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[ontac-forum] different histories about category theory

To: "ONTAC-WG General Discussion" <ontac-forum@xxxxxxxxxxxxxx>
Cc: "Paul J. Werbos" <pwerbos@xxxxxxx>
From: "Paul S Prueitt" <psp@xxxxxxxxxxxxxxxxxx>
Date: Sat, 10 Dec 2005 16:10:03 -0700
Message-id: <CBEELNOPAHIKDGBGICBGKEJCGPAA.psp@xxxxxxxxxxxxxxxxxx>

Arun,    (01)

I have read and re-read several of your papers with John Sowa.    (02)

We share an interest in formalization of analogy, but feel that analogy
cannot be fully formalized – because the formalization process itself must
use induction to create a set of axioms.  So the relationship to the algebra
of toposes (for those that are familiar with it) and axiomatic theories is
one where induction is critical.  The work on similarity measure by Douglas
Hofstadter has strongly influenced my thinking on the notion of analogy.  It
was certain not his position that similarity was something that one could
render computationally.    (03)

I am sure that John Sowa would want me to agree with him about the
definition of induction.  However, I must say that induction is not
something on which there is a lot of agreement.  My take is that natural
induction is part of the process involved in the formation of any mental
event.  So “interpretation” and “induction” have certain strong
relationships.    (04)

This “take” upsets many people, who want induction to be something
formalizable.  I once had a long debate with Joe Firestone, founder of KMCI
(a knowledge management organization in Washington DC) where he wanted the
statement that induction could be reduced to mere deduction to be true.
This type of desire is common.  I realize that I am not mainstream.    (05)

I know that John has specific thoughts about deduction, induction and
abduction.  But his belief system is really different in fundamental ways
from mine and from those I have studied and known.    (06)

This is ok, except that my group takes the position that the mainstream (of
which you work with John really is in the center of) is incorrect in certain
specific ways, and is approaching the formalization of human knowledge in a
way that will not be successful (ever) in our opinion.    (07)

I have said all this before, so I need not say it again.    (08)

Quasi Axiomatic Theory approached this entire subject in a way that is
similar to but with considerable differences to the work developed by
Alexander Grothendiech, and later extended in your own work and to work on
toposes and Chu spaces.  But these are still acting as if formalism can
solve more of the problem that formalism has been shown to solve.  And
pushes the knowledge technology further out into a esoteric discipline that
only a few can understand.  I often say that th complications of many
things – particularly computer science – is due to an inability of the
mainstream to recognize the limits of formalism and the key role that
complexity plays in everyday everything.    (09)

Category theory is also developed by Robert Rosen and his mentor N.
Rashensky (mathematical biology, theoretical biology).    (010)

This type of category theory, Rosen’s, is present in a large percentage of
the Grossberg type biologically feasible models of neural architecture.  The
category theory of Rosen focuses on anticipatory mechanisms  (one of his
books was titled “Anticipatory Systems”).  The point is that this literature
and discipline is quite different from the Chi Space and toposes
constructions.    (011)

The foundations to the notion of representational basis for knowledge
representation can be traced to C S Peirce, but not in the way that Sowa
interprets Peirce.  John knows Peirce as well as anyone, but the
interpretation is different form that developed by the Russians.    (012)

I have developed an appreciation for the way that the Soviet school headed
by Pospelov and Finn interpreted Peirce.  I meet both Pospelov and Finn
several times, including during by visit to Moscow in 1997, where I gave a
lecture on quasi axiomatic theory.    (013)

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