with an example, and later even LU decomposition is defined). Also the mathematics shifted to a higher gear: introduction of real and complex numbers, the cardinality of an (infinite) set (Cantor), computability (Gödel, Turing) and Boolean algebra. With Turing, a previous topic is picked up again: cryptography and the decoding of the German <em>Enigma</em> machine during WW II. Both the <em>Enigma</em> and the decoding machine (the <em>Bombe</em>) at Bletchley Park were still electromechanical but soon the first electronic computers appeared: the ENIAC and several others, while Von Neumann, Turing and others developed theoretical models. Soon numerical analysis became a flourishing discipline as a follow up of methods like Gauss elimination and Newton's root finding.</p>
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</p>
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The enumeration above illustrates that the spectrum covered is really broad. Not only are the topics placed in their historical context, also the persons who were the key players in the evolution are briefly sketched. The concepts are explained and should be understandable by a general public, although some chapters may require more effort than others. According to the authors the primary purpose of the book is to provide additional reading for science and engineering students. There are few illustrations and some of them could even be left out. A pictogram-like illustration of a glass tube or a plot of 4 by 4 grid do not add much to the understanding of the concept. With so many topics covered, the extensive subject index is very useful. Of course, it is impossible to cover all the topics in detail and the specialist will not find many things that are new to him (except maybe some of the history). Practically nothing is found about compilers, multi-media, human-computer interaction, genetic algorithms, etc. However, for a layperson, this an excellent introduction to the subject. Probably if such a person needs an explanation about some term in the domain of computing he or she would probably look it up in Wikipedia. Every paper in the book has a list of references, many of which are pointing to Wikipedia pages anyway. But if the book is not used as a lead and the Internet is explored freely, then a possible danger is that there is an overdose of information, sometimes too specialised and the explorer may wander off and get lost or drift away from the subject. This book gives a valuable compact, focused, and coherent alternative.</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">A. Bultheel</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">KU Leuven</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>
n 31 short (10-15 pages or less) stand-alone papers, the four authors bring a survey of many aspects related to computing. The papers place the topics in their historical context and also the sequential chapters are somewhat ordered along a timeline: from original number systems to quantum computing. Many terms and topics are briefly explained, sometimes an example is worked out or even a proof of a theorem is included. Thus not only the historical, but also the technical aspect is well represented. It may be considered a compact encyclopedia of computing for a general public.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/yoshihide-igarashi" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Yoshihide Igarashi</a></li><li class="vocabulary-links field-item odd"><a href="/author/tom-altman" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Tom Altman</a></li><li class="vocabulary-links field-item even"><a href="/author/mariko-funada" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mariko Funada</a></li><li class="vocabulary-links field-item odd"><a href="/author/barbara-kamiyama" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Barbara Kamiyama</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/chapman-and-hallcrc-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Chapman and Hall/CRC Press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-4822-2741-3 (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">$59.95</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">350</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.crcpress.com/product/isbn/9781482227413" title="Link to web page">http://www.crcpress.com/product/isbn/9781482227413</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/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</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/01-08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-08</a></li></ul></span>Thu, 12 Jun 2014 09:05:46 +0000Adhemar Bultheel45566 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/computing-historical-and-technical-perspective#comments