FOM: For categorical foundations II, replies
Torkel Franzen
torkel at sm.luth.se
Thu Nov 20 03:11:57 EST 1997
Colin says:
>He [me] says that axioms of set theory too are meaningful to us only
>after we are acquainted with informal mathematical examples:
Well, not exactly. In order to understand why these particular
axioms should be considered crucial or sufficient, one may well need
to know a bit about the use of sets and set-theoretic constructions in
mathematics. But the basic notion of collection is all that we invoke
to explain and justify the axioms in an intrinsic sense.
In the case of category theory, you say that there is a similar
primitive notion to be had:
>Oh yes, I think there is. I think the notion of "operation" (including
>"symmetry" and "growth" and more as special cases) is just as primitive as
>the notion of "collection". I take it that Feferman agrees with me on this.
>Where he
>disagrees with me is when I say the category axioms (which I gave as C1-3)
>mathematize this primitive notion just as foundationally as set theory
>mathematizes collections.
Here again I've expressed myself poorly. I have no objection at all
to the observation that there is a primitive notion of
operation. Clearly it's not through sets that we arrive at this
notion. But the few and abstract axioms of category theory are justified
by the fact that they summarize (what experience has shown to be)
crucial properties of a large number of constructs, mostly mappings of
various kinds, living in various environments. They are the result of
abstraction. I simply can't separate my appreciation of these axioms
from my knowledge of the mathematical concepts on which they are based.
They are more like the group axioms than they are like the axioms of set
theory.
Let me repeat my earlier caveats. This whole (philosophical)
discussion is somewhat dubious, because it so closely concerns what is
the basis for our understanding of mathematical notions. To avoid
slipping into pointless observations about our individual
understanding of this or that, we need to be able to point to some
public circumstances that can be adduced in evidence for this or that
being more basic to our understanding. Here the problem is that many,
perhaps most people, have grown up with a set-theoretic view of things,
and so are disinclined to suppose that categories could be just as
basic. On the other hand, I have seen category theory enthusiasts (not
Colin) wax quite unrealistically enthusiastic, talking as though
the ordinary extensional view of functions as graphs had now been
revealed to be a complicated construct based on this or that
construction in category theory. It's a general problem when we
propose new foundations or new systems of logic as basic to our
thinking that there may be no people who have in fact learned
mathematics through these new foundations, so the claim that they
are in fact basic may sound a bit hollow. But then, who knows, maybe
the next generation will in fact base their understanding on these
concepts.
More information about the FOM
mailing list