# FOM: Chu spaces, information, and FOM

Jon Barwise barwise at phil.indiana.edu
Tue Nov 11 09:23:15 EST 1997

A follow-up on Vaughan's note on Chu spaces:

A couple of examples of Chu transform might help.

1) Recall the notion of a theory-interpretation.  Recall the standard
interpretation of PA into ZF, where one has a map F of the sentences of
number theory into, say, the sentences of set theory, and a map G in the
other direction of the interpretations of ZF into those of PA.  Note that M
|= F(phi) iff G(M) |= phi, for all phi and M.  This is the basic condition
in the definition of Chu transform.

2) Consider a map G of one topological space X into another X.  Let F be
inverse of G as defined on the open sets O of X'.  Then we have x \in F(O)
iff G(x) \in O for all x \in X and O \in the topology of X'.

The simplest notion that generalizes these two examples is that of Chu
transform.  It seems to me a remarkably simple notion whose importance was
overlooked for a surprisingly long time.  Vaughan has done a good job of
bringing it to the attention of computer scientists.

This past summer Jerry Seligman and I published a book (#44 in the
Cambridge Tracts in Theoretical Computer Science) where we use the notions
of Chu space and Chu transform to develop a mathematical/logical theory of
information and use it to investigate various topics.  Our basic idea is to
use Chu transforms (which we rename "infomorphism" to better fit its role
in our theory) to model the part-whole relationship between the parts of a
distributed system and the whole system.  (In the above example, think of
the infmorphism (F,G) as showing a sense in which PA is a part of ZF.)
Using families of such morphisms one can see how various parts of a system
can carry information about other parts, and develop the notion of the
"local logic" of a distributed system.  We claim that the local logic of a
system captures the information implicit in the system.

After developing the theory of local logics, we go on to apply the theory
to a number of topics, including vagueness, common sense reasoning, and
quantum logic.  The book is called "Information Flow: The Logic of
Distributed Systems."
found at http://www.phil.indiana.edu/~barwise/kjbbooks.html.

This book does not show that the notions of Chu space and Chu transform
have significance for the philosophy of mathematics but there is an
indirect argument that they do.  After all, it is clear that mathematical
proofs do carry information about mathematical objects.  So one might try
to understand mathematics itself as a complex distributed system, with
proofs as one part of the system, various kinds of mathematical objects as
other parts, and so on.  Then mathematics would end up being modeled by a
family of infomorphisms (Chu transforms) linking these various domains (Chu
spaces).  I have toyed with this idea but have not yet succeeded in
developing it in what I consider a satisfactory manner.  If I had, there
would have been a chapter about it in the book.

Jon