popper-logic-scientific-discovery

degrees of testability 109
demand that the degree of precision in measurement should be raised
as much as possible.
It is often said that all measurement consists in the determination of
coincidences of points. But any such determination can only be correct
within limits. There are no coincidences of points in a strict sense.* 2
Two physical ‘points’—a mark, say, on the measuring-rod, and another
on a body to be measured—can at best be brought into close proximity;
they cannot coincide, that is, coalesce into one point. However trite
this remark might be in another context, it is important for the question
of precision in measurement. For it reminds us that measurement
should be described in the following terms. We find that the point of
the body to be measured lies between two gradations or marks on the
measuring-rod or, say, that the pointer of our measuring apparatus lies
between two gradations on the scale. We can then either regard these
gradations or marks as our two optimal limits of error, or proceed to
estimate the position of (say) the pointer within the interval of the
gradations, and so obtain a more accurate result. One may describe this
latter case by saying that we take the pointer to lie between two
imaginary gradation marks. Thus an interval, a range, always remains. It
is the custom of physicists to estimate this interval for every
measurement. (Thus following Millikan they give, for example, the
elementary charge of the electron, measured in electrostatic units,
as e = 4.774.10 − 10 , adding that the range of imprecision
is ± 0.005.10 − 10 .) But this raises a problem. What can be the purpose
of replacing, as it were, one mark on a scale by two—to wit, the two
bounds of the interval—when for each of these two bounds there must
again arise the same question: what are the limits of accuracy for the
bounds of the interval?
Giving the bounds of the interval is clearly useless unless these two
bounds in turn can be fixed with a degree of precision greatly exceeding
what we can hope to attain for the original measurement; fixed,
that is, within their own intervals of imprecision which should thus
be smaller, by several orders of magnitude, than the interval they
determine for the value of the original measurement. In other words,
* 2 Note that I am speaking here of measuring, not of counting. (The difference between
these two is closely related to that between real numbers and rational numbers.)