How good were you in Mathematics in schools ! : a²+b² = (a+b)² - 2ab !!

World Health Organisation came into being in 1948 and is now headquartered in Geneva.   Year 2020 started normally but has been proceeding on a wrong note. On January 31, the first 2 novel coronavirus cases in the UK,  the first 2 cases in Russia,  and the first case in Sweden and in Spain were reported. Canada reported its 4th case.  On that day US  declared Coronavirus a Public Health Emergency.  Wuhan (the city where the virus originated) is the largest city in Central China, with a population of over 11 million people. The city, on January 23, shut down transport links. Following Wuhan lock down, the city of Huanggang was also placed in quarantine, and the city of Ezhou closed its train stations. That meant 18 million people had been placed in isolation. The World Health Organization (WHO) said cutting off a city as large as Wuhan is "unprecedented in public health history" and praised China for its incredible commitment to isolate the virus and minimize the spread to other countries.

By every scale, WHO failed in realizing the potent danger; WHO however ensured that Corona virus was named Covid-19 not reflecting the place or country of its origin.  Covid has now affected   213 countries and territories around the world and 2 international conveyances.  Reportedly   22,718,030 people have been affected with death toll @ 793,648.  Somehow statistical references never refer to the no. of recovered persons @ 15,405,274.  

How good were you in Mathematics in schools !  :   a²+b² = (a+b)² - 2ab  !!  ~ in my X standard I had Maths tuition by Mr V Chandrasekharan alongside close to 50 students – it looked a virtual classroom atmosphere and the Master taught Algebra so well.  Ever thought of this angle ? – Schools which would have reopened by June would perhaps now open by Dec or by Jan 2021 – and how would students learn the subjects, especially Mathematics.

A similar conflict occurred perhaps during World Wars.  When the smoke cleared after World War 1, European academia was in disarray. Many scholars had been recruited to the frontlines, where they faced disease, famine, and enemy soldiers. France like every other country  was  hit hard, with Aubin and Goldstein claiming that “from the class entering in 1910, more than 6 out of 10 science graduates never came back from the front.” This left French academia fractured, convoluted, and stagnate for a generation. Mathematicians in particular struggled for decades to rebuild their field. By the 1930s, lack of coordination in French mathematics-and much of the western world-produced different methods and terminology, and lack of consensus made writing a new textbook infeasible. In fact, a mathematics textbook had not been published since Goursat published his in 1904. The field was languishing and the future of mathematics was hazy. There was a noticeable impasse and increasing uncertainty around the study of mathematics in France. 

Éléments de mathématique is a treatise on mathematics  authored by  Nicolas Bourbaki, composed of twelve books (each divided into one or more chapters). The first volumes were published   from 1939 initially in the form of booklets and then as bound volumes.  The strange singular "mathématique" in the title is deliberate, to convey the authors' belief that the material is a unity, contrary to what conventional form of the title might suggest.

By many measures, Nicolas Bourbaki ranks among the greatest mathematicians of the 20th century. Largely unknown today, Bourbaki is likely the last mathematician to master nearly all aspects of the field. A consummate collaborator, he made fundamental contributions to important mathematical fields such as set theory and functional analysis. He also revolutionized mathematics by emphasizing rigor in place of conjecture.  History of that period has  wedding announcements for his daughter Betty, a baptismal certificate in his name and an impressive family lineage extending back to an ancestor Napoleon raised as his own son.

.. .. … b u t  -  factually –  Nicolas Bourbaki   never existed !!!  

Nicolas Bourbaki is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics.   The group's core founders were Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné and Weil; others participated briefly during the group's early years, and membership has changed gradually over time.

When the World War imposed such a challenge,  5 mathematicians (Henri Cartan, Claude Chevalley, Jean Delsarte, André Weil, and Jean Dieudonné) formed a group to not only repair the fractured community but to set the curriculum for the next 2 decades. However, they didn’t realize they were setting in motion a multi-generational endeavor to rethink the foundations of mathematics. They also delighted and annoyed the rest of the mathematical community along the way. They  comically named  their group after Charles-Denis Bourbaki, a French general famous for his spectacular failures during the Franco-Prussian War. Nicolas is thought by some to refer to St. Nicolas, perhaps suggesting the group knew the gifts they were about to bring to the struggling mathematics community.  The founding members and a later addition, René de Possel, first met at Café Grill-Room A. Capoulade, a small cafe in Paris’ Latin Quarter. Their first project was to find a more rigorous method of handling Stokes’ theorem, a fundamental tool in higher dimensional calculus, as new methods were largely unknown to French mathematicians. Over a lunch of cabbage soup and grilled meats, they also agreed to work as a collective, to have regular conferences, and to incorporate the work of mathematicians from other countries, namely Germany.

In July of 1935, the first formal Bourbaki conference was held, with the addition of 3 well-known French mathematicians. It was at this meeting that they decided to publish all work under the name Bourbaki and to expand the scope of their work to reinventing other mathematical subfields, such as topology, set theory, abstract algebra, and Lie groups. That is, Bourbaki did not set out to invent new mathematics. Rather, they wanted to streamline, organize, create better definitions, incorporate modern techniques, and create a set of fundamental axioms which all of mathematics could be based on. Shortly after the meeting, Possel’s wife baptized Nicolas Bourbaki, allowing the group to submit articles under his name. Their first article, “Sur un théorème de Carathéodory et la mesure dans les espaces topologiques,” was submitted and quickly accepted by the mathematics community.

Within the next few years, the Bourbaki conference was being held 3 times a year and some of the continent’s most prominent mathematicians had gotten involved. The lucky people who have been invited to spectate these spirited and often chaotic meetings would “always come out with the impression that it is a gathering of madmen.” Dieudonné claimed there was no formal structure, and it was common for conferences to involve viscous, unprompted criticism from other members, some of which might be 20 or 30 years apart. He believed that the group offered criticism harsher than anything found on the outside. In fact, the only official rule was that members had to retire by age 50 because older mathematicians, or so it was thought, would be unlikely to adapt new methods and ways of approaching problems.

These methods resulted in the group creating a plethora of work, which was eventually published as a collected volume in 1939 as Éléments de mathématique. In the following decades, this book became standard reading, expanded over time to 12 books with 6,000 pages, and has been republished numerous times, the last of which was in 2016.

However, in 1940, Germany invaded France, drastically slowing the group’s work. Some members were recruited into the military, while others fled Nazi persecution, leaving French mathematics again at risk of collapse.

Bourbaki founder André Weil remarked in his memoir Apprenticeship of a Mathematician that France and Germany took different approaches with their intellegentsia during the war: while Germany protected its young students and scientists, France instead committed them to the front, owing to the French culture of egalitarianism.  Weil had strong interests in languages and Indian culture, having learned Sanskrit and read the Bhagavad Gita.  After graduating from the ENS and obtaining his doctorate, Weil took a teaching stint  in India. While there, Weil met the mathematician Damodar Kosambi, who contributed an article and attributed it to "the little-known Russian mathematician D. Bourbaki, who was poisoned during the Revolution." It was the first article in the mathematical literature with material attributed to the eponymous "Bourbaki”.

Damodar Dharmananda Kosambi (31 July 1907 – 29 June 1966) was an Indian mathematician, statistician, philologist, historian and polymath who contributed to genetics by introducing Kosambi map function.  He is well known for his work in numismatics and for compiling critical editions of ancient Sanskrit texts.   Damodar Kosambi emulated his father by developing a keen interest in his country's ancient history.  Kosambi was critical of the policies of then prime minister Jawaharlal Nehru, which, according to him, promoted capitalism in the guise of democratic socialism.  

Nicolas Bourbaki was influential in 20th century mathematics, particularly during the middle of the century when volumes of the Éléments appeared frequently. The group is noted among mathematicians for its rigorous presentation and for introducing the notion of a mathematical structure, an idea related to the broader, interdisciplinary concept of structuralism. Bourbaki's work informed the New Math, a trend in elementary math education during the 1960s. Although the group remains active, its influence is considered to have declined due to infrequent publication of new volumes of the Éléments. However the collective's most recent publication appeared in 2016, treating algebraic topology.

Charles Denis Sauter Bourbaki (1816 – 1897) was a French general. He was born at Pau, the son of Greek colonel Constantin Denis Bourbaki, who died in the War of Independence in 1827.    In the Crimean War he commanded a portion of the Algerian troops; and at the Alma, Inkerman and Sevastopol Bourbaki's name became famous. In 1857 he was made general of division, commanding in 1859 at Lyon. His success in the war in Italy was second only to that of MacMahon, and in 1862 he was proposed as a candidate for the vacant Greek throne, but declined the proffered honour. In 1870 the Emperor Napoleon III entrusted him with the command of the Imperial Guard, and he played an important part in the fighting round Metz. His conduct at Gravelotte however was questionable as with the Prussians exhausted from the fighting and heavy casualties, the French were poised to mount a counter-attack but for Bourbaki refusing to commit the reserves of the French Imperial Guard to the battle because he considered it a defeat.

With regards – S. Sampathkumar

20.08.2020.