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S15a. How precise can physical language be?
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The relation between theory and reality necessarily uses ordinary
language and is therefore somewhat fuzzy. If one insists on 100%
unambiguous statements, one is on the level of pure mathematics or
mathematical physics (platonic reality), and cannot have any contact
with (physical) reality.
The best one can do is to have completely precise concepts on the
theoretical level and a description in ordinary, informal language
that relates theory to reality. In the formal theory, all concepts
can be precisely defined, and get names corresponding to their intended
use in reality. This ensures that one knows precisely what one talks
about - on the conceptual level.
In this informal language there must be room for linguistic
approximations without specifying their quality more than by
fuzzy words interpreted by the circumstances, since this is the way
we necessarily perceive reality.
When formulating the interface between theory and reality,
one must use the formulations people use who are using this interface,
They know how 'large' something must be to be taken as 'infinite'.
They estimate limits from finite sequences (most of numerical
analysis would be void if we couldn't...), usually quite successfully
- although this is meaningless mathematically.
A mathematical limit in theory does _not_ translate into a mathematical
limit in reality.
This is necessary since all our observations are finite, and most of
them are noisy. As there are approximate ways of determining the mass
of the Moon, but no exact methods, so there are approximate methods
for determining probabilities, but no exact ones. Exact real numbers
belong to theory, not to reality. (Even counting is not sure to result
in an integer. What about the number of people in a room when just
someone enters?)
Careful protocols for experimentation and measurement are useful to
achieve a certain amount of objectivity and repeatability, but even the
best protocols cannot reduce the level of fuzziness in the interface
between theory and reality to zero. I recommend
Experimentation and Measurement, by W.J. Youden,
reprinted 1997 by the National Institute of Standards and Technology
http://ts.nist.gov/ts/htdocs/230/233/calibrations/Publications/exp_meas.pdf
Although a very old paper (from 1961), it is still considered by NIST
to be up to date and exemplary in its lessons about measurements.
Among other things, it discusses on pp. 26ff in greatest detail
how to measure the thickness of a sheet of paper in an ensemble of
sheets typically called a thick book.
If one follows his argument closely, one finds that even classically,
observables such as the 'thickness of a sheet of paper' are
probabilistic only, notwithstanding that probably everything relevant
about paper can be understood by classical mechanics and
thermodynamics.
Thus there are no exact concepts in observed Nature.
But in a good theory of Nature, all concepts should be exact.