[FOM] Scope questions

[Vaughan Pratt]

Having participated in this list some years ago, and having returned this
year, largely in observer mode, I'm prompted to ask to what extent the
subject of foundations of mathematics is defined by the interests and
technical specialties of its contributors.

A similar question can be asked of philosophy.  Religion, ethics, paradoxes,
and logic are staples of philosophy, but literature, art, music, etc. are
traditionally studied elsewhere, being viewed as simply outside the scope
of philosophy.  Some philosophers however have viewed color vision and
quantum mechanics as raising philosophically interesting questions, but
only after gaining sufficient competence in those respective areas to make
such judgments.

Physics is not terribly different here.  Given the list: orbital mechanics,
quantum mechanics, relativity, Bose condensates, atomic structure,
cold fusion, molecular structure, salts, organic compounds, the aether,
flogiston, epicycles, living cells, nuclear reactors, automobiles, robots,
and mechanical engineering, the typical physicist is likely to draw a line
after atomic structure and classify the rest under chemistry, history of
physics, engineering, bunk, etc.  However physicists have occasionally applied
their expertise to investigating such ostensibly out-of-scope phenomena
as the trajectories and dynamics of baseballs and boomerangs, because it
appeared to them that they raised questions of interest to physics, or at
least questions best answered by a physicist.

Returning to FOM, what is the place in the foundations of mathematics
of each of: integers, rationals, reals, complex numbers, polynomials,
relational structures, algebras, topological spaces, manifolds, buildings
(in the sense of K-theory), coalgebras, categories, and toposes?  Are they
all equally important foundationally, or are some of these out of scope?

In particular (reflecting my own interests), what about coalgebras?
Do these have equal foundational significance to algebras, more, less, ...?
And to what extent does the answer depend on the number of people who both
work with coalgebras and are interested in foundations?

Coalgebras have been mentioned only twice on FOM, once in a posting by me
dated 1/13/00 on defining the continuum from scratch as a final coalgebra
(analogously to defining the natural numbers from scratch as an initial
algebra), which I felt was a very foundationally significant observation
about the continuum but which no one responded to; and once (tangentially)
in an advertisement posted by a workshop organizer for a CONCUR'02 workshop
mentioning coalgebras as being within its scope.

More recently I have worked on comonoids, which are to coalgebras as
monoids are to algebras, modulo some nice twists.  In the course of this
work it seemed to me once again that the material raised some foundationally
significant issues.  This formed the topic of my invited talk at a coalgebra
workshop in Warsaw two weeks ago.  Associated with this talk is an open
problem I'd worked on without success some years ago that I was originally
offering $500 for in January and which in the following months edged up
to $2000.  Quite apart from any foundational significance, this problem
would appear to be quite difficult (I won't embarrass those whom I know to
be working on it).

>From FOM's essentially complete neglect of coalgebras one could infer
that coalgebras are of exactly zero relevance to FOM, my argument to the
contrary notwithstanding.  Or perhaps it is just that coalgebras have the
same status for FOM as quantum mechanics has for philosophy: they will be
studied for their foundational relevance after someone in foundations (this
does not describe me, I know relatively little about the subject judged by
the standards of this list) has spent some time with them, much as Popper
felt equipped to write about philosophical aspects of quantum mechanics
once he had worked with the subject for some time.

Vaughan Pratt