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.
If the inline PDF is not rendering correctly, you can download the PDF file here.
 Batitsky, V., Domotor, Z. (2007). When good theories make bad predictions. Synthese, 157, 79–103.
 de Boer, J. (1994/95). On the history of quantity calculus and the international system. Metrologia 31 (6), 405–429.
 Domotor, Z. (2008). Structure and indeterminacy in dynamical systems. In Indeterminacy: The Mapped, the Navigable, and the Uncharted. MIT Press, 171–193.
 Domotor, Z., Batitsky, V. (2010). An algebraic-analytic framework for measurement theory. Measurement, 43 (9), 1142–1164.
 Domotor, Z. (2012). Algebraic frameworks for measurement in the natural sciences. Measurement Science Review, 12, 213–233.
 Finkelstein, L. (2003). Widely, strongly, and weakly defined measurement. Measurement, 34, 39–48.
 Griesel, H. (1969). Algebra und Analysis der Größensysteme. Mathematisch-Physikalische Semesterberichte, 16 (1), 56–93 + 189–224.
 Hölder, O. (1901). Die Axiome der Quantität und die Lehre vom Mass. Berichte über die Verhandlungen der Königlichen Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematische-Physische Klasse, 53, 1–64.
 International Standards Organization. (1992). Quantities and Units – Part 0: General Principles. ISO 31-0:1992.