Nicholas Bourbaki was one of the most influential mathematicians of the twentieth century whose contribution is cardinal to how mathematics is done today. However, there was just one problem – he did not exist. Who or what was Nicholas Bourbaki?

There is much to learn from the case study of Nicholas Bourbaki. One big takeaway is to identify and learn from general structures. Another would be emphasis on rigor. Here is a blog that I had written years ago which also cites some good references for further reading about the history of Bourbaki. In particular, this podcast and these articles provide a good start on the history of Nicholas Bourbaki:
http://mathshistory.st-andrews.ac.uk/HistTopics/Bourbaki_1.htmlhttp://mathshistory.st-andrews.ac.uk/HistTopics/Bourbaki_2.html
However, Bourbaki’s was not the only approach of pedagogy in mathematics. For example, see Vladimir Arnold’s essay ‘On Teaching Mathematics’. Arnold famously had anti-Bourbaki views about teaching mathematics.

Bourbaki’s approach was somewhat of a reaction to Henry Poincare’s reliance on intuition. To read Poincare’s philosophy of mathematics read:
https://plato.stanford.edu/entries/poincare/http://mathshistory.st-andrews.ac.uk/Extras/Poincare_Intuition.html-http://vigeland.caltech.edu/ist4/lectures/Poincare%20Reflections.pdf.
The philosophies of Arnold and Poincare, accomplished mathematicians/physicists themselves, show that Bourbaki’s method should not be the only way. Some critics of Bourbaki, such as Vladimir Arnold, have found their adherence to fundamental mathematics limiting.

In any case, intuition (as advocated by Poincare and Arnold) and rigor (as advocated by Bourbaki) are both right in their own ways and both are important in making progress in mathematics and in general, sciences.