[Top] [All Lists]

Re: [ontolog-forum] Ontological Assumptions of FOL

To: Pat Hayes <phayes@xxxxxxx>
Cc: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Sun, 18 Mar 2007 09:25:56 -0500
Message-id: <45FD4BF4.3020406@xxxxxxxxxxx>
Pat,    (01)

The point I was trying to make is that there are an
open-ended number of possible representations of anything,
including the universe.  And I don't believe that any
particular method is ideal for all purposes for all time.    (02)

JFS>> Many people prefer to use mereology instead of set theory.    (03)

PH> Use for what, John?    (04)

As a formalism for representing claims about some domain.    (05)

PH> ... if we go around saying that the universe "is" a
 > mereological sum, we are in exactly the same danger.    (06)

Of course.  I would never make either claim.    (07)

The critical distinction between set theory and mereology
is that the axioms of set theory generate something new:    (08)

    For any x, the set {x} is distinct from x.    (09)

But the formalism of mereology makes no such assumption:    (010)

    For any x, the sum of all the parts of x is identical to x.    (011)

A typical example is to consider the old way of dividing the
territory of France into provinces and the newer way of
dividing it into departments:    (012)

  1. Using sets, the set of all provinces, the set of all
     departments, and the territory of France are treated
     as three distinct entities (or things or whatever you
     want to call them).    (013)

  2. Using mereology, the sum of all provinces, the sum of
     all departments, and the territory of France are one
     and the same.    (014)

PH> Not all things have parts (what is a part of the number 17?)    (015)

I would prefer to use some symbol instead of the word "part"
because the common word "part" is used in many different ways.
But whatever you want to call it, the fundamental axiom of
mereology is that everything (including the number 17) is
a "part" (or whatever) of itself.    (016)

PH> And set theory does support virtually all of mathematics,
 > whereas mereology supports virtually none.    (017)

That is a critical distinction, but it is not a criticism.
It simply means that set theory, by itself, can generate
new entities (or things), but mereology cannot.  Therefore,
an additional generative axiom must be coupled with mereology
in order to build up new structures.    (018)

For example, set theory has two operators, memberOf and
subsetOf, but mereology has only one operator, partOf.
Both subsetOf and partOf obey the following axioms:    (019)

  1. Every set/sum is a subset/part of itself.    (020)

  2. The subset/partOf relation is a partial ordering.    (021)

The memberOf relation is Cantor's most significant
innovation.  That is what gives set theory its generative
power:    (022)

    For all x, there exists a y such that memberOf(x,y).    (023)

In particular, the simplest such y is {x}.  To start,
all you need is the empty set {}, and you can build your
way up to infinities of infinities.    (024)

I admire Cantor's magnificent creation, but I don't see
any reason to prefer it.  If you use mereology as your
theory of collections, you need to add some additional
generative operator.  My preference is Peano's successor
function:  start with 0, and build up the integers as
s(0), s(s(0)), s(s(s(0))), etc.    (025)

Starting with mereology plus s(), you can build up all
of arithmetic, including Dedekind cuts for the reals.    (026)

For geometry, Whitehead chose mereology (actually, he
invented his own version) for the planned fourth volume
of the Principia.  His primitive elements were blobs
of 4D space-time, but I don't remember what generative
axioms he used.    (027)

Tarski independently built up a theory of solid geometry
based on mereology, in which the only primitive elements
were spheres.  He defined a point as a "fictitious" limit
of a descending sequence of nested spheres.  Lines and
planes are also fictitious limits.  Then he showed that
Euclid's axioms could be stated in terms of those
fictitious things.  Tarski's approach is actually a more
realistic model of physical solids than Euclid's.    (028)

I personally prefer the Peano-Whitehead-Tarski approach of
assuming domain-dependent generative mechanisms instead
of building up all of mathematics by repeatedly nesting
sets of sets of empty sets.  All those nested empty sets
are impressive, but not very perspicuous.    (029)

John    (030)

Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/  
Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/  
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/ 
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (031)

<Prev in Thread] Current Thread [Next in Thread>