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particularly simple; but a law which can be represented by a logarithmic
function is usually regarded as a simple one. Similarly a sine
function is commonly said to be simple, even though the geometrical
shape of the sine curve is perhaps not so very simple.
Difficulties like this can be cleared up if we remember the connection
between the number of parameters and the degree of falsifiability,
and if we distinguish between the formal and the material reduction of
dimensions. (We must also remember the rôle of invariance with
respect to transformations of the co-ordinate systems.) If we speak of
the geometrical form or shape of a curve, then what we demand is invariance
with respect to all transformations belonging to the group of displacements,
and we may demand invariance with respect to similarity
transformations; for we do not think of a geometrical figure or shape as
being tied to a definite position. Consequently, if we think of the shape of
a one-parametric logarithmic curve (y = log ax) as lying anywhere in a
plane, then it would have five parameters (if we allow for similarity
transformations). It would thus be by no means a particularly simple
curve. If, on the other hand, a theory or law is represented by a logarithmic
curve, then co-ordinate transformations of the kind described
are irrelevant. In such cases, there is no point in either rotations or
parallel displacements or similarity transformations. For a logarithmic
curve as a rule is a graphic representation in which the co-ordinates
cannot be interchanged. (For example, the x-axis might represent
atmospheric pressure, and the y-axis height above sea-level.) For this
reason, similarity transformations are equally without any significance
here. Analogous considerations hold for sine oscillations along a
particular axis, for example, the time axis; and for many other cases.
45 THE SIMPLICITY OF EUCLIDEAN GEOMETRY
simplicity 129
One of the issues which played a major rôle in most of the discussions
of the theory of relativity was the simplicity of Euclidean geometry.
Nobody ever doubted that Euclidean geometry as such was simpler
than any non-Euclidean geometry with given constant curvature—not
to mention non-Euclidean geometries with curvatures varying from
place to place.
At first sight the kind of simplicity here involved seems to have little