Bourbaki and foundations

[Timothy Y. Chow]

Michael Sheard wrote:

> (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?

In the past, there have been many---shall we say---spirited discussions on 
FOM on this topic.  You can search the FOM archives circa January 1998 and 
circa March 2016 for examples of such discussions.

Nowadays we have the advantage of being able to point to Penelope Maddy's 
article, "What do we want a foundation to do?" as a starting point.

http://www.socsci.uci.edu/%7Epjmaddy/bio/What%20do%20we%20want%20-%20final

Depending on what you mean by a "foundation" and depending on what you 
want a foundation to do, the answer may vary.  Proponents of category 
theory often lay great emphasis on what Maddy would call "Essential 
Guidance," and less emphasis on (for example) "Risk Assessment."

You might want to take a look at the following MathOverflow questions for 
some thoughts on the topic (including my own).

https://mathoverflow.net/questions/360578
https://mathoverflow.net/questions/376839

Tim