Re: (biassed) summary of the argument so far.
Message-id: <>
Date: Fri, 4 Jun 1993 14:32:49 +0000
To: interlingua@ISI.EDU,,
        sowa <>
Subject: Re:  (biassed) summary of the argument so far.

We seem to be nearly at a state of mutual comprehension, if not 
agreement. A few answers to your questions.

>.... pure mathematics isn't about anything
>until you make a decision to apply it to something. To me, this
>distinction between pure & applied mathematics seems so fundamental
>and so elegant that I can't see why anyone would object to it.

Of course, this is almost tautologous.  I would just say 'mathematics' 
rather than 'pure mathematics', however: you usage makes it sound like 
there are two kinds of maths, which is pedagogically useful but not 
really correct. But I suspect we agree here, in fact.

Our difference lies in what we think it is to 'apply' mathematics. You 
seem to take this to involve far more than I take it to. All one needs 
to do, in order to apply some mathematical ideas, in my view, is to 
ensure that the area of application satisfies the assumptions made by 
those ideas (which for set, and hence model, theory, is not much). You 
seem to take it to involve far more than this.

>Please excuse me for bringing it up again, but you made the comment
>that tensor calculus was "applied mathematics". 

That was before I fully understood your usage, which I now have 
clear (thanks to your reply to this comment). 

>But I said that it
>is just as pure as set theory.  Originally, the subject was developed
>for the purpose of applying it to stresses and strains in solid objects
>(whence the name "tensor").  But then the same mathematical structure
>was applied to relativity.  If you make the distinction, tensor calculus
>is pure and it has two very different major applications.  But if you
>don't make the distinction, it becomes very difficult to talk separately
>about the mathematical operations and the domain of application.

This distinction is exactly that between use and mention. In applying 
a piece of matrhematics one is using it as a language to talk about 
something or other. Since this way of speaking is so convenient we 
often speak as though there really were abstract things that the 
'pure' mathematics is about: 'things' like the Klein group, say (as 
opposed to something which 
displays that group structure, like the set of rotations and 
reflections of a rectangle.)  Whether or not one wishes to allow such 
abstractions in the universe is to some extent a matter of taste, but 
the use/mention distinction keeps track of the difference between the 
math itself and what the math is supposed to be about.

>> ... I find your insistence that 
>> lexical representations must refer only via a computational 
>> 'simalcrum' of something real, which itself is only related 
>> vaguely to the actual world, to be alarmingly confusing on 
>> precisely this issue....
>Fine, we seem to agree about the distinction between lexical and
>nonlexical items.  In an earlier note, I believe that you were
>also willing to accept the meaning triangle, which puts concepts
>in the middle between words and things.  Is that correct?
>If you don't accept the meaning triangle, then I can understand
>your objecting to my computational surrogates.

Well, the meaning triangle is fine when the LHS is taken to mean 
language, ie human, natural language. As I said in earlier notes, 
then the middle of it becomes the represented knowledge, in whatever 
form it might be encoded. For example, all of CYC's knowledge base 
would be attached to the center node of that triangle, not the 
'language' node.

>But if you do accept the meaning triangle as a reasonable hypothesis
>about intervening mental states in human use of language, I cannot
>understand why you would object to an intervening level of surrogates
>in a computational system that is used to relate a language (natural
>or artificial) to the world.

That suggests where we disagree. The use of 'language' to refer both 
to natural language (which is used by humans to communicate) and 
(hypothesised) representational languages (which are used by agents 
to represent and infer) is little more than a pun, in my view, and 
potentially very misleading. 

The meaning triangle says nothing about surrogates. It is only a way 
of saying that (natural) language refers to the world via thoughts 
or concepts, which seems like an insightful point. It says nothing 
at all about how such concepts should be implemented or whether there 
should be two 'levels' of mental representation.

I chiefly object to the notion that invoking a level of 'surrogates' 
is somehow necessary or compulsory in order to apply model theory to 
the analysis of logical representations, ie that logical representations 
can refer only to other computational structures. This seems wrong on 
two points. First, if we replace it with a notion of 'abstract' or 
'mathematical' surrogate, I can see how it could follow from your 
views on the nature of pure mathematics. But even then, there would 
seem to be no justification for thinking that such things would (or 
even could) take the form of actual computational or mental structures. 
This just seems obviously wrong. For example, I believe that there 
are hundreds of billions of stars. Does this mean that I must have 
a surrogate universe with hundreds of billions of surrogate stars 
in my head somewhere?

>And I am not sure about your use of the word "vaguely" in your
>comment above.  Did you mean that in a deprecating sense as some
>sort of criticism of my point?  

Yes, I confess, although it does reflect the fact that I don't 
understand quite what this relationship is supposed to amount to, 
and don't believe you have explained it clearly. Again I wnat to
make a distinction which you seem to be blurring(?), between 
an analogical representation which is 'similar' to what it 
represents (like, say, a floor plan of a room), and something 
like a detailed description of what it represents, which a 
database would be. But this is a minor point.
>The notion of surrogate gives me a very clear way of distinguishing
>the perceived reality (either in my head or in a computer simulation)
>from the actual reality, which may in fact be quite different.

This is easy to distinguish. What you mean by 'perceived reality' 
is something like 'the way the world would (or might) be if my 
perceptions were correct', I presume? That is a model of my 
(perception-governed) representation of it.

Just a side comment: there is only one reality, of course, so 
'the perceived reality' is only a metaphorical usage. I hope we 
agree on this.

>To me, it seems to be a very natural way of talking about things.
>I really don't understand why you object to it so strenuously.
>Are you objecting because you want to distinguish some abstract
>theory of meaning from some computational implementation?

Just say 'theory of meaning', and the answer is yes. Of course I 
distinguish the *theory* of meaning from anything computational: 
again, that is a use/mention distinction, like the distinction 
between the theory of internal combustion and a V-6 Ford engine. 
The theory of meaning isnt the kind of thing one would (usually) 
seek to implement, its a theory of what such implemented 
things might mean. 

>  When I see the phrase "there exists"
>in a mathematical textbook, I interpret it in a metaphorical
>sense, similar to the way I would interpret a novel or a myth.

I agree with this general idea, although I am quite willing to 
countenance a more platonic approach. (And agree with Quine that 
it is very hard to avoid allowing some abstract entities into 
ones ontology, such as the integers. Even the intuitionists 
allowed the integers.) Part of the utility of set (and hence 
model) theory is precisely that it is completely agnostic on 
this issue.

I would distinguish a philosophical position on this stuff from 
a pragmatic one. I lean towards nominalism, but also admit that 
actual working mathematics requires one to appear to adopt a 
much more platonist view. The strenousness of my objections to 
your messages has been motivated more by my strong belief that 
it is necessary to maintain this freedom to be ontologically 
promiscuous than by my disagreement with your particular views. 

To insist that logical representations must refer only 
via some kind of computational surrogate seems to me to be an 
unwarranted intrusion of a very particular philosophical stance on 
semantic, and even computational, practice. It seems to me to 
be mistaken (surrogate galaxies??), confusing (use/mention, 
deductive roles of databases, distinctions in earlier messages), 
and unnecessary (model theory relates sentences to things 
perfectly well). 

But you know, the real reason for pursuing this so relentlessly 
has been the sheer pleasure of trying to discover what on earth 
you are saying, and Ive had to adopt some debating tricks to 
dig this out. I feel like I understand you now, so thanks for 
the edification (but don't go saying that I agree with you, 

Best wishes

Pat Hayes

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