Re: What happens to the midpoint when a line is cut in two?
In general, this mathematical issue mirrors the more general issue of
the existence of "discrete" and "continuous" duality of the nature.
-- "John F. Sowa" <sowa@bestweb.net> wrote:
I received an offline question about the following passage
from pp. 109-110 of my Knowledge Representation book:
> The representation of space as a set of points introduces
> other puzzles. At the top of Fig. 2.16 is a line segment
> that is being sliced in two equal halves, which are separated
> at the bottom [of this fig.]. A question arises about what
> happens to the midpoint. If the original lines were considered
> a set of points, it would have just one midpoint. The two
> halves of the line would not be identical, since only one of
> them would contain the former midpoint. The other part would
> contain an uncountable infinite set of points leading up to,
> but not including the old midpoint.... The paradox of Fig. 2.16
> did not occur to Aristotle or Euclid who never said that a
> line consisted of points....
Since this issue has many implications for ontology, especially
an ontology of time, I'm resending the following response to
these mailing lists.
John Sowa
_________________________________________________________________
> I do not agree that there is a paradox. The misleading term
> is ‘identical’. For me ‘identical’ means ‘being exactly the
> same’. Now a left half of a line is ‘by definition’ not exactly
> the same as a right half. So, you do not mean ‘identical’: you
> mean something like ‘can be considered to be the same within
> the structure I consider’.
By identical, I meant that there is an isomorphism between
the two sides. In Euclidean geometry, it was reasonable
to assume that any line segment could be cut in two parts
that were identical in that sense.
But in point-set topology, one half of the cut would be a
closed set at both ends, but the other would be open at one
end and closed at the other. They would not be identical
in the sense above.
> But, you introduce the so-called paradox as an argument
> against set point theory. So the structure within which
> you work is set theory.
There is a difference between set theory and the use of set
theory in point-set topology. I never said that set theory
was "wrong" -- merely that point-set topology is not the best
system to use as a foundation for geometry.
Many mathematicians have recognized that fact, including Peirce,
Whitehead, Tarski, Goedel, and others. Whitehead was planning
to write a fourth volume on geometry for _Principia Mathematica_,
and he invented a version of mereology (which he called
"extensive abstraction") in order to avoid that problem.
Tarski had another solution, which I summarize in the KR book
on pp. 110-112: define a geometry in which the only primitive
construct is a sphere. In Tarski's geometry, a point not a "part"
of space, but a fictitious limit of a convergent sequence of
spheres. This actually corresponds very closely to Aristotle's
view: a point is not part of a line, but a location on the line.
Tarski's solution has a strong similarity with Whitehead's
approach. For two dimensions, a sphere corresponds to a circle,
and in one dimension, it corresponds to a line segment. In
such a system, the "parts" of a line segment are smaller line
segments, and a point is a fictitious limit of a convergent
sequence of line segments.
When you cut a line segment in two halves, both halves are
isomorphic. The former midpoint corresponds to exactly
equivalent limiting sequences on both halves.
This approach is widely used for an ontology of time, in which
instants are defined as fictitious limits of convergent sequences
of smaller and smaller time intervals. This solution solves
Zeno's paradox about motion: at an instant, time stands still
and motion is impossible. If there are no true instants, there
is no paradox.
John Sowa