## History, Mathematics and Self-Improvement

# What Can We Learn from a Mathematician Who Never Existed?

## The interesting history and contributions of Nicolas Bourbaki

A few students and professors studying at an esteemed university in France — the Ecole Normale Supérieure — were extremely unhappy with their mathematics textbooks, especially their calculus textbook. The primary reasons for this dissatisfaction were the inconsistency and disorganization in the subject at the time, especially regarding formulae, functions, and notations, caused by the disarray of the mathematical community after World War I.

After much debate and deliberation, they decided to write a textbook on their own or a consistent doctrine that untangled the vast web of mathematical content. Regular congresses were held to discuss the content of this treatise.

They were concerned with functions which, according to the Oxford Dictionary’s Lexico, are mathematically defined as — “relations or expressions involving one or more variables.” Since there was a difference in the interpretations of these functions, the group started defining them to maintain a worldwide standard. Thus, the group decided to standardize their works under the pseudonym, Nicolas Bourbaki (named after a general as a joke), to maintain a unifying factor for the group members.

They also began devising new mathematical notations. A group member had even created the symbol for an empty set (∅), which is quite commonplace now, using a letter in the Danish and Norwegian alphabet. The group also used the dangerous bend icon to indicate difficult material. They also intended to make mathematics easy to understand, one example being the replacement of long geometrical terms with everyday words.

Despite these efforts to make their ideas more simple, other mathematicians criticized the Bourbaki collective of utilizing formal jargon and overemphasizing on algebra. This methodology of solving abstract problems differed from the longstanding belief that mathematics was intuitive and relying too much on logic destroyed creativity.

There was some dissension amongst some group members against the usage of set theory instead of category theory. Set theory studies collections of objects using Venn diagrams, whereas category theory uses graphs with arrows, called categories.

Instead of using the commonly used term Mathematiques (mathematics in English), the Bourbaki collective used the word Mathematique (equivalent to mathematic), to promote unity between all fields of the subject.

After they published their first treatise — Éléments de mathématique — and subsequent works, Nicolas Bourbaki became famous worldwide. Most mathematicians did not doubt his existence, believing that he was a reclusive Russian scholar, however those that did, were shunned by the rest of the mathematical community. They even announced the wedding of Betty Bourbaki, his alleged daughter, and forged his ancestry, connecting him to someone extremely dear to Napoleon. However, in 1968, the group could not maintain the secret, and they revealed the truth.

Many of the Bourbaki proofs and theorems, earlier found not appliable, are now finding their significance in computer science, for example, the Bourbaki-Witt theorem. Concepts such as structuralism, which is a topic in psychology that studies the relationships between objects rather than objects themselves, found inspiration in the work of the Bourbaki collective.

# Conclusion

I found two things quite interesting about the Bourbaki collective:

- Their method of working — chaotic, disorganized, and loud. Anyone could interrupt anyone. While this might seem ineffective and unproductive to some, world-changing ideas developed in this way. Anarchy and disorder might be fruits of discord and conflict, but that does not mean they cannot themselves bear positive fruits, if channelled in the right manner. The successfulness and impact of Nicolas Bourbaki is a testimony to the same. Quoting André Weil, a founding member of the Bourbaki collective,

“A good organization would have no doubt required that everyone be assigned a topic or a chapter, but the idea to do this never occurred to us.…What is to be learned concretely from that experience is that any effort at organization would have ended up with a treatise like any other.”

- The Bourbaki Collective gave up their chance for individual credit so that. they could have unity and enhance the subject that they enjoyed. They worked under a pseudonym to unify their work, but also partly as a practical joke. We learn that sometimes we need to sacrifice certain things to promote something we are passionate about.

One can apply these principles in their own lives as well. Nicolas Bourbaki’s name is still present in theorems like the Bourbaki-Witt Theorem and the Jacobson-Bourbaki Theorem. Therefore, in consideration of his/their contributions and pseudonymous nature, some call him the Greatest Mathematician Who Never Lived.

Note: This is article was earlier submitted as a Mathematics project for the Grade 9 ICSE Board Examinations. It has been abridged, merged, and edited to make it suitable for publication. Here is a link to the second part of the project:

# References

- The Prehistory of Mathematical Structuralism by Erich H. Reck and Georg Schiemer (2020)
- “Twenty-Five Years with Nicolas Bourbaki, (1949–1973)” by Armand Borel (1998)
- “The Withering Immortality of Nicolas Bourbaki” by David Aubin (1997)
- Bourbaki: A Secret Society of Mathematicians by Maurice Mashaal (2006)
- https://www.britannica.com/topic/Nicolas-Bourbaki
- https://scroll.in/article/946194/nicolas-bourbaki-one-of-the-greatest-mathematicians-of-20th-century-never-really-existed
- Mathematical Apocrypha by Steven G. Krantz (2002)
- https://www.youtube.com/watch?v=0O_boW9YA7I