popper-logic-scientific-discovery

370
new appendices
between rain and Sundays, or whether a certain given formula for
predicting rain on Sundays works, such as ‘at least once in three weeks’;
but though we may reject this formula in view of our tests, we cannot
determine, by our tests, whether or not there exists some better
formula.
(4) Under these circumstances, it seems tempting to say that randomness
or disorder is not a type of order which can be described
objectively and that it must be interpreted as our lack of knowledge as to
the order prevailing, if any order prevails. I think that this temptation
should be resisted, and that we can develop a theory which
allows us actually to construct ideal types of disorder (and of course
also ideal types of order, and of all degrees in between these
extremes).
(5) The simplest problem in this field, and the one which, I believe,
I have solved, is the construction of a one-dimensional ideal type of disorder—
an ideally disordered sequence.
The problem of constructing a sequence of this kind arises immediately
from any frequency theory of probability which operates with
infinite sequences. This may be shown as follows.
(6) According to von Mises, a sequence of 0’s and 1’s with equidistribution
is random if it admits of no gambling system, that is to say, of
no system which would allow us to select in advance a sub-sequence in
which the distribution is unequal. But of course, von Mises admits that
any gambling system may, ‘accidentally’, work for some time; it is only
postulated that it will break down in the long run—or more precisely, in
an infinite number of trials.
Accordingly, a Mises collective may be extremely regular in its commencing
segment: provided they become irregular in the end, von Mises’s
rule is incapable of excluding collectives which start off very regularly,
say with
00 11 00 11 00 11 . . .
and so on, for the first five hundred million places.
(7) It is clear that we cannot empirically test this kind of deferred
randomness; and it is clear that whenever we do test randomness in a
sequence, we have a different type of randomness in mind: a sequence