On Jan 27, 2009, at 8:57 AM, John F. Sowa wrote: Pat, I agree with the technical points you've made, but the following claim is still controversial: PH> But set theory is not particularly discrete. Sets are the basis for all mathematics, including continuous mathematics.
I really don't think this is controversial any longer. Maybe it was in 1920, but not now. For example, what happens to the midpoint when you break a line
segment in two equal parts?
As you probably know, Ive been thinking and writing about this particular issue for years. 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. There isn't anything particularly non-set-ish about conventional real analysis, of course.
1. In classical Euclidean geometry, the two halves are isomorphic.
the modern notion of isomorphism wasn't even formulated until many centuries after Euclid.
2. But if the line is considered isomorphic to a set of points
that are in a one-to-one correspondence with the real numbers,
the former midpoint must belong to one side or the other.
Assumption #2 implies that the two "halves" cannot be isomorphic
because the one that contains the former midpoint will be closed
at that end. But the one from which the former midpoint was
removed will be open at that end.
It depends on exactly how you define 'isomorphic'. The two intervals have the same Lebesgue measure, for example. One would not expect that one-to-one mappings between points would yield very insightful results on the classical continuum, since real intervals can be put into 1:1 correspondence with subsets of themselves.
This puzzle has troubled many brilliant mathematicians, including
Kurt Gödel.
Tarski's solution to that puzzle (and related issues) was to use
a version of mereology combined with an ontology whose basic
constructs were not points, but 3-dimensional spheres of arbitrary
finite size.
A similar technique was used (and I believe invented) by Whitehead. Its now a standard piece of textbook mathematics, called variously the method of ultrafilters or ideals. I used it recently to show that most context logics can be reduced to quantification over point-like 'contexts' which are transparent to negation. Tony Cohn used Tarski's method directly in his spatial ontologies. But this is all rooted (or certainly can be rooted) in set theory. Mereology is only an 'alternative' to set theory if one is a committed philosophical mereologist, and they have been a near-extinct species since the 1930s. All the axiomatizations of mereology have models described in set theory.
Tarski then showed that it was possible to define a Euclidean point
as the limit of an infinite sequence of decreasing concentric
spheres. This solution implies that points are not truly "parts"
of a line (or any other geometric structure). Instead, they are
limits and each half of the line segment can have its own limit
point.
Tarski's construction doesn't get rid of the puzzle, it only reproduces the conventional continuum, so one still gets open and closed intervals. Of course, one can take the view that each half-interval also contains its own limits, ie is a closed interval, but then the midpoint gets 'counted' 'twice' (neither of these terms really makes sense) when the two 'halves' are 're-joined'. Computer graphic systems often use this as an abstraction, where a 'shape' is defined to be the closure of the interior of a set of points. This makes all shapes contain their limit points, but also gets rid of stray isolated points (and in 3-d, lines and planes) which are liable to appear when shapes are 'subtracted' from one another. This (now classical) technique seems to solve the old puzzle quite well. IN the time catalog the title 'glass continuum' is given to the class of theories which allow an interval to be divided into two exactly similar pieces; the simplest model of these theories uses this closure-of-interior construction.
NOne of this stuff, while fascinating, has anything to do with claims for or against set theory. These constructions, like all mathematical constructions described since around 1914, are understood as reducible to constructions involving sets.
)(For that matter, even lines and planes are limits of
constructions made of spheres.)
For the fourth volume on geometry of the _Principia Mathematica_.
Whitehead independently developed a version of mereology in which
the basic construct was a 4-D blob and points were decreasing
sequences of nested blobs. He never completed that volume, but
he published some of the material in other writings.
It's significant that Gödel couldn't find a solution that used
only set theory and that Tarski and Whitehead adopted mereology
rather than set theory to address such issues.
No, they both used set theory. Principia was the magnum opus of early set theory. Neither Tarski nor Whitehead adopted mereology rather than set theory.
And by the way, the Whitehead-Tarski approach can be adapted
to relate a time ontology based on intervals to one that is
based on instants. In effect, an instant is the limit of a
decreasing nest of intervals.
I know. I have written several papers on the topic, and its reviewed in one chapter of the time catalog. But it should be noted that classical mathematics also defines the continuum as a limit construction. This has been the standard way to describe continuous structures in (what we would now call) set-theoretic language since the late 19th century.
Pat
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