popper-logic-scientific-discovery

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some structural components of a theory of experience
to do with degrees of falsifiability. But if the statements at issue
are formulated as empirical hypotheses, then we find that the two
concepts, simplicity and falsifiability, coincide in this case also.
Let us consider what experiments may help us to test the hypothesis,
‘In our world, we have to employ a certain metrical geometry with
such and such a radius of curvature.’ A test will be possible only if we
identify certain geometrical entities with certain physical objects—for
instance straight lines with light rays; or points with the intersection of
threads. If such an identification (a correlating definition, or perhaps an
ostensive definition; cf. section 17) is adopted, then it can be shown
that the hypothesis of the validity of an Euclidean light-ray-geometry is
falsifiable to a higher degree than any of the competing hypotheses
which assert the validity of some non-Euclidean geometry. For if we
measure the sum of the angles of a light-ray triangle, then any significant
deviation from 180 degrees will falsify the Euclidean hypothesis.
The hypothesis of a Bolyai-Lobatschewski geometry with given curvature,
on the other hand, would be compatible with any particular
measurement not exceeding 180 degrees. Moreover, to falsify this
hypothesis it would be necessary to measure not only the sum of the
angles, but also the (absolute) size of the triangle; and this means that
in addition to angles, a further unit of measurement, such as a unit of
area, would have to be defined. Thus we see that more measurements
are needed for a falsification; that the hypothesis is compatible with
greater variations in the results of measurements; and that it is therefore
more difficult to falsify: it is falsifiable to a lesser degree. To put it
in another way, Euclidean geometry is the only metric geometry with a
definite curvature in which similarity transformations are possible. In
consequence, Euclidean geometrical figures can be invariant with
respect to more transformations; that is, they can be of lower
dimension: they can be simpler.
46 CONVENTIONALISM AND THE
CONCEPT OF SIMPLICITY
What the conventionalist calls ‘simplicity’ does not correspond to what
I call ‘simplicity’. It is the central idea of the conventionalist, and also
his starting point, that no theory is unambiguously determined by