Measurement is one of the most important elements of modern science. However, there is a great deal of controversy concerning how to define measurement. What is measurable? What can only be understood in terms of hermeneutic or phenomenological inquiry? Most philosophers agree that “measurement is an activity that involves interaction with a concrete system with the aim of representing aspects of that system in abstract terms (e.g., in terms of classes, numbers, vectors etc.)”, but the consensus more or less ends here. Indeed, such a description can be predicated of linguistic and perceptual activities which would not normally be considered examples of “measurement,” so even this basic definition is controversial. To even describe measurement as “concrete” in this way is questionable, since constructs such as the “average household” or an electron at complete rest are not things that we have right in front of us.

There are five primary accounts of measuremenet:

1) Operationalism and conventionalism – “measurement [is] a set of operations that shape the meaning and/or regulate the use of a quantity term.”

2) Mathematical theories of measurement – measurement is “the mapping of qualitative empirical relations to relations among numbers (or other mathematicial entities.”

3) Information-theoretic accounts – measurement is “the gathering and interpretation of information about a system.”

4) Realism – “measurement [is] the estimation of mind-independent properties and/or relations.”

5) Model-based accounts – measurement is “the coherent assignment of values to parameters in a theoretical and/or statistical model of a process.”

These accounts do not necessarily contradict with one another. One may believe that more than one account is correct or accurate.

“While mathematical theories of measurement deal with the mathematical foundations of measurement scales, operationalism and conventionalism are primarily concerned with the semantics of quantity terms, realism is concerned with the metaphysical status of measurable quantities, and information-theoretic and model-based accounts are concerned with the epistemological aspects of measuring.”

Different accounts of measurement are linked with other philosophical positions. Conventionalists and operationalists, for example, tend to have anti-realist views of measurement. Model-based accounts tend to contradict certain interpretations of the mathematical account of measurements as well.

One of the earliest accounts of magnitude comes from Euclid:

“According to Euclid’s Elements, a magnitude—such as a line, a surface or a solid—measures another when the latter is a whole multiple of the former (Book V, def. 1 & 2). Two magnitudes have a common measure when they are both whole multiples of some magnitude, and are incommensurable otherwise (Book X, def. 1). The discovery of incommensurable magnitudes allowed Euclid and his contemporaries to develop the notion of a ratio of magnitudes. Ratios can be either rational or irrational, and therefore the concept of ratio is more general than that of measure (Michell 2003, 2004; Grattan-Guinness 1996).”

Another ancient thinker, Aristotle, distinguished between quantitites and qualities. Qualities could not be measured could not admit of either inequality or equality, but differed by degrees, whereas the reverse is true of quantities. According to Duns Scotus, changes in quality can be understood in terms of either the subtraction or addition of smaller degrees of this quality, The theory was

“later refined by Nicole Oresme, who used geometrical figures to represent changes in the intensity of qualities such as velocity (Clagett 1968; Sylla 1971). Oresme’s geometrical representations established a subset of qualities that were amenable to quantitative treatment, thereby challenging the strict Aristotelian dichotomy between quantities and qualities.”

According to Leibniz’s principle of continuity, all change in nature, without exception, is produced by degrees. He believed that this was true of both extended magnitudes such as duration and length, as well as experiences, such as the hearing of sounds. Kant likewise articulated a “Distinction betweene xtensive and intensive magnitudes,” according to which magnitudes “are those “in which the representation of the parts makes possible the representation of the whole”; for example, a line can only be represented by “a sucessive synthesis in which parts of the line oin to form the whole.” Kant believed that this synthesis resulted from the intuitions of space and time. Intensive magnitudes, on the other hand, such as color or warmth, exist in continuous degrees, and they are perceived “in an instant rather than through a successive synthesis of parts.” “The degrees of intensive magnitudes “can only be represented through approximation to negation” (1787: A 168/B210), that is, by imagining their gradual diminution until their complete absence.”

The distinction between intensive and extensive magnitudes were challenged throughout the 19th century.

“Thermodynamics and wave optics showed that differences in temperature and hue corresponded to differences in spatio-temporal magnitudes such as velocity and wavelength. Electrical magnitudes such as resistance and conductance were shown to be capable of addition and division despite not being extensive in the Kantian sense, i.e., not synthesized from spatial or temporal parts. Moreover, early experiments in psychophysics suggested that intensities of sensation such as brightness and loudness could be represented as sums of “just noticeable differences” among stimuli, and could therefore be thought of as composed of parts (see Section 3.3). These findings, along with advances in the axiomatization of branches of mathematics, motivated some of the leading scientists of the late nineteenth century to attempt to clarify the mathematical foundations of measurement (Maxwell 1873; von Kries 1882; Helmholtz 1887; Mach 1896; Poincaré 1898; Hölder 1901; for historical surveys see Darrigol 2003; Michell 1993, 2003; Cantù and Schlaudt 2013). These works are viewed today as precursors to the body of scholarship known as “measurement theory”.”

Tal, Eran, “Measurement in Science”, The Stanford Encyclopedia of Philosophy (Summer 2015 Edition), Edward N. Zalta (ed.), URL = <plato.stanford.edu/archives/sum2015/entries/measurement-science/>.