Re: Getting back to the notes of May 10th
Message-id: <>
Date: Tue, 1 Jun 1993 01:58:44 +0000
        sowa <>
Subject: Re: Getting back to the notes of May 10th
Cc:, interlingua@ISI.EDU
At 11:20 PM 5/15/93 -0400, sowa wrote:
>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.

No, there is no muddle. Sets can contain anything you like. There is 
nothing in either the formal basis or the philosophy of set theory to 
suggest that it cannot be appplied to any domain of individuals whatever. 
Obvious connections between set-theoretical accounts of analysis, and 
routine applications of analytical ideas in engineering and physics, show 
that such interpretations are in fact routinely assumed in everyday 
mathmatical usages. A distinction between the abstract stuff of pure 
mathematics and the real applications of applied mathematics is an 
illusion. Pure mathematics is simply  mathematics that hasn't been 
applied yet.

>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.

Yes, and I can mention several others who think that way. I disagree 
with them and have yet to hear any convincing argument for their 
position. The reply to Barwise's rhetoric is to simply remark that 
sets arent things that one would be able to see, since they don't 
reflect light, not being physical entities. I defy you to see a 
day or a room (as opposed to the contents of a room) or a political 
party or Julius Caesar either.

>> ... 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.

This is a complete myth. Whitehead and Russell set out to reconstruct 
conventional mathematics using only set theory for foundational reasons, 
but this is by no means required by the semantic framework that they or 
others used. Quine, Hilbert, Carnap and others took it as simply obvious 
that their formalisms refer equally well to sets of physical entities as 
sets of anything else. This idea of pure mathematics being about a special 
abstract ontology is not justified by historical precedent or current 
usage, and is close to incoherency.

>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.

I entirely agree! The disagreement between us is not here, but comes 
>From my being happy to use set-theoretical terminology to talk about 
the world, and you refusing to permit this until all these other 
subjects are somehow finished. I am happy to talk of a set of treetrunks, 
but you will not allow this, as you think it presupposes some account 
to be given of how treetrunks can be recognised (or something). But it 
does not: it's just a way of talking about the world. 

>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. 

Perhaps this is the problem. In allowing physical things into
my models, I am not claiming that all this work on 'grounding' has 
been solved. I am not trying to get away with intellectual robbery. 
Watch my lips: I am claiming that we do not NEED to do this in order 
to have a useful semantic theory. So invoking this semantic theory 
is not claiming to have done it!

>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.

If by 'actually mapped' you mean an account of how representations come 
to be accurately attached to our physical environment, what has been called 
the 'grounding problem', then of course I agree this has not even begun to 
be addressed by model theory. Nor should it, for this is not the business 
of semantics. So why are you arguing with me?


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