A Richer Picture of Mathematics – The Göttingen Tradition and Beyond

March Book of the Month is A Richer Picture of Mathematics – The Göttingen Tradition and Beyond by David E. Rowe.

Historian David E. Rowe (Prof. em. for History of Natural Sciences at the University of Mainz) captures the rich tapestry of mathematical creativity in this collection of essays from the “Years Ago” column of The Mathematical Intelligencer. With topics ranging from ancient Greek mathematics to modern relativistic cosmology, this collection conveys the impetus and spirit of Rowe’s various and many-faceted contributions to the history of mathematics. Centered on the Göttingen mathematical tradition, these stories illuminate important facets of mathematical activity often overlooked in other accounts.

Six sections place the essays in chronological and thematic order, beginning with new introductions that contextualize each section. The essays that follow recount episodes relating to the section’s overall theme. All of the essays (one about “The young Felix Klein”) in this collection, with the exception of two, appeared over the course of more than 30 years in The Mathematical Intelligencer. Based largely on archival and primary sources, these vignettes offer unusual insights into behind-the-scenes events. Taken together, they aim to show how Göttingen managed to attract an extraordinary array of talented individuals, several of whom contributed to the development of a new mathematical culture during the first decades of the twentieth century.

See: https://www.springer.com/de/book/9783319678184

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Klein Concert

Maria Mannone from the University of Palermo is a theoretical physicist and a musician (composer and conductor). Her field of interest is mathematical music theory. She and her students have created music based on the Klein bottle. In May 2019 the first Klein Concert was performed at the Music Conservatory of Palermo. For more about the concert, including a video, see

“Klein Concert”: a Report on a Geometric Journey

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The Enormous Theorem

Originating author is Nitsa Movshovitz-Hadar.


This article is an English translation with emendations of a Hebrew paper written in 2014 about two years after the untimely passing of the illustrious lecturer and group theory expert, Professor David Chillag. It is dedicated to commemorating him. He entrusted me with the transparencies that accompanied his lecture on the enormous theorem presented in 2011 at the Technion Mathematical Club, hoping to produce a paper for teachers and teachers’ teachers on the subject. Unfortunately, we did not manage to do it together.
I thank those of our mutual colleagues who helped me with this^1, and at the same time, of course, I take full responsibility for things that are not entirely clear or even incorrect.
Nitsa Movshovitz-Hadar

Introduction – What is this article all about?

The definition of a group in mathematics is an abstraction of the set of integers with the familiar operation of addition. A group is a set (such as the integers) with a binary operation which enables combining any two elements in the set and get as a result one element in the set (such as the sum of two integers), that has a few simple properties such as the existence of a unique “neutral element” (zero in the case of the integers). There is a large wealth of mathematical groups, differing from one another in composition and size. Some of them consist of a finite set (such as the set \{0,1,2,3,4,5,6,7,8,9,10,11\}), others consist of an infinite one (such as the integers). The binary operations vary as well (think, for example, about addition-modulo-12 defined on the finite set mentioned earlier^2, or about addition defined on the integers). The wealth of mathematical groups is so large that there is no chance of classifying even the finite ones of them. Fortunately, there is a special family of finite groups called simple finite groups, which can be classified. It turns out that this special family of finite groups is of great importance because, in some sense, all finite groups can be built from simple finite groups in a way that is somewhat analogous to the fact that all the positive integers can be built from the prime ones. The “enormous theorem,” the subject of this article, gives a complete classification of the simple finite groups. In this way, it provides tools for analyzing the structure and features of all finite groups. The theorem received its “enormous” nickname for some very good reason, to be explained below.
Let us meet the people behind the scene.

Continue reading
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Site of the Month: 3Blue1Brown

3Blue1Brown presents animated videos about mathematics. It was created by Grant Sanderson, a graduate student from Stanford University who worked for Khan Academy. He created a YouTube-channel with videos on calculus, linear algebra, geometry, topology,  and many special topics such as Fourier transformations or the Riemann hypothesis. He answers questions like “But why is a sphere’s surface area four times its shadow?” or “Ever wondered why slicing a cone gives an ellipse?”. The videos use 3D-animations and they explain mathematical problems from a visual perspective.

The channel name and logo reference the colour of Grant’s right eye, which has blue-brown sectoral heterochromia.

See also: https://en.wikipedia.org/wiki/3Blue1Brown

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Developing Research in Mathematics Education

9781138080294August’s book of the month is Developing Research in Mathematics Education: Twenty Years of Communication, Cooperation and Collaboration in Europe, edited by Tommy Dreyfus, Michèle Artigue, Despina Potari, Susanne Prediger, and Kenneth Ruthven.

It is the first book in the series New Perspectives on Research in Mathematics Education, to be produced in association with the prestigious European Society for Research in Mathematics Education. This inaugural volume sets out broad advances in research in mathematics education which have accumulated over the last 20 years through the sustained exchange of ideas and collaboration between researchers in the field.

An impressive range of contributors provide specifically European and complementary global perspectives on major areas of research in the field on topics that include:

  • the content domains of arithmetic, geometry, algebra, statistics, and probability;
  • the mathematical processes of proving and modeling;
  • teaching and learning at specific age levels from early years to university;
  • teacher education, teaching and classroom practices;
  • special aspects of teaching and learning mathematics such as creativity, affect, diversity, technology and history;
  • theoretical perspectives and comparative approaches in mathematics education research.

This book is a fascinating compendium of state-of-the-art knowledge for all mathematics education researchers, graduate students, teacher educators and curriculum developers worldwide.

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The Legacy of Felix Klein

The Legacy of Felix KleinDecember’s Book of the Month is The Legacy of Felix Klein by Hans-Georg Weigand, William McCallum, Marta Menghini, Michael Neubrand and Gert Schubring (Eds.) (2018).

Throughout his professional life, Felix Klein emphasised the importance of reflecting upon mathematics teaching and learning from both a mathematical and a psychological or educational point of view, and he strongly promoted the modernisation of mathematics in the classroom. Felix Klein developed ideas on university lectures for student teachers, which he later consolidated at the beginning of the last century in the three books Elementary Mathematics from a higher standpoint. At the 13th International Congress on Mathematical Education (ICME-13) 2016 in Hamburg, the “Thematic Afternoon” with the The Legacy of Felix Klein as one major theme, provided an overview of Felix Klein’s ideas. It highlighted some developments in university teaching and school mathematics related to Felix Klein’s thoughts stemming from the last century. Moreover, it discussed the meaning, the importance and the legacy of Klein’s ideas nowadays and in the future in an international, global context.
The whole book intends to show that many ideas of Felix Klein can be reinterpreted in the context of the current situation, and give some hints and advice for dealing today with current problems in teacher education and teaching mathematics in secondary schools. In this spirit, old ideas stay young, but it needs competent, committed and assertive people to bring these ideas to life.

(–> Link to Springer)

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Three Volumes of Felix Klein’s “Elementary Mathematics from a Higher Standpoint”

Felix Klein Mathematics 1The November Book of the Month is Elementary Mathematics from a Higher Standpoint by Felix Klein.

The volumes, first published between 1902 and 1908, are lecture notes of courses that Klein offered to future mathematics teachers, realizing a new form of teacher training that remained valid and effective until today: Klein leads the students to gain a more comprehensive and methodological point of view on school mathematics. The volumes enable us to understand Klein’s far-reaching conception of elementarisation, of the “elementary from a higher standpoint”, in its implementation for school mathematics.

These three volumes constitute the first complete English translation of Felix Klein’s seminal series “Elementarmathematik vom höheren Standpunkte aus”. “Complete” has a twofold meaning here: First, there now exists a translation of volume III into English, while until today the only translation had been into Chinese. Second, the English versions of volume I and II had omitted several, even extended parts of the original, while we now present a complete revised translation into modern English. These translations were done bei Marta Menghini (University of Rome) und Gert Schubring (University of Bielefeld and University of Rio de Janeiro).

(–> Link to Springer)

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Journey Through Genius

The May Book of the Month is Journey Through Genius: The great theorems of mathematics by William Dunham, first published by John Wiley in 1990, and republished in Penguin Paperback in 1991.

Dunham places each theorem within its historical context and explores the life of the creator — from Archimedes to Gerolamo Cardano to Georg Cantor. He also provides step-by-step proofs for the theorems, each easily accessible to readers with no more than a knowledge of high school mathematics. His approach is to present this material as a work of art like music or literature.

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Mathematics News

ScienceDailyMay’s Site of the Month is: ScienceDaily – Mathematics News

This site is part of the ScienceDaily website. Here are updated articles that give short overviews of recent applications of mathematics. The entries are backed up with references for further reading.

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Unsolved Problems in Number Theory

NumberTheoryMarch Book of the Month is Unsolved Problems in Number Theory by Richard Guy, first published by Springer Verlag in 1981. Third edition (nearly three times the size!!) published in 2004.

These problems are mostly very easy to understand, but are as yet unsolved. Guy gives an account of the problems, and the progress made on them. He does this in such a way that they provide food for thought and avenues for exploration for mathematicians at varying levels of maturity in number theory.

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