# Search Results

## The Analytic Versus Representational Theory of Measurement: A Philosophy of Science Perspective: (Invited Article)

In this paper we motivate and develop the analytic theory of measurement, in which autonomously specified algebras of quantities (together with the resources of mathematical analysis) are used as a unified mathematical framework for modeling (a) the time-dependent behavior of natural systems, (b) interactions between natural systems and measuring instruments, (c) error and uncertainty in measurement, and (d) the formal propositional language for describing and reasoning about measurement results. We also discuss how a celebrated theorem in analysis, known as *Gelfand representation*, guarantees that autonomously specified algebras of quantities can be interpreted as algebras of observables on a suitable state space. Such an interpretation is then used to support (i) a realist conception of quantities as objective characteristics of natural systems, and (ii) a realist conception of measurement results (evaluations of quantities) as determined by and descriptive of the states of a target natural system. As a way of motivating the analytic approach to measurement, we begin with a discussion of some serious philosophical and theoretical problems facing the well-known representational theory of measurement. We then explain why we consider the analytic approach, which avoids all these problems, to be far more attractive on both philosophical and theoretical grounds.

## Measurement, Information Channels, and Discretization: Exploring the Links

The goal of this paper is to present a unified algebraic-analytic framework for (static and dynamic) deterministic measurement theory, which we find to be fully adequate in engineering and natural science applications. The starting point of this paradigm is the notion of a *quantity algebra* of a measured system and that of a measuring instrument, underlying the causal linkages in classical *‘system + instrument’* interactions. This approach is then further enriched by providing a superimposed *data lattice* of measurement outcomes, intended to handle the information flow from the measured system to its measurand's designated instrument.

We argue that the language of *Banach* and *von Neumann algebras* is ideally suited for the treatment of quantities, encountered in theoretical and experimental science. These algebras and convex spaces of *expectation functionals* thereon together with *information (co)channels* between them provide a comprehensive information-theoretic framework for measurement theory. Concrete examples and applications to length and position measurements are also discussed and rigorously framed within the proposed quantity algebra and associated information channel paradigms.

In modeling physical systems, investigators routinely rely on the assumption that state spaces and time domains form a *continuum* (locally homeomorphic to the real line or its Cartesian powers). But in sharp contrast, measurement and prediction outcomes pertaining to physical systems under consideration tend to be presented in terms of small *discrete* sets of rational numbers. We investigate this conceptual gap between theoretical and finitary data models from the perspectives of temporal, spatial and algebraic *discretization* schemes.

The principal innovation in our approach to classical measurement theory is the representation of interactive instrument-based measurement processes in terms of channel-cochannel pairs constructed between dynamical quantity algebras of a target system and its measurand's measuring instrument.

## Abstract

There are several methods of automotive diagnostics used in services to detect a large variety of faults and damages of various parts of engines of internal combustion. Undoubtedly, they are effective, but they are simply unable to find all types of mechanical faults occurring during the operation. This is the reason why authors of this paper tried to use a special tool, which has been proven for years for detecting faults of rolling element bearing in rotating machinery. During their research, the authors tried to find valuable results by measuring vibration of various parts of engines. Three items were tested, a Diesel engine and two Otto motors. A large number of measurements have been taken at various speed, at different points, in different directions, with different parameter setup, etc. However, there was one setup which has been applied to all three engines. It is the measurement setup of vibration velocity, in the frequency range of 2 Hz-300 Hz. Valuable consequences have been found regarding the clogging of the air filters and the exhaust systems. As a conclusion the authors expressed their opinion, that, apart from the traditional diagnostic methods used in services, vibration measurements can also be useful, especially for detecting faults of rolling element bearings.

The goals of this paper fall into three related areas: (i) we present an overview of a universal algebraic paradigm in which measurement specialists can construct formal models of measurement in a unified manner and systematically reason about a large class classical measurement operations, (ii) we construct convenient von Neumann quantity algebras and quantity-channels between them to represent measurements, and introduce the dual framework of state spaces and state-channels between them to investigate the statistical structure of measurements, and (iii) we provide several detailed examples that illustrate the power and versatility of algebraic approaches to measurement procedures.

## Abstract

The principal objective of this paper is to provide a torsor theory of physical quantities and basic operations thereon. Torsors are introduced in a bottom-up fashion as actions of scale transformation groups on spaces of unitized quantities. In contrast, the shortcomings of other accounts of quantities that proceed in a top-down axiomatic manner are also discussed. In this paper, quantities are presented as dual counterparts of physical states. States serve as truth-makers of metrological statements about quantity values and are crucial in specifying alternative measurement units for base quantities. For illustration and ease of presentation, the classical notions of length, time, and instantaneous velocity are used as primordial examples. It is shown how torsors provide an effective description of the structure of quantities, systems of quantities, and transformations between them. Using the torsor framework, time-dependent quantities and their unitized derivatives are also investigated. Lastly, the torsor apparatus is applied to deterministic measurement of quantities.

## 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.