popper-logic-scientific-discovery

APPENDIX i
Definition of the Dimension of a
Theory (cf. sections 38 and 39)
The definition which follows here should be regarded as only provisional.*
1 It is an attempt to define the dimension of a theory so as to
make it agree with the dimension of the set of curves which results if
the field of application of the theory is represented by a graph paper. A
difficulty arises from the fact that we should not assume that either a
metric or even a topology is defined for the field, to begin with; in
particular, we should not assume that any neighbourhood relations are
defined. And I admit that this difficulty is circumvented rather than
overcome by the definition proposed. The possibility of circumventing
* 1 A simplified and slightly more general definition is this. Let A and X be two sets of
statements. (Intuitively, A is a set of universal laws, X a set—usually infinite—of singular
test statements.) Then we say that X is a (homogeneous) field of application with respect
to A (in symbols: X = F A) if and only if for every statement a in A, there exists a natural
number d(a) = n which satisfies the following two conditions: (i) any conjunction c n of n
different statements of X is compatible with a; (ii) for any such conjunction c n there exist
two statements x and y in X such that x.c n is incompatible with a and y.c n is derivable from
a.c n, but neither from a nor from c n.
d(a) is called the dimension of a, or the degree of composition of a, with respect to
X = F A; and 1/d(a) or, say, 1/(d(a) + 1), may be taken as a measure of the simplicity of a.
The problem is further developed in appendix *viii.