<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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.</p></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.

An Algebraic Approach to Unital Quantities and their Measurement

Departments of Philosophy, Biochemistry and Biophysics, University of Pennsylvania, 433 Cohen Hall, 249 South 36 Street, Philadelphia, PA 19104-6304, USA

Department of Philosophy, St John’s University, 800 Utopia Parkway, Queens, NY 11439, USA

Measurement Science Review

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"An Algebraic Approach to Unital Quantities and their Measurement"}

The goals of this paper fall into two closely related areas. First, we develop a formal framework for deterministic unital quantities in which measurement...

An Algebraic Approach to Unital Quantities and their Measurement

Published Online: Jun 11, 2016

Page range: 103 - 126

Received: Mar 06, 2016

Accepted: May 26, 2016

DOI: https://doi.org/10.1515/msr-2016-0014

Keywords
Deterministic measurement, measurement unit, measurement uncertainty, pointer quantity, pointer state, quantity calculus, quantization, state space, unital quantity

© 2016 Zoltan Domotor et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

An Algebraic Approach to Unital Quantities and their Measurement

Published Online: Jun 11, 2016

Page range: 103 - 126

Received: Mar 06, 2016

Accepted: May 26, 2016

DOI: https://doi.org/10.1515/msr-2016-0014

KeywordsDeterministic measurement, measurement unit, measurement uncertainty, pointer quantity, pointer state, quantity calculus, quantization, state space, unital quantity

© 2016 Zoltan Domotor et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Deterministic measurement, measurement unit, measurement uncertainty, pointer quantity, pointer state, quantity calculus, quantization, state space, unital quantity