FOM: CATvsSET

[John Mayberry]

	Robert Black's excellent posting suggests to me that it might 
helpful to clarify our discussion if we distinguish what is 
*foundational* from what is *fundamental*. Although whatever is 
foundational must be fundamental, the converse clearly does not hold. 
Group theory is surely fundamental, but no one would argue that it 
provides a foundation for mathematics. And, to use Black's example, 
general topology and its basic notions (open set, closure, compact set, 
. . . ) are obviously fundamental and, equally obviously, not 
foundational.
	The ideas of category theory are so general, so powerful, and 
so beautiful that it is difficult for me to see how anyone could deny 
that they are fundamental - here my sympathies lie entirely with Awodey 
and McLarty. But neither category theory nor topos theory is a 
*foundational* theory.
	A foundational theory deals with those "self-evident" 
propositions upon which, ultimately, all mathematical argument rests, 
but which we accept without proper mathematical proof, and with those 
basic concepts in terms of which, ultimately, all mathematical 
concepts are defined, but which we grasp and accept without a proper 
mathematical definition.  

--------------------------
John Mayberry
J.P.Mayberry at bristol.ac.uk
--------------------------