pragmatic foundation

[Andre Joyal <joyal.andre@uqam.ca>]

Dear category theorists,

I invite everyone to read the interesting interview of Yuri Manin 
published in the November issue of the Notices of the AMS:

http://www.ams.org/notices/200910

http://www.ams.org/notices/200910/rtx091001268p.pdf

One the ideas discussed by Manin is that of a "pragmatic foundation" of
mathematics as opposed to a "normative foundation" by logicists or constructivists. 
He attributes the former to Bourbaki.

I disagree.

The foundational framework of Bourbaki is very much in the tradition 
of Zermelo-Fraenkel, Godel-Bernays and Russell.
I am aware that Bourbaki was more interested in the development of 
mathematics than in its foundation. 
My guess is that the foundation was too problematic to be given a proeminent place 
in the treaty, not for logical reasons but for conceptual reasons.
I claim that nobody truly understand set theory, even today!
The emperor has no clothes!
I mean that the hierarchy of infinite cardinals is so profoundly mysterious 
that it looks pathological.
What is the value of a theory if it leads to meaningless problems and structures? 
Having no good answer to offer, Bourbaki decided to diminish the importance of 
foundation rather than leaving it open. 
It may explain why category theory was not incorporated in the foundation later.

In the interview, Manin also said that:

>And so I don’t foresee anything extraordinary 
>in the next twenty years. Probably, a rebuilding of 
>what I call the “pragmatic foundations of math- 
>ematics” will continue. By this I mean simply a 
>codification of efficient new intuitive tools, such 
>as Feynman path integrals, higher categories, the 
>“brave new algebra” of homotopy theorists, as 
>well as emerging new value systems and accepted 
>forms of presenting results that exist in the minds 
>and research papers of working mathematicians 
>here and now, at each particular time. 

Any comments?


AJ


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