Subject: categories: Re: Michael Healy's question on math and AI

From: Peter McBurney <[email-masked]>
Date: Fri, 26 Jan 2001 21:37:46 -0000

Professor Lawvere's message included the following sentence:

On Wednesday, January 24, 2001 11:26 PM, F. William Lawvere 
[SMTP:[email-masked]] wrote:

>For > example, some say that logic is more general than mathematics, 
partly
> because of ignoring the strongly qualitative aspect of modern mathematics 
> and partly because of the philosophical tradition of hiding the fact that 
no
> logic other than mathematical logic has had any significant real-world
> applications.

It's not entirely clear to me what is being asserted in the second part of 
this sentence.  If what is being asserted includes the statement that the 
only logic which has had any significant real-world applications is 
mathematical logic, then this assertion is incorrect.  To give just one 
example, over the last decade the Advanced Computation Laboratory of the 
Imperial Cancer Research Fund (ICRF) in London, UK, has built intelligent 
computer decision-support systems for medical applications using logics of 
argumentation.  These logics typically use non-deductive modes of 
reasoning, and are based on the work of philosophers of argumentation 
dating from the 1950s; this work in philosophy was undertaken outside, and 
in strong opposition to, the tradition of mathematical logic.  A 
category-theoretic semantics has been provided for some of these logics of 
argumentation.    The resulting decision-support systems have found 
real-world application in cancer treatment advice, in drug prescription and 
in the automated assessment of chemical properties, such as toxicity and 
carcinogenicity.   Moreover, current research in Artificial Intelligence is 
developing the use of non-deductive argumentation formalisms for automated 
dialogues between autonomous software agents in multi-agent systems, work 
that is likely to form the basis of next-generation e-commerce systems.

Peter McBurney

************************************************************************  
**********
Peter McBurney
Agent Applications, Research and Technologies (Agent ART) Group
Department of Computer Science
University of Liverpool
Liverpool L69 7ZF
U. K.

Email:  [email-masked]
Web-page:  www.csc.liv.ac.uk/~peter

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-----Original Message-----
From:	F. William Lawvere [SMTP:[email-masked]]
Sent:	Wednesday, January 24, 2001 11:26 PM
To:	[email-masked]
Subject:	categories: Michael Healy's question on math and AI

Re : Michael Healy's question on math and AI

This is to answer Mike and also several other people who have contacted me
recently asking how I would respond to queries about

(1)	Artificial Intelligence, cognitive science, linguistic engineering,
knowledge representation, and related attempts at creating modern subjects, 
and

(2)	the relevance of category theory and of mathematics in general to 
these.

     My basic response is strong advice to actually learn some category
theory, rather than resting content with slinging back and forth 
ill-defined
epithets like "set theory", "contingency", etc..

So much confusion has been accumulated that an opposition of the form
"set-theoretical versus non-set-theoretical" has at least seven wholly
distinct meanings, hence billions of electrons and drops of ink can be
spilled by surreptitiously identifying any two of these.  For example, the
opposition can concern whether or not large cardinal assumptions are needed 
for a certain result, which is mathematically meaningful and hence
independent of whether or not the ZFvN rigidification of Cantor is being
used as a framework.  Another example is the opposition habitually used in
geometry between properties of spaces which can be explained in terms of
arbitrary mappings versus those which depend on the cohesion being studied
(e.g. "the underlying abstract group vs. the Lie group"). Obviously these
two oppositions are not the same although they may be related.

One of the oppositions which I have emphasized since 1964 is
    the ZFvN rigid hierarchy based on galactically "meaningful"
inclusion, requiring the totally arbitrary "singleton" operation of Peano
with the resulting chains of mathematically spurious rigidified membership, 
on the one hand,
                             versus
    the category of abstract sets, involving many potential universes of
discourse and arbitrary specific relations between them, on the other hand.
(Abstract sets can CARRY structures of mathematical interest, but precisely 
because of the need of flexibility in the latter, they themselves have only 
very few properties, unlike the ZFvN "sets").

Within Cantor's original conception itself there is a fundamentally
important opposition: the abstract sets, which he called "Kardinalzahlen",
versus the cohesive and variable sets which he called "Mengen".  (An
additional confusion stems from the use, by nearly all of Cantor's
followers, of the term "cardinal number" to mean (not a
Kardinalzahl=abstract set, but) a label for an isomorphism class of 
abstract
sets, an invariant which Cantor of course also studied, but which is too
abstract to support the specific relations between abstract sets 
themselves,
the mappings, and hence cannot carry the needed mathematical structures).

(A)    The real issue is that for purposes of pure AND applied mathematics, 
we need to be able to represent (without spurious ingredients) these
cohesive and variable sets (or "spaces") and their relationships.  The ZFvN 
rigidification fails so miserably in doing this that even those geometers
and analysts who pay lip service to it as a "foundation" never in practice
use its formalism.

(B)   Category theory made explicit some universal features of the
relationship between quantity and quality whose fundamental importance had
been forced into consciousness by the work of Volterra and Hurewicz (both 
of
whom made basic contributions to both functional analysis and algebraic
topology) and of many others. This relationship between quantitative and
qualitative aspects concerns cohesive and variable sets and structures 
built
on such spaces.  For example, Volterra already recognized that spaces have
"elements" other than points, and Hurewicz recognized the need for
cartesian-closed categories (even before the lambda-calculus formalism, or
category theory, was devised); moreover, the original fiber bundles were
explicitly modeling dynamical situations, etc.

Many people working in the new fields, striving to realize the dream of a
theoretical computer science, do not seem to be aware of points like  (A)
and (B). It would certainly be a bad strategy for the advancement of 
science
to "hide" the fact that category theory belongs to the background of a new
result and thus to help perpetuate that sort of ignorance.

The role of mathematics in general (not only of category theory) also
seems to be widely misunderstood, even in those fields which definitely 
need
more mathematics in order to mature and make a real contribution.  For
example, some say that logic is more general than mathematics, partly
because of ignoring the strongly qualitative aspect of modern mathematics
and partly because of the philosophical tradition of hiding the fact that 
no
logic other than mathematical logic has had any significant real-world
applications. Because of the minimal
mathematical education required of students of philosophy, the claim is too 
easily accepted in many philosophical circles that "mathematics is
unsuitable" for some given issue of conceptual analysis; this conclusion
seems to be based on the syllogism:
        mathematics is set theory (a misconception which the philosophers
themselves have done much to disseminate),
        set theory is clearly not suitable (actually because of the defects 
of the ZFvN rigidification, which make it ill-suited for mathematics as
well)
        hence ......
This syllogism serves as an excuse to indefinitely postpone learning
mathematics (and category theory in particular).

An older sort of excuse is the assertion that the proposed science should
concern the REAL WORLD, not pure mathematics. This superficially appealing
truism has frequently been used to mask the fact that comparing reality 
with
existing concepts does not alone suffice to produce the level of
understanding required to change the world; a capacity for constructing
flexible yet reliable SYSTEMS of
concepts is needed to guide the process. This realization (not Platonism)
was the basis of the supreme respect for mathematics expressed by champions 
of reality like Galileo, Maxwell, and Heaviside. For example, the
differential calculus provides the capacity to construct systems 
descriptive
of celestial motions, fluid interactions, electromagnetic radiation fields, 
etc., and therefore engineers have learned it. The functorial calculus 
helps
to provide a similar capacity adequate to the requirements, not only of the 
older sciences,
but of the newer would-be sciences as well. Hence my response.

                 Bill Lawvere