popper-logic-scientific-discovery

394
new appendices
each entry represents one of the relative-atomic statements. (We can, of
course, make the representation three-dimensional, by marking the
position with a pin whose length represents the time, measured from
some assumed zero instant; and variations in the colour of the pinhead
may be used to indicate the names of the various planets.)
It has been explained, mainly in sections 40 to 46, and in my old
appendix i, how the minimum number of the relative-atomic statements
needed to refute a certain theory could be used as a measure of
the complexity of the theory. And it was shown that the formal simplicity
of a theory might be measured by the paucity of its parameters, in so far as
this paucity was not the result of a ‘formal’ rather than a ‘material’
reduction in the number of the parameters. (Cf. especially sections 40,
44, f., and appendix i.)
Now all these comparisons of the simplicity of theories, or of their
contents, will, clearly, amount to comparisons of the ‘fine-structure’ of
their contents, in the sense explained in the preceding appendix,
because their absolute probabilities will all be equal (that is, equal to
zero). And I wish first to show that the number of the parameters of a
theory (with respect to a field of application) can indeed be interpreted
as measuring the fine-structure of its content.
What I have to show, to this end, is that for a sufficiently large finite universe,
the theory with the greater number of parameters will always be more probable (in the
classical sense) than the theory with the smaller number of parameters.
This can be shown as follows. In the case of a continuous geometrical
field of applications, our universe of possible events, each
described by a possible relative-atomic statement, is of course infinite.
As shown in sections 38 f., we can in this case compare two theories
with respect to the dimension, rather than the number, of the possibilities
which they leave open; that is, the possibilities which are favourable to
them. The dimension of these possibilities turns out to be equal to the
number of parameters. We now replace the infinite universe of relativeatomic
statements by a finite (although very large) universe of relativeatomic
statements, corresponding to the chessboard example in the
preceding appendix. 2 That is to say, we assume that every relativeatomic
statement refers to a little square with the side ε as the position of
2 Cf. appendix *vii, text to note 12.