popper-logic-scientific-discovery

a planet rather than to a point of the plane, and that the possible
positions do not overlap. 3 Somewhat differently from the example of
the preceding appendix, we now replace the various curves which are
the usual geometrical representations of our theories by ‘quasi curves’
(of a width approximately equal to ε); that is to say, by sets, or chains,
of squares. As a result of all this, the number of the possible theories
becomes finite.
We now consider the representation of a theory with d parameters
which in the continuous case was represented by a d-dimensional continuum
whose points (d-tuples) each represented a curve. We find that
we can still use a similar representation, except that our d-dimensional
continuum will be replaced by a d-dimensional arrangement of ddimensional
‘cubes’ (with the side ε). Each chain of these cubes will
now represent one ‘quasi curve’, and thus one of the possibilities
favourable to the theory; and the d-dimensional arrangement will represent
the set of all ‘quasi-curves’ compatible with, or favourable to, the
theory.
But we can now say that the theory with fewer parameters—that is
to say, the set of quasi curves which is represented by an arrangement
of fewer dimensions—will not only have fewer dimensions, but will
also contain a smaller number of ‘cubes’; that is, of favourable
possibilities.
Thus we are justified in applying the results of the preceding section:
if a 1 has fewer parameters than a 2, we can assert that, in a sufficiently
large but finite universe, we shall have
and therefore
(*)
p(a 1)