The nature of measurement

Some theoreticians define ‘measurement’ as the assignment of numbers to prop­erties in such a way that the traits of the property are adequately represented by mathematical entities.

when measuring some attribute of a class of objects or events, we associate numbers (or other familiar mathematical entities, such as vectors) with the objects in such a way that the properties of the attribute are faithfully repre­sented as numerical properties.

Another definition, that agrees with the former one, is due to Finkelstein:

Measurement is the process of empirical, objective assignment of numbers to properties of objects or events of the real world in such a way as to describe them.

(Finkelstein 1982: 6)

Nevertheless, in economics, it is sometimes necessary to measure properties of idealized objects, and so I would rather prefer to characterize measurement as the representation of an empirical or idealized property by means of a mathematical object belonging to some sort of mathematical system. Numbers, vectors, line segments, and so on can be and have been used for the purpose of measuring some object or magnitude. Actually, Eudoxo’s theory of proportions represented irrational numbers by means of ratios of geometric segments. The methodology to inverse the procedure and represent ratios of segments by means of real numbers is intimated by Newton in his Arithmetica universalis (1761). But it is due to Holder (1901) the rigorous axiomatization of the classical concept of magnitude in such a way that ratios of magnitudes (of which segments are an instance) could be expressed as positive real numbers.²

Historically, the first example of a numerical representation was analytic geom­etry, “which provides coordinate-vector representations for qualitative geometri­cal structures formulated in terms of such primitives as points, lines, comparative distances, and angles” (klst 1989: 1).

a representation theorem shows how to embed a qualitative structure isomor­phically into some family of numerical structures, and the corresponding uniqueness theorem describes the different ways that the embedding is pos­sible.

(Ibid.: 2)

Naturally, not any assignment of numbers to properties counts as measurement. The existence of a numerical representation requires that the properties satisfy certain conditions. The aim of the theory known as ‘representational theory of measurement’ (rtm) is to establish necessary and jointly sufficient conditions guaranteeing the existence and uniqueness (up to a certain class of transforma­tions) of a numerical representation for a wide variety of possible measurement situations. These situations are represented by means of set-theoretical structures, and the conditions are formulated as axioms that these structures should satisfy in order to ensure the existence of the numerical representation. When the set-the­oretical structures are entirely qualitative, in the sense that they do not contain results of any previous measurement (numbers), the resulting representation is called ‘fundamental measurement’. Sometimes the representation is proven over structures that are defined in terms of numbers or other mathematical enti­ties, as when the existence of utility functions is proven over sets of vectors intended to represent consumption menus. This one is a case of representational measurement which is not fundamental.

Some philosophers, like Brian Ellis (1966), believe that measurement is the assignment of numbers to quantities. He describes a quantity as a property with respect to which things can be compared in some way. And he clarifies:

two things A and B are comparable in respect of a given quantity q if and only if one of the following relationships connects them:

(i) A is greater in q than B (that is A >_q B);

(ii) A is equal in q to B (that is A =_q B);

(iii) A is less in q than B (that is A b 2 A; in other words, if (A, ≥) is a weak order.

Sydenham distinguishes a property from its “manifestations” and defines

^q = ^fq1^, q2^, ■ ■ ■^, q_i^, ∙ ∙ ∙}

as the set of all its manifestations,

Ω = ^fw1 ^{, w}2^, ■ ■ ■ ^, w_i, ■■ ∙^g

as the class of all objects manifesting elements of Q, and

R = {^r1, ^r2, ■ ■ ■, R, ■ ■ ■, ^rN}

as a set of empirical relations over Q. In order to provide a formal definition of measurement, Sydenham introduces class N of numbers, a set of relations

^P = ^{P1, ^P2, ' ' ', ^Pi, ' ' ', ^PN^},

and defines measurement as an “objective empirical operation” composed of two functions, M:Q → N and F: R → P, such that P_i = F(R_i) and

^Ri(of catalogue of mathematical theorems.

Actually, some have even come to say that the set-theoretical structures rtm makes use of, far from representing empirical instrumentalist measurement pro­cedures, are not even about the ‘natural world'. Michell (2007: 36), for one, writes that rtm

is based upon an inconsistent triad: first, there is the idea that mathematical structures, including numerical ones, are about abstract entities and not about the natural world; second, there is the idea that representation requires at least a partial identity of structure between the system represented and the system representing it; and third, there is the idea that measurement is the numerical representation of natural systems.

As I said above, by ‘metrization of property P' I understand a demonstration, out of empirically meaningful conditions, of the fact that P is measurable. rtm accomplishes the metrization of P by building suitable set-theoretical structures and proving the existence of a certain homomorphism, which is a mathematical representation of P. By ‘measuring procedure’ I understand a procedure for actu­ally finding the value of P for given objects, with respect to a unit of reference. It seems to me that RTM does not claim that every measuring procedure consists of explicitly constructing some homomorphism. What it does claim is that any (putative) magnitude P must satisfy certain conditions in order for it to be metriz- able (Diez 2000: 20, n. 5). The task of the theory of fundamental metrization is to probe into the conditions that P must satisfy in order to guarantee the existence of such representation when the same does not presuppose any other previous metri- zations. The fundamental measuring procedures determine specific empirical pro­cedures for qualitative comparison of the specific property involved, and chooses a standard with which the assignment begins.

It is certain that rtm has never maintained the claim that “mathematical struc­tures, including numerical ones, are about abstract entities and not about the natural world”. Clearly, mathematical structures are abstract entities, insofar as they are set-theoretical structures. But I have explained in Chapter 4 how math­ematical structures relate to real-concrete objects and empirical structures.

It is important at this point to stress that the RTM does not claim that every actual measurement procedure has to be performed by means of the proof of the existence of a homomorphism from a measurement structure into a numerical one.

5.2