FOM: Category or collection; chicken or egg?
sf at Csli.Stanford.EDU
Thu Nov 20 02:46:02 EST 1997
Which comes first?
I think the argument which I made in "Categorical foundations and
foundations of category theory" [CFFCT], and which McLarty
strenuously objects to, has not been properly understood, either by him or
by Pratt. To elaborate a bit, let me quote from CFFCT, p.150: "...when
explaining the general notion of structure and of particular kinds of
structures such as groups, rings, categories, etc. we implicitly *presume
as understood* the ideas of *operation* and *collection*; e.g. we say that
a group consists of a collection of objects together with a binary
operation satisfying such and such conditions. Next, when explaining the
notion of *homomorphism* for groups or *functor* for categories, etc., we
must again understand the concept of operation. Then to follow category
theory beyond the basic definitions, we must deal with questions of
completeness, which are formulated in terms of collections of morphisms.
Further to verify completeness in concrete categories, we must be able to
form the operation of Cartesian product over collections of its
structures. Thus at each step we must make use of the unstructured
notions of operation and collection to explain the structural notions to
be studied. The *logical* and *psychological priority* if not primacy of
the notions of operation and collection is thus evident."
These questions of priority take place in ordinary mathematical parlance
and understanding. What do you have to understand first before you
explain the next thing? You can't explain what a linear transformation is
before you've explained what a linear space is. You can explain what a
Boolean ring is before you explain what a Boolean algebra is; no logical
priority for one over the other there, but I would give the edge to the
latter for psychological priority.
McLarty wants to say in place of "What is a category?"--"What are the
first order axioms of categories?" and likewise for sets, etc., thus
seemingly putting everything on a par: no priorities at all, and no need
to posit collections and operations. Similarly, supposedly, for groups,
rings, topological spaces, etc. But if we want to be able to say: the
set of all permutations of a set of n elements forms a group under
composition, where are we? This is a specific structure. What are its
elements? What is the operation?
I think the above position about logical and psychological priority would
be regarded as innocuous, even banal, if it didn't mention categories at all.
It's just because of the program of re-expressing all mathematical notions
in terms of categorical notions that led to saying: "Well, sets can be
explained in terms of the category SET, so you see, categories are more
basic than sets" that has led to the objections. Much as those who pursue
this program (the categorical ideologues) would like
to think that they have kicked away the traces, they only have to face up
to the fact in example after example that there is an ineliminable residue
of the notions of collection and operation.
And, I must repeat for Vaughn's sake, I do not mean 'collection' in the
sense of 'set' or 'class' in an axiomatic theory like ZF. I mean it in an
informal sense pre-axiomatic sense, which could be clarified in various
ways by axiomatic foundations, but not necessarily founded in terms of
something more basic. The general notions of collection and operation
seem to me to constitute an irreducible minimum of abstract mathematical
thought. It is quite another matter to ask: "Which collections exist?"
and "Which operations exist?" and nothing in the above pretends to answer
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