Mereology and (not vs.) set theory
sowa@watson.ibm.com
Message-id: <199205161253.AA00350@venera.isi.edu>
Date: Sat, 16 May 92 08:25:57 EDT
From: sowa@watson.ibm.com
To: CG@CS.UMN.EDU, INTERLINGUA@ISI.EDU
Cc: CPM5749@RIGEL.TAMU.EDU
Subject: Mereology and (not vs.) set theory
Chris Menzel commented on my comments:
>> ...you can get a theory of mass terms by avoiding element-of and
>> adding different operators for measuring and subdividing. An
>> operator that allows unlimited subdividing would be inconsistent
>> with set theory, but it is compatible with mereology.
>
> I don't see this. We could imagine adding such an operator to set
> theory in at least two ways: we could postulate subdividable
> urelements, and introduce an operator that just applies to them.
> Indeed, there is nothing preventing us from having mereology apply
> to the domain of urelements and building sets over them. Second, we
> could *model* infinite subdivision by defining the right sorts of
> algebraic structures with the right kinds of elements to model the
> world according to mereology (such as the lattice theoretic
> structures of Link's account of mass terms) and define a subdivision
> operator accordingly.
Yes, but what I'm asking for is not to add more bells and whistles
to set theory, but to step back to a simpler theory.
As an analogy, consider the algebraic theories of groups, rings,
and fields. The usual textbooks proceed in the following steps:
1. First define a group as a collection of elements with an
operator called multiplication and an identity element 1.
2. Then define a ring as a group with a second operator called
addition and an additive identity element 0.
3. Then define a field as a ring where you can divide by any element
except 0.
Somebody might object to this procedure by asking "Why do I need to
distinguish three different kinds of structures, when I could just
take the rational numbers as a field, and they would automatically
give me a model for groups and rings as well?"
The answer is that a simpler theory with less structure actually
gives you more models: there are groups that cannot be extended
to rings, and rings that cannot be extended to fields. If you
start with the rational numbers as your only model for groups,
rings, and fields, there is too much built-in structure to let
you reconstruct all the wierd kinds of groups and rings that you
might want to consider.
To show that the axioms for mereology are at least as consistent
as set theory, I showed how you could use any model for sets as a
model for mereology. But the converse does not hold. If I have
an infinitely divisible fluid, the axioms for part-of apply to it,
but the axioms for element-of do not.
The point is that the theory with simpler structure actually gives
you more -- it allows you to handle things that create problems for
set theory, such as infinitely divisible fluids or the collection of
redwood trees in Muir Woods. What I am advocating is the following
procedure:
1. Start with a base theory that has only one operator, part-of.
That gives you basic mereology.
2. Then allow that base theory to be extended in different ways
by adding different operators and axioms.
3. By extending mereology with element-of, you get conventional set
theory. By extending it with unlimited divisibility, you get a
a theory of fluids. Or you could invent other kinds of extensions
to handle the redwood trees that have a common root system.
I'm not saying that we should throw out set theory. All I'm saying
is that we should start with a simpler theory as base and treat
set theory as just one possible extension. See David Lewis's book
for more discussion along these lines.
> Two things. First, although mereology doesn't seem to *commit* one
> to Platonic objects the way set theory arguably does, the issue is
> still wide open. It is a separate philosophical step to say that
> none of the objects in mereology's intended domain are abstract. It
> is perfectly consistent with basic mereology to assume there are
> abstract objects, and mereological sums of such. (Glenn Kessler
> developed an account of the ontology of arithmetic that made such a
> move ten or so years ago.)
Unlike most mathematicians, I am definitely not a Platonist. I am
much more sympathetic to Wittgenstein's view of mathematics. But I
hesitate to summarize it here because I'm sure that I'd be deluged with
irate notes from mathematical Platonists. For an amusing account of
"Mathematical Platonist Meets Wittgenstein", see what happened when
Alan Turing audited Wittgenstein's course on the philosophy of
mathematics -- in Ray Monk's _Ludwig Wittgenstein_, The Free Press,
New York, 1990. Just one quote from Wittgenstein (p. 418):
Turing doesn't object to anything I say. He agrees with every word.
He objects to the idea he thinks underlies it. He thinks we're
undermining mathematics, introducing Bolshevism into mathematics.
But not at all.
Similarly, I'm not trying to undermine mathematics by throwing away
set theory. I'm just trying to build it up in two stages, where the
first stage is a simpler theory, mereology, that has applications to
problems for which set theory is unsuitable.
> Second, though, it is a dubious virtue for a theory to rule out (or
> even simply be somewhat inhospitable toward) Platonic objects. The
> most natural ontologies of most, e.g., engineering and manufacturing
> domains are replete with abstract objects. Consider for example a
> new product design, a car body design, say, or a database schema, or
> the objects found in a general, type level description of a
> manufacturing process. All are paradigms of abstract objects, and
> we find ourselves referring to them, quantifying over them, etc.
> like any other objects. We just can't trip over them, but other
> than that, if we take our language at face value they're as real as
> rocks.
I completely agree with you about the importance of dealing with models
of things that do not happen to exist in the real world. But I don't
like the word "abstract object". I prefer to use the term "data
structure" because it suggests something that I could actually write
a program to represent (if I had enough time and space).
I'm perfectly happy to quantify over data structures that represent
unicorns, a new car design, or the Olympian gods. But I don't believe
that those data structures exist in a Platonic "realm of ideas". If
I use the phrase "there exists x", I mean it in the sense that I could
write a program that would create a representation of x in some computer
storage somewhere. If x is too big to be represented completely, then
my program should be able to exhibit any part of x that I have the time
and inclination to examine. I don't have enough time and energy to
examine an infinite structure, but I do like the reassurance that my
program is always capable of going just one more step, if I want it to.
But you might ask about uncountable sets. I would answer that the
word "exists" when applied to such things is simply a metaphor that
extends a language game we use with ordinary objects to a new domain.
But that opens up a whole new discussion topic that the computers on
this network do not have the time and space to represent. If anyone
is interested, please read Wittgenstein.
> The role of abstract objects in our representations of the
> world--both in the representational apparatus and in the world
> represented--has always been an important issue, but it becomes even
> more pressing and relevant these days as representational standards
> are being developed (KIF, PDES SUMM, ANSI IRDS, etc.) and as the
> task of capturing domain ontologies is coming to play a central role
> in distributed knowledge sharing and enterprise integration.
Yes, I believe that the subject matter is very important. But I
also believe that if you do a global change of "data structure" for
"abstract object", you will get a clearer statement that we can both
agree with.
John