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It is nicely written and contains historical references that I assume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 summarized an unpublished letter: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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> (04)
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