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Re: [ontolog-forum] Ontological Means for Systems Engineering

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Ron Wheeler <rwheeler@xxxxxxxxxxxxxxxxxxxxx>
Date: Wed, 28 Jan 2009 13:28:41 -0500
Message-id: <4980A3D9.5000008@xxxxxxxxxxxxxxxxxxxxx>
Could you put this on the wiki changing the quotes to questions or 
observations that you are supporting or refuting.    (01)

It is nicely written and contains historical references that I assume 
are indisputable as references although someone may later add 
counterarguments to your interpretation.    (02)

Ron    (03)

John F. Sowa wrote:
> Pat,
> Any formal system that is widely used and taught has proved its
> value, and it should be considered a basic tool in the toolkit
> of any scientist or engineer.  Set theory certainly belongs there.
> But if you're looking for promising new insights, you're not likely
> to find them in the well-worn areas.  There are two kinds of sources
> that are worth searching:  recent publications that few people have
> had a chance to read and/or assimilate; and obscure publications
> that most people have either never seen or have not had the
> background to relate to current problems.
> Examples of the latter are the 19th geometrical algebras, which
> David Hestenes rescued and revived.  Another example, which we
> at VivoMind have used extensively, includes methods of processing
> graphs which chemists and architects have developed and applied,
> but which the comp. sci. gang has ignored.
> JFS>> Anything that Gödel considered puzzling about set theory can
>  >> hardly be considered a closed subject.
> PH> I have never seen any evidence that Goedel did find it puzzling,
>  > but in any case I disagree.
> Hilary Putnam discussed that topic in his introduction to Peirce's
> book, _Reasoning and the Logic of Things_.  On p. 38, Putnam
> summarized an unpublished letter:
>     "Kurt Gödel remarked that, at least intuitively, if you divide
>     the geometrical line at a point you would expect the two halves
>     to be mirror images of each other.  Yet this is not the case if
>     the geometrical line is isomorphic to the real numbers."
> PH> But for sure, this is not the right forum to be discussing
>  > byways in the foundations of mathematics....
> But which method is the byway?  Peter Simons, for example, wrote
> the book _Parts:  A Study in Ontology_, in which he summarized
> all the theories of ontology he could find in 1987, analyzed the
> relationships among them, and recommended mereology as a good
> foundation for ontology.  Many linguists maintain that mereology
> is a better representation than set theory for natural language
> plurals.  Barry Smith also recommends mereology for ontology.
> PH> Nothing written on this topic before about 1870 is of anything
>  > but historical interest, as the modern notions of limits and
>  > continuum were not available then.
> Leibniz understood the continuum well enough to invent differential
> and integral calculus, his method of differentials was proved to
> be sound, and it is an excellent basis for teaching.  The great
> mathematicians and physicists from Gauss to Laplace to Hamilton
> and Maxwell learned the subject from differentials.  Faraday
> never studied any calculus, but he had an excellent intuition
> about the nature of fields, which Maxwell formalized in his
> famous equations.
> In the 20th century, most textbooks avoided the "discredited"
> differentials.  Then in the late 1960s, Abraham Robinson
> proved that differentials are a sound basis for calculus,
> and he recommended that textbooks return to that method.
> I strongly agree that differentials are a far better basis for
> teaching calculus.  When I was in high school, I learned calculus
> by reading the book _Calculus for the Practical Man_, which used
> differentials.  With that background, I passed the advanced
> placement test and skipped the freshman year of calculus at MIT.
> When I later took the analysis course, I learned the method of
> epsilons and deltas, which is easy to understand if you first
> learn differentials and continue to think in terms of differentials.
> JFS>> Whitehead had published two earlier books on geometry, and he
>  >> planned to write the fourth volume of PM on geometry.  However,
>  >> he did not consider point sets to be an adequate foundation for
>  >> geometry.  For the foundations, he developed a version of mereology
>  >> he called *extensive abstraction* .
> PH> Why do you call this a version of mereology? It seems to me to
>  > have nothing at all to do with mereology.
> That bio of Whitehead emphasized the philosophical motivation for
> the name 'extensive abstraction'.  The formalism that W. presented
> in his 1919 book is a version of mereology, which Peter Simons
> analyzed in his compendium of theories.
> JFS>> Tarski showed that points can be considered the limit of
>  >> a sequence of spheres.  That conclusion can be considered a
>  >> formalization of Aristotle's position #2 above:  points are
>  >> designated locations on a line, not parts of it.
> PH> It is ironic, then, that this very construction shows that
>  > any axiomatic mereology has a model in set theory.
> Actually, it shows the opposite.  Tarski began with Lesniewski's
> mereology.  His version of geometry can be used as the basis for
> constructing 3-D physical shapes from tiny spheres such as atoms.
> To show the relationship to Euclidean geometry, he defined points
> as limits of nested spheres.  In effect, he modeled Euclidean
> point sets in terms of mereology.
> PH> This was indeed Whitehead's own main point, as I understand him:
>  > not to eliminate set theory, but to show how it can be understood
>  > as an idealized abstraction at the infinite 'limit' of finite
>  > experience.
> I never said that Whitehead wanted to "eliminate set theory".  He
> certainly did not repudiate it as a basis for the first 3 volumes
> of PM.  But for his volume on geometry, he developed a version
> of mereology, which he used in much the same way that Tarski
> used Lesniewski's mereology for a similar version of geometry.
> JFS>> I agree that you can use an ontology based on set theory as
>  >> a basis for analysis, but it seems ad hoc,
> PH> It has been the accepted standard foundation for FOM for over
>  > 40 years. The only serious alternative that has been suggested
>  > has been category theory...
> Robinson's differentials are another alternative for analysis.
> PH> ... mereology is not an alternative to set theory as a
>  > foundation for mathematics.
> Lesniewski developed a basis for mathematics based on mereology
> as an alternative to PM.  Peano's axioms do not depend on set
> theory, and you can construct the reals by Dedekind's method
> without requiring anything beyond mereology.  Unfortunately,
> Lesniewski died in May 1939, and most of his manuscripts were
> lost in the war.  See
>     Luschei, E. C. (1962) The Logical Systems of Lesniewski,
>     North-Holland, Amsterdam.
> PH> And by the way, I don't believe anyone has exhibited models
>  > of ZF in mereology, and I do not think it is possible. Can you
>  > cite any such construction?
> I have Simons' book somewhere in my house, but I can't find it
> at the moment.  In any case, it is easy to use mereology to
> define the equivalent of Zermelo's iterative construction:
>   1. First, note that mereology has only one operator, 'partOf',
>      but set theory has two operators, 'subsetOf' and 'elementOf'.
>   2. For any application of set theory that does not use sets
>      of sets, you can identify 'partOf' with 'subsetOf' and
>      identify 'elementOf' with partOf applied to singletons.
>   3. You can use FOL + mereology to define the notions 'function'
>      and 'relation' whose arguments range over the parts of a
>      mereological collection.
>   4. To construct the ZF hierarchy, the starting set of individuals
>      S can be treated as a mereological collection as in #2 above.
>   5. To construct a hierarchy above the starting collection S,
>      you need to assume an axiom of the following kind:
>      For any well-defined structure x, it is permissible to assume
>      the existence of an entity y that is distinct from x and
>      from any other previously defined entity.
>   6. Given the starting collection S of #4, use the axiom #5 to
>      create a new mereological collection S':
>      By hypothesis, each part x of S is well defined.  Therefore,
>      by #5, there must be an entity y that is distinct from all
>      previously constructed entities.  Let y be a part of S'.
>      Then S' is the collection of all such parts, and a function
>      f:S->S' can be defined by f(x}=y for all parts x of S.
>   7. Repeat the method #6 to construct S'' from S' and a function
>      f':S'->S'', and so on for S''' with f'', etc.
> This construction can be identified with Zermelo's construction
> by calling each collection S', S'', ... the set of all subsets
> of the previous collection and identifying all the f functions
> with the set theoretic elementOf.
> PH> Do you want us to become involved with the foundations of
>  > mathematics?
> Not particularly.  But this forum is definitely involved with
> the foundations of ontology, which requires choices about many
> related areas such as space-time, set theory, mereology, etc.
> Just because something has been widely taught for 40 years or
> even a century does not imply that it is an ideal foundation
> for the future.  And some things that had been widely taught
> a few centuries ago (such as Leibniz's method of differentials)
> may become more important than a recently popular approach.
> John
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