Re: Getting back to the notes of May 10th

sowa <>
Date: Sat, 15 May 93 23:20:41 EDT
From: sowa <>
Message-id: <>
Subject: Re: Getting back to the notes of May 10th
Cc:, interlingua@ISI.EDU,

Some comments:

> ... your approach to understanding reference to cats and mats
> would require a complete theory of cat/mat imagery. 

That is not just my approach.  Any theory of reference -- psychological,
philosophical, or computational -- must eventually address the question
of how things in the world are related to symbols.  The point that I
was trying to make in the notes to Pat and Len is very simple:

 1. Model theory, as defined by Tarski and the mathematical logicians,
    relates formal languages to mathematical constructions.  It is a
    branch of pure mathematics, not a theory of perception nor a theory
    of physics.

 2. In order to use model theory to relate a language (formal or natural)
    to the world, some very complex issues must be addressed about the
    way that the mathematical constructions relate to the world.

According to Pat, there is a mythical TMT, which is able to deal with
models whose individuals ARE real world objects.  I admit that there is
a lot of loose talk in elementary textbooks about sets whose elements
ARE people, trees, cats, and dogs.  But those books mix mathematical
constructions and physical objects in a very muddled way.

In one of his earlier notes, Pat mentioned an anecdote where he was
talking with Jon Barwise about a similar issue, and Barwise said
"Look around this office.  You see a lot of objects.  But how many
SETS do you see?  Where are the sets?"  Barwise was trying to make
a similar distinction between sets as mathematical constructions
and the things in the world.  Last November, Barwise invited me to
give a talk to his seminar in Indiana, and I showed my four-part
diagram that I described in an earlier note to Pat:

   Natural language <--> Logic <--> Models <--> World

Barwise very much approved of that approach, as did Godehard Link,
a Montegovian, who was at the ontology workshop in Padova and who
came up to me after my talk (in which I also showed the four-part
diagram) and said that he completely agreed with the need to make
such a distinction.

> ... Taking a relation
> in the usual model theoretic way as a set of ordered pairs, surely it
> is false that "the only thing [model theory] can do is relate
> mathematical symbols to mathematical things."  If I've got the symbol
> `John' in my language, then if the interpretation function (taking it
> to be a kind of relation) for my language includes the pair <`John',
> John Sowa>, then I have related a symbol to a nonmathematical thing.

You're using mathematical language in the same way that I just objected to.
You have to distinguish pure mathematics from applied mathematics.  In pure
mathematics, there is never such a thing as an "ordered pair" consisting
of a symbol and a person, nor an ordered pair consisting of two physical
objects of any kind.  When you develop a theory of pure mathematics
(as Whitehead, Russell, Tarski, and others did), every variable and
every construction consists of purely mathematical entities:  numbers,
points, spheres, the empty set {}, and combinations of such things.

After you have built such a theory, then you can talk about applying
it to the real world.  In that case, you associate with each formal
symbol in the mathematical construction some physical object in the
world.  But that association is not part of the formal theory; it is
part of the methodology for applying the theory.

That is the crux of the disagreement between Pat and me.  I maintain
that model theory says nothing about how it can be applied to the
world.  That application is not part of model theory, but of some
other subject, which could be psychology, AI, or philosophy of science.

Pat mentioned Kripke, Montague, and others who wanted to use model
theory to relate to the world.  But Kripke's famous work on model theory
for modal logic was just as much a theory of pure mathematics as Tarksi's.
Montague did assume certain "functions" that could recognize trees and
dogs.  But his formal construction stopped short of specifying those
functions; his actual accomplishment was not to relate language to
the world, but to relate it to the names of certain functions that
he never defined.  He is entitled to do so, but it is not thereby
correct to claim that he "solved" the problem of relating language to
the world.  His undefined function symbols were just as "ungrounded"
as any GENSYM used in any AI program.

I do not mean to belittle the work of Tarski, Kripke, or Montague.
They accomplished some brilliant work that solved a very important
part of the problem:  relate a complex language with quantifiers,
Boolean operators, and other things like modalities to much simpler
model-like constructions.  But I want to make it very clear that they
did not even begin to address the question of how those models actually
mapped to the physical world.