popper-logic-scientific-discovery

degrees of testability 119
common, then we must operate with an equation which is not
invariant with respect to the transformations of the Euclidean group,
but relates to a singular, i.e. an individually or ostensively specified,
co-ordinate system. Thus it is connected with individual names. 2
The transformations can be arranged in a hierarchy. A definition
which is invariant with respect to a more general group of transformations
is also invariant with respect to more special ones. For each
definition of a set of curves, there is one—the most general—
transformation group which is characteristic of it. Now we can say: The
definition D 1 of a set of curves is called ‘equally general’ to (or more
general than) a definition D 2 of a set of curves if it is invariant with
respect to the same transformation group as is D 2 (or a more general
one). A reduction of the dimension of a set of curves may now be
called formal if the reduction does not diminish the generality of the
definition; otherwise it may be called material.
If we compare the degree of falsifiability of two theories by considering
their dimensions, we shall clearly have to take into account
their generality, i.e. their invariance with respect to co-ordinate
transformations, along with their dimensions.
The procedure will, of course, have to be different according to
whether the theory, like Kepler’s theory, in fact makes geometrical
statements about the world or whether it is ‘geometrical’ only in that it
may be represented by a graph—such as, for example, the graph which
represents the dependence of pressure upon temperature. It would be
inappropriate to require of this latter kind of theory, or of the corresponding
set of curves, that its definition should be invariant with
respect to, say, rotations of the co-ordinate system; for in these cases,
the different co-ordinates may represent entirely different things (the
one pressure and the other temperature).
This concludes my exposition of the methods whereby degrees of
falsifiability are to be compared. I believe that these methods can help
us to elucidate epistemological questions, such as the problem of simplicity
which will be our next concern. But there are other problems which
2 On the relations between transformation groups and ‘individualization’ cf. Weyl,
Philosophie der Mathematik u. Naturwissenschaft, 1927, p. 59, English edition pp. 73 f.,
where reference is made to Klein’s Erlanger Programm.