by adrian_b | 2025-06-14 11:08:12+00:00
The set of values of any physical quantity must have an algebraic structure that satisfies a set of axioms that include the axioms of the Archimedean group (which include the requirements that it must be possible to compare, add and subtract the values of that physical quantity).

This requirement is necessary to allow the definition of a division operation, which has as operands a pair of values of that physical quantity, and as result a scalar a.k.a. "real" number. This division operation, as you have noticed, is called "measurement" of that physical quantity. A value of some physical quantity, i.e. the dividend in the measurement operation, is specified by writing the quotient and the divisor of the measurement, e.g. in "6 inches", "6" is the quotient and "inch" is the divisor.

In principle, this kind of division operation, like any division, could have its digit-generating steps executed infinitely, producing an approximation as close as desired for the value of the measured quantity, which is supposed to be an arbitrary scalar, a.k.a. "real" number. Halting the division after a finite number of steps will produce a rational number.

In practice, as you have described, the desire to execute the division in a finite time is not the only thing that limits the precision of the measured values, but there are many more constraints, caused by the noise that could need longer and longer times to be filtered, by external influences that become harder and harder to be suppressed or accounted for, by ever greater cost of the components of the measurement apparatus, by the growing energy required to perform the measurement, and so on.

Nevertheless, despite the fact that the results of all practical measurements are rational numbers of low precision, normally representable as FP32, with only measurements done in a few laboratories around the world, which use extremely expensive equipment, requiring an FP64 or an extended precision representation, it is still preferable to model the set of scalars using the traditional axioms of the continuous straight line, i.e. of the "real" numbers.

The reason is that this mathematical model of a continuous set is actually much simpler than attempting to model the sets of values of physical quantities as discrete sets. An obvious reason why the continuous model is simpler is that you cannot find discretization steps that are good both for the side and for the diagonal of a square, which has stopped the attempts of the Ancient Greeks to describe all quantities as discrete. Already Aristotle was making a clear distinction between discrete quantities and continuous quantities. Working around the Ancient Greek paradox requires lack of isotropy of the space, i.e. discretization also of the angles, which brings a lot of complications, e.g. things like rigid squares or circles cannot exist.

The base continuous dynamical quantities are the space and time, together with a third quantity, which today is really the electric voltage (because of the convenient existence of the Josephson voltage-frequency converters), even if the documents of the International System of Units are written in an obfuscated way that hides this, in an attempt to preserve the illusion that the mass might be a base quantity, like in the older systems of units.

In any theory where some physical quantities that are now modeled as continuous, were modeled as discrete instead, the space and time would also be discrete. There have been many attempts to model the space-time as a discrete lattice, but none of them has produced any useful result. Unless something revolutionary will be discovered, all such attempts appear to be just a big waste of time.