European Mathematical Society - 26-03
https://euro-math-soc.eu/msc-full/26-03
enWhere do numbers come from
https://euro-math-soc.eu/review/where-do-numbers-come
<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>There is a modern trend in calculus courses to start from application examples and practical problem solving, and from there come to some abstraction and theorems. In this book, Körner presents some basic results in a way that is the opposite of this. It is a strict mathematical top-down approach, formulating definitions, properties and theorems with hard proofs starting from a minimal set of axioms. As the title suggests, his guiding idea is to introduce the number systems on a pure axiomatic basis, loosely following the historical evolution. In this sense it is not really a replacement for a calculus course, but rather a complement to it. It does have the structure of a course text, with a sequence of definitions, theorems and proofs that is interrupted with many exercises and challenges for the reader-student.</p>
<p>One would expect that the topics to be covered and the order in which they are introduced are more or less clear. Nevertheless it is somewhat surprising that rational numbers come first in part 1, and natural numbers are a "special case" in part 2, and finally the reals and complex numbers in part 3. The quaternions and polynomials over a field follow as some kind of "encores".</p>
<p>Rationals without integers sounds almost impossible, but here the historical context comes in as a motivation. Originally people counted quantities and that does not really involve numbers in the sense that one added 3 sheep ad 4 sheep to get 7 sheep, but adding 3 sheep to 4 apples was not something to consider. So, to come to abstract counting numbers, the first thing to do in a Greek tradition is to find an axiomatic system for what we call now the positive natural numbers $\mathbb{N}^+$. The Greek had rationals disguised as ratios of lengths or areas but the Indians and Chinese properly considered rationals as numbers. In this book these are introduced as equivalence classes of (numerator, denominator) pairs where numerator and denominator are elements from the previous set with their rules for adding and multiplication. This gives the strictly positive rationals $\mathbb{Q}^+$. Then zero is introduced first as a place holder, and eventually as a number, and this entails the definition of the negative rational numbers and thus besides inverses for multiplication now also inverses for addition will exist. So we now have the structure of $\mathbb{Q}$. This idea of introducing a new mathematical concept as a couple of items from a previous structure with their known composition rules (adding and multiplication) and then identifying them in equivalence classes, is a technique that is used at several places in this book.</p>
<p>In part 2 the natural numbers are derived via the introduction of 1 as the least positive rational whose (reduced) denominator is 1 and then using an induction to generate all the positive integers as part of the rationals, and eventually zero and the negative integers to give $\mathbb{Z}$. Along the way we learn about long division, Bezout's theorem, and prime numbers. Modular arithmetic is introduced via finite fields leading to Fermat's little theorem, coding theory, the Chinese remainder theorem and encryption. The Peano axioms to define the rational numbers $\mathbb{Q}$ as an ordered field is the ultimate conclusion of part 2. It includes some philosophical considerations about the existence of numbers and the idea of an axiomatic approach to mathematics in general. That includes the Russell paradox, Gödel's theorem, and the consistency problem of mathematics.</p>
<p>Historically, mathematics became a profession somewhere in the 17th century. First Körner introduces extensions like $\mathbb{Q}[\sqrt{2}]$ using again the technique of defining addition, multiplication and an order relation on couples of rational numbers. To come to analysis, it requires the fundamental axiom of an intermediate value and hence the existence of a limit for bounded sequences. Equivalence classes of converging sequences are introduced by identifying sequences with the same limit as equivalent. Pointwise operations can be defined for the sequences and the real numbers in the set $\mathbb{R}$ are then identified with these equivalence classes. The complex numbers are then easily introduced as couples of reals (using again the same trick of defining operations for couples of reals) and introducing limits and continuity in $\mathbb{C}$ is relatively easy. This paves the way to define polynomials over a field and to derive the fundamental theorem of algebra. The main reason for considering zeros of polynomials over a field, in particular with integer or rational coefficients, is that this allows to distinguish between the algebraic and the transcendental numbers. Integral domains and quaternions as generalizations of complex numbers are the remaining items with a short discussion.</p>
<p>Clearly this book is probing the fundamentals of mathematical analysis and will be useful as an extra reading for an introductory calculus course. It will certainly satisfy those readers who are looking for abstraction and who want to extract the maximal number of results from the minimal set of axioms. The historical elements on the side are entertaining but not essential. It is not exactly recreational mathematics, but the text is nicely written. It has many explicit proofs and many exercises and invitations to think about a statement. No solutions are provided though. It is an excellent way to get in touch with the foundations of mathematics at a relatively elementary level.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book gives an axiomatic approach at a first year university level to number systems and some elements from calculus. It could be a complement for a freshmen's 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/tw-k%C3%B6rner" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">T.W. Körner</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/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge 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">978-1-1084-8806-8 (hbk), 978-1-1087-3838-5 (pbk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 59.99 (hbk), £ 24.99 (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">268</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/logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Logic and Foundations</a></li><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/where-do-numbers-come" title="Link to web page">https://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/where-do-numbers-come</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/26-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-01</a></li><li class="vocabulary-links field-item odd"><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 even"><a href="/msc-full/97f30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F30</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97f40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F40</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97f50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F50</a></li></ul></span>Mon, 25 Nov 2019 09:26:01 +0000Adhemar Bultheel49948 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/where-do-numbers-come#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#commentsThe History of the Priority Dispute between Newton and Leibniz
https://euro-math-soc.eu/review/history-priority-dispute-between-newton-and-leibniz
<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>
Priority disputes among mathematicians are from all times, but the one between Newton and Leibniz about the discovery of calculus is notorious. Many authors, and historians have written about it. Even during the lifetime of the protagonists, the Royal Society had a commission to investigate the matter. Their conclusion was that Newton was the first, but since at that time Newton was the president of the Royal Society, this conclusion may have been a bit biased.</p>
<p>
The supporters of Leibniz whose home base was Hanover were mainly from the continent, while most of the British defended their national hero. In those days mechanics, mathematics, optics, chemistry, alchemy, astronomy, and history were all part of the job of a prominent scientist. In Newton's case certainly also theology, history, and monetary politics. While Leibniz started as a lawyer and published on palaeontology. So the whole scientific (and political) community was involved.</p>
<p>
Newton introduced his fluxions inspired by physics. A fluxion is the instantaneous change in a fluent. We now say that it is the time derivative of a function of time (the fluent). The problem was that the notion of limit was still unknown, so his peers had problems with computations that used infinitesimal small (but nonzero) quantities, that seemed to vanish when appropriate and remained nonzero at other instances. This was directly connected to the construction of a tangent and what was called a quadrature, which is the computation of the area under a curve, thus what we now call an integral. Newton's great insights happened mainly during the period of the Great Plague in 1995-1667 when he retreated to Woolthorpe Manor to live with his mother. In that time he also developed his theory of gravitation, laid the foundation of classical mechanics, and explained the planetary motion. None of this was however published until much later. The mechanics were published for the first time in his <em>Philosophiæ Naturalis Principia Mathematica</em> in 1687 and two other editions in 1713 and 1726. His book <em>The method of fluxions</em> was only written in 1671 and published in 1736.</p>
<p>
Leibniz was educated as a lawyer ans got only interested in mathematics later in 1672 when he visited Paris and meets Huygens. He was mainly concerned with quadrature. The approximate length of a curve $ds$ could be considered as the hypotenuse of a rectangular triangle with sides $dx$ and $dy$. Using geometrical arguments and similarities of triangles he obtained a method to compute the quadrature of an arbitrary curve. This was around 1674, but it was not published before 1684. He used the notation $dy/dx$ for the derivative, which was conceptually much easier to work with than Newton's fluxion notation which used the dot atop the fluent variable. This of course becomes problematic for higher order derivatives. Leibniz also introduced the integral sign ∫ as a elongated 'S' for sum, that we are still using today and which is included in the title of this book by writing "Dispute" as "Di∫pute". It is clear, and generally agreed by now, that Leibniz and Newton developed their theory independently by following different methods. However in the heat of the controversy Leibniz was accused of blatant plagiarism. Strangely enough, it were not Newton and Leibniz that stood in the barricades most of the time. In fact they exchanged polite and friendly letters. It were their followers, friends, and believers who did all the fighting on the front line, although they were of course backed up and sometimes directed by the protagonists. Newton remained more on the background, but when accusations became too direct, Leibniz had no choice but to protest against an open insult by a warrior from the opposite camp.</p>
<p>
Among the historical defenders of Leibniz were Jacob and John Bernoulli. Among Newton's warriors were John Collins, John Wallis, and Nicolas Fatio de Duillier, which is called Newton's monkey by Sonar. This Fatio has put the fuse that lit the powder keg by openly accusing Leibniz of plagiarism. At a later stage John Keill became the `army commander' of the group defending Newton. Some of the problems arose because the first correspondence was not directly between Newton and Leibniz but passed via others like Henry Oldenburg, the secretary of the Royal Society, who was not a mathematician. Oldenburg was advised on matters of mathematics by Collins, an outspoken nationalist, who was naturally opposing anything that came from the continent. There were misunderstandings, half spoken truths, and hesitation to disclose results that oxygenated the fire. The war went on, even beyond the grave. Clearly the new calculus found applications, and because Leibniz's formalism was easier, his calculus was the eventual winner. In fact it caused a drop back of the English mathematical scenery. While they were at a comparable level with the mathematics on the continent when the controversy started, they were not able to keep up with the development of calculus and analysis for a while in the eighteenth to nineteenth century post-Newton era.</p>
<p>
This fight may be well known, but disputes in those days were very common among others as well. Newton and Hooke became personal enemies over a priority dispute in optics (Newton did not want to publish his <em>Opticks</em> until after Hooke died), Huygens rejected Newton's corpuscular theory of light. He also fought with Heuraet over the rectification of curves, and he quarrelled with Hooke over a clock mechanism. Newton and Flamsteed, the Astronomer Royal, were fighting over the trajectory of the Great Comet of 1680, which Newton explained with gravity. And there were other such disputes that are also described by Sonar in this book.</p>
<p>
Thomas Sonar is from Hanover and before he engaged in the study of this history, he was rather convinced that it was a good-hearted Leibniz that was the one who was maltreated and unjustly accused by a quarrelsome and short-tempered Newton and his disciples. Sonar may have started his research with the idea of defending Leibniz, when he finished the original German version of this book in 2016, his conclusion was much more mollified. Leibniz also had not always told the truth and he wasn't the saint attacked by the devil Newton. He also had his pawns in the war and used them. This conclusion becomes clear only after meticulously investigating all the original correspondence of the seventeenth century and of all the books and papers that were published about the matter. This is the most thorough discussion of the matter that has been published so far and that still remains very readable with a minimum of mathematical knowledge, hence available for a general readership. In fact Sonar starts with an elementary introduction like a modern introductory calculus book would, so that the reader should know what calculus is about, or at least grab the meaning of derivative and integral. Then he introduces the `giants on whose shoulders Newton claimed to stand': John Wallis, Isaac Barrow, Blaise Pascal, Christiaan Huygens. So we find a biography of these people, and what they did for mathematics. In retrospect it is clear that calculus was on the doorstep, and that it only took some great minds to bring it in the open.</p>
<p>
But Sonar also gives a detailed description of the political situation and events of those days in England, France, Spain, and the Netherlands. Of course these are not really essential for the mathematics, but it sketches the framework in which scientists were working. It were usually political leaders that employed the top scientists and they made the start of academies financially possible. This part of the book has several very useful timelines, and there are many beautiful pictures throughout the book. Just reading this political prequel to the main dish is already a wonderful experience. Then of course we meet both Newton and Leibniz, how they grew up and studied and how they arrived at the discovery of their new calculus. At first there is some friction (Sonar calls it a cold war) between the two, then there is a period of relaxation, but when things get published the smoldering fire becomes a real war. Sonar includes many quotes from the letters that go back and forth about the matter with precise dates of when they were written, whether it was as an impulsive reaction to a previous message or it was written only after a long time of postponing it, possibly who was the messenger, and, not unimportant, when the letters arrived. With every new player we are given his or her background and some biography.</p>
<p>
Fortunately the excellent and smoothly reading English translation comes so shortly after the German original and was done by Sonar himself with the help of Keith Morton, his Oxford thesis advisor and later by his advisor's wife Patricia Morton. I can highly recommend this book if you have just a slight interest in history and/or mathematics. Perhaps the professional mathematical historians may not find much new or innovative material, since this 'cold case' has long been settled and solved, I believe they will still enjoy reading this book.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is the English translation of the German original that appeared in 2016. It is a detailed analysis of this famous controversy that is brought in an easily accessible format for a general readership. It starts with a brief introduction to differentiation and integration (that can be skipped if you don't need it), then sketches the political situation in England, France, Spain and the Netherlands of the 17th century, en finally elaborates on the rise and decline of the controversy backed up by many quotations from the letters that were mailed back and forth on this matter.</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/thomas-sonar" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Thomas Sonar</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-international-publishing-birkh%C3%A4user-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer International Publishing / 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">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-72561-1 (hbk); 978-3-319-72563-5 (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">137.79 € (hbk); 107.09 € (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">576</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://www.springer.com/gp/book/9783319725611" title="Link to web page">https://www.springer.com/gp/book/9783319725611</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/01-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-02</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/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A45</a></li><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></ul></span>Tue, 29 May 2018 06:18:34 +0000Adhemar Bultheel48508 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/history-priority-dispute-between-newton-and-leibniz#commentsHow Euler Did Even More
https://euro-math-soc.eu/review/how-euler-did-even-more
<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>
Sandifer is a renowned expert on Euler. In 2001 he was co-founder and later secretary of the Euler Society. He wrote several books and contributed and promoted the Euler centenary year in 2007. Since November 2003, he also had a monthly column called <em>How Euler Did It</em> hosted by the Euler Archive website in collaboration with the Mathematical Association of America (MAA). These columns are stand-alone items elucidating a particular aspect of Euler's work. They are available online at the <a href="http://eulerarchive.maa.org/hedi/" target="_blank">Euler Archive</a>. A first collection of these columns appeared as a book <em>How Euler Did it</em> (MAA, 2007). Unfortunately in 2009, Sandifer had to recover from a severe stroke, but some colleagues filled up some of the gap so that the column kept appearing until early 2010. This book contains 35 of these columns from March 2007 till February 2010. September 2009 does not exist for the reason just explained. October, November, and December 2009 were filled up by guest author Rob Bradley.</p>
<p>
In the previous collection, the items were ordered chronologically, but in this volume, they are grouped by topic: Geometry, Number Theory, Combinatorics, Analysis (the largest part), Applied Mathematics, and some miscellaneous part called <em>Euleriana</em>. The columns are explaining indeed how Euler did prove some of his results. Euler got his problems for example from marginal notes by Fermat who, as we know, announced theorems by scribbling some notes in the margin of a book, or some problems were formulated by Euler himself and there are many other sources. Some historical background is given, but the main contribution is just explaining how Euler indeed constructed his proof. Sometimes original drawings are reproduced. These columns are certainly welcomed by readers not familiar with the original language of the papers or the correspondence which was often Latin, German, or French. Some translations of the original texts by Euler can be found at the <a href="http://eulerarchive.maa.org/" target="_blank">Euler Archive</a> but many are not translated yet.</p>
<p>
There are too many different topics to be discussed in detail in this review. They include prime numbers, trigonometry, probability theory, mortality tables and actuarial science, the zeta and gamma functions, formulas to approximate π, partial fractions, complex analysis, optics, fluid dynamics, gravity and many more. The <em>Euleriana</em> part deals with Euler as a teacher, but there are also contributions about Euler and the hollow earth, errors that Euler made, and about Euler and pirates. I leave it to your imagination what these are all about. You will have to read the column to know.</p>
<p>
Let me take just one example from the applied mathematics chapter to illustrate how the items are treated by Sandifer in his blog. In 1756 Euler while working in Berlin publishes a paper on modeling of saws. The column starts by explaining that Euler worked in 1728 as a physician for the Russian Navy in St. Petersburg where he learned the importance on lumber and the operation of sawmills. When 30 years later the Prussian King Frederik II was about to embark in a war, Euler, remembering the importance of lumber, wrote his paper. The saw model is about a vertical blade that cuts when moving down. Each tooth should cut the same amount of wood, which means the teeth side of the saw should be slanted, the next tooth cutting another peel where the previous one had just removed its part. This models the shape of the saw. Then the motion is formalized, the middle part of the saw blade that really cuts the wood, the speed and the energy needed, and finally the manpower needed to lift the saw (it was supposed to move down by gravity), and the amount that was cut per worker and per hour. It beautifully illustrates the genius of Euler in an easily understandable mathematical language as it is brought to us by Sandifer.</p>
<p>
As you can see from my previous enumeration, there are enough topics to make the book of interest to many. Simple mathematics suffice to illustrate the brilliant mind of Euler. They are also mathematical gems as a column: well written and documented, mainly addressing mathematicians or teachers, but understandable with a minimum of training. If fits perfectly well in the MAA Spectrum series that targets the general mathematically-interested reader. Given the dynamics and volatility of websites, it is a good initiative that the MAA has made these columns available as a book.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of monthly columns that Sandifer wrote on the work of Euler. It contains 35 of the last columns that appeared in 2007-2010 after previously a similar collection of <em>How Euler Did It</em> was published in book form by the MAA in 2007.</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/c-edward-sandifer" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">C. Edward Sandifer</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/maa-cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MAA; Cambridge 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">9780883855843 (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">£ 23.00</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">247</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/how-euler-did-even-more" title="Link to web page">http://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/how-euler-did-even-more</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/01-06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-06</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/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/05-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05-03</a></li><li class="vocabulary-links field-item odd"><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 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/51-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-03</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span>Mon, 20 Jul 2015 16:06:22 +0000Adhemar Bultheel46314 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/how-euler-did-even-more#comments