FOM: dualities and morphisms

[Robert Black]

I'd like to take up another of the questions posed in Mic Detlefsen's
posting of 2 October, and again suggest that once the question is
modernized it has what I think should be an uncontroversial answer.  The
question was:

3: Is the existence of the dualities a fact of fundamental importance in
understanding the nature of mathematics? Should foundational research today
be just as concerned with it as Hilbert was? Can the phenomenon of the
dualities be adequately accounted for by anything other than a 'formalist'
philosophy like Hilbert's?

Dualities like the point/line duality of plane projective geometry are
automorphisms of a structure.  Mic also mentions modelling one geometry in
another, which is basically a matter of finding an isomorphism (or at least
a monomorphism).  So it seems to me the modernization of Mic's question is:
Is the notion of a morphism of fundamental importance in understanding the
nature of (modern) mathematics?

Now the answer to that seems to me to be uncontroversially yes.  Since
Bourbaki, but going back to Emmy Noether, the first question one asks about
any mathematical structure has tended to be:  what are the morphisms?
There is a sense in which it is (or ought to be) completely uncontroversial
that modern mathematics is structuralist in style, and that structures get
studied together with their morphisms.  Category theory is the general
presentation of this point of view, and any philosopher of mathematics
ignorant of it is ignorant of his own subject.

That is not, however, to say that these uncontroversial facts have any very
powerful immediate philosophical consequences.  What I've just said leaves
entirely open, for example, the question much discussed on the FOM list of
whether category theory can provide *foundations* for mathematics.  And
nothing much about ontology seems to follow either - e.g. from the claim
that a number is 'just a place in a structure' it's not going to follow
that numbers aren't objects until you've decided whether or not places in
structures are objects.  Also nothing immediately follows about how (if at
all) talk about structures and their morphisms should be reduced to
manipulations in ZFC.  Questions like these are central to what we're
debating now, but they assume as a given the centrality of the notions of
structure and morphism.

Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845