FOM: Citation for argument against categorical foundations.

Solomon Feferman sf at Csli.Stanford.EDU
Mon Nov 17 18:06:54 EST 1997

The argument was made in the paper "Categorical foundations and
foundations of category theory" in the volume _Logic, Foundations of
Mathematics and Computability Theory_ (Proc. LMPS V conference),
Vol. 1, Reidel, 1997, pp.149-169.  Basically, the argument is that the
notion of category is defined in terms of the notions of collection and
operation, viz. the collection of all the "objects" of the category, the
collection of all its "morphisms", and the operations of domain,
co-domain, and composition applied to morphisms and of 1_a applied to 
objects a.  (Alternatively, one can deal with just the collection of
morphisms alone.).  Thus the foundations of category theory must be given
in terms of some sort of theory of operations (or functions) and
collections (or classes), though there are alternatives to standard
set-theoretical formalisms for that purpose (one such is given op.cit.).
In the informal foundations, one does not distinguish between large and
small categories; that depends on working specifically within a
set-theoretical framework.  

In practice, one defines concrete categories, such as the one in McLarty's
message, in terms of classes and functions, namely as the class of all
linear spaces for the objects and the class of all linear transformations 
between specified linear spaces as the morphisms.  Moreover, each object
(in this case, linear spaces) is itself a structure, which is defined using
the notions of set or class and function.  

On Mon, 17 Nov 1997, Colin McLarty wrote:

>         I believe Sol Feferman somewhere argued against categorical
> foundations for math, saying something like: it is impossible to define
> linear transformations before defining linear spaces. (I had thought this
> was in the Midwest Category Theory Seminar of 1969, in his paper with an
> appendix by Kreisel, but it is not.) Can anyone help me with a citation?
> Colin McLarty

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