popper-logic-scientific-discovery

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some structural components of a theory of experience
the bounds of the interval are not sharp bounds but are really very
small intervals, the bounds of which are in their turn still much
smaller intervals, and so on. In this way we arrive at the idea of what
may be called the ‘unsharp bounds’ or ‘condensation bounds’ of the
interval.
These considerations do not presuppose the mathematical theory of
errors, nor the theory of probability. It is rather the other way round;
by analysing the idea of a measuring interval they furnish a background
without which the statistical theory of errors makes very little
sense. If we measure a magnitude many times, we obtain values which
are distributed with different densities over an interval—the interval of
precision depending upon the prevailing measuring technique. Only if
we know what we are seeking—namely the condensation bounds of
this interval—can we apply to these values the theory of errors, and
determine the bounds of the interval.* 3
Now all this sheds some light, I think, on the superiority of methods that
employ measurements over purely qualitative methods. It is true that even in the
case of qualitative estimates, such as an estimate of the pitch of a
musical sound, it may sometimes be possible to give an interval of
accuracy for the estimates; but in the absence of measurements, any
such interval can be only very vague, since in such cases the concept of
condensation bounds cannot be applied. This concept is applicable
only where we can speak of orders of magnitude, and therefore only
where methods of measurement are defined. I shall make further use of
the concept of condensation bounds of intervals of precision in section
68, in connection with the theory of probability.
38 DEGREES OF TESTABILITY COMPARED BY
REFERENCE TO DIMENSIONS
Till now we have discussed the comparison of theories with respect to
their degrees of testability only in so far as they can be compared with
the help of the subclass-relation. In some cases this method is quite
* 3 These considerations are closely connected with, and supported by, some of the results
discussed under points 8 ff. of my ‘Third Note’, reprinted in appendix *ix. See also
section *15 of the Postscript for the significance of measurement for the ‘depth’ of
theories.