European Mathematical Society - 01a05
https://euro-math-soc.eu/msc-full/01a05
enRepublic of Numbers
https://euro-math-soc.eu/review/republic-numbers
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In twenty short biographical chapters it is sketched how the role of mathematics in the American society and its educational system has evolved from the early 19th till the late 20th century. That is one chapter per decade, but the life span of the individual mathematicians is of course wider: From Nathaniel Bowditch (1773-1838) to John Nash (1928-2015). In the 19th century, the US were expanding and fighting for independence. Importing slaves was gradually abolished which entailed a civil war between the Northern and Southern states. In the 20th century they participated in global conflicts and survived a cold war. Around 1800, there were only nine colonial colleges, (for white men only), and they mainly trained lawyers, physicians, and clergy. In rural regions teaching to read and write was for the lucky ones and it was forbidden to teach slaves. In the 1990's, there are numerous renowned universities and a regular educational system was established, with mathematics taking an important place at all levels of education. How did this come about? That is what Roberts is illustrating with this selection of 23 biographical sketches (some chapters treat two persons simultaneously). He did not take the leading mathematicians to illustrate the evolution (only few were famous) but there is a diversity of characters and people who were in some sense related to mathematics, and often they were involved in educational issues.</p>
<p>
Here are some names from the first of these two centuries. Simple calculations were sufficient for every day life in 1800, except for navigation which required some knowledge of celestial mechanics. Nathaniel Bowditch taught himself mathematics which he needed as a sailor and wrote a book on navigation and later translated work of Laplace. Sylvanus Taylor had some education when entering the military. Later he became the director of West Point, the US military academy that he modeled after the Ecole Polytechnique in Paris and whose alumni played an important role in professionalizing mathematics in other places. Abraham Lincoln did not become a mathematician, but in his youth, he maintained a scrap book with elementary mathematical problems. Only some of its pages have been recovered. Catherine Beecher and Joseph Ray were authors of popular math text books, and Daniel Hill was a popular educator at West Point. J.W. Gibbs became famous as a mathematical physicist with his work on thermodynamics. Charles Davis was a naval officer who supervised the computation of the <em>Nautical Almanac</em>. and was later superintendent of the Naval Observatory. After the civil war (1861-1865), the educational system became more tolerant for women, Christine Ladd was one of the first women to become a researcher at John Hopkins University. She fulfilled all the requirements for a PhD but it was only awarded 44 years later in 1927 when she turned 80. Kelly Miller is an example of an African American who attended the "black" Howard University, and wrote a math textbook and essays on popular mathematics. H. Hollorith, known from the punch cards named after him, was also founder of the Tabulating Machine Company, which later grew into IBM and E.H, Moore is a mathematician known for several things like the Moore-Penrose inverse. He had some students that became famous mathematicians: G. Birkhoff, L. Dickson, and O. Veblen. Those names bring us to the end of the 19th century, with data processing on the horizon and mathematics and mathematicians being imported from Europe on a larger scale raising mathematics to a higher level.</p>
<p>
The list of names from the 20th century is started with E.T. Bell, a popularizer of mathematics whose <em>Men of mathematics</em> became a classic. By this time, education had been formalized. Classes were split according to the age of the pupils, lessons were separated by a bell signal, and schools had a non-teaching management. The <em>Mathematical Association of America</em> (MAA) was established in 1915 as an offspring of the <em>American Mathematical Society</em> (AMS). The <em>National Council of Teachers of Mathematics</em> (NCTM) with the first president Charles Austin was founded in 1920 as a follow up for the <em>Men's Mathematics Club</em> of the greater Chicago area. Edwin B. Wilson was a student of Gibbs and became mainly involved with statistics. The couple Liliane and Hugh Lieber are known for their series of booklets popularizing math and science with text in free verse format for easy reading by Liliane (maiden name Rosanoff, an emigrate from Ukraine) and drawings by Hugh. Their best known title is <em>The education of T.C. MITS: what modern mathematics means to you</em>. T.C MITS stands for The Celebrated Man In The Street. With WW II, computers came into vision and Grace Hopper designed a computer language that was a precursor of what later became COBOL. Izaak Wirszup studied mathematics under Zygmund in Poland, and survived a Nazi concentration camp. Zygmund, who had escaped the Nazis, invited him to the US where Wirszup became mainly involved in math education. The 1960's was the period where African Americans were fighting racial segregation and Edgar L. Edwards, Jr., was one of the first black teachers at the University of Virginia. Also Joaquin Diaz, although an American citizen from Puerto Rico, was subject of racial discrimination because he was considered Hispanic, and non-American. As an applied mathematician working on fluid dynamics, he was involved in NACA (precursor of NASA). The <em>math wars</em> of the 1980's was the fight over traditional versus "new" mathematics that was abruptly introduced in the US, a reform supported by the NCTM. Frank Allen, who was a believer in the original ideas of <em>New Math</em>, and who had been involved in NCTM became an active polemicist in the debate. The last man in the row is John Nash whose life is well known because of the biography <em>A beautiful mind</em> by Sylvia Nasar and the eponymous film.</p>
<p>
This enumeration of names shows that Roberts is not focussing on mathematical research at university level, but rather at the historical evolution of mathematical education at a lower level, which is of course not independent of what happens at the universities. Why these names? I guess any list of names can be criticized, but I think Roberts chose a good mixture of sex and race, that somehow represents how political and social circumstances have influenced the mathematical education. In the beginning, navigation and the military interest were stimulations for doing mathematics. The military definitely remained to have an important influence and WW II has given a boost to the development of math and science in the US because of the many scientists that fled Europe for the Nazis, which made a <em>Space Race</em> possible during the <em>Cold War</em> period. The latter events are however more important at a research level, and that is not so present in this book. Nevertheless the USSR having Sputnik first is related to the forcing initiative to introduce the <em>New Math</em>.</p>
<p>
Roberts has assigned one particular year to every chapter. Each chapter starts with an epigraph and the description of a particular event that happened in that year to the person that is going to be discussed. That introduction takes only one to three pages and should serve as an appetizer for the longer biography that is following. There is some discussion of the mathematics but it is nowhere technical (no formulas), and there is a photo of each of the mathematicians discussed (except for Abraham Lincoln who is not a mathematician anyway). There are notes and references for what is mentioned in the text but no extensive list for further reading. The book is a very readable survey that will be of interest to any mathematician and non-mathematician alike, but maybe more so for those who are particularly interested in the history of math education.</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>
With 20 short biographical chapters, this book illustrates how the US evolved from the early 19th century with schools where children learned to read and write while mathematics was mainly of interest to navigators and astronomers to the end of the 20th century where mathematics had become a main ingredient at all levels of education.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-lindsay-roberts" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Lindsay Roberts</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">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">9781421433080 (hbk), 9781421433097 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 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">252</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="https://jhupbooks.press.jhu.edu/title/republic-numbers" title="Link to web page">https://jhupbooks.press.jhu.edu/title/republic-numbers</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/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</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/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a60</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</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>Fri, 20 Dec 2019 14:36:53 +0000Adhemar Bultheel50113 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/republic-numbers#commentsTales of Impossibility
https://euro-math-soc.eu/review/tales-impossibility
<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>Squaring the circle (i.e. finding the side of a square whose area equals the area of a circle using only compass and straightedge) is probably the best known problem that became iconic for representing something that is impossible to do. But the classical geometric problems from Greek antiquity, equally impossible to solve with these instruments are doubling the cube, trisection a general angle, and constructing a general regular polygon. For the latter two problems it could be done in some cases, but not for a general one. These problems have attracted many cranks who claimed to have solved the problem until Pierre Wantzel published his theorem in 1837 proving that these problems were impossible to solve (which by the way did not stop cranks to continue spamming editors with their solutions).</p>
<p>Richeson describes in this book the history of these problems, and it is quite interesting to read that just trying to solve these problems gave rise to a lot of investigation that resulted in proper theorems. Progress towards the solution was only possible when algebra was introduced and linked to geometry and number theory. But of course the story starts with the definition of the problems and it is illustrated that solving them corresponds to the following problems respectively: given a segment of length 1, construct one with a length $\pi, \sqrt[3]{2}, \cos(\theta/3)$ for arbitrary $\theta$, or $\cos(360^\circ/n)$ for arbitrary $n$. There is a lot to tell about the mathematics as it was practised in Greek antiquity. They had natural numbers of course, but they did not consider other numbers as such. In their geometric framework, it were rather ratios of lengths of line segments or of areas of geometric figures. The number $\pi$ as the ratio of the circumference of a circle over its diameter is an universally known example. However this did not work so well with the area of a circle and its radius since the dimensions of an area and a length did not match. What made sense was comparing the area of a circle and the area of a square, which explains the problem of squaring the circle. Sticking to these ratios suggests that they considered all quantities as being commensurable, that is multiples of a fundamental unit, so that their number concept is essentially one of rational numbers. However $\pi$ for the circle and $\sqrt{2}$ in the Pythagoras theorem were known examples of non-rational numbers which they tried to approximate.</p>
<p>Richeson skims the most important mathematicians of antiquity to illustrate how they dealt with $\pi$ and how they attempted to find quadratures, that is to find squares or rectangles that had the same area as the area of another geometric figure. That could be a circle, but also other ones that showed up in their quest to solve the circle problem, like lunes or some parts of a circle or part of another conic. But it was Archimedes who came up with many formulas and with bounds for $\pi$. Since the compass and straightedge were not able to solve the problem, people tried to relax on these restrictions. With neusis constructions or marked straightedge some of the unsolvable problems became solvable. In the centuries that followed, many ingenious instruments were designed to produce all kinds of curves (quadratrix, conchoid, limaçon of Pascal, spirals, carpenter's square curve, ...). On the other hand, one could try to find out what could be done with less than compass and straightedge. For example what if the compass is rusty and has only a fixed angle, or what if we only had a compass. Georg Mohr was the first to prove (1672) that with only a compass one can do everything that can be done with compass and straightedge. He was forgotten and Mascheroni re-discovered this much later. A straightedge alone however can not do the same job.</p>
<p>When algebra was introduced in Europe by the Arabs, mathematicians concentrated on solving equations using formulas, rather than by geometric constructions. When people started to represent curves by algebraic formulas (traditionally attributed to Descartes), the idea of constructible numbers was born and the first impossibility claims emerged. More complicated quadratures and better approximations for $\pi$ were computed, certainly with the newly invented calculus. With complex numbers new construction methods for regular $n$-gons were produced. They also gave formulas for solving polynomial equations and that paved the way for Pierre Wantzel (1814-1848) to eventually come up with his theorem about the degree of the minimal polynomial of a constructable number from which followed a precise statement about what was possible and what was impossible to produce with compass and straightedge.</p>
<p>Richeson is able to bring the story as a popularizing book about mathematical history with a brief characterization or biography of the mathematicians involved and of course the evolution of mathematics from geometrical ideas of antiquity to the algebraic number theory of Wantzel. It is however far from a "hard core" history book. It has many citations (all in English) and there are many notes at the end, but these are all informative and do not disrupt the reading. The impossible problems discussed are easy to understand and have attracted many mathematical hobbyists in the past. So the discussion in this book is easily accessible from a mathematical point of view. Although every chapter is somehow related to the four problems, Richeson takes a broad view and the computation of $\pi$ or the constructions of curves with mechanical instruments can hardly be called diversions from the main theme. Most amusing are the intermezzo's that he calls "tangents" after each chapter. These give some diversions of all sorts. The first one about how to recognize a mathematical crank is particularly amusing. There are others like what geometry can be done using toothpicks (line segments with a fixed length), how to compute $\pi$ at home, what geometric constructions are possible using square origami paper, there is the story of the Indiana $\pi$-bill in which Edwin Goodwin tried to pass his circle squaring by law, etc. The whole book, both informative and amusing, is a highly recommended read.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>In this book Richeson explores the history of the classical Greek (geometric) problems that are impossible to solve with compass and straightedge: trisecting a general angle, squaring the circle, doubling the cube, and constructing any regular polygon. The stories of these problems and related ones are traced until they were proved to be impossible by Pierre Wantzel in the 19th century.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-s-richeson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David S. Richeson</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">9780691192963 (hbk), 9780691194233 (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">£ 24.99 (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">456</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="https://press.princeton.edu/books/hardcover/9780691192963/tales-of-impossibility" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691192963/tales-of-impossibility</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/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</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/01a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A20</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/51-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-03</a></li></ul></span>Mon, 25 Nov 2019 09:10:21 +0000Adhemar Bultheel49946 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/tales-impossibility#commentsDo dice play god?
https://euro-math-soc.eu/review/do-dice-play-god
<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>Stewart considers six ages of uncertainty. Clearly people have always been fascinated with the future and have tried in many ways to remove its inherent uncertainty. So there was an age of belief in external powers such as gods, oracles, horoscopes and reading the future from the bowels of a slaughtered goat.</p>
<p>With the development of mathematical instruments like Newton's laws of motion it was possible to predict the trajectories of the planets and to describe the dynamical behaviour of objects on earth. This triggered the conviction that we were living in a deterministic mechanical world organized like a clockwork and that everything was predictable provided that we could measure all initial conditions and all the parameters involved. Uncertainty still existed but only as the consequence of our inability to measure everything with sufficient precision.</p>
<p>Gambling is also something of all ages, but since around the sixteenth century, patterns were observed and ideas of frequencies of random events and the probability of how often they can be expected became useful tools for gamblers and they were successfully applied. Cardano, Fermat, Pascal, Huygens, and Jakob Bernoulli developed the basics of the theory and coin tossing and rolling dice became common instruments to generate random sequences. Observation errors were also considered to be a random phenomenon. Their analysis showed the bell shape of the normal distribution when sufficiently many are accumulated. Linear regression and least squares fitting were born. Quetelet started to apply this kind of analysis, originally used by astronomers and physicists, to social and other data, and this became the origin of expectations and an abstract, non-existing, "average person" was distilled from the data. This is how gradually statistics came about. But there were problems since probabilities seemed to change depending on prior knowledge leading to fallacies and paradoxes contradicting common sense. Bayes eventually formalized all this with formulas. Ardent discussions about Bayesian versus frequentist interpretations were the result. Still today many counter-intuitive results can be the origin of a lot of Fake News.</p>
<p>The fourth age started at the beginning of the twentieth century when mathematicians thought to have uncertainty well under control. But then nature forced quantum mechanical mysteries upon the experimental physicists, and uncertainty became an inherent property of the world we live in.</p>
<p>But also in mathematics, uncertainty was reintroduced when mathematicians started to model nonlinear dynamical systems. These are deterministic, but they can be supersensitive to tiny perturbations, a phenomenon popularized as the butterfly effect. Thus in this fifth age even deterministic systems became unpredictable.</p>
<p>The sixth age is the age we are living in today. Since uncertainty is not going to go away, mathematicians and scientists are trying to manage uncertainty. Sometimes we can even use it to our advantage, but there are still many open problems to solve.</p>
<p>It is clear that there is no crisp boundary between these periods. There are for example even today still people believing they can read the future from tea leaves or they think they get messages from "the other side". Therefore also Stewart cannot separate and treat these six ages in a strict chronological order. The eighteen chapters are more thematic and some topics may require to go way back in time to trace the origins. However, as we read on, we see how insights into uncertainty is growing and how we can bring it somewhat under control.</p>
<p>Stewart has written many books already and knows better than anyone else how to bring a story about mathematics to a broad audience. So the mathematics are painlessly made crystal clear. What is most interesting here are the side tracks. Among these I count the fact that physically throwing a dice or tossing a coin is not as random as one would expect. There are also these seemingly impossible results like if a family has two children and you know one of them is a girl, what is the probability that they have two girls, which is very different from the problem where you know that the eldest is a girl. Stewart also gives some confronting examples of people found guilty in court based on wrong statistics.</p>
<p>On a more theoretical side there is a good discussion of difficult concepts like entropy, information and the arrow of time. Of course quantum physics is more difficult and requires a rather extensive discussion. Entropy as well as quantum theory is still today subject to different interpretations and Stewart adds his own vision to the discussion. He is also explicit about Bell's theorem (1964) which shows that the EPR (Einstein-Podolsky-Rosen) paradox is inconsistent with the theory. Stewart explains some loopholes in Bell's theorem that have been raised and adds some of his own.</p>
<p>That known uncertainties are most influential on our modern society is illustrated with other examples. Strange attractors and the dynamics of weather forecasting are explained, and how climate change cannot be denied, and what mechanism is playing in the consequential disasters, the so called extreme events. We are still suffering from the consequences of the 2008 crisis in the bank sector, and there are heated discussions by people objecting against vaccination. These are two other important topical issues of our society today. Finally it is shown that for simulations it is important to generate random sequences, but it is a complicated problem to generate one that is "as random as can be". And how should randomness be measured anyway?</p>
<p>So this is far from a dull introduction to probability theory and statistics. It is a lively story with historical roots but with many relevant references to how managing uncertainty is important for our everyday life, as well as for the big challenges that our society is facing today.</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>Ian Stewart deals in his characteristic way with the history of uncertainty. This starts with the belief in gods, ghosts or horoscopes to deal with an uncertain future. Then probability and statistics were developed to measure the amount of uncertainties about the future as it is computed in simulations, but eventually it turns out that we live in an inherently uncertain world of quantum physics and chaotic dynamical systems where we have to learn to manage uncertainty and even employ it to our advantage where possible.</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/ian-stewart" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ian stewart</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/profile-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">profile books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781781259436 (hbk), 9781782834014 (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">£20 (hbk), £15.80 (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">304</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/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><li class="vocabulary-links field-item even"><a href="/imu/probability-and-statistics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Probability and Statistics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://profilebooks.com/do-dice-play-god.html" title="Link to web page">https://profilebooks.com/do-dice-play-god.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/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/60c05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">60C05</a></li></ul></span>Mon, 25 Nov 2019 08:52:18 +0000Adhemar Bultheel49945 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/do-dice-play-god#commentsCalculus Reordered
https://euro-math-soc.eu/review/calculus-reordered
<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>
For reasons of sequentiality, an elementary calculus course is usually organising the topics that are discussed in the order where first limit is defined, which is needed to define continuity, then the derivative as the limit of a ratio, and only then the integral as the limit of a Riemann sum. The same order can be repeated for functions of a discrete variable (sequences) where the variable is usually denoted as an index. The limit can only be considered when the index goes to infinity (the only accumulation point), continuity and derivative don't make much sense (unless one wants to discuss finite differences) and (improper) integrals correspond to (infinite) sums, and the convergence of series. At the beginning some functions are assumed known: algebraic functions are a minimum. Sometimes also goniometric functions, and possibly the exponential and logarithm, but in principle all of these and the more "advanced" ones can be defined later. Functions are "defined" when no name exists for a converging series or a primitive function of an integral. The logarithm for example is the integral of 1/x, and the exponential function is its inverse. At least this is how I organized my lecture notes, but this is not always the order used, and it is also not in this order that all these concepts were developed historically.</p>
<p>
What Bressoud does in this book is looking at the contents of a calculus course from an historical perspective. In what order were all these concepts developed, and where do all the well known theorems come from? In fact analysis came relatively late. The Greek were mainly doing geometry using rational numbers, and yet they computed the volume of a sphere and the area of a circle. Then the algebra came to Western Europe through the Arabs, and only then analysis took off with Newton and Leibniz who were already manipulating series or at least truncated series as interpolating approximations. It is only when analytic geometry bridged the gap between algebra and geometry, that analysis took over as the dominant tool for solving the practical analytic problems. The limit and its geometric interpretation came only very late.</p>
<p>
The first chapter of this book covers the period up to and including Newton and Leibniz. The second is about the further evolution of calculus, analytic geometry, the logarithm, differential equations, waves and field theory, culminating in the Maxwell equations. The emergence of Taylor and Fourier series are covered in chapter three, but only in chapter four, we find convergence criteria for series. This convergence, just like the proper definition of a derivative, and other concepts that are defined as the result of a limit, can only come to a conclusion by including the limiting value by upper and lower bounds approaching each other. Only using these bounding inequalities, will eventually lead to the proper concept of a limit. This new concept allowed a previously unseen expansion of analysis, requiring to rethink the concept of a function, since it was realized how exotic some functions can be, like everywhere continuous and nowhere differentiable. Bressoud's historical survey illustrates that the order of our calculus course is inverting history. First came accumulation (integral), then ratios of change (derivative), then sequences of partial sums (series) and only in the end the algebra of inequalities (limit).</p>
<p>
In an appendix, Bressoud adds his thoughts about how calculus should be taught. It is often the case that because of time restrictions that the integral is introduced as an anti-derivative, leading to cookbook recipes to compute integrals. Given the current technology, this is indeed a waste of time when there is no insight of the integral as a summation brought to a limit. Similarly teaching formulas to compute derivatives without seeing the derivative as a limiting process of ratios of change doesn't make much sense either, and series should be introduced as the limit of the sequence of its partial sums. And finally the limit is the result of a sandwich principle where upper and lower bounds approach each other. All of these insights are essential when looking at numerical analysis where exactly these insights are the elements that compute integrals, derivatives, and evaluate transcendental functions.</p>
<p>
As I have been teaching elementary courses in algebra, analysis, and numerical analysis, I can fully appreciate Bressoud's conclusions that I described in the previous paragraph and I fully realize the importance of algebra as an essential element in the development of calculus, certainly when one moves to functions of several variables, (linear) algebra, matrices and vector spaces become essential. However, it is somewhat unclear if Bressoud is promoting to keep also the historical order as he describes it in this book: integration, differentiation, series, limit. In my opinion the limit is the missing link that lets the whole building of calculus fall into a logical sequence and hence it should come first. It would be a waste of time to re-live the trial and errors of history in a calculus course. Apart from the pedagogical conclusions, the book gives a nice survey of how the main achievements and theorems that any student meets in a calculus course came about. Moreover, it is shown that the historical approach is sometimes quite different from what is written in the lecture notes, because the mathematical tools and the objectives of the person developing them were in those days quite different from what is available to a student in the 21st century.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The book is a survey of how the main ideas that underpin a modern calculus course were developed in their historical context. Based on this, Bressoud draws some conclusions about how we should teach a calculus course. An approach following the historical origin will be much closer to intuition and have didactic advantages.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-m-bressoud" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David M. Bressoud</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">9780691181318 (hbk), 9780691189161 (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">£ 24.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">242</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/13397.html" title="Link to web page">https://press.princeton.edu/titles/13397.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/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/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</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/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</a></li><li class="vocabulary-links field-item odd"><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 even"><a href="/msc-full/97d40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97D40</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97i99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97I99</a></li></ul></span>Mon, 05 Aug 2019 09:54:15 +0000Adhemar Bultheel49603 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/calculus-reordered#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#commentsInfinity: A Very Short Introduction
https://euro-math-soc.eu/review/infinity-very-short-introduction
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is one of the small booklets (literally pocket books: 174 x 111 mm) appearing in the Oxford series <em>Very Short Introductions</em> that treat diverse subjects from accounting to zionism. Infinity, is a concept mainly of importance and practically useful in mathematics, but it has also philosophical and even religious aspects. Stewart is as broad as "a very short introduction" allows and adds a lot of history to his discussion. So much is to be told on only 143 small pages. Although there is obviously a lot of overlap, Stewart's treatment is wider than Marcus Du Sautoy's <a href="/review/how-count-infinity" target="_blank">How to Count to Infinity</a> and Eugenia Chen's <a href="/review/beyond-infinity-expedition-outer-limits-mathematics" target="_blank">Beyond Infinity</a> who stay more on a mathematical playground.</p>
<p>
Infinity, and certainly the infinitely large, has long been something fuzzy that was discussed on a philosophical basis. The Greek were arguing over a distinction between an actual (existing) infinity and a potential version, i.e. that "something" that is beyond all natural numbers, which is never reached by enumeration. They got away with the infinitely small by their concept of commensurability in what was mainly a geometric approach to mathematics. The infinitely small was beyond any possible subdivision of a finite length. Their fundamental common measure was thus finite and that led Zeno to his paradoxes. The infinitely small was somehow tackled when calculus was developed by Newton and Leibniz in the eighteenth century introducing infinitesimals. They represented something almost zero but not quite. When used in calculations one could divide by them, since they were not zero, but at some point, when suitable for the result, they were assumed to be zero. Not very rigorous mathematics that was. It was not until towards the end of the nineteenth century that Georg Cantor brought more insight into the nature of the infinitely large. Stewart guides us through this history and illustrates how the concept of infinity has played a role in several disciplines that all have somehow contributed to how we think of the concept today.</p>
<p>
With a first chapter, Stewart puts forward some puzzles or paradoxes that involve infinity to illustrate that it is not sufficient to say that infinity is that "something" that is beyond all numbers. More precise definitions are needed for the infinitely large as well as for the infinitely small. Examples are the processes that hide irrational numbers like a staircase approximating the diagonal of a square converging to a straight line when its steps become finer and finer, and the regular polygon converging to the circle as it gets more edges. These demonstrate the problem of evaluating $0\times\infty$ in a sensible way. Hilbert's hotel is illustrating that a more precise definition of the infinitely large is required, and Stewart gives some other examples. These puzzles and paradoxes are first raised as questions for the reader to think about. Stewart's explanations of all these confusing statements are given afterwards.</p>
<p>
The second chapter illustrates that infinity is not hidden away in higher mathematics but that it is also embedded into elementary calculus. Gabriel's horn is obtained by revolving $1/x$ for $x>1$ around the $x$-axis. This has the surprising property that its volume is finite even though the surface is infinite. Of course infinity is also hidden in 0.9999... being equal to 1, a fact that astonishes many an undergraduate student, and of course infinity resonates in the decimal representation of irrational numbers. Distinguishing discrete from continuous would not be possible without infinity. Here as in the other chapters Stewart gives quite some attention to history: Dedekind defining the real numbers as sections which are essentially infinite objects, Lambert who proved the irrationality of $\pi$. In the Jain religion of India (600 BCE), people distinguished infinity from enormously large numbers, etc.</p>
<p>
Chapter three is further exploring the historical views of infinity. Space and time were traditionally assumed to be infinite, but when looking at the infinitely small, the situation is different. People had difficulty in dealing properly with infinitely small things. Zeno's paradoxes are examples that illustrate that a sum of infinitely many nonzero numbers can be finite. Since the ancient Greek there has been a distinction between an actual infinity and a potential infinity, a discussion that has continued throughout the centuries among philosophers. Even some theologians claimed that God was the only existing impersonation of something infinite. Some proofs for the existence of God were based on this belief. For mathematics, this distinction is not essential. Mathematical existence is abstract and does not coincide with physical or actual existence.</p>
<p>
The next chapter is a discussion of the infinitely small and how this has triggered the development of calculus. The original historical concept of infinitesimals is now replaced by the concept of a limit. The infinitesimals where revived when in the 1960's Abraham Robinson developed non-standard analysis.</p>
<p>
In geometry, infinity is where the horizon is. It led to the development of perspective in the Renaissance. This is extensively discussed in chapter six, explaining why a ship seems to become smaller as it approaches the horizon, and how this has led to the concept of a point or a line at infinity. The Euclidean plane can be modelled as a disk where infinity is represented by its boundary. More concretely, the line at infinity makes it easy to produce perspective drawings. Eventually this discussion ends in ideas of projective geometry and the mapping of the plane to a sphere and vice versa by stereographic projection, the point at infinity corresponding to the North Pole on the sphere.</p>
<p>
Infinity is a useful concept in mathematics, but how does it appear in a physical world? That is what the next chapter is about. In physical sciences, infinity often leads to a nasty singularity. Stewart discusses three examples. The analysis of the rainbow phenomenon is an optical example. If light is incident at a certain angle, then the intensity of the rainbow would be infinite according to ray optics. This singularity entailed that light had to be reconsidered as a wave. In Newton's gravitation theory a singularity occurs when the distance between particles becomes zero and the potential becomes infinite. For example Zhihong Xia proved in 1988 that by solving equations in a five-body problem, dramatically non-physical solutions are obtained after a singularity. Black holes are singularities in general relativity theory and in cosmology the Big Bang is obviously a singularity. Stewart also explains here why cosmologists are wrong when they use curvature as a parameter that determines whether our universe is finite or not.</p>
<p>
Te last chapter is the discussion of how Cantor came to his proof that the real number are not countable and how this has led to set theory and his transfinite numbers, and how this resulted in a revision of the foundations of mathematics. This story is best known by mathematicians or anyone who is a bit familiar with this kind of mathematical background literature. But again here Stewart follows the historical evolution of who did what and why in brewing up the eventual result.</p>
<p>
This is a lot of information and because of the compact presentation, it will not always be casual reading for a general reader. There are a few references provided per chapter, which might be of interest if the reader wants to look up more details. Some aspects are elaborated more than what is needed for explaining the impact of infinity (e.g. the computation of the angle of the rainbow, the geometry of perspective) but these topics are of course interesting in their on right, and they are usually not found in other treatments of infinity. If you are interested in only the strict mathematical concept of infinity, then Du Sautoy's or Chen's treatises that were mentioned above might be simpler alternatives. But in this booklet, even the experienced reader may have more occasion to learn something new. Some of these non-essential but nevertheless flashes of a that's-interesting-I-didn't-know-that experience will make it worthwhile 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 booklet wants to introduce a general reader to the concept of infinity. With a lot of historical, philosophical, and occasionally theological background Stewart shows how the concepts of the infinitely small and the infinitely large were eventually settled in a mathematical setting towards the end of the nineteenth and early twentieth century when the current foundations of mathematics were established.<br />
</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/ian-stewart" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">ian stewart</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">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-1987-5523-4 (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">£7.99 (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">154</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/infinity-a-very-short-introduction-9780198755234" title="Link to web page">https://global.oup.com/academic/product/infinity-a-very-short-introduction-9780198755234</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A30</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></ul></span>Tue, 03 Apr 2018 06:35:02 +0000Adhemar Bultheel48365 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/infinity-very-short-introduction#commentsSingle Digits: In Praise of Small Numbers
https://euro-math-soc.eu/review/single-digits-praise-small-numbers
<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 caleidoscopic collection of mathematical short stories. Such a story is only a couple of pages long and can explain a concept, some historical fact or mathematician, a puzzle, an open problem, or a simple mathematical fact. These stories are somehow related to each of the digits 1-9. A short tenth chapter of only 2 pages gives solutions to 3 of the problems that were formulated in earlier chapters. The author has played with the idea to also include zero. However zero has such a particular position among the digits that it takes up long chapters in books on the history of mathematics and several books have been devoted to zero alone (e.g. <em>Zero: The Biography of a Dangerous Idea</em> (C. Seife, 2000), <em>The Nothing that Is: A Natural History of Zero</em> (R. Kaplan, 2000), <em>Finding Zero: A Mathematician's Odyssey to Uncover the Origins of Numbers</em> (A.D. Aczel, 2015),...).</p>
<p>
The connection with the digit that forms the title of the chapters is rather loose. For example digit 1 stands for `unique', but the items covered in this chapter include origami, the golden ratio, number systems, factoring knots, countability, fractals, Sierpinski's gasket, Benford's law, Brouwer fixed point theorem, perfect squares, gamma function, Picard theorems and many others. Under the umbrella ot 2 we find Jordan curves, symmetry, the Pythagorian theorem, Euler's formula for polytopes, several conjectures related to prime numbers, the Towers of Hanoi pizzle, formulas for π, Apollonian circle packing, arithmetic and geometric means, Newton's method for root finding, etc. This illustrates the diversity of subjects that are involved. A minimal knowledge of mathematics suffices to understand and appreciate the items discussed. It may be already illustrated by the previous enumeration that the subjects become somewhat more advanced towards the end of each chapter, and there is also a graduate increase in complexity over the different chapters towards the end of the book.</p>
<p>
The table of contents lists 118 of these short stories (some may cover more than a single item). Each one is a brief excursion in the world of mathematics, a gem that illustrates what keeps mathematicians from the past present and future fascinated about their subject. It is always summarized in an accessible way. Other books that have similar collections exist, and there is certainly some overlap with those books, but it is remarkable that many of the stories here contain facts that I have not seen in other books before. There are no proper mathematical proofs but the problem, puzzle, theorem, conjecture, open problem, result, or just a simple fact is made very clear without ever needing deep mathematics.</p>
<p>
A similar idea of linking such a collection to numbers was used by Ian Stewart in his book <em><a href="/review/professor-stewarts-incredible-numbers">Professor Stewart's Incredible Numbers</a></em> (Profile Books, 2015). He however includes zero and negative integers, rationals, irrationals, and even infinity. Stewart's items are in general less advanced on a mathematical level. There are fewer topics but they are more elaborated.</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 mathematics related items: problems, puzzles, theorems, conjectures, open problems, results, or just simple facts. They are loosely related to the digits 1-9 and accordingly grouped in chapters.</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/marc-chamberland" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Marc Chamberland</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">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">9780691161143 (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 26.95</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">240</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/10437.html" title="Link to web page">http://press.princeton.edu/titles/10437.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/00-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A80</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/00a35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A35</a></li><li class="vocabulary-links field-item odd"><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 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/11a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11A99</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/11-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11-03</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></ul></span>Tue, 21 Jul 2015 14:57:25 +0000Adhemar Bultheel46318 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/single-digits-praise-small-numbers#commentsAnd yet it is heard. Musical, Multilingual and Polycultural History of Mathematics (2 vols.)
https://euro-math-soc.eu/review/and-yet-it-heard-musical-multilingual-and-polycultural-history-mathematics-2-vols
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In these two volumes of a thousand pages, Tonietti gives a very personal selection of the history of mathematics, and in particular the pieces where music and mathematics meet. He has some strong views on certain aspects that are not always mainstream and these are strongly put forward. One of his pet peeves is that many of his colleagues approached the subject with an eurocentric bias. Another one is that mathematics and science in general is too often considered to be an abstract universal entity besides or above the socio-cultural soil in which they are rooted. That means for example that also the language spoken and or even more so, the <em>lingua franca</em> used by the scientists and philosophers has had its influence which often goes hand in hand with cultural and religious foundations. This is his contribution to prove the Sapir–Whorf hypothesis in the case of mathematical sciences. Every culture generates its own science, and qualifications like superior and inferior or questions of precedence are often absurd. And of course, probably the main reason for writing this book, is his conviction that the mutual influence of music on the development of mathematics and vice versa is grossly underestimated. In this context he brings several contributions to the forefront that were wrongfully neglected (Aristoxenus, Vincenzo Galilei, Simon Stevin, Kepler,...). Thus there are many provocative viewpoints that some historians will disagree with, yet his arguments are extensively documented. Note however that this is not really a book on the history of mathematics, and neither is it a history of music. The reader is supposed to be familiar with music theory and should have some background in mathematics too. The book is a long plea and an extensive argumentation to underpin the viewpoints of the author, like those just mentioned. There are practically no formulas in the text, and relatively few illustrations, but the number of citations is overwhelming. These are almost always given in the original language with translation in brackets. The Chinese citations are written in pinyin, but full Chines characters are added in an appendix. This illustrates the importance that Tonietti is attaching to the language, since indeed, the translation is always an approximation and often an interpretation of what the original text is meant to say.</p>
<p>
Let's go quickly though some of the contents to illustrate what has been said above. Volume 1 contains Part I: The ancient world, and Volume 2 consists mainly of Part II: The scientific revolution, and a shorter Part III: It is not even heard.<br />
Part I treads the ancient cultures united around their language used: The Greek, Chinese, Sanskrit, Arabic, and Latin. For the Greek, music was part of the <em>quadrivium</em> and hence coexisted at the same level as mathematics and astronomy in the schools of Pythagoras, Euclid, Plato and Ptolemy. The search for harmony in music was reflected in the music of the spheres, all based on an orthodoxy of commensurability, hence integers and rationals. The music theory was developed on the basis of length of strings. The often neglected unorthodox outsiders here are Aristoxenus (who does not restrict to rationals) and Lucretius (although the latter wrote in Latin, he is Greek in spirit) who get special attention.<br />
The Chines on the other hand developed a theory of music studying the length of pipes (the <em>lülü</em>), bells, and chime stones. The cultural essense of <em>qi</em> is an energetic flow, a continuum which is apposite to the discrete orthodoxy of the Greek. Another difference is the lack of an equivalence for the verb "to be". This implies a different way of doing mathematics like for example the way in which they proved the Pythagoras theorem.<br />
Indian rules and regulations stem from religion. Precise prescriptions of how to build an altar show mathematical knowledge. Of course there was music, mostly by singing mathras, but most curiously, there is no trace of a music theory left. Musicians had `to trust their ears'.<br />
The Arabs are the saviors of the Greek culture. Most of what we know about the Greeks comes to us through them. The <em>Syntaxis mathematica</em> of Ptolemy came to us in Arabic as the <em>Almagest</em>: `the greatest' Greek collection of astronomical data. So they inherited the orthodoxy in music and mathematics from the Greek. They brought us our number system, but also terms like algorithm and algebra.<br />
Meanwhile in Europe, Latin had conquered the scientific scenary. This brought about a clash between the people, like Fibonacci promoting the introduction of the new Indo-Arabic number system against the Roman numerals. The Greek orthodoxy prevails, with Euclid being the reference for mathematics. Music theory florishes (Beothius, Guido D'Arezzo, Maurolico, Cardano,...). Tonietti gives special attention to Vicenzo Galilei, the father of Galileo, who picked up some ideas of Aristoxenus again.<br />
Besides the appendix with Chinese characters mentioned above, three other appendices are texts related to music translated from Chinese, Arabic, and Latin.</p>
<p>
In Part II chapters are named again afer the main languages (mostly European) used to disseminate scientific results. The interplay between geometry, astronomy, and music becomes explicit in work by Stevin and Galileo, but most of all in Kepler's <em>Harmonices Mundi Libri Quinque [Five books on the harmony of the world]</em> in which he completed Ptolemy's <em>APMONIKA</em> and interwaves geometry, astronomy, music and geometry, reflecting the music of the spheres. Tonietti does not shy away from critique on colleagues who had different interpretations of Kepler's work. People started using national languages besides Latin in their writings and (perhaps because of that) mathematical symbolism increases like writing music on staves was adopted before. Transcendent symbolism was mixed with music, God, and natural phenomena in work of Mersenne, Descartes, Wallis, and Huygens (Constantijn and Christiaan). The latter was not only a musician and composer, he used the newly invented logarithms and Leibnitz's differential calculus in his music theory. All this, according to Tonietti, shows that the status of music should be reinstalled as an essential element that contributed to the development of mathematics. Also Leibnitz and Newton worked on music for some time, but of course their main contribution here is the mathematical symbolism that allowed to deal with the infinite and the infinitesimal. With the use of the twelfth root of two in the equable temperament, the Pythagorean-Plato orthodoxy was definitely finished. The music of the spheres had degenerated and became intense discussions about God and creation. Again Tonietti analyzes interpretations of other historians, philosophers, or theologians on these topics sometimes rebutting them with his own vision. While in the 18th century French became the prominent language in Europe, music theory received its last flares. The French composer Rameau wrote a treatise on harmony using a theoreical basis, but he was opposed by Euler who declared tones more pleasing when they could be represented more simply. He based his analysis on prime numbers, with reminiscences of Pythagoras. Another opponent was d'Alembert discussing the harmonics of the vibrating string and also the other illuminists compiling the <em>Encyclopédie</em> had their explanation for musical terms. Still musicians wrote of science and scientists wrote of music. Entering the 19th century, Lagrange and Fourier, and later Riemann entered the discussion about vibrations, while von Helmholtz also did the physical experiments to analyze sound. Max Planck not only wrote about music but even composed some and also Einstein loved playing the violin and had correspondence with the composer Schönberg.</p>
<p>
Part III is very short. It gives a short round-up of things not discussed like Africa, Cental and South America, and more extensively the music and navigation skills of Polynesians. A last chapter briefly touches upon the science of acoustics, in which music is largely neglected. A quote from one of the final paragraphs that renders explicitly what Tonietti has allowed to emerge in his book:</p>
<p>
<em>The decision to move away from musicians and their music impoverished both natural philosophers, first of all, and then mathematicians and physicists. This influenced and facilitated the development of their research in those main directions which are known to everybody, but which continue to deserve to be criticised for their limitations and their (negative?) effects on our life.</em></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 history of music theory and its relation with mathematics, placed in a general cultural framework, and by doing so, giving a less usual, less eurocentric approach. Besides the well known historical framework, Tonietti selects and extensively discusses some less known sources and gives arguments for his critique on the views or interpretations of some of his colleagues. He comes to the conclusion that although mathematics and natural sciences has taken big steps forward in recent history, music theory was detached and has lost the interest of mathematicians, and this is a regrettable impoverishment for natural philosophers, mathematicians and physicists.</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/tito-m-tonietti" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Tito M. Tonietti</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Birkhäuser Verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-0348-0667-0 (hbk - vols.)</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">253,34 €</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">1020</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/birkhauser/mathematics/book/978-3-0348-0677-0" title="Link to web page">http://www.springer.com/birkhauser/mathematics/book/978-3-0348-0677-0</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/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</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/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</a></li></ul></span>Wed, 13 Aug 2014 07:09:30 +0000Adhemar Bultheel45674 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/and-yet-it-heard-musical-multilingual-and-polycultural-history-mathematics-2-vols#commentsMath and the Mona Lisa. The Art and Science of Leonardo da Vinci
https://euro-math-soc.eu/review/math-and-mona-lisa-art-and-science-leonardo-da-vinci
<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 Mona Lisa is probably one of the best known paintings our collective memory. Leonardo da Vinci was presumably appointed to paint a portrait of the wife of Francesco del Giocondo around 1503 but the painting was never delivered since he may have continued working on it till 1516 when he took it with him to France. After his death in 1519 in Amboise it was bought by the king Francis I, it subsequently hung in several castles in France, and ended up in the collection of Louis XIV. After the revolution it moved to the Louvre, with a short interruption spent in Napleon's bedroom. It was stolen in 1911 but recovered in Florence. It was attacked by acid in 1956 and all kind of objects and paint were thrown at it in the museums, but these mostly left it undamaged since it was protected behind bullet proof glass.</p>
<p>
The Mona Lisa is the last in a row of three women portraits painted by Leonardo with an interval of 15 years, and besides the painting of <em>The Last Supper</em>, that is almost as famous as the Mona Lisa, it is just one item in the abundant artistic and scientific production of Leonardo da Vinci. And viewed at a greater distance, Leonardo da Vinci is just an island in the ever broadening stream of artistic and scientific evolution. Although there is obviously some focus on Leonardo da Vinci, he can only be understood when placed among his contemporaries, who are the result of a long history, and are themselves a stepping stone for future achievements. Atalay, being himself a scientist and artist, is placing da Vinci's creativity in this evolutionary stream from its source in the Mesopotamian number system till modern string theory and cosmology.</p>
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Thus the title of the book may be a bit misleading since the contents is much much broader that it may suggest. It was originally written in 2004 and has been translated in 12 languages since. That it is reprinted in this paperback version shows that it is still of interest and indeed it is one of the most pleasing, entertaining, and accessible surveys that I have ever read about this ongoing quest of humankind trying to unravel the secrets of nature. Atalay's love for beauty, mathematics and science, brilliantly reflects the genius of Leonardo and his age of Renaissance.</p>
<p>
From the the mathematical precursors, Atalay brings Fibonacci, the other Leonardo, to the forefront. In particular the number sequence named after him and the ratio known as the golden ratio. These numbers and the ratio, represented as φ, shows up in nature more than often, but also in mathematics as the golden rectangle and the golden triangle, and the golden spiral. Atalay gives many examples of where this occurs in nature, from nautilus shells over phyllotaxis, to the DNA helix and galaxies. The ratio shows up when judging aesthetics of the human face and the human body. It is no wonder that da Vinci dissected bodies and drew the <em>Vitruvian man</em>, another of these da Vinci immortal images that live in the memory of mankind. But the golden mathematical configurations, deliberately or not, were also implemented in the layout of paintings, and in architecture, of which the pyramids are the most famous examples. After introducing the rules of perspective, we arrive at da Vinci's work of art with the portraits of the three women (Genivra de' Benci, Cecilia Gallerani, and the Mona Lisa) forming a center since it is more or less the middle of the book. The second part is more related to da Vinci as a scientist and how that science evolved later. Da Vinci was famous for his many unfinished projects, many scale models were produced much later and hobbyists can buy kits on the Internet to build them: catapult, helicopter, bicycle, odometer, parachute, tank, etc., many are remarkably close to our modern design. But science did not stop with da Vinci. Shortly after came other polymaths like Galileo, Kepler, Copernicus, Newton, and later Maxwell, Einstein, Schrödinger, etc. who formulated the basic laws of nature, and the planetary system, and we are still unraveling the laws of the cosmos and of the tiniest of its building blocks.</p>
<p>
Atalay succeeds in taking the reader along on this journey, not as a detached guide or journalist, but showing also some personal involvement. He includes some of his own artistic work, shows pictures he took as a child while visiting the pyramids with his parents, he gives details of why and how Schrödinger came to his equations, sketches the shy and timid personality of Dirac, etc. Chapter titles like "the nature of science", "the nature of art", "the art of nature", and "the science of art", he makes it crystal clear how much he believes in the confluence of all three: nature, science, and art. If the book, after 10 year in print still needs recommendation, I happily ratify a confirmation.</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 reprint of the original much acclaimed book from 2004. Atalay describes the symbiosis of art and science from the dawn of civilization to our current state of knowledge, placing the the work and the personality of Leonardo da Vinci at the center.</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/b%C3%BClent-atalay" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Bülent Atalay</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/smithsonian-books-random-house" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Smithsonian Books / Random House</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">978-1-58834-493-9 (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">$15.95 (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">352</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.randomhouse.com/book/206390/math-and-the-mona-lisa-by-bulent-atalay/978158834493" title="Link to web page">http://www.randomhouse.com/book/206390/math-and-the-mona-lisa-by-bulent-atalay/978158834493</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a67" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a67</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a99</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/85-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">85-03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97m50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97M50</a></li></ul></span>Wed, 13 Aug 2014 06:38:13 +0000Adhemar Bultheel45672 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/math-and-mona-lisa-art-and-science-leonardo-da-vinci#comments