Witti-Buch2 2001.qxd - Austrian Ludwig Wittgenstein Society

Jonas Larsson
Assume that Lobachevsky one day exclaimed to his euclidic-minded colleague:
"The three lines that intersect in this circle are parallel to the line at the top of the circle!"
Perhaps his colleague starts to contemplate why Lobachevsky, a man of reason that
usually passes intelligible judgements, now makes a seemingly absurd statement. Now,
it does not help if Lobachevsky insists that the lines really are parallel. Rather, he has to
explain his judgement by showing that a plane is understood as a set of points that lie
in the interior of a circle, that a line is taken as a chord of a circle, and that parallel lines
are defined as lines that never intersect. This conception explains why Lobachevsky
passed the judgement that the lines are parallel, in the sense that it explains what one
needs to know in order to identify the lines as parallel. Put differently, this conception
explains Lobachevsky's judgement with reference to the conceptual route one has to
embark upon-the geometry one needs to be familiar with-in order to understand the
statement that the lines are parallel. Even though I agree with Bloor that the concept of
'parallel line' does not explain geometrical reasoning, the explanation above is not a
psychological or sociological but a cognitive explanation. As I have shown, no expertise
in psychology and sociology is needed to explain why Lobachevsky judged the lines
parallel; what is required is expertise in the intellectual activity itself, i.e. in hyperbolic
geometry.
Summary and Conclusion
The advocates of the Strong Programme use finitism and the concomitant idea that
usage determines meaning in order to justify the symmetry requirement. I have
explained how finitism replaces meaning determinism and conceptual compulsion with
psychological and social contingencies that are assumed operate causally on the
episode of application. However, by investigating the actual usage of the concept of
'parallel line' in geometry I have showed that it is a mistake to believe that the priority of
practice over meaning implies the priority of psychological and sociological explanations.
Even though the meaning of the concept of 'parallel line' is dependent on its usage in
different geometries, we still need expertise in hyperbolic geometry, rather than in
psychology and sociology, in order to explain judgement passed within different
geometries. While the advocates of the Strong Programme attempt to justify the
symmetry requirement on the basis of finitism and the idea that usage determines
meaning, I have drawn attention to a case where usage determines meaning and to an
ordinary explanation that contradicts the symmetry requirement.
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