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# K. Popper - Logic scientific discovery

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```of probability is a theory of certain chance-like or
random sequences of events or occurrences, i.e. of repetitive events
such as a series of throws with a die. These sequences are deﬁned as
‘chance-like’ or ‘random’ by means of two axiomatic conditions: the
axiom of convergence (or the limit-axiom) and the axiom of randomness. If a
sequence of events satisﬁes both of these conditions it is called by von
Mises a ‘collective’.

A collective is, roughly speaking, a sequence of events or occur-
rences which is capable in principle of being continued indeﬁnitely;
for example a sequence of throws made with a supposedly indestruct-
ible die. Each of these events has a certain character or property; for
example, the throw may show a ﬁve and so have the property ﬁve. If we
take all those throws having the property ﬁve which have appeared up
to a certain element of the sequence, and divide their number by the
total number of throws up to that element (i.e. its ordinal number in
the sequence) then we obtain the relative frequency of ﬁves up to that
element. If we determine the relative frequency of ﬁves up to every
element of the sequence, then we obtain in this way a new sequence—
the sequence of the relative frequencies of ﬁves. This sequence of frequencies is
distinct from the original sequence of events to which it corresponds,

1 R. von Mises, Fundamentalsätze der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift 4, 1919,
p. 1; Grundlagen der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift 5, 1919, p. 52; Wahrschein-
lichkeit, Statistik, und Wahrheit (1928), 2nd edition 1936, English translation by J. Neyman,
D. Sholl, and E. Rabinowitsch: Probability, Statistics and Truth, 1939; Wahrscheinlichkeitsrechnung
und ihre Anwendung in der Statistik und theoretischen Physik (Vorlesungen über angewandte Mathematik
1), 1931.

probability 139

and which may be called the ‘event-sequence’ or the ‘property-
sequence’.

As a simple example of a collective I choose what we may call an
‘alternative’. By this term we denote a sequence of events supposed to
have two properties only—such as a sequence of tosses of a coin. The one
property (heads) will be denoted by ‘1’, and the other (tails) by ‘0’.
A sequence of events (or sequence of properties) may then be
represented as follows:

0 1 1 0 0 0 1 1 1 0 1 0 1 0 . . . .(A)

Corresponding to this ‘alternative’—or, more precisely, correlated
with the property ‘1’ of this alternative—is the following sequence of
relative frequencies, or ‘frequency-sequence’:2

O
1

2

2

3

2

4

2

5

2

6

3

7

4

8

5

9

5

10

6

11

6

12

7

13

7

14
. . . .(A′)

Now the axiom of convergence (or ‘limit-axiom’) postulates that, as the
event-sequence becomes longer and longer, the frequency-sequence
shall tend towards a deﬁnite limit. This axiom is used by von Mises
because we have to make sure of one ﬁxed frequency value with which we
can work (even though the actual frequencies have ﬂuctuating values).
In any collective there are at least two properties; and if we are given
the limits of the frequencies corresponding to all the properties of a
collective, then we are given what is called its ‘distribution’.

The axiom of randomness or, as it is sometimes called, ‘the principle of the
excluded gambling system’, is designed to give mathematical expres-
sion to the chance-like character of the sequence. Clearly, a gambler
would be able to improve his chances by the use of a gambling system

2 We can correlate with every sequence of properties as many distinct sequences of
relative frequencies as there are properties deﬁned in the sequence. Thus in the case of an
alternative there will be two distinct sequences. Yet these two sequences are derivable
from one another, since they are complementary (corresponding terms add up to 1). For
this reason I shall, for brevity, refer to ‘the (one) sequence of relative frequencies correl-
ated with the alternative (α)’, by which I shall always mean the sequence of frequencies
correlated with the property ‘1’ of this alternative (α).

some structural components of a theory of experience140

if sequences of penny tosses showed regularities such as, say, a fairly
regular appearance of tails after every run of three heads. Now the
axiom of randomness postulates of all collectives that there does not
exist a gambling system that can be successfully applied to them. It
postulates that, whatever gambling system we may choose for selecting
supposedly favourable tosses, we shall ﬁnd that, if gambling is con-
tinued long enough, the relative frequencies in the sequence of tosses
supposed to be favourable will approach the same limit as those in the
sequence of all tosses. Thus a sequence for which there exists a gambling
system by means of which the gambler can improve his chances is not
a collective in the sense of von Mises.

Probability, for von Mises, is thus another term for ‘limit of relative
frequency in a collective’. The idea of probability is therefore applic-
able only to sequences of events; a restriction likely to be quite unacceptable
from a point of view such as Keynes’s. To critics objecting to the
narrowness of his interpretation, von Mises replied by stressing the
diﬀerence between the scientiﬁc use of probability, for example in
physics, and the popular uses of it. He pointed out that it would be
a mistake to demand that a properly deﬁned scientiﬁc term has to
correspond in all respects to inexact, pre-scientiﬁc usage.

The task of the calculus of probability consists, according to von Mises,
simply and solely in this: to infer certain ‘derived collectives’ with
‘derived distributions’ from certain given ‘initial collectives’ with cer-
tain given ‘initial distributions’; in short, to calculate probabilities
which are not given from probabilities which are given.

The distinctive features of his theory are summarized by von Mises
in four points:3 the concept of the collective precedes that of prob-
ability; the latter is deﬁned as the limit of the relative frequencies; an
axiom of randomness is formulated; and the task of the calculus of
probability is deﬁned.

51 PLAN FOR A NEW THEORY OF PROBABILITY

The two axioms or postulates formulated by von Mises in order to
deﬁne the concept of a collective have met with strong criticism—

3 Cf. von Mises, Wahrscheinlichkeitsrechnung, 1931, p. 22.

probability 141

criticism which is not, I think, without some justiﬁcation. In particular,
objections have been raised against combining the axiom of con-
vergence with the axiom of randomness1 on the ground that it is
inadmissible to apply the mathematical concept of a limit, or of con-
vergence, to a sequence which by deﬁnition (that is, because of the
axiom of randomness) must not be subject to any mathematical rule or
law. For the mathematical limit is nothing but a characteristic property of the
mathematical rule or law by which the sequence is determined. It is merely a prop-
erty of this rule or law if, for any chosen fraction arbitrarily close to
zero, there is an element in the sequence such that all elements follow-
ing it deviate by less than that fraction from some deﬁnite value—
which is then called their limit.

To meet such objections it has been proposed to refrain from com-
bining the axiom of convergence with that of randomness, and to
postulate only convergence, i.e. the existence of a limit. As to the axiom
of randomness, the proposal was either to abandon it altogether
(Kamke) or to replace it by a weaker requirement (Reichenbach).
These suggestions presuppose that it is the axiom of randomness
which is the cause of the trouble.

In contrast to these views, I am inclined to blame the axiom of
convergence no less than the axiom of randomness. Thus I think that
there are two tasks to be performed: the improvement of the```