# Bourbaki Mathematics: Part 2

In continuation to my previous post on Bourbaki Mathematics which focused on the history, I thought I would make a post that focused more on the maths.

As mentioned in my first post, the Bourbaki group’s main work was the series entitled ‘Elements of Mathematics, originally written in French. It aimed to be a “self-contained treatment of the core areas on modern mathematics”. The books in the series were as follows:

• I: Set Theory
• II: Algebra
• III: Topology
• IV: Functions of one real variable
• V: Topological Vector Spaces
• VI: Integration
• VII: Commutative Algebra
• VIII: Differential manifolds (summary of results)
• IV: Lie Groups and algebras
• V: Spectral Theory
• VI: History of Mathematics

In publishing the series, Bourbaki achieved the aim of providing mathematics with new and broad foundations. Additionally, these new foundations “lay in the notion of ‘structure’, illustrated by the now common word ‘isomorphism’“. The Bourbaki group was largely successful in introducing an emphasis on rigour. Its clear emphasis on structure produced the right idea at the right time and changed the way most mathematicians thought.

One of the greatest aim of the Bourbaki group was to make their textbooks as ‘modern as possible’, meaning that it was in line with the current trends of German mathematical research of the time. There was a perception that French mathematicians needed to absorb the best ideas of the Gottingen school, particularly Hilbert and the modern algebra school of Emmy Noether, Emil Artin and Bartel van der Waerden. As a consequence, the Bourbaki group’s point of view, whilst being ‘encyclopedic’, was not neutral.

Criticism of their work includes (taken from wikipedia):

• Algorithmic content is not on topic and is almost completely omitted;
• More emphasis is placed on axiomatic theory-building than the heuristic technique of problem solving;
• Analysis is treated ‘softly’, with a lack of ‘hard’ estimates;
• Measure theory is approached using functional analysis, which may be cumbersome in some cases;
• Combinatorics is not discussed;
• Logic is mentioned minimally;
• The applications of this mathematics is not considered;
• No images are used in these textbooks, and hence geometry is reduced as a whole to abstract algebra and soft analysis.

(*Note that Soft analysis often asks qualitative questions where hard analysis asks more quantitative questions.)