popper-logic-scientific-discovery

probability 165
binomial formula (3), it is sufficient to assume that α is chance-like or
random. 2 (Our task is equivalent to that of showing that the special
theorem of multiplication holds for the sequence of adjoining
segments of a random sequence α.)
The proof* 1 of formula (3) may be carried out in two steps. First we
show that formula (2) holds not only for sequences of overlapping
segments α (n), but also for sequences of adjoining sequences α n. Secondly,
we show that the latter are ‘absolutely free’. (The order of these
steps cannot be reversed, because a sequence of overlapping segments
α (n) is definitely not ‘absolutely free’; in fact, a sequence of this kind
provides a typical example of what may be called ‘sequences with
after-effects’. 3 )
First step. Sequences of adjoining segments α n are sub-sequences of
α (n). They can be obtained from these by normal ordinal selection. Thus
if we can show that the limits of the frequencies in overlapping
sequences α(n) F′(m) are insensitive to normal ordinal selection, we
have taken our first step (and even gone a little farther); for we shall
have proved the formula:
α n F′ (m) = α(n) F′ (m) (4)
I shall first sketch this proof in the case of n = 2; i.e. I shall show that
α 2 F′(m) = α(2) F′(m) (m � 2) (4a)
is true; it will then be easy to generalize this formula for every n.
From the sequence of overlapping segments α (2) we can select two
2 Reichenbach (Axiomatik der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift 34, 1932,
p. 603) implicitly contests this when he writes, ‘ . . . normal sequences are also free
from after-effect, whilst the converse does not necessarily hold’. But Reichenbach’s normal
sequences are those for which (3) holds. (My proof is made possible by the fact that I
have departed from previous procedure, by defining the concept ‘freedom from aftereffect’
not directly, but with the help of ‘n-freedom from after-effect’, thus making it
accessible to the procedure of mathematical induction.)
* 1 Only a sketch of the proof is here given. Readers not interested in the proof may turn
to the last paragraph of the present section.
3 Von Smoluchowski based his theory of the Brownian movement on after-effect
sequences, i.e. on sequences of overlapping segments.