Measurability

Abstract

We discuss measurability, intended as the fundamental problem of assessing what can be measured.

We review some of the main ideas that have been historically proposed on this subject, including the positions of Helmoltz, Campbell and Stevens, the representational viewpoint and some criticism of it.

Then we propose an approach that attempts to combine scientific rigour with openness to innovation and we discuss its applicability to both fundamental and derived quantities, in virtually any scientific domain.

Such an approach may be particularly suited for addressing the measurement of quantities related to human perception.

Introduction

What can be measured? This is a key question in measurement science and it is closely related to another fundamental one: what really is measurement or, in other words, what is a good definition of measurement?

Increasing the number of measurable characteristics was a part of Galileo’s scientific programme for modern science, according to the motto: “count what is countable, measure what is measurable, and what is not measurable make measurable” [19]. Interestingly enough, two centuries later, Fechner (1801–87), made a similar statement for psychophysics, writing: “as an exact science psychophysics, like physics, must rest on experience and the mathematical connection of those empirical facts that demand a measure of what is experienced or, when such a measure is not available, a search for it” [9].

As noted by Finkelstein [25], “the true foundations of the modern theory of measurement were laid by Helmoltz”, in a paper published in 1887: in this work [1], as well as in a later and similarly important one, by Campbell [2], the problem of measurability was posed explicitly.

In the first half of the 20th century the question was considered so important that the British Association for the Advancement of Science appointed a Committee composed of physicists and psychologists, to consider and report upon the possibility of providing quantitative estimates of sensory events. The final report of the Committee (1939, cf. [4]) showed the impossibility of reconciling the positions of the two parts towards a common understanding of measurement. The physicists, in particular, took a strong stance against the possibility of actually making measurements in the behavioural sciences. This led to an essentially parallel development of measurement science in physical science on one side and in behavioural sciences on the other, with consequences up to the present days.

Yet the problem of quantities expressing the human response to some external stimuli is of great interest for metrology. This is reflected, for example, in the inclusion, among the base units of the International System (SI), of the candela, which quantifies the response of the human eye to light and is therefore a measurement of a biological response to an optical radiation [17], [26]. More generally, there is an interest in quantities related to human perception, since they are suited to describing quantitatively the interaction of persons with their living or working environment. This kind of study is also collectively referred to as soft metrology, which may be defined as “measurement techniques and models which enable the objective quantification of properties which are determined by human perception”, where “the human response may be in any of the five senses: sight, smell, sound, taste and touch” [24]. An example of the interest in these themes is provided by some recent Calls of the European Community, entitled “Measuring the impossible”, as a part of the New and emerging science and technology (NEST) programme [32]. As a rationale for the Calls, it is observed that “many phenomena of significant interest to contemporary science are intrinsically multidimensional and multi-disciplinary, with strong cross-over between physical, biological and social sciences”. Moreover, “products and services appeal to the consumers according to parameters of quality, beauty, comfort, etc., which are mediated by human perception” and also “public authorities, and quasi public bodies such as hospitals provide citizens with support and services whose performance is measured according to parameters of life quality, security or well being”. We may thus say that new needs suggest a reconsideration of the problem of measurability, taking into account new findings in the specific disciplines involved (physical metrology, measurement theory, experimental and cognitive psychology, neurophysiology, to name but some).

So in this paper we first present a review and a discussion of the main ideas that have been historically proposed on measurability, including the positions of Helmoltz and Campbell, the Report of the British Association, the position of Stevens, the representational viewpoint and some criticism of it. Then we propose an approach that attempt to combine scientific rigour with openness to novelty, trying to provide an answer to some of the main questions that have been historically raised in this regard. Such an approach may be particularly suited for addressing the measurement of quantities related to human perception and for providing a foundation to soft metrology.

Prior to entering into the discussion, let us consider a simple and classical introductory example, the scale for the hardness of minerals, proposed by Mohs (1773–1839) in 1812. The key idea of his method was that the hardness of a mineral may be characterised by its ability to scratch other minerals. He identified a series of 10 reference materials, collectively suited to express the different degrees of hardness we may encounter in nature and he assigned the numbers from 1 to 10 to them, so fully defining a reference scale. We will call a standard each element in the scale and measure the corresponding number. On the basis of it, it is possible to determine the hardness of any piece of mineral a by comparing it to the scale in order to identify the standard to which it is equivalent, in the sense that it neither scratches nor is scratched by it.

Consider now some questions.

Is the technique just described really a measurement, since it is different to some extent – rougher we would say – from other types of more familiar measurements, such as mass, length or electrical resistance, to name some? More specifically, in what way is it similar to other, more traditional, measurements and in what way does it differ? This simple example leads us to the problem of measurability that we will treat in the following sections.

Before proceeding, let us introduce some language.

What we measure is called an attribute or characteristic or property of an object (or event or body or system). We prefer the terms characteristic and object, respectively. A measurable characteristic is called quantity or magnitude – we prefer the former. A characteristic for which measurability has not yet been assessed is sometimes called a quality. We will simply call it characteristic, since the term quality is also used in contrast to quantity to denote a characteristic which is purely ordinal.

What we use for expressing the result of a measurement is considered to be a number, a numeral or a symbol. We prefer to consider it a number but we will also call it a measure, as we have just done with the Mohs scale. Despite our preferences, we will occasionally use other terms, especially when quoting other authors, in order to maintain their language. Definitions and notation conventions are summarised in Appendix 1.

We are now ready to start a brief excursus in the history of measurement science.