Bourbaki and foundations

[Sam Sanders]

Dear Michael,

>   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)?

What do you mean by “X is an obviously better foundation for math than Y”?    This is a crucial question.  

Often, “better” is in the eye of the beholder and even the subject of religious wars (in the FOM). 

Even “foundation” is subjective to some extent: what should this construct be able to do (see my answer to your (3))? 

> (2) How widespread is this belief today?

I have no idea.  

> (3) What is your position on this belief?

Set theory is crucial for building models (as far as I know) of e.g. type theory, which includes systems underlying Coq.  

Hence, set theory is an important foundational paradigm.

Alternative approaches are (definitely in certain areas) closer to the way mathematics is actually done.  

Hence, alternative approaches are an important foundational paradigm.

My opinion/belief is that this “dual” state of foundations is fine, even an enrichment. We will never be able to evolve beyond it.  

In fact, we should be wondering/investigating why this dual nature exists, and not try to deny half of our mathematical reality.   

Best,

Sam