Witti-Buch2 2001.qxd - Austrian Ludwig Wittgenstein Society

Finitism and Symmetry
determining its proper application, then determinism is certainly wrong. As I have shown
above, the concept of 'parallel line' does not have a use-independent meaning. In this
respect my investigation supports finitism. However, 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. Contrary to the advocates of finitism, I would claim that the
concept of 'parallel line' has a determined use in this context and, henceforth, that
Riemann's denial of the existence of parallel lines has a determined meaning. Why? The
meaning is determined in the sense that the example describes-rather than leaves
indeterminate-what we need to know in order to understand the sentence "there are no
lines that do not intersect" in elliptic geometry.
It is important to notice that the advocates of finitism cannot but reject the idea that
usage is determined in this sense. They deny not only that the concept of 'parallel line'
has a meaning that determines its proper usage in geometry, but also that the concept
has a determined use within the different geometries. For the finitist, it is the community
and their negotiations over correct usage, which is of fundamental importance in order
to understand what proper usage is, not the proper usage itself. Contrary to the
advocates of finitism, I have given priority to the practice of geometry and drawn
attention to the fact that the concept of 'parallel line' has a determined usage in the
relevant geometries.
What can we say about explanations on the basis of these examples? One thing is
clear: the concept of 'parallel line' does not explain the way we reason about parallel
lines in geometry, rather the meaning of the concept is invented as different geometries
are invented. Consequently, Lobachevsky-the founder of hyperbolic geometry-did not
discover that more than one line can be drawn parallel to a given line; he invented a
conception of 'parallel line' that permitted more than one parallel line to be drawn through
such a point. This undoubtedly means that we cannot explain judgements in geometry
with reference to the concept of parallel lines. In this respect Bloor is correct in rejecting
the idea that concepts explains usage. However, does this rather simple conceptual fact
mean that psychological and sociological factors needs to be invoked to explain why a
mathematician judges the lines intersecting below to be parallel to the line at the top?
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