An Algebraic Approach to Unital Quantities and their Measurement

Abstract

The goals of this paper fall into two closely related areas. First, we develop a formal framework for deterministic unital quantities in which measurement unitization is understood to be a built-in feature of quantities rather than a mere annotation of their numerical values with convenient units. We introduce this idea within the setting of certain ordered semigroups of physical-geometric states of classical physical systems. States are assumed to serve as truth makers of metrological statements about quantity values. A unital quantity is presented as an isomorphism from the target system's ordered semigroup of states to that of positive reals. This framework allows us to include various derived and variable quantities, encountered in engineering and the natural sciences. For illustration and ease of presentation, we use the classical notions of length, time, electric current and mean velocity as primordial examples. The most important application of the resulting unital quantity calculus is in dimensional analysis. Second, in evaluating measurement uncertainty due to the analog-to-digital conversion of the measured quantity's value into its measuring instrument's pointer quantity value, we employ an ordered semigroup framework of pointer states. Pointer states encode the measuring instrument's indiscernibility relation, manifested by not being able to distinguish the measured system's topologically proximal states. Once again, we focus mainly on the measurement of length and electric current quantities as our motivating examples. Our approach to quantities and their measurement is strictly state-based and algebraic in flavor, rather than that of a representationalist-style structure-preserving numerical assignment.

Torsor Theory of Physical Quantities and their Measurement

“…In simple idealized deterministic situations, a single geometric measurement reading from the instrument includes the available significant digits and one estimated digit, given by a tolerance interval or round-off to the nearest significant digit. 4 With regard to validating the deterministic length-theoretic metrological proposition "Length(f) = 5.3470 meters," if the measurement is made with a perfectly calibrated metric ruler that has a 1 millimeter precision, specified by uniform least count marks spaced 1 millimeter apart, then the measurement statement about the flagpole's length in a single measurement operation acquires a new form…”

“…of unital electric current quantities over the automorphism group Aut R >0 of scale transformations. The structure of state spaces for mass and electric current is discussed in [4]. In this way we obtain the required torsor-theoretic background for the MKSA system of quantities.…”

“…However, there will be other examples as we proceed 3. We revisit the ontology and role of real numbers in quantity calculus and measurement below 4. Note that here the choice of the 1 meter measurement unit precedes the consideration of deterministic uncertainties expressed in millimeter or other measurement subunits.…”