On Jan 27, 2009, at 1:57 PM, John F. Sowa wrote: (01)
> The question results from two distinct ontologies for a line, each of
> which can be formalized mathematically in different ways. (02)
Yes, I do know this, John. I survey these alternative theories in the
'time catalog' and give formalizations of them. (03)
> They were
> mentioned by Aristotle (04)
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. Infinite series were confusing the very best
mathematical thinkers until the late 19th century. (05)
who probably summarized earlier discussions
> in Plato's Academy:
> 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.
> 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.
> PH> I really don't think this is controversial any longer. Maybe it
> > was in 1920, but not now.
> Anything that Gödel considered puzzling about set theory can hardly
> be considered a closed subject. (06)
I have never seen any evidence that Goedel did find it puzzling, but
in any case I disagree. But for sure, this is not the right forum to
be discussing byways in the foundations of mathematics. (07)
> > Principia was the magnum opus of early set theory. Neither Tarski
> > nor Whitehead adopted mereology rather than set theory.
> 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* . (08)
Why do you call this a version of mereology? It seems to me to have
nothing at all to do with mereology (09)
> See below for an excerpt from a biography of Whitehead.
> For Tarski's use of mereology in defining a 3-D theory of geometry
> based on finite spheres instead of points, see
> Tarski, Alfred (1929) “Foundations of the geometry of solids,”
> in Tarski (1982) _Logic, Semantics, Metamathematics_, Second ed.,
> Hackett Publishing Co., Indianapolis, pp. 24-29.
> 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. (010)
It is ironic, then, that this very construction shows that any
axiomatic mereology has a model in set theory. What ideal theory shows
is that (under quite generous assumptions) any mereological entity can
be treated as the set of its 'atoms', where by 'atom' I mean the
limits of Whitehead's construction. 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. (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.
> I agree that you can use an ontology based on set theory as
> a basis for analysis, but it seems ad hoc, (012)
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, and then only because it is seen as providing a
cleaner way to handle very large transfinite cardinals, which is not
of the slightest importance to continuous mathematics. (013)
> there are alternative theories that preserve the intuition
> that dissecting a line creates two identical halves. (014)
First, mereology is not an alternative to set theory as a foundation
for mathematics; and second, this intuition, for what it is worth, can
be accommodated within set theory in many ways. I outlined some of
them in my previous message. (015)
> 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:
> 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.
> > All the axiomatizations of mereology have models described in
> > set theory.
> So what? You can form models of set theory in mereology or
> category theory. (016)
BUt neither of these is widely accepted as an adequate foundation. 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
most that can be claimed for set theory
> is that it's more widely taught than the other alternatives.
> 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)
Well bully for him. What has this got to do with any conceivable topic
of interest to this forum? Do you want us to become involved with the
foundations of mathematics? There is a FOM email forum for people
interested in that topic. (019)
> Source: http://www.iep.utm.edu/w/whitehed.htm
> 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.
> 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).
> 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.
> 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.
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