European Mathematical Society - 01A55
https://euro-math-soc.eu/msc-full/01a55
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#commentsFrom Servant to Queen: A Journey through Victorian Mathematics
https://euro-math-soc.eu/review/servant-queen-journey-through-victorian-mathematics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The title of this book is an echo of a very similar title used by E.T. Bell for his book Mathematics: Queen and Servant of Science (MAA, 1997). Bell explains the unreasonable effectiveness of mathematics in the physical sciences. The meaning of this title is that mathematics, has the historic role of being a tool, a servant, for other more applied sciences, but at some point, what is usually called pure mathematics, evolved into a separate discipline, where mathematics is studied for its own right, a science that is the queen of intellect and a source of beauty. And yet, as we have experienced on several occasions already, what was originally assumed to be pure game of abstraction, devoid of any practical use, turned out later to have very practical applications.</p>
<p>In this book, Heard describes how this transition happened in England during the Victorian period (approximately 1830-1900). He sketches how mathematics, traditionally in its servant role for the other sciences, like physics, astronomy, economics, and even the invention of practical mechanical machinery, gradually appeared in its queen role of pure mathematics.</p>
<p>The book starts with sketching the situation in England in the 18th and first half of the 19th century. Since calculus was introduced by Newton and Leibniz, mathematics started to take off in a different direction. A mathematician used to be someone who applied mathematics to practical use, without being a "professional mathematician" in the modern sense of the word. It could be anyone occasionally using some computational technique, and the plural in mathematics may have referred to all these applications. But gradually mathematical knowledge became the ruler of all other sciences, just like queen Victoria was ruling the British Empire.</p>
<p>Britain was still under the spell of Newton, and the controversy with Leibniz had started an aversion for the continental approach to mathematics. There were two main scientific centres in Britain. Oxford which was considered to be the university of choice if you wanted to specialize in the classics. Theology and classics had been the dominant studies for many centuries. And the alternative was Cambridge with its system of of tripos that generated the wrangles, a prestigious honours degree in mathematics, that opened many doors to public positions. So this was the place to be if you were interested in mathematics. Perhaps because of Newton's spirit still dwelling in the premises there, it had more prestige in the eye of some beholders.</p>
<p>Pure mathematics and the mathematical profession was definitely much more accepted on the continent. There was much more exchange of ideas and results were published in professional mathematical journals. England had a tradition of publishing popular science magazines with puzzle sections, but no proper mathematical journals. The more practical notation in the Leibniz approach to calculus, which was more popular on the continent, may have given an advantage. So British mathematics lingered behind, and a lack of communication made that they had difficulties understanding the more advanced continental mathematics. In Britain, only from around 1830, a similar movement of pure mathematicians started to emerge. It became gradually accepted that the square root of a negative number could be studied for its mathematical properties, without the necessity of it representing some physical quantity. Although it was still generally belief that even pure mathematics was developed to the benefit and the advantage of science and technology.</p>
<p>Starting the London Mathematical Society has been a strong driving force in this evolution. Founded in 1964 at the University College London, it officially started a year later with August De Morgan as its first president. At first it was just a local community, but from the beginning it was keeping up a high standard for its members, as well as for the publications in its Proceedings and for the winners of the De Morgan medal that they awarded. The number or members was relatively low, although slowly growing, but its high standard eventually attracted many foreign members. In fact, here the British took a leading role, because the LMS became an example for other societies abroad (SMF, DMV, AMS....).</p>
<p>In the remaining chapters, Heard explains what it actually meant to be a "professional mathematician" for Victorians. In the chapter with the title "The pure mathematician as hero", he introduces the biographies of several British mathematicians who produced some "pure mathematics" and that usually were somehow linked to the LMS. James Whitbread Lee Glaisher who worked on number theory and who was editor of the Messenger of Mathematics (now Quarterly Journal of Mathematics); Henry J.S. Smith (known for example for the Smith normal form of a matrix, and the first to introduce the Cantor set); Percy MacMahon (combinatorics); and others. The British did not have the tradition of seminars as that was usual on the continent. Perhaps for this reason, several of the missionaries of pure mathematics were not the "heroes", the charismatic leaders, that had a school of followers like their continental counterparts had.</p>
<p>Then there is a chapter about the mathematics that was required to solve the problem of light and electromagnetism. Unlike Newton's corpuscular approach to optics, light (and later other electromagnetic quantities) seemed to be propagating like waves. But waves had to propagate in some medium that was termed aether. So there was much ado about the mathematics of the aether. This is a somewhat strange subject to be discussed extensively in the context of this book, but it definitely was a hot topic in those days, and it illustrates that in applied mathematics, Britain was not at all a backwater area. It shows that "pure mathematics" can also be produced by engineers, and non-professional mathematicians. Here we meet big names like Faraday, Stokes, Maxwell, Heaviside, W. Thomson (lord Kelvin), but also mathematicians got involved like Airy, and Clifford. The latter is known from the Clifford algebra, but he is also the one who linked gravity to non-Euclidean geometry which later inspired Einstein for his relativity theory.</p>
<p>The last two chapters are more of a social nature. With G.H. Hardy's A mathematician's apology in mind, (Hardy is the standard example of a thoroughbred pure mathematician abhorring any practical application) there is a discussion about mathematics as a profession, and what it meant to be a professional mathematician. It was quite different from a scientist. A typical British scientist was supposed to be a gentleman, free and independent, whose conclusions were not questionable. If you had a profession, then that was supposed to be at public service (which was not exactly what pure mathematics was pursuing), and you had an organisation that defended your rights (which was not the role of the LMS). So the idea came up that also a pure mathematician can be creative and explore unknown domains, which was also to the eventual benefit of society. This transformed a pure mathematician into an artist striving for beauty. Heard gives an account of the geometer George Salmon who in Nature praises the work of Cayley as an artist which was quite opposite the ideas about aesthetics of Walter Pater (an Oxford essayist).</p>
<p>The book includes many short (in line) and long (displayed) quotes and a few illustrations. There is also a long list of extra literature at the end of the book and many references at the end of each chapter to refer to the source of the quotes in the text. However if you are not an historian you can safely ignore these and keep on reading, because the story is told in a smooth and entertaining way. It is really interesting to read how long the Leibniz-Newton dispute had serious consequences, how ideas changed in about seventy years, and the important role that was played by the LMS in this process. It also illustrates that Britain was (and it still is) an island and that British were in many ways different from the rest of the world. They are probably less so today than they were in the Victorian period. Globalisation has made national identities more fuzzy worldwide, but it will take many more generations before this will be erased completely.</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 Heard illustrates how during the Victorian period, pure mathematicians emerged and separated from the practitioners. Scientists who started practising mathematics for the mathematics, and not for its direct application. Especially the role of the London Mathematical Society in this process is highlighted. He also explains what it meant to be a professional mathematician in those days.</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/john-heard" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John Heard</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">9781107124134 (hbk), 9781108604178 (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">£34.99 (hbk), £36.00 (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">277</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="http://www.cambridge.org/academic/subjects/mathematics/history-mathematics/servant-queen-journey-through-victorian-mathematics" title="Link to web page">http://www.cambridge.org/academic/subjects/mathematics/history-mathematics/servant-queen-journey-through-victorian-mathematics</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/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</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></ul></span>Mon, 01 Jul 2019 11:03:28 +0000Adhemar Bultheel49492 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/servant-queen-journey-through-victorian-mathematics#commentsBernard Bolzano: His Life and Work
https://euro-math-soc.eu/review/bernard-bolzano-his-life-and-work
<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>Bernard Bolzano (1781-1848) is well known among mathematicians. He was however a philosopher and logician in the first place. As a product of the Enlightenment, he had very modern, almost revolutionary, ideas about science, church, and society. This was however not appreciated by the ruling political upper class, since these ideas endangered their absolute power over people.</p>
<p>The first author of this book is specialised in the philosophy of mathematics and the second is a philosopher, both ardently interested in Bolzano and his work. They started working together at the end of the 1990's, which resulted in this massive book project. They could complete it because the majority of the work of Bolzano became available as it is compiled in <a target="_blank" href="https://www.frommann-holzboog.de/editionen/20"><em>Bernard Bolzano: Gesamtausgabe</em></a> (Fromann-Holzboog, 1969ff), which is still an ongoing project since 102 volumes are currently (2019) published, and 29 are still in preparation.</p>
<p>Bernard Bolzano was born in Prague in Bohemia, which the authors want to explicitly distinguish from (modern) Czechia, and not only for geographical reasons. More important in this context is its political history. The fact that this region was part of the Holy Roman Empire, and later of the Austrian Empire with oppressive rulers resulted in a smoldering anti-catholic and anti-German sentiment. By the end of the 18th century, the Czech language and culture experienced a revival as a consequence of widespread romantic nationalism, and Bolzano was an exponent of this movement.</p>
<p>Bolzano was raised in a pious catholic family (12 children but only two reached adulthood). His health has been weak throughout his life. Although his father wanted him to become a merchant like he was himself, Bolzano started studying mathematics and theology and became a catholic priest in 1804. He started teaching the philosophy of religion at the Charles University in Prague where he became a popular teacher, loved by his students. However, Bolzano's rather liberal opinions about the church and about politics, made him not popular among his superiors. The Austrian government however was suspicious of his ideas spreading among his students. They put some pressure on the local authorities, which led to Bolzano's suspension as a professor in 1819 and he was put under house arrest. Later he was also tried by the church and, since he refused to recant his "heresy", he retired from his chair and spent the summer with his friends the Hoffmanns outside Prague, and only returned during the winter period. This gave him ample time to work on his mathematical and philosophical texts. However many of his writings had only limited distribution because of his conviction official publication was almost impossible. Other texts remained unpublished until 1962 or even much later. This explains why he had relatively little direct impact and some of his original ideas were later rediscovered by others.</p>
<p>Among his major publications were his Rein analytischer Beweis. Here he tried to remove infinity from calculus, and hence had to define limit, continuity, derivative, and convergence without it. In this context we find his treatment of the intermediate value theorem and a definition of a Cauchy sequence. So he developed these ideas some years before Cauchy. He used the Bolzano-Weierstrass theorem many years before Weierstrass did. His Grossenlehre is his attempt to start setting up a logical foundation of mathematics, which he generalized in his Wissenschaftslehre to a complete theory of knowledge. In his Paradoxien des Unendlichen (Paradoxes of the infinite) the word "set" is used and he also has the bijection between the elements of an infinite set and an infinite subset. Many other results stayed unpublished for a long while until in the 20th century, and hence are attributed to other mathematicians. For example in his discussion of continuous functions, he had some fractal-like sawtooth monster-function, that became known as a Weierstrass function, and there are several other examples discussed in the book.</p>
<p>Those strict mathematical subjects in this book are discussed in a relatively short chapter, but the other aspects of Bolzano's work, mostly philosophical, are even more thoroughly discussed in the other chapters. That includes his opinions about ethics, political philosophy, philosophy of religion and the catholic church, about aesthetics and a science of beauty, as well as about ontology and metaphysics. A large chapter is devoted to logic and another one to his Wissenschaftslehre (Theory of science). Concerning logic, Bolzano was of the same idea as Leibniz who was convinced that logic had an important part in the philosophy of science, and he is at the origin of the interaction between logic and mathematics. This was opposed to Kant, who thought there was no role for logic in philosophy, and that what was used in mathematics was a completely different thing. Logic became a core aspect of Bolzano's philosophical work and it is a hidden precursor of what later became known as analytical philosophy, for which Gottlog Frege is usually considered to be the founding father.</p>
<p>His Theory of Knowledge, just like his logic, starts from the concept of truth by which he means the "truth in itself". Several such propositions do exist outside our mind. We do not have to reason about these. This sounds Cartesian, but he considered it not to be fundamental but as a way to refute scepticism. Then these propositions are brought in relations, inductions, etc., which is the logic in a narrow sense. Only after eleven hundred pages he comes to the theory of knowledge: ideas that can be conceptual or empirical and they are subject to judgement. New propositions can be obtained by logic, probability, or what is called an entailment relation (a set of propositions can entail a new proposition).</p>
<p>This book contains a thorough analysis of the philosophical work of Bernard Bolzano. Much of his work, has for a long time been unpublished, so that this book comes at a good time with the complete and very extensive scientific and philosophical production of Bolzano becoming more generally available in the Gesamtausgabe. The discussion of his mathematics and of his ideas on all aspects of science and society, are strictly documented with many citations from the original sources (in English translation) and from other authors that have studied his work. It is also placed in relation with other philosophers and mathematicians. This book is in the first place about the philosophy (of mathematics) as it can be found in Bolzano's writings. The strict mathematics in a narrow sense are omnipresent but at a second 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>After a sketch of the time and the life of Bernard Bolzano, the book is mainly an analysis of his work, which is mostly of a philosophical nature, even the chapter on mathematics. The chapters on logic and on his theory of knowledge are the most extensive ones, but also his philosophical ideas about religion, ethics and aesthetics, ontology and metaphysics, and politics, are amply discussed.</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/paul-rusnock" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Paul Rusnock</a></li><li class="vocabulary-links field-item odd"><a href="/author/jan-sebestik" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jan Sebestik</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">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">978-0198823681 (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">£ 80 (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">704</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://global.oup.com/academic/product/bernard-bolzano-9780198823681" title="Link to web page">https://global.oup.com/academic/product/bernard-bolzano-9780198823681</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/03-mathematical-logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03 Mathematical logic and foundations</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/03a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03A30</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/03a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03A05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/03a10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03A10</a></li><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/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</a></li></ul></span>Mon, 03 Jun 2019 09:01:48 +0000Adhemar Bultheel49422 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/bernard-bolzano-his-life-and-work#commentsThe Mathematical World of Charles L. Dodgson (Lewis Carroll)
https://euro-math-soc.eu/review/mathematical-world-charles-l-dodgson-lewis-carroll
<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>
Charles Lutwidge Dodgson (1832-1898) is the real name of Lewis Carroll, the author of <em>Alice's Adventures in Wonderland, Through the Looking Glass</em> and of other such books. He wrote these for Alice Liddell, the daughter of Henry Liddell, the dean of Christ Church in Oxford and he became forever famous as `the man who wrote <em>Alice</em>' and for a long time this was the only way he was remembered. Yet he was also a mathematician with very original ideas and a creditable photographer. These aspects were only properly recognized since the 1980s. His diaries and his collected publications were becoming available since the 1990s and with that material, more studies of his non-fiction work was possible. This book gives a survey of what is known so far.</p>
<p>
The book starts with a (short and in some respect only partial) biography, concentrating on the mathematician in him. Introduced by his father (a country parson) to mathematics as a young child, he excelled in this subject at school. Later, just like his father, he studied at Christ Church in Oxford and got a master degree in mathematics. While studying he gave private lessons to students. He was ordained a deacon when he was 29, but never became a priest. When Liddell became the dean of Christ Church, Dodgson was appointed as 'Master of the House' which gave him a reasonable income but also a large teaching load. He starts publishing his pamphlets as aids for teaching, and some work on the evaluation of determinants. He took on photographing as a hobby. He became rather good at it, using it as an art form, rather than as just a way to catch reality. Urged by Alice Liddell to write up the stories he told during their boat trips, he started working on <em>Alice's Adventures</em> which was published under his pen name Lewis Carroll. Teaching was his main occupation besides his writing and the photography. He became (reluctantly) Curator of the Christ Church Common Room for ten years when he was 50. This was a time-consuming burden requiring management and many decisions to take. In that context he could use his already existing interest in voting systems. He died a week before his 66th birthday.</p>
<p>
The next chapters are written by specialists and discuss in more detail Dodgson's contributions to different mathematical subjects. The first one deals with geometry. Dodgson was teaching geometry following Euclid's books as it was usual in those days. However there were some new ideas, among others by Sylvester, criticizing Euclid's approach. Euclid's arguments were not always waterproof in a mathematical sense, and there was the emerging hyperbolic geometry of Lobachevsky. Dodgson was defending however Euclid in his booklet <em>Euclid and his Modern Rivals</em>. The discussion about the parallel postulate involved either infinitely long lines or infinitely small quantities. So he tried to replace it by a finite alternative using an "obvious" property about areas outside and inside an hexagon inscribed in a circle. He thought of non-Euclidean geometry as nonsensical.</p>
<p>
In the next chapter on algebra, it is explained what his condensation method for the evaluation of determinants is. This may be his most useful original contribution to mathematics. He was also opposed to the name 'matrix' with the meaning we give it today. He called it a 'block' because a matrix refers to the mould, rather than the object, which is the mould filled with numbers. To denote the elements in a matrix, which we denote by a letter with two indices, he had his own strange notation. His work on determinants was published but (like many of his mathematical publications) remained largely unnoticed for a long time.</p>
<p>
Logic has been one of Dodgson's favourite subjects and he wrote several texts about it. In those days, as we still have today, there existed several proposals for an approach to, and the notation of, formal logic. The one from Boole was, and still is, a very useful one. Dodgson adhered the formal approach which was not easily accepted by classical logicians, and required a lot of dispute. He developed several tools to deal with logic problems: a method with diagrams, he had his own formal notation, a method of trees, and an algorithm to solve syllogisms. Bertrand Russell once said about Dodgson' work that it was brilliant but largely useless.</p>
<p>
Dodgson also wrote several texts on voting systems. The problem of cycles (where each candidate can win from the next candidate in a cyclic way) was known several centuries before, but Dodgson was probably not aware of that. He detected it on his own and made proposals for a correct voting system, for assigning seats to parties in a political election, or for a correct outcome of a tournament. He wrote about these problems in terms of game theory, an approach that John Nash would bring to a culmination only much later.</p>
<p>
It is clear that the author of the <em>Alice</em> books would also be interested in recreational mathematics, puzzles, riddles, and games. Many examples are discussed as well the way in which Dodgson solved them. He also had techniques to remember dates and numerical data, and techniques to check divisibility which could be smuggled into a number game.</p>
<p>
It is strange that Dodgson was so little recognized for his mathematical work. Most of his, sometimes original, approaches were only discovered at the end of the 20th century. He was not the best salesman for his results. Perhaps he didn't take it seriously enough, and maybe he was too quarrelsome (sarcasm indeed happened sometimes), or perhaps he was rather obscure when sticking to his own notation and ideas. He had for example his own symbolic notation for the trigonometric functions. He also was stammering a bit, but that did not seem to be a serious hinder to him. Moreover his contributions are very diverse. He did not have a single field in which he became the renowned expert and finally perhaps he was also in the shadow of Lewis Carroll. The legend goes that Queen Victoria, charmed by reading <em>Alice's Adventures in Wonderland</em>, asked for his next book and promptly received his <em>An Elementary Treatise on Determinants</em>. This story is however a hoax. He may not have received during his life the recognition by mathematicians he deserved, but nevertheless this book has an extensive chapter discussing his mathematical legacy in geometry, trigonometry, algebra, logic, voting, probability, and cryptology as it became clear only recently.</p>
<p>
The book ends with a complete bibliography listing all his publications and a list of references for additional reading. With that this book completes a survey of the life and work of a (perhaps somewhat dull) mathematician that is in many ways the opposite (but in as many ways also the complement) of the light-hearted Lewis Carroll that authored the books witnessing of such a rich fantasy dedicated to a little girl called Alice. A man well known as Lewis Carroll, yet perhaps too long underestimated as Charles Dodgson. This nicely edited book with many illustrations and written by experts on the subject will certainly help to turn the tide by adjusting the image and allow you to form a proper opinion about Charles Dodgson.</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 starts with a biography of Charles Dodgson, the mathematician, best known for his books like <em>Alice's Adventures in Wonderland</em> under his pen name Lewis Carroll. Its main purpose however is to discuss his work and influence as a mathematician.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/robin-wilson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robin Wilson</a></li><li class="vocabulary-links field-item odd"><a href="/author/amirouche-moktefi" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Amirouche Moktefi</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</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">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">9780198817000 (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">£29.99</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">288</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://global.oup.com/academic/product/the-mathematical-world-of-charles-l-dodgson-lewis-carroll-9780198817000" title="Link to web page">https://global.oup.com/academic/product/the-mathematical-world-of-charles-l-dodgson-lewis-carroll-9780198817000</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/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</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></ul></span>Wed, 20 Mar 2019 09:56:06 +0000Adhemar Bultheel49213 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/mathematical-world-charles-l-dodgson-lewis-carroll#commentsGiovanni Battista Guccia. Pioneer of International Cooperation in Mathematics
https://euro-math-soc.eu/review/giovanni-battista-guccia-pioneer-international-cooperation-mathematics
<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">Ángeles Prieto</div></div></div><div class="field field-name-field-review-appendix field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://euro-math-soc.eu/sites/default/files/book-review/GucciaBongiornoCurbera_1.pdf" type="application/pdf; length=73197" title="GucciaBongiornoCurbera.pdf">This book examines the life and work of the geometer G.B. Guccia and the first decades of the Circolo Matematico di Palermo.</a></span></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/benedetto-bongiorno" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Benedetto Bongiorno</a></li><li class="vocabulary-links field-item odd"><a href="/author/guillermo-p-curbera" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Guillermo P. Curbera</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" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer International Publishing</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">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">978-3-319-78666-7</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">34,31€</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">316</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/9783319786667" title="Link to web page">https://www.springer.com/gp/book/9783319786667</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/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</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/01a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a60</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</a></li></ul></span>Tue, 29 Jan 2019 16:49:09 +0000Ángeles Prieto49057 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/giovanni-battista-guccia-pioneer-international-cooperation-mathematics#commentsThe Forgotten Genius of Oliver Heaviside: A Maverick of Electrical Science
https://euro-math-soc.eu/review/forgotten-genius-oliver-heaviside-maverick-electrical-science
<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>
Oliver Heaviside (1850-1925) was a British self-made electrical engineer and mathematician. He is probably best known as an electrical engineer, although his name is not explicitly attached to many terms. There is a Heaviside condition for transmission lines, but maybe less known is that he also coined terms such as conductance, inductance, impedance, and many more. In mathematics, his name is explicitly attached to the Heaviside step function usually denoted as $H(x)$ in his honour, although he himself preferred the notation $\mathbf{1}$ instead. Sometimes his name is also attached to a method to compute the partial fraction expansion of a rational function. However, probably his most important contribution to science is that he reformulated the Maxwell equations in the form as we know them today. As a side product he introduced vector calculus and the associated (force)fields. It also brought complex numbers and complex analysis into electro-technical formulas. He is also the originator of operational calculus. The mathematics community was originally reluctant to accept it because it lacked fundamental rigidity. But it worked so well that it could not be ignored and others provided the necessary rigidity. It allowed to transform a differential equation into an algebraic one, which is much easier to solve. The letter <em>p</em> that is often used as the variable in the Laplace domain was his notation. In the time domain, it is a differential operator. The square root of $-1$, which mathematicians usually denote as $i$ is denoted as $j$ in (electrical) engineering because $i$ or $I$ was used for current (although Heaviside preferred to use $C$ for current). He considered $j$ as an operator that had the effect of delaying the signal with a quarter of a cycle, just as mathematicians see a multiplication with $i$ as a rotation over 90 degrees. These are but a few illustrations to show that his field was electrical engineering, and that he is more recognized for his legacy in that domain, it can be said that his influence on mathematics, although less known, is equally important.</p>
<p>
Mahon has used the term "maverick" in the subtitle which is most appropriate in the case of Heaviside. Now one can be a maverick in different ways. For example Richard Feynman was really unconventional and did not pander to the customs attached to his status, but it was all in a playful and friendly way. Heaviside on the other hand was a grumpy, stubborn, loner, who did not shy away from aggressive reactions and including insulting remarks in his scientific papers about people he did not agree with. He was for sure not the most likeable person. He had only few friends and admirers who tried to mediate between him and the scientific community. Among them were George FitzGerald, Oliver Lodge, and Heinrich Hertz, who together with Heaviside became known as <em>The Maxwellians</em> by the book of Bruce Hunt (1991). No wonder that such an outspoken character has inspired other biographers to write about Heaviside. Fortunately much of his publications, notes and letters are at their disposal to reconstruct his personality. Paul J Nahin's <em>Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age</em> (John Hopkins University Press, 2002), is a basic reference, and there is an account of Heaviside's character by his friend G.F.C. Searle described in the booklet <em>Oliver Heaviside, the Man</em> (1987). Also the book under review was published earlier in 2009 by a different publisher and with a slightly different title <em>Oliver Heaviside: Maverick mastermind of electricity</em>. Not much is changed in this edition except the spelling and the notes at the end which have been extended.</p>
<p>
As a child Oliver Heaviside had scarlet fever which left him partially deaf. This may in part explain his tendency to withdraw from crowds and prefer solitude. He suffered of several illnesses throughout his life, some were due to poverty and negligence. His uncle was Charles Wheatstone (from the Wheatstone Bridge known in electricity) and an expert in telegraphy. To understand Heaviside's scientific breakthrough, one should think back mid 19th century. James Maxwell just discovered the relation between electricity and magnetism which he presented as a complex system of 24 equations. Submarine cables for telegraphy were laid with a lot of experimentation and disastrous failures. Wheatstone looked after Heaviside's education, but when his parents could not afford the studies anymore he studied on his own. When working for Wheatstone's telegraph company he trained himself as an electrician and published a paper about the Wheatstone Bridge, that was received well by, among others, William Thomson (Lord Kelvin) and James Maxwell. His next publication earned him his first enemy: R.S. Culley, the engineer-in-chief of the nationalized telegraph company. Heaviside's view on the duplex method was opposed to Culley's and he ridiculed the man for his short-sightedness. His next achievement was a development of a theory for transmission lines (mathematically these are the telegraphers equations) which are of course tremendously important for telegraphy (and telephony) cables. The speed of light popped up in his equations, which pointed already to the electromagnetic interpretation of light.</p>
<p>
Heaviside has been poor throughout his whole life. He got a small income form his scientific contributions to <em>The Electrician</em> in the period 1882-1902. They also published his 3 volume work <em>Electromagnetic Theory</em> (1893-1912) but that didn't earn him much money. He had moved to London in 1882. It was there that he reduced twelve of Maxwell's equations to just four using vector calculus. They are in their simplest form the following relations: $\mathrm{curl}\,\mathbf{E}=−\frac{\partial \mathbf{B}}{\partial t}$, $\mathrm{curl}\,\mathbf{B}=\frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}$, $\mathrm{div}\,\mathbf{E}=0$, and $\mathrm{div}\,\mathbf{B}=0$, where $\mathbf{B}$ is the magnetic field and $\mathbf{E}$ the electric field, and $c$ the speed of light. A mathematical beauty by its simplicity and its symmetry. Previously people had tried to deal with them using quaternions, invented by Hamilton in 1843. Together with his brother Arthur, Oliver had worked on the design of a distortion free transmission line. One had to arrange that $G/C=L/R$, which is known as the Heaviside condition ($C=$ capacitance, $R=$ resistance, $L=$ inductance, and $G=$ shunt inductance). This was his gift to society that would make long-distance telephony possible. However Arthur and Oliver had to ask for publishing permission from their employer the Post Office, but that was surprisingly refused. The bad omen was William Preece, who had opposing views on the solution and happened to be the engineer-in-chief of the Post Office then, and this resulted in a lifelong and bitter battle between him and Oliver. Oliver referred to Preece's ideas as the "drain-pipe theory". Oliver's results were eventually published with some delay, which brought him new fame. Unfortunately for Oliver, Preece was well respected and the next year he became president of the IEE and in this position he could thwart Oliver some more. In that period the Maxwellians became friends and collaborators. First FitzGerald and Lodge, and later Hertz from Germany.</p>
<p>
Approaching the turn of the century, fate still haunted Oliver. His mother died in 1894, his father in 1896. Poorer than ever and plagued by illness his friends organized a pension for him and he moved to live on his own. Pupin (and AT&T) in the U.S. got rich on a patent for distortion-free transmission lines based on a formula that Heaviside published 3 years earlier. When the Royal Society wanted to award him the Hughes Medal, he refused because they had rejected the last part of his paper on operators in physics saying that the mathematics were not rigorous enough. Nevertheless, later he accepted a pension, just to survive. But recognition started to emerge. At some point he was even shortlisted for the Nobel Prize (like Einstein and Planck, but the 1912 Prize went to Niels Dalen). By then he had published his three volumes of his <em>Electromagnetic Theory</em>. He got an honorary membership of the American IEE (1918) and he was awarded the Faraday Medal of the IEE (1921). When he was found unconscious in 1925 he was moved to a home, but he died shortly after.</p>
<p>
Mahon goes through the life of Heaviside in twelve chapters, each one corresponding to a place where Heaviside lived and covering successive time periods. However each chapter is also the starting point to discuss one of his achievements. Because that requires to explain what preceded, how he came to his conclusion, and how and by whom the idea was picked up, and what its eventual fate was, the period covered in a chapter is much broader than announced in the title. Hence, the text is not always strictly chronological. Therefore the chronology summarised in the time-line in the beginning of the book comes in handy. There are also, sometimes relatively extensive, biographies of the persons who played a role in Heaviside's life, and then the list of the main characters inserted after the time-line is also useful. It is clear that Mahon, who also authored a book on Maxwell and co-authored another one on Maxwell and Faraday, is an admirer of Heaviside's electrical contributions, but he also gives credit to his significance for mathematics. The book is obviously written for a general public, so the discussions about Heaviside's results are only slightly technical. There are many quotations from texts written by Heaviside, which explains a lot of how his character was, and what he thought and expected from others. The way he wrote about his housekeeper at a later age is hilarious. He must have been a very difficult man to live with. There are also 33 pages with extra notes giving additional explanations. The book gives some insight in his work and the man that is behind it. Heaviside's very peculiar and enigmatic character will forever remain inscrutable, but Mahon does a really good attempt to understand what was driving this lone genius and to give him some of the respect that he rightfully deserves and that he had to miss during most of his life.</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 second edition (only slightly modified) of a biography of Oliver Heaviside (1850-1925). Heaviside was a self-taught electrical engineer and mathematician. He had an obtrusive character and an unconventional approach of doing research. Therefore it took a while before the geniality of his work was recognized.</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/basil-mahon" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Basil Mahon</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781633883314 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 29.00 (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">336</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</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.penguinrandomhouse.com/books/556320/the-forgotten-genius-of-oliver-heaviside-by-basil-mahon/9781633883314/" title="Link to web page">https://www.penguinrandomhouse.com/books/556320/the-forgotten-genius-of-oliver-heaviside-by-basil-mahon/9781633883314/</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/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</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/35q61" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35Q61</a></li></ul></span>Mon, 08 Jan 2018 19:59:11 +0000Adhemar Bultheel48153 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/forgotten-genius-oliver-heaviside-maverick-electrical-science#commentsAn Imaginary Tale: The Story of √-1
https://euro-math-soc.eu/review/imaginary-tale-story-%E2%88%9A-1
<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 reprint in the <em>New Princeton Science Library</em> of a classic. The series brings reprints in cheap paperback and eBook format of classics, written by major scientists and makes them available for a new generation of the broad public. The series includes not only math books but covers a broader area, although there are several mathematics classics in the catalog written by E. Maor, J. Havil, and R. Rucker, but also J. Napier, A. Einstein, O. Toeplitz, R. Feynman, S. Hawking, R. Penrose, W. Heisenberg, etc. So if you missed out on some of the original editions, or were not even born at that time, this is a chance to get one of these more recent reprints. Another classic, reprinted at about the same time is Maor's <a href="/review/e-story-number"><em>e: the story of a number</em></a> which follows a similar idea that may have inspired the current author.</p>
<p>
The current reprint is of the first paperback edition of 2007 which is an updated version of the original from 1998. Paul Nahin is an electrical engineer who wrote several successful popular science books. His first one was a biography of Heaviside, and this book about complex numbers (it contains even an introduction to complex functions) was his second. Several other were to follow, some of which have been reviewed in this EMS database: <a href="/review/chases-and-escapes-mathematics-pursuit-and-evasion"><em>Chases and Escapes. The Mathematics of Pursuit and Evasion</em></a> (2007), <a href="/review/digital-dice-computational-solutions-practical-probability-problems"><em>Digital Dice. Computational Solutions to Practical Probability Problems</em></a> (2008), <a href="/review/number-crunching"><em>Number Crunching</em></a> (2011), <a href="/review/logician-and-engineer-how-george-boole-and-claude-shannon-created-information-age"><em>The Logician and the Engineer. How George Boole and Claude Shannon Created the Information Age</em></a> (2012), <a href="/review/holy-sci-fi-where-science-fiction-and-religion-intersect"><em>Holy Sci-Fi! Where Science Fiction and Religion Intersect</em></a> (2014), Of course complex numbers and functions are important tools in electrical engineering. The book has a strong historical component of course, but, unlike the book by Maor about the history of the number e, this book has much more mathematics in it. Hence it requires some mathematical affinity to understand much of what is presented here. It requires the knowledge of advanced secondary school or even freshman's university level, in particular when it turns into an introductory course on complex functions in the trailing chapter. Nevertheless, Nahin avoids a textbook structure of definitions, theorems and proofs, but keeps the level of a casual account, cheering up the reader with some witty remarks now and then.</p>
<p>
The historical background starts with the solution of the cubic equation and the search for a formula that gives its roots, which was a hot topic in the 16th century. Knowing such a formula was a strong weapon for `mathematicians' that made a living as (human) computers, so it was important not to share it with competitors. Obviously this includes the well known story of del Ferro who knew how to solve the cubics and who told it to Antonio Fior before he died. Niccolo Fontana, better known as Tartaglia, also knew how to solve them. It came to a public duel between the latter two to solve the most equations in a given time. Tartaglia won much to Fior's surprise. Cardano who was a well respected mathematician in those days, stalked Tartaglia to tell him the magic formula, and after much pressure, Tartaglia eventually told Cardano, but made him promise to keep it a secret. However Cardano could consult the letters by del Ferro and considered his promise to Tartaglia not valid anymore and published the result anyway, which resulted in a vigorous priority fight.</p>
<p>
Solving cubic equations is important for the history of complex numbers because the square roots in the relatively complex formulas could give complex conjugate solutions. However, even though the square roots of negative numbers made no sense to them, it turned out that when computations were performed as if these were genuine numbers, this could lead to valid results, which was most puzzling at first. The formulas are known as Cardano's but historically this is clearly a mistake. They were re-discovered a couple of times by others (e.g. Leibniz).</p>
<p>
Since in antiquity (think of the Greek) many computations were done by geometric constructions. Descartes, Wallis, Newton, and others thought about a construction of the square root with compass and ruler. In geometric constructions it is difficult to give a meaning to a negative number when it concerns the length of a line segment or the area of a polygon. It was not before Bombelli had the idea of drawing a line with marks for the numbers (which we now call the real line) that negative numbers finally made sense. It was even more staggering in those days to make geometric sense of an imaginary number. Some possible interpretation was that a line intersected another one outside an certain interval, which made the intersection `imaginary'.</p>
<p>
Of course, as we know, the proper interpretation of a complex number should be a point in the complex plane. That idea and the modulus-argument representation of the complex numbers with a complex exponential was first given by the Norwegian Caspar Wessel in 1797. He was not even a professional mathematician and succeeded where many great minds had failed before him. His finding went unnoticed though, until much later. Argand and Buée came to the same solution about a decade later and published their findings almost simultaneously which started another row between them. Again these names do not sound very familiar to us. Argand was a French amateur mathematician and Buée was a French priest who published a confusing, almost mystical paper on the subject. Just like Wessel's, their results faded away and were only re-discovered much later. Of course once the complex plane is accepted, one gets the goniometric representation, the formulas of De Moivre, the multiplication with i corresponds to a rotation over 90 degrees, etc. This is close to the vector interpretation by Hamilton, who finally gave a formal definition of the complex number field where a complex number was a couple of reals with vector addition and scaling and with a particular way of multiplying the vectors.</p>
<p>
Once the complex playground has been fixed in the first three chapters, the next three chapters deal with applications of complex numbers. Since the complex numbers are like vectors in a plane, some geometric problems can be easily solved with complex arithmetic. A less known theorem of Cotes and a puzzle problem from Gamow's book <em>One, Two, Three... Infinity</em> are given as examples. Other applications discussed include the `imaginary' time axis in the space-time indefinite metric of relativity theory, the maximal distance of a random walk with decreasing step sizes, Kepler's laws, and electrical circuits. More mathematics is found in the chapter on Euler and the famous Euler formula (exp(ix) + 1 = 0) but also infinite series, infinite products, the calculation of i to the power i, and even the gamma and zeta functions. The final chapter is an introduction to complex functions, derivatives, contour integration, Cauchy integral theorem, Green's theorem, analytic and harmonic functions.</p>
<p>
Concerning the structure of the text, I can mention that it is occasionally interrupted by `boxes' that discuss some topic closely related to the surrounding text, but that is not essential and could be skipped without a problem. On the other hand, some technical material is moved to appendices. The reprint is the unaltered version of 2007. That means that no additions or corrections are added since and the little defects that remained are still there. For example, the strange looking capital gamma symbol on page 177 (it is at least a different font from the surrounding pages) is still there. The preface of 2007 does explain what corrections and additions were made on that occasion. This book is not recommended if you are allergic to formulas, but if you want to peek behind the formulas and theorems in a textbook (a textbook is more `to the point' and hence necessarily `duller'), this is a the book that I recommend to read. You will definitely enjoy it. In fact it clearly reflects the the joy and delight that the author experienced when he was confronted with complex analysis during his engineering studies. </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 in the New Princeton Science Library of the bestselling original from 1998. It tells the story of the square root of –1, and that includes complex numbers, how they came about and what they can be used for. At the end there is even a brief introduction to complex functions. </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/paul-j-nahin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">paul j. nahin</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">2016</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">9780691169248 (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">£11.95</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">296</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/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://press.princeton.edu/titles/9259.html" title="Link to web page">http://press.princeton.edu/titles/9259.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/01a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A40</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/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/30-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30-03</a></li></ul></span>Wed, 27 Apr 2016 09:27:50 +0000Adhemar Bultheel46900 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/imaginary-tale-story-%E2%88%9A-1#commentsThe Real and the Complex: A History of Analysis in the 19th Century
https://euro-math-soc.eu/review/real-and-complex-history-analysis-19th-century
<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 history book on the development of mathematics in the 19th century. Each chapter is built up around one or a few mathematicians. First a short bio is sketched, often embedded in the political context of their time, but the more important part is where it is shown what theory the person has developed, in what context it was done, hence why he (there are unfortunately no she's) did it. It shows that even the big shots of mathematics that contributed to the tremendous expansion of mathematical knowledge in the 1800s were searching, sometimes uncertain or even mistaken. With several of the emerging concepts that we are now familiar with today still being shaped and reshaped, these were first formulated, not taking care of the tiny details, which needed a revision or even rethinking the concept later on. It was also the century in which analysis came to the foreground as one of the main mathematical topics next to geometry and algebra that had dominated before. Analysis became a working tool that was used in other domains of mathematics, it became essential in modeling physical phenomena and it was intensively used to solve many applied problems. However, mathematics gradually evolves towards a more abstract subject and is developed more independently from the applications. How all this came about becomes clear by reading this book.</p>
<p>The book is written as a textbook on the history of mathematics, and hence it is assumed that the reader has attended some analysis courses: real and preferably also complex analysis. There are also (few) end-of-chapter exercises which are clearly pointing to history students. That means that some topics are suggested to investigate and to report on them in an essay. There is also an end-of-course chapter giving advise of how to choose a topic for an essay and what kind of content it should be given (the mathematical technicalities are less important, as long as they are correct, but concentrate on why and how mathematical concepts grew into the ones that we know today).</p>
<p>The organization of the chapters is more or less chronological. One may recognize three parts: the first one sketches the situation in the early and the first half of the 19th century; the second part deals with the middle of the century, when complex analysis comes more to the foreground; and the third part is then announcing the transition to the 20th century, the foundations of mathematics are questioned, set theory, the real number system, topology all push mathematics into a more abstract framework.</p>
<p>Often the mathematician's findings were written down in books that grew out of lecture notes, which of course forced them to reflect upon the foundations of what they were teaching and these books were obviously very influential. The concepts of course still survive, but we would not always be satisfied with the way they were originally described.</p>
<p>The initial setting is made in the first three chapters with Lagrange, Fourier and Legendre. It is clear that these were interested in developing new ideas, not worrying too much about the basics or the finer details. Lagrange struggled with the foundations of analysis, his approach being basically algebraic without infinitesimals. Fourier had proposed his trigonometric series, and Legendre's contribution was to set up a theory of elliptic integrals. These elliptic functions form a recurrent topic in the next chapters as it was further developed by Abel who was the first to place them in the realm of complex functions, as did also Jacobi, Gauss, Liouville, and Hermite. In fact, Gray uses them as a kind of case study that extends over a large part of the book. Cauchy of course was influential on many other domains as well: continuity, series, differentiation and integration, and complex functions. He was the first to introduce more structure and rigor in real analysis. Equally productive was the master calculator Gauss with contributions on integration and complex analysis. These chapters span approximately the first half of the century. Gray interrupts here the development to give a reflection on what has been achieved so far.</p>
<p>The next half of the century starts with a new topic: potential theory (Green, Cauchy, Dirichlet). Riemann is given somewhat more attention in several chapters with his Riemann function, elliptic functions (the bread crumbs in this historical expedition), but of course also complex analysis and his conformal mapping theorem. There is also a discussion of how his work was received by his contemporaries. An alternative for Riemann's geometric function theory was provided by Weierstrass who initiated the concept of an analytic function. Here Gray reflects again on the past chapters. Still real analysis, and certainly complex analysis had not reached the rigor that we are used to.</p>
<p>This rigor was only starting to develop in what is the third part of this book. For example, it was only noted 20 years after its original formulation that Cauchy's theorem stating that the sum of (infinitely many) continuous functions was continuous did not always hold true. Only then, it was realized that functions could be much more exotic objects than the smooth curves that they were originally thought of. Mending Cauchy's theorem led to the concept of uniform convergence, non-differentiable and non-integrable functions (Bolzano, Cantor, Schwarz, Heine, Dini,...). This entailed Lebesgue's integration theory and the unavoidable rigorous definition of real number system, set theory, and topology.</p>
<p>It is also interesting to see what triggered the development of mathematics. Fourier used his series in heat diffusion problems, Legendre used elliptic integrals to study the mechanics of a pendulum, potential theory grew out of electromagnetic problems. However with rigor came also abstraction and, although still applicable, mathematics itself became the driving force for its own expansion.</p>
<p>The text is illustrated with portraits of the key mathematicians and where appropriate, plots to visualize some of the interesting functions or graphs are included. In several appendices, we find some translated papers (Fourier, Dirichlet, Riemann, Schwarz), and some more technical mathematical ones on series of functions and their convergence and on potential theory. Obviously a list of references (many of them being the original publications that were discussed, while others are more recent historical studies) as well as a mixed index of names and subjects are indispensable in a book like the present one.</p>
<p>It is very well known that history is of utmost importance and that we should learn from it. However, it is astonishing how fast even recent history is completely forgotten. This book certainly learns something to students but also to professional mathematicians, something that is too often neglected. Mathematics develops not by adding some epsilon improvements to an existing sequence of definitions, properties, theorems and proofs. The majority of the papers that are published are in that vein of thinking. With each one the boundaries of our mathematical knowledge crawls a bit forward. But what is actually progressing mathematics is the exploration of an unknown mine field. The explorers that venture there are the ones whose names will be printed in bold face in future history books. This book learns that these explorers of the past may have felt uncertain, made mistakes, and even used a trial and error approach. Only when the right track has been flagged, the road can be paved with the proper rigor needed to move on. So this book is not only an interesting read for the students who (have to) study it, but equally valuable for professional mathematicians. This is if they are prepared to take the time and reflect on the not so distant past of their beloved subject to which they want to contribute.</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 textbook for a course on the history of mathematics. In three parts Gray sketches the evolution of the shaky emerging of analysis in the beginning of the 19th century, its growth into a more rigorous subject and the extension to complex analysis, and finally how near the end of the century the foundations of mathematical analysis were revised, resulting in the definition of the real number system and new subjects such as topology entering mathematics.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/jeremy-gray" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jeremy Gray</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-verlag-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-23714-5 (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">37,09 € (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">366</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/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li><li class="vocabulary-links field-item odd"><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="http://www.springer.com/gp/book/9783319237145" title="Link to web page">http://www.springer.com/gp/book/9783319237145</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/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</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></ul></span>Tue, 09 Feb 2016 06:16:38 +0000Adhemar Bultheel46704 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/real-and-complex-history-analysis-19th-century#commentsAndré-Louis Cholesky
https://euro-math-soc.eu/review/andr%C3%A9-louis-cholesky
<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>
André-Louis Cholesky (1875-1918) was a French military officer who served as a topographer in the army. He studied at the <em>École Polytechnique</em> and was sent to Crete, Tunisia and Algeria for measurements. He was also participating in correspondence courses at the <em>École Spéciale des Travaux Publics</em> founded by <em>Léon Eyrolles</em> in 1891. During the war he was assigned to the artillery but in the period 1916-1918 he was also director of the Geographical Service in Romania. After his return to France he took part in the second Battle of Picardy where he was fatally wounded. He died in Bagneux near Soissons at the end of WW I. Cholesky is famous among topographers, certainly in France, while most mathematicians will be familiar with the Cholesky method for the factorization of symmetric matrices. Cholesky never published the method himself, but Ernest Benoît, a military colleague, published the method in 1924. Benoît had also written a short biography before (an English translation of 1975 is included in this book).</p>
<p>
Claude Brezinski is professor emeritus of numerical analysis, but he has been keen on the history of sciences and has published papers and books in both domains. Already in 1996 he wrote about Cholesky. That was when he got access to the documents at the <em>École Polytechnique</em> that, according to the French law, became public 120 years after Cholesky's birth. His research got an enormous momentum when in 2003 he was contacted to help classify Cholesky's archives that the Cholesky family wanted to donate to the <em>École Polytechnique</em>. The result is reflected in this book.</p>
<p>
The logical start is a detailed biography of Cholesky. It has many illustrations, mostly pictures of Cholesky, and (translated) citations from several documents.<br />
Raymond Nuvet, the vice-mayor of Montguyon, Cholesky's birth place, is largely responsible for a second chapter that sketches the family history from Cholesky's great great grandfather till his grandchildren. The roots lay most probably somewhere in Poland, but the precise origin is uncertain.<br />
Chapter 3 explains some elements from topography. That involves triangulation: one has to measure all the angles of the connected triangles, but only the length of one side of a starting triangle. All unknown lengths, and hence the coordinates of the vertices, can be computed by solving a system of equations. To deal with measurement errors, a least squares solution is computed. In this context the system is usually underdetermined. Another aspect of topography is leveling. Because not all measurement points will be in the same horizontal plane, their relative elevation has to be measured and taken into account. Cholesky developed a method of double-run leveling in 1910, which is still used today.</p>
<p>
The following chapter is more extensive and deals with Cholesky's method to solve (symmetric) linear systems. The history starts with the least squares technique attributed to Gauss, the method of Gaussian elimination, variants by Doolittle and others, and of course a discussion (and a translation) of Cholesky's unpublished notes (the original French version appears as an appendix). An analysis of the notes shows the skills of Cholesky. He discussed the computational complexity, the convergence of the square root computation, and gives a rounding error analysis. Brezinski writes <em>"If this paper was submitted today to a numerical analysis journal, it would be recommended for publication without any hesitation".</em> The chapter continues by explaining how the method was re-invented, and how it gradually was spread among the computing community and what current research is dealing with (iterative methods, preconditioning, etc.).</p>
<p>
Other work of Cholesky (military and topographical) is surveyed in a short chapter, and another chapter sketches a biography of Léon Eyrolles, especially the early evolution of the <em>École Spéciale des Travaux Publics</em> where Cholesky was a professor. Eyrolles is also the founder of the publishing company <em>Éditions Eyrolles</em>. In the archives of Cholesky, also an unpublished book was found about graphical computation. It is typeset in the original French version in an appendix and it is placed in its historical context and extensively discussed in a separate chapter by Dominique Tournès who is professor of mathematics and the history of mathematics.</p>
<p>
The next chapter is devoted to Ernest Benoît. The authors had a hard time to find the correct information. In all reports he was mentioned as <em>Commandant Benoît</em> (or with another military title as appropriate). Not even an initial for his first name was known and the name Benoît is a common name in France. Nevertheless his history is tracked down and a translation of his eulogy of Cholesky is included in English translation.<br />
The last chapter is an inventory of other documents from the Cholesky archive. It contains translations of military reports about Cholesky and of diary booklets written by Cholesky during his field work in France in 1905.</p>
<p>
This book is the only book about Cholesky that is currently available. After the little bits that were available in the few publications before, the wealth of facts that is brought here about Cholesky's life and work is overwhelming and will be almost impossible to surpass. Whatever the reader did not find here can be found in the archives in the <em>Fonds Cholesky</em> at the <em>Éole Polytechnique</em> to which the authors refer if needed. The style is rather factual and there are many illustrations. All the historical and mathematical context that a reader can wish for is provided. Sometimes pushed to the extreme. For example, every name of a person occurring in the text is followed by place and date (day, month, and year!) of birth and death (if known of course). This is clearly very informative, but you can imagine that it sometimes hinders readability a bit when several names occur in the same sentence.</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>
Cholesky (1875-1918) was a military topographer and well known as such, but he is also known among mathematicians because of his method for the factorization of symmetric matrices. The book contains his biography and a family history, but it also discusses the history of his contributions to science, and a biography of other important persons connected with Cholesky. The book is the result of the inventory of the archives that became publicly available only recently. His original (in French) unpublished paper on his factorization method and a course book on graphical computation are included. Both are extensively commented and analyzed.</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/claude-brezinski" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">claude brezinski</a></li><li class="vocabulary-links field-item odd"><a href="/author/dominique-tourn%C3%A8s" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dominique Tournès</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-basel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser basel</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">978-3-319-08134-2 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">100,69 € (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">344</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="http://www.springer.com/978-3-319-08134-2" title="Link to web page">http://www.springer.com/978-3-319-08134-2</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/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/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</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/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/01a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a60</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a90" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A90</a></li></ul></span>Mon, 20 Oct 2014 07:20:08 +0000Adhemar Bultheel45782 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/andr%C3%A9-louis-cholesky#comments