Pat, (01)
The question results from two distinct ontologies for a line, each of
which can be formalized mathematically in different ways. They were
mentioned by Aristotle, who probably summarized earlier discussions
in Plato's Academy: (02)
1. Points are parts of a line, and the line itself or any shorter
segment is a collection of points. This was Zeno's hypothesis,
which he used to state his famous paradox about motion. (03)
2. The only parts of a line are smaller lines of non-zero length.
In this ontology, points are designated locations on the line,
but not parts of the line. This was Aristotle's preferred
hypothesis, which he used to counter Zeno. (04)
PH> I really don't think this is controversial any longer. Maybe it
> was in 1920, but not now. (05)
Anything that Gödel considered puzzling about set theory can hardly
be considered a closed subject. (06)
> Principia was the magnum opus of early set theory. Neither Tarski
> nor Whitehead adopted mereology rather than set theory. (07)
Only the first three volumes of the PM were completed. 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* .
See below for an excerpt from a biography of Whitehead. (08)
For Tarski's use of mereology in defining a 3-D theory of geometry
based on finite spheres instead of points, see (09)
Tarski, Alfred (1929) “Foundations of the geometry of solids,”
in Tarski (1982) _Logic, Semantics, Metamathematics_, Second ed.,
Hackett Publishing Co., Indianapolis, pp. 24-29. (010)
Tarski published this after Whitehead, but there is no evidence that
he was aware of Whitehead's 1919 book. Tarski used the version of
mereology that had been developed by Lesniewski. In this paper,
T. 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. (011)
PH> First, its important to note that while it is an (ancient)
> puzzle, its not a well-defined mathematical question, as "break"
> isn't a well-defined mathematical operation. The standard
> mathematical answer gets you into distinguishing open and
> closed intervals, and conventional real analysis. (012)
I agree that you can use an ontology based on set theory as
a basis for analysis, but it seems ad hoc, especially since
there are alternative theories that preserve the intuition
that dissecting a line creates two identical halves. (013)
For more discussion of this puzzle and various ways of thinking
about it, see the discussion by Hilary Putnam in his introduction
to the following book: (014)
Peirce, Charles Sanders (1898) Reasoning and the Logic of Things,
The Cambridge Conferences Lectures of 1898, ed. by K. L. Ketner,
Harvard University Press, Cambridge, MA, 1992. (015)
> All the axiomatizations of mereology have models described in
> set theory. (016)
So what? You can form models of set theory in mereology or
category theory. The most that can be claimed for set theory
is that it's more widely taught than the other alternatives. (017)
A similar argument can be made for Gibb's theory of vectors,
which is much more widely taught than geometrical algebras.
David Hestenes had a long uphill battle getting physicists to
recognize that Clifford algebra is a far simpler mathematical
foundation for physics. Finally, Hestenes was awarded the
Oersted Medal for his work, and in his acceptance speech, he
argued that Clifford algebra should be taught to physicists
at the beginning of their studies. (018)
John
_________________________________________________________________ (019)
Source: http://www.iep.utm.edu/w/whitehed.htm (020)
Excerpt: (021)
By 1910, when the first volume of the Principia Mathematica was being
published, Hermann Minkowski had reorganized the mathematics of
Einstein’s special relativity into a four-dimensional non-Euclidean
manifold. By 1914, two years before the publication of Einstein’s paper
on general relativity, theoretical developments had advanced to the
extent that an expedition to the Crimea was planned to observe the
predicted bending of stellar light around the sun during an eclipse.
This expedition was canceled with the eruption of the First World War. (022)
These developments helped conspire to prevent Whitehead’s planned fourth
volume of the Principia from ever appearing. A few papers appeared
during the war years, in which a relational theory of space begins to
emerge. What is perhaps most notable about these papers is that they are
no longer specifically mathematical in nature, but are explicitly
philosophical. Finally, in 1919 and 1920, Whitehead’s thought appeared
in print with the publications of two books, An Enquiry into the
Principles of Natural Knowledge (“PNK,” 1919) and The Concept of Nature
(“CN,” 1920). (023)
While PNK is much more formally technical than CN, both books share a
common and radical view of nature and science that rejects the
identification of nature with the mathematical tools used to
characterize its relational structures. Nature for Whitehead is that
which is experienced through the senses. For this reason, Whitehead
argues that there are no such things as “points” of either time or
space. An infinitesimal point is a high abstraction with no experiential
reality, while time and space are irreducibly extensional in character. (024)
To account for the effectiveness of mathematical abstractions in their
application to natural knowledge, Whitehead introduced his theory of
“extensive abstraction.” By using the logical and topological structures
of concentric part-whole relations, Whitehead argued that abstract
entities such as geometric points could be derived from the concrete,
extensive relations of space and time. These abstract entities, in their
turn, could be shown to be significant of the nature they had been
abstractively derived from. Moreover, since these abstract entities were
formally easier to use, their significance of nature could be retained
through their various deductive relations, thereby giving evidence for
further natural significances by this detour through purely abstract
relations. (025)
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