K. Popper - Logic scientific discovery
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K. Popper - Logic scientific discovery

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By contrast, a rule for constructing an empirical sequence would be, for
instance: ‘The nth element of the sequence α shall be 0 if, and only if,
the nth toss of the coin c shows tails.’ But empirical rules need not
always define sequences of a random character. For example, I should
describe the following rule as empirical: ‘The nth element of the
sequence shall be 1 if, and only if, the nth second (counting from some
zero instant) finds the pendulum p to the left of this mark.’

The example shows that it may sometimes be possible to replace an
empirical rule by a mathematical one—for example on the basis of
certain hypotheses and measurements relating to some pendulum. In
this way, we may find a mathematical sequence approximating to our
empirical sequence with a degree of precision which may or may not
satisfy us, according to our purposes. Of particular interest in our
present context is the possibility (which our example could be used
to establish) of obtaining a mathematical sequence whose various
frequencies approximate to those of a certain empirical sequence.

In dividing sequences into mathematical and empirical ones I am
making use of a distinction that may be called ‘intensional’ rather than
‘extensional’. For if we are given a sequence ‘extensionally’, i.e. by
listing its elements singly, one after the other—so that we can only
know a finite piece of it, a finite segment, however long—then it is
impossible to determine, from the properties of this segment, whether
the sequence of which it is a part is a mathematical or an empirical
sequence. Only when a rule of construction is given—that is, an ‘inten-
sional’ rule—can we decide whether a sequence is mathematical or
empirical.

Since we wish to tackle our infinite sequences with the help of the
concept of a limit (of relative frequencies), we must restrict our
investigation to mathematical sequences, and indeed to those for

some structural components of a theory of experience156

which the corresponding sequence of relative frequencies is con-
vergent. This restriction amounts to introducing an axiom of con-
vergence. (The problems connected with this axiom will not be dealt
with until sections 63 to 66, since it turns out to be convenient to
discuss them along with the ‘law of great numbers’.)

Thus we shall be concerned only with mathematical sequences. Yet we
shall be concerned only with those mathematical sequences of which
we expect, or conjecture, that they approximate, as regards frequencies,
to empirical sequences of a chance-like or random character; for these are our main
interest. But to expect, or to conjecture, of a mathematical sequence
that it will, as regards frequencies, approximate to an empirical
one is nothing else than to frame a hypothesis—a hypothesis about the
frequencies of the empirical sequence.1

The fact that our estimates of the frequencies in empirical random
sequences are hypotheses is without any influence on the way we may
calculate these frequencies. Clearly, in connection with finite classes, it
does not matter in the least how we obtain the frequencies from which
we start our calculations. These frequencies may be obtained by actual
counting, or from a mathematical rule, or from a hypothesis of some
kind or other. Or we may simply invent them. In calculating frequen-
cies we accept some frequencies as given, and derive other frequencies
from them.

The same is true of estimates of frequencies in infinite sequences.
Thus the question as to the ‘sources’ of our frequency estimates is
not a problem of the calculus of probability; which, however, does not
mean that it will be excluded from our discussion of the problems of
probability theory.

In the case of infinite empirical sequences we can distinguish two
main ‘sources’ of our hypothetical estimates of frequencies—that is to
say, two ways in which they may suggest themselves to us. One is an
estimate based upon an ‘equal-chance hypothesis’ (or equi-probability
hypothesis), the other is an estimate based upon an extrapolation of
statistical findings.

1 Later, in sections 65 to 68, I will discuss the problem of decidability of frequency hypoth-
eses, that is to say, the problem whether a conjecture or hypothesis of this kind can be
tested; and if so, how; whether it can be corroborated in any way; and whether it is
falsifiable. *Cf. also appendix *ix.

probability 157

By an ‘equal-chance hypothesis’ I mean a hypothesis asserting that the
probabilities of the various primary properties are equal: it is a hypoth-
esis asserting equal distribution. Equal-chance hypotheses are usually based
upon considerations of symmetry.2 A highly typical example is the con-
jecture of equal frequencies in dicing, based upon the symmetry and
geometrical equivalence of the six faces of the cube.

For frequency hypotheses based on statistical extrapolation, estimates of
rates of mortality provide a good example. Here statistical data about
mortality are empirically ascertained; and upon the hypothesis that past trends
will continue to be very nearly stable, or that they will not change much—at
least during the period immediately ahead—an extrapolation to
unknown cases is made from known cases, i.e. from occurrences which
have been empirically classified, and counted.

People with inductivist leanings may tend to overlook the hypo-
thetical character of these estimates: they may confuse a hypothetical
estimate, i.e. a frequency-prediction based on statistical extrapolation,
with one of its empirical ‘sources’—the classifying and actual counting
of past occurrences and sequences of occurrences. The claim is often
made that we ‘derive’ estimates of probabilities—that is, predictions of
frequencies—from past occurrences which have been classified and
counted (such as mortality statistics). But from a logical point of view
there is no justification for this claim. We have made no logical deriv-
ation at all. What we may have done is to advance a non-verifiable
hypothesis which nothing can ever justify logically: the conjecture that
frequencies will remain constant, and so permit of extrapolation. Even
equal-chance hypotheses are held to be ‘empirically derivable’ or ‘empiric-
ally explicable’ by some believers in inductive logic who suppose them
to be based upon statistical experience, that is, upon empirically
observed frequencies. For my own part I believe, however, that in mak-
ing this kind of hypothetical estimate of frequency we are often guided
solely by our reflections about the significance of symmetry, and by
similar considerations. I do not see any reason why such conjectures
should be inspired only by the accumulation of a large mass of induct-
ive observations. However, I do not attach much importance to these

2 Keynes deals with such questions in his analysis of the principle of indifference. Cf. op. cit.,
Chapter IV, pp. 41–64.

some structural components of a theory of experience158

questions about the origins or ‘sources’ of our estimates. (Cf. section
2.) It is more important, in my opinion, to be quite clear about the fact
that every predictive estimate of frequencies, including one which we
may get from statistical extrapolation—and certainly all those that refer
to infinite empirical sequences—will always be pure conjecture since it
will always go far beyond anything which we are entitled to affirm on
the basis of observations.

My distinction between equal-chance hypotheses and statistical
extrapolations corresponds fairly well to the classical distinction
between ‘a priori’ and ‘a posteriori’ probabilities. But since these terms
are used in so many different senses,3 and since they are, moreover,
heavily tainted with philosophical associations, they are better
avoided.

In the following examination of the axiom of randomness, I shall
attempt to find mathematical sequences which approximate to random
empirical sequences; which means that I shall be examining
frequency-hypotheses.*2

58 AN EXAMINATION OF THE AXIOM
OF RANDOMNESS

The concept of an ordinal selection