Witti-Buch2 2001.qxd - Austrian Ludwig Wittgenstein Society

Jonas Larsson
In Euclidic geometry, the axiom that through one point not on a given line, only one
line can be drawn parallel to the given line, is set up on the following basis:
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Imagine a two-dimensional indefinite plane and an indefinite line of which only a
segment can be perceived. In this case, only one line can be drawn through the point
parallel to the given line. However, think about hyperbolic geometry where a plane is
defined as a set of points that lie in the interior of a circle, a line is defined as a chord of
a circle, and parallel lines are defined as lines that never intersect. In circle below, the
three lines that intersect are parallel to the line at the top of the circle:
Hence, more than one line can be drawn parallel to the given line. Now consider
elliptical geometry, where a line is defined as the great circle of a sphere and a great
circle is any circle that divides a sphere into equal halves. Consequently, any two such
circles intersect:
So for every one point not on a given line, there are no lines that do not intersect
with the given line. Thus, there are no parallel lines.
What can the actual usage of the concept of 'parallel lines' teach us about finitism
and determinism? If 'determinism' is the view that a concept is endowed with a meaning
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