popper-logic-scientific-discovery

162
some structural components of a theory of experience
It may seem odd, perhaps, that we should try to trace the highly
irregular features of chance sequences by means of mathematical
sequences which must conform to the strictest rules. Von Mises’s
axiom of randomness may seem at first to be more satisfying to our
intuitions. It seems quite satisfying to learn that a chance sequence
must be completely irregular, so that every conjectured regularity will
be found to fail, in some later part of the sequence, if only we keep on
trying hard to falsify the conjecture by continuing the sequence long
enough. But this intuitive argument benefits my proposal also. For if
chance sequences are irregular, then, a fortiori, they will not be regular
sequences of one particular type. And our requirement of ‘absolute
freedom’ does no more than exclude one particular type of regular
sequence, though an important one.
That it is an important type may be seen from the fact that by our
requirement we implicitly exclude the following three types of gambling
systems (cf. the next section). First we exclude ‘normal’ or
‘pure’* 4 neighbourhood selections, i.e. those in which we select
according to some constant characteristic of the neighbourhood. Secondly we
exclude ‘normal’ ordinal selection which picks out elements whose
distance apart is constant, such as the elements numbered k, n + k,
2n + k . . . and so on. And finally, we exclude [many] combinations of
these two types of selection (for example the selection of every nth
element, provided its neighbourhood has certain specified [constant]
characteristics). A characteristic property of all these selections is that
they do not refer to an absolute first element of the sequence; they may
thus yield the same selected sub-sequence if the numbering of the
original sequence begins with another (appropriate) element. Thus the
gambling systems which are excluded by my requirement are those
which could be used without knowing the first element of the
sequence: the systems excluded are invariant with respect to certain
(linear) transformations: they are the simple gambling systems (cf.
section 43). Only* 5 gambling systems which refer to the absolute
* 4 Cf. the last paragraph of section 60, below.
* 5 The word ‘only’ is only correct if we speak of (predictive) gambling systems; cf. note *3
to section 60, below, and note 6 to section *54 of my Postscript.