Pat, (01)
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. (02)
But if you're looking for promising new insights, you're not likely
to find them in the wellworn 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. (03)
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. (04)
JFS>> Anything that Gödel considered puzzling about set theory can
>> hardly be considered a closed subject. (05)
PH> I have never seen any evidence that Goedel did find it puzzling,
> but in any case I disagree. (06)
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summarized an unpublished letter: (07)
"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." (08)
PH> But for sure, this is not the right forum to be discussing
> byways in the foundations of mathematics.... (09)
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. (010)
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. (011)
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. (012)
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. (013)
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. (014)
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* . (015)
PH> Why do you call this a version of mereology? It seems to me to
> have nothing at all to do with mereology. (016)
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. (017)
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. (018)
PH> It is ironic, then, that this very construction shows that
> any axiomatic mereology has a model in set theory. (019)
Actually, it shows the opposite. Tarski began with Lesniewski's
mereology. His version of geometry can be used as the basis for
constructing 3D 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. (020)
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. (021)
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. (022)
JFS>> I agree that you can use an ontology based on set theory as
>> a basis for analysis, but it seems ad hoc, (023)
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... (024)
Robinson's differentials are another alternative for analysis. (025)
PH> ... mereology is not an alternative to set theory as a
> foundation for mathematics. (026)
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 (027)
Luschei, E. C. (1962) The Logical Systems of Lesniewski,
NorthHolland, Amsterdam. (028)
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? (029)
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: (030)
1. First, note that mereology has only one operator, 'partOf',
but set theory has two operators, 'subsetOf' and 'elementOf'. (031)
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. (032)
3. You can use FOL + mereology to define the notions 'function'
and 'relation' whose arguments range over the parts of a
mereological collection. (033)
4. To construct the ZF hierarchy, the starting set of individuals
S can be treated as a mereological collection as in #2 above. (034)
5. To construct a hierarchy above the starting collection S,
you need to assume an axiom of the following kind: (035)
For any welldefined 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. (036)
6. Given the starting collection S of #4, use the axiom #5 to
create a new mereological collection S': (037)
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. (038)
7. Repeat the method #6 to construct S'' from S' and a function
f':S'>S'', and so on for S''' with f'', etc. (039)
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. (040)
PH> Do you want us to become involved with the foundations of
> mathematics? (041)
Not particularly. But this forum is definitely involved with
the foundations of ontology, which requires choices about many
related areas such as spacetime, set theory, mereology, etc. (042)
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. (043)
John (044)
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