Bourbaki and foundations

[Michael Sheard]

I recently read the book "The Artist and the Mathematician," which is Amir Aczel's "biography" of Nicolas Bourbaki.  In the concluding chapter, we find this passage:


"Bourbaki lost an incredibly important opportunity to remake its oeuvre in the new form of the theory of categories, something that would have better suited the study of structures than did the old theory of sets with its myriad problems and inadequacies. ... Bourbaki had a chance, through the work of Grothendieck and his students, to refound modern mathematics on the theory of categories, but Bourbaki missed that chance.  In part, this missed opportunity led to the demise of Bourbaki.  For mathematics remained based on a flawed system, set theory, rather than something that would have been much more appropriate. ..."


Earlier, Aczel had already said that Bourbaki's "greatest error" was "letting Grothendieck go and disagreeing with his vision for the future of mathematics" -- specifically, its unwillingness to revise its earlier volumes in order to replace set theory, a "discipline full of theoretical holes," with category theory, which "does not suffer from the inherent limitations of set theory," as the foundation of mathematics.


Although much of my work is in set theory, I am open to the possibility that category theory could serve as a better foundation for mathematics in its full scope than set theory.  I just do not feel that I have ever seen much evidence for that proposition.  So, I would be very interested to hear reactions to any or all of the following questions:


(1) How widespread was the belief that category theory is an obviously better foundation for mathematics than set theory, among mathematicians in 2006 (when the book was published)?


(2) How widespread is this belief today?


(3) What is your position on this belief?


Thanks and best wishes,
Mike







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