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 welldefined mathematical question, as "break" isn't a welldefined mathematical operation. The standard mathematical answer gets you into distinguishing open and closed intervals, and conventional real analysis. There isn't anything particularly nonsetish 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 onetoone 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 onetoone 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 3dimensional 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 pointlike '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 nearextinct 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 halfinterval 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 'rejoined'. 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 3d, 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 closureofinterior 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 4D 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 WhiteheadTarski 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) settheoretic language since the late 19th century.
Pat
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