Bourbaki and foundations

[JOSEPH SHIPMAN]

So far I have not seen anyone give an explanation that makes sense of what is WRONG with set theory. The explanations why you might want to use category theory as a basis for formal proof assistant systems are fine, and the explanations why you might want to LEARN category theory because it is closer to mathematics practice in many areas of mathematics are fine, but the fact that you need sets to do certain types of mainstream math even if you get them FROM a category-theoretical foundation, and the fact that you can formalize all of mathematics in set theory if you want to, are so straightforward and easy to understand that I don’t grasp where the negativity is coming from!

WHO is forcing anyone to use set theory in a way that requires any significant and unnecessary effort?

Sent from my iPhone

> On May 9, 2022, at 4:27 PM, Timothy Y. Chow <tchow at math.princeton.edu> wrote:
> 
> 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