Re: Practical effects of all this firstname.lastname@example.org
Date: Wed, 28 Apr 1993 23:31:09 +0000
To: sowa <email@example.com>
Subject: Re: Practical effects of all this discussion
Cc: interlingua@ISI.EDU, firstname.lastname@example.org,
Hi, John. I expected an answer. Sorry to keep ON about this, but it is important. (In what follows I have elided sections of text, indicated by dots....., to keep the length down.)
>I haven't "confidently tossed it aside". As an impressionable undergraduate
>math major, I was amazed and enthralled by "the paradise that Cantor opened
>up to us" as Hilbert put it. But I now sympathize with Wittgenstein that
>the entire body of work on transfinite set theory is a "swamp of confusions."
I'm not impressed with the later Wittgenstein (or indeed the early Wittgenstein) as a philosopher of mathematics. But my point was only that your views on this issue are not those of the majority of philosophers of mathematics, and certainly not of almost all mathematicians, who are swimming happily in the 'swamp'. Thats fine, everyone is entitled to their views, but in specifying a standard we must be careful not to be too peculiar.
>I have no intention of stopping anyone from working in that swamp if they
>want to. But I believe that an agnostic attitude towards uncountable sets
>is a much sounder, more realistic attitude. There is nothing in either
>conceptual graphs or KIF that commits us to assuming uncountable sets,
>nor is there any feature of either one that prevents us from talking about
>them if we want to. All that I am saying is that if we want a solid
>foundation for mathematics, knowledge representation, or anything else,
>we should avoid talk about uncountable structures unless we explicitly
>intend to venture into the swamp.
Think of me as an alligator :-)
>.......... I distinguish Tarski's actual writings from so-called
>"Tarskian" or "Tarski-style" metaphysics that goes far beyond anything that
>Tarski himself dared to say.
Im not really interested in what Tarski said. Model theory - what I christened TMT - is defined in any logic textbook.
>> Tarskian model theory - TMT - defines how the denotations of symbols
>> interact to give the denotations of more complex expressions. Thus for
>> example it says that if we interpret a relation symbol 'Bigger' to refer to
>> a certain relation between things, and interpret the name 'New York' to
>> refer to a thing, and 'Arthur' to refer to another, then 'Bigger(Arthur,
>> NewYork)' is true just if that relation holds between those things; and so
>> forth. That is all it says, simple things like that. A 'model' of some
>> collection of symbols is simply a complete specification of appropriate
>> denotations for them all.
>This statement sounds innocuous enough until you get to the phrase "just if
>that relation holds between those things." Tarski never addressed the issue
>of how the symbols in his "formal languages" ever related to real "things."
>His models were always set theoretic constructions.
Theres a basic mistake here. I was careful in my little toy example not to use set-theoretic language, because it often misleads. If we say that a TMT model is a set U and a subset of U for each predicate letter... and so forth, that indeed sounds like we are making a 'set-theoretic construction', and that in turn might sound like making something out of sets in much the way we might make something out of Lego. So one can be led to think that a model is apparently MADE OF sets, as opposed to being made of real thing. But this is just a slip of the mental tongue. Sets aren't a kind of stuff, even an abstract stuff. The language of sets is just a mathematical technique for talking about collections of things. Its the standard language, in our mathematical 'swamp', for making general statements and specifying general constructions. But the sets might well be sets of such things as people, or marks on windows, or ocean liners (or any other damn thing). If I were to try to state !
TMT in general terms I would want
to say that the domain could be any collection of things, and the denotation of a name could be any thing in that collection, and so forth; and the natural way to do this is to talk of sets. But, to repeat, that does not interpose a new kind of abstract entity between the language and the world it denotes. 'Set theoretic constructions' should not be contrasted with Brooklyn Bridge.
Now, with a finitist philosophy of mathematics you will probably disagree, because you can't allow talk of these collections, which are liable to rapidly get too big for you to be happy with. But that, if you will forgive my directness, is your problem. Theres no reason why our Krep formalism should be distorted by your taste for quaint positions in the philosophy of mathematics.
>.... When you go beyond
>Tarski to his students like Montague, you find that they sweep all those
>issues under the rug....
What issues? There are no issues to be swept. Montague was trying to give a TMT for English, which is a very ambitious plan, probably doomed to failure because English is not a formal language and is not used only to make assertions. But here we are talking about Krep formalisms.
>...... by saying that the intension of "tree" is a "function"
>that yields the value "true" when applied to a tree, and yields the value
>"false" when applied to a non-tree. Never do they say how those functions
Another mistake. (Probably the one that leads to confusing sets with data structures.) TMT does not require that the relations and functions denoted are computable. TMT doesn't expect that inference processes will have to manipulate these models. [Aside: the full story is richer, illustrated by completeness theorems and such results as Herbrand's theorem, but the basic point is valid: models are not required to be computable or manipulable.]
> People like Carnap in his _Logische Aufbau_ and Gooodman in
>his _Structure of Appearance_ wrestled with those ideas without solving
>them even to their own satisfaction. Wittgenstein believed that he had
>solved that problem in his _Tractatus_, but he spent the rest of his life
>explaining the difficulties and limitations of his earlier views.
These philosophers wrestled with various problems, but none of them have direct relevance to the matter in hand. Wittgenstein wrote the _Tractatus_ years before Tarski published his papers on model theory, and spent his later years alone in a small, bare room refusing to let anyone read what he was writing.
>............ Sets and data structures are neat, clean
>manageable entities where the basic elements can be clearly distinguished
>from one another.
Sets are just collections. If you can talk or think of things, then you can talk or think of collections of them.
>...But if you take a walk in Muir woods, you'll discover
>that it is very hard, or even impossible to say clearly when two redwood
>trees are distinct or when they happen to be the same individual. For
>aspen forests, the problem is even worse, because you might have an entire
>"forest" that is actually a single individual with one root system and
>hundreds of trunks.
Again you confuse ontological issues with semantic ones. One might find all kinds of individuals in Muir woods. Just in talking of "redwood trees" you have made some ontological assumptions about separation of the world into individuals. There is no reason why a TMT formalism might not reason about trees, trunks, root networks, forests, degrees of separation, rates of flow along capillaries, one root network being part of another, etc. etc.. None of this kind of intricacy is in any way incompatible with TMT. Think about graph theory and the notion of a subgraph to see how one might approach an adequate way of describing an aspen forest. These examples are complex (and untypical) but pose no intrinsic difficulties.
For a wierder example, look at the last section of my old paper on liquids, in which the same piece of liquid can be looked at as one thing or as several things, with different temporal critera of individuation. This is quite compatible with TMT, although one has to start being very careful in using equality.
>The point is that Tarski-style model theory can only be applied with any
>degree of rigor to set theoretic constructions, data structures in a
>digital computer, or similar abstractions.
As emphasised earlier, to talk of 'set theoretic constructions' as though they were a kind of object is a mistake.
>>........... But Chris Menzel was citing David Lewis, who
>keeps making statements that "possible worlds are real" -- something that I
>find totally unintelligible except as a metaphor.
I tend to agree. Most writers agree with you that Lewis' ontology is strange, but don't find that a reason to reject Kripke semantics.
>There are two uses for the word "model": (1) in logic, a structure for
>which some axioms have denotation "true"; (2) in engineering, a structure
>that has parts and relationships that correspond to some aspects of a real
>world system or subsystem. Although these two uses arose from different
>academic disciplines, they are not inconsistent. I prefer, in fact, to
>identify these two kinds of models: the denotation of any formula in
>any formal language can only be computed in terms of some abstract model
>of the world, never in terms of the world itself.
I know this is what you prefer to do, but my point in these argumentative messages is only to emphasise that this is not what most people prefer to do, and certainly not what MUST or SHOULD be done.
> A civil engineer who
>is building a road and a mechanical engineer who is building a car to
>drive on that road may have very different models of exactly the same
>road. Both may be good approximations for their purposes, but the formulas
>used by the two engineers may be inconsistent or incompatible with one
>another. Yet for each engineering model, the formulas for that discipline
>may have denotation "true".
This kind of observation is only relevant to TMT if we want to combine both these views into one representation. There are many strategies that can be used to do that. We would probably want to keep the two engineers' terminologies distinct in any case, and if we maintain a distinction (perhaps sugared for user convenience) between, say, civileng-slope and autoeng-slope, there is no problem of inconsistency. Or McCarthy's 'context' mechanism will handle this perfectly, for example. Or we might introduce propositions called something like civileng-assume and autoeng-assume, and qualify all the assertions from each theory by adding these appropriately as extra antecedents.
I don't mean to claim that these issues of representation are all trivial or even all fully understood. But TMT (and suitable extensions of it) is an aid in clear thinking about them, not a bar to progress in solving them.
>> TMT does assume that the concept of 'thing' is somehow specified, and the
>Aye, there's the rub! "Somehow specified"! My complaint is that the
>theorists like Montague never do any actual specification. And the
>engineers who really do specifications always do them in terms of an
>abstract model of the world -- never the world itself.
No they don't! If an engineer calculates the stresses in the girders of a bridge, he is reasoning about that bridge, not some 'abstract model' of the bridge. If he were not, he (or anyone) would never be able to relate his calculations to the actual bridge. If he makes a mistake, its the bridge that collapses, not an abstract model of anything.
>> is merely to observe that TMT assumes that ontological issues are somehow
>Yes. And I am simply insisting that those assumptions be made explicit.
You keep insisting this, but I am pointing out that this insistence of yours is misplaced.
>Whenever anyone attempts to make them explicit, it becomes very clear
>that they are only talking about a very limited, finite abstraction
>from the world -- more like a Barwise & Perry "situation type" than
>anything that resembles the real world in its entirety.
No, it becomes clear that they are talking about a (not necessarily finite) part of the world in a limited way. Again, look at my toy example. Its impossible to talk about the world 'in its entirety' with confidence in any language, even English.
>>.......... John's reaction to this point is to
>> simply reject all of modern mathematics since Hilbert. This reveals
>> considerable selfconfidence, but I do not find it very convincing. Most
>> rebels in mathematical philosophy - even the intuitionists - have agreed
>> that there are infinitely many integers.
>If anyone wants to use KIF or CGs to represent transfinite set theory,
>they are welcome to do so.
Which is all that matters for discussing a Krep standard.
>But I claim that no mathematical theory with
>application to anything in the real world need ever talk about an infinite
>set. I am willing, as the 19th century mathematicians did, to say that
>the integers are infinite. By that, I mean that for any integer you can
>name, I can find another one that is larger. But I also agree with the
>19th century mathematicians that it is inappropriate to talk of an infinite
>set as a completed whole. Instead of saying, let x be a member of the
>set of integers, I would simply say "let x be an integer".
OK, you adopt a finitist position. Thats a coherent, albeit awkward, position to maintain. (I have some sympathy with it.) But my only point here is that a Krep standard should not be influenced in any way by thinking that arises from, or is influenced by, such an idiosyncratic 19th-century position.
>........... All I am saying is that every model
>is an abstract construction. It may have parts that are in correspondence
>with things in the real world, but the model itself is not the world.
If the correspondence is sufficiently close to be an isomorphism, then it is easy to PROVE that, if the abstract construction is a model, then that part of the physical world is one also. If not, then your notion of 'correspondence' needs to be more carefully explained. If it can be explained in anything like a precise way, then that explanation will probably make the very assumptions of individuation which would enable one to regard that part of the real world as a TMT model directly. If not, you will somehow have to give that precise explanation (without using 20th-century mathematical ideas).
>> centrally concerned with this case, has philosophical objections to the
>> modal logic semantics, and wishes to modify the semantics of KIF to make
>> this case easier. His proposed solution is idiosyncratic and has not been
>> fully worked out, but in any case is too eccentric for a proposed knowledge
>> representation standard.
>On the contrary, there is no "fully worked out theory" that relates
>symbols in a formal language to things in the real world. As I point out
>above, Montague just postulates "tree" functions that are magically able
>to distinguish trees.
There is no 'magic' involved. A semantics is free to postulate that there are individuals. Your 'correspondences' between abstract or mental models and the real world - or indeed ANY account of how language eventually comes to say something about the world - will also make ontological assumptions of this kind. That TMT makes no assumptions of how this is to be done means that it is compatible with (almost) any way of doing it, which is exactly what one would expect a general-purpose semantic theory to be.
>......By emphasizing the model as the mediating structure
>between word and symbol, I am simply returning to Aristotle's three-way
>distinction between words, things, and "experiences in the psyche".
>That was the established wisdom of the "perennial philosophy", and if
>anything is eccentric, it is the attempt to postulate magic denotation
>functions that ignore the essential role of the intervening model.
I can't help noting that you seem to constantly be 'returning' to something. After rejecting a century of mathematics, you now seem to be rejecting two millenia of linguistic philosophy. You illustrate the point nicely: I don't want to be stuck with Aristotle's primitive idea of the psyche as an essential part of my representational formalism.
Beckman Institute (217)244 1616 office
405 North Mathews Avenue (217)328 3947 or (415)855 9043 home
Urbana, IL. 61801 (217)244 8371 fax
email@example.com or Phayes@cs.uiuc.edu