popper-logic-scientific-discovery

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some structural components of a theory of experience
the following: ‘The nth element of the sequence α shall be 0 if, and
only if, n is divisible by four’. This defines the infinite alternative
(α)
1 1 1 0 1 1 1 0 . . .
with the limits of the relative frequencies: αF′ (1) = 3/4; and αF′
(0) = 1/4. Sequences which are defined in this way by means of a
mathematical rule I shall call, for brevity, ‘mathematical sequences’.
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 ‘intensional’
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