Measurement as representation and coordination

[This is a reflection on Johanna Wolff, “The Philosophical Significance of the Representational Theory of Measurement – RTM as Semantic Foundations” (2023).]

The outcome of a measurement is a representation of the item measured.  As an example take an echocardiogram: the monitor display is an image of the heart which shows how the blood flows through the heart and heart valves.  An image is a representation, but so is the mechanical display, in which a needle points to 17, on an old-fashioned tire pressure gauge. And so is the list of numbers a carpenter writes down to record his measurements before he initiates his repairs.

This simple point, though it suffices to subsume measurement as a topic under the heading of representation, is not nearly enough to answer questions about what measurement is, let alone to constitute a theory of measurement.

A foundational account?

As Wolff points out, when Krantz, Suppes, and Luce introduced the Representational Theory of Measurement (RTM) there is every sign that it is meant as providing a foundational account of measurement.  But what foundations does it provide; indeed, how are we to understand just what can be meant by foundations of measurement?

A look at what Krantz et al. do suggests an analogy to foundations of mathematics, as in Principia Mathematica.  But it also suggests something like a pun: their main achievement is a series of representation theorems, in the sense that term has in mathematics.  The representation theorem most familiar to philosophers is that every Boolean algebra can be represented as an algebra of sets.  “Can be represented as” amounts here to just “is isomorphic to”.  In other cases the representation meets a still lower bar. Suppose we rank players on a football team with the relation is at least as good as (a ‘weak ordering’)  Then we can represent this relationship numerically, with ≤ representing that relationship.  But two players may then have to be assigned the same number:  a  homomorphism rather than an isomorphism.  (Every finite weak ordering is homomorphic to a numerical weak ordering.)   

But is there something more to it?  If we have ranked the players, with that weak ordering, we have already measured them, our judgments of form “X is at least as good as Y” are our initial measurement results.  We can take the further step to represent the team numerically, changing our one-step measurement procedure into a two-step procedure.  The added second step is a ‘paper and pencil’ step. Any truly foundational question about measurement must pertain already to the initial ranking.

Wolff explains the predicament in this way:

Given a suitable axiomatisation of the empirical relational structure, RTM shows in a formally rigorous way that a numerical representation of that empirical structure is possible. […]  So, once the axiomatic characterisation of the empirical structure in question is in place, RTM provides us with solid, formal tools for finding suitable numerical representations for the relational structure in question. But here is the one-million-dollar question: what justifies a particular axiomatization of the empirical relational structure?  (p. 86)

Failure of operationalism

There is barely any guidance for this in Krantz et al., and what guidance there is consists mainly in nods in the direction of operationalism. One demand, they write, “is for the axioms to have a direct and easily understood meaning in terms of empirical operations” (Krantz vol. 1: 25).   One example: a collection of rods, with two procedures, ordering by length and concatenation.  If the former is executed repeatedly, until it results in no further change, then the collection becomes weakly ordered.  Read backward: the weak ordering postulated in the axioms has as  its meaning that if  a ≤ b then the ordering procedure would, if it were executed, place a below b.  

The second part of this guidance imposes a demand for descriptive adequacy.  That implies in this example that if a, b, c are submitted to the ordering procedure pairwise, the result will be such that if a is placed below b, and b is placed below c, then an independent such operation would place a below c.

Note well that these explanations are rife with counterfactuals.  The facts are only that certain rods have actually been shuffled into a relative place arrangement.  The axiom says something much more far-reaching, namely that the relation of being at least as long as is a weak ordering.

Meaning? There is certainly a connection between what the theoretical terms mean and the operations that count as measurement procedures.  But it is not the connection between definiendum and definiens.

Coordination, empirical grounding

Operationalists wanted to write the story as beginning with a game with rods which arranges and re-arranges them until equilibrium is reached, segueing into an axiomatic description of the result.  

We should read it backward: the game begins with an axiomatic ‘rational reconstruction’ of the intuitive notion of length, and 

that axiomatic description is ‘empirically grounded’ by specifying how certain procedures, which are executable under certain conditions, establish whether or not a ≤ b. 

Perhaps this is clear enough for such unrealistically simple examples about rods and balances.  The important further point is that this view of the relation between the resultant numerical structure of the measurement outcomes extends smoothly to more advanced theories and their theoretical quantities.

To take the theoreticity of the quantities involved just one step further, the above goes for the axioms of mechanics about mass and force, empirically grounded in the experimental procedures described by Ernst Mach as his “definition” of mass.

Mach introduced his approach in his 1868 article “Über die Definition der Masse”, and elaborated it in his 1883 book The Science of Mechanics.  The direct measurement of acceleration is taken as given, and Mach does not hesitate to use the term “definition” for the combination of two principles:

All those bodies are of equal mass which, mutually acting on each other, produce in each other equal and opposite accelerations.

If we take A as our unit, we assign to that body the mass m which imparts to A the acceleration that A in the reaction imparts to it. (1974 Dover print: 266)

This is not left abstract: Mach gives examples of simple machines that can perform the required experiments – or, as he later also said, measurements of mass.

It was Suppes who elsewhere insisted strongly on the point that Mach’s ‘definition’ of mass was not a definition, in our current sense of the term.  (The same complaint applies to Reichenbach’s notion of a coordinative definition.)  We have the criterion that a defined term is one that can everywhere be replaced without loss by its definiens.  

Mach’s principles enable no such innocuous rewriting of mechanics. The argument is simple: if a body were to be unaccelerated (throughout its existence), because the (total) force on it is zero, then the laws of mechanics tell us only that its mass multiplied by zero equals zero. So mass cannot be defined in terms of actual behavior.  Counterfactuals about behavior, on the other hand, we can only derive from the theory itself, and are not independently ascertainable.  

Nevertheless, it is those mechanical, implementable procedures, in which bodies are made to interact so as to induce mutual accelerations, that give empirical significance to the theory of masses and forces.  The term “mass” is not defined, the quantity is not derivative, but in the theory the role of mass is such that its value can determined, in principle, in specified realizable conditions (from data plus equations provided by the theory).

Some quantities are directly measurable: Krantz et al.’s examples, such as arrangement of rods by length, are thus.  In contrast, a theoretical quantity is one that can be measured only by procedures which the theory in question, itself, counts as measurements of that quantity.  Mass and force are cases in point.  

But those procedures, which can be implemented by us, and are therefore describable also in non-theoretical language, are precisely what anchors the theory to what the theory is about.  In the time of Schlick and the early Reichenbach, the philosophical term for this connection was “coordination”.  Currently the more favored is “theory-mediated measurement”.  I have spelled it out as “empirical grounding”.  Whatever the terminology, seeing matters in this light we are not prey to a naïve foundationalism, but also not bereft of links of our theories to the world we live in.

Wolff’s proposal: RTM as semantics of measurement

Since, as Wolff points out, Krantz et al. give only hints about the way the axiomatic characterization of the empirical structure gets in place, does their work have value except as a library of mathematical theorems?

Wolff argues that it does, and offers the original proposal that RTM provides a semantic foundations of measurement.

In the more usual narrow sense, the topic of semantics is the relation between a language and what that language is about.  But language is only our main means of communication, by representing things to each other:  we paint word pictures.  So in the broader sense, the topic of semantics is meaning überhaupt, about the relation between any representation and what it represents.  

“Representation” in the sense of representation theorems in mathematics refers to certain straightforward types of mappings: isomorphism, homomorphism ….  Focusing there, Krantz et al. say, somewhat myopically, that measurement is 

 Wolff quotes this, and points out that it is at least incomplete as a statement of what measurement is, by omitting any reference to the experimental or observational procedures that lead to a characterization of those empirical structures of interest.  It is incomplete also in omitting any attention to the theory-mediated character of all but direct measurement: that the pointer is at 17 signifies according to, or relative to, the theory that dictated the design of the manometer, that the tire’s pressure is 17psi. Outcomes of measurement are not simply physical events, they have meaning, in the way that words and pictures do.

the construction of homomorphisms (scales) from empirical structures of interest into numerical relational structures that are useful. (p. 9, quoted Wolff p. 96)

The achievement and value of RTM, with respect to the aspect of measurement on which it focuses, Wolff argues, is that it brings us a structuralist view which goes far beyond the naïve idea that measurement is an assignment of numbers, let alone the reading of a book of nature written in arithmetic. 

Take again the example of the soccer team weakly ordered by our judgments of how good the players are.  This can be represented numerically.  An assignment of numbers which reflects that ordering is in effect a homomorphism of the team, thus viewed, into the real number continuum.  And any such assignment of numbers is adequate to that task: the ‘scale’ is highly non-unique.  So such a measurement has outcomes that have a lot of surplus structure: to understand it properly we need to know what transformations any such adequate numerical representation remains adequate.  And what we know, if we know that, is the invariant structure common to all such adequate assignments of numbers: that is the real outcome!

My example is too simple to show how crucially this matters.  To see better, think of how temperature measurement outcomes are different from height measurement outcomes.  In the case of height, we can meaningfully say that Paul is twice as tall as his son Peter.  Whether we use inches or centimeters as our scale, the result is the same, for here the relevant transformations are linear.  But if we compare the temperatures of two cups of tea we cannot do the same:  the transformation of the Celsius or Kelvin scale to Fahrenheit will not agree on “twice as hot”.  For the relevant transformations of temperature scales, it is only the ratios of intervals that are invariant.

Elicitation of the invariant structure is the subject of the uniqueness theorems.  A representation theorem needs to be followed by a uniqueness theorem: the adequate numerical representation is unique up to transformations of type such and such.  

Wolff makes the point in strong terms: 

RTM shows us, what numerical presentations, and measurement representations in particular, tell us about the phenomena: they tell us something about the structure of phenomena, and nothing else. (Wolff 2023: 102)

The semantic foundations provided by RTM, then, are not primarily about reference … but rather about inference – how do we ensure our indirect reasoning about the phenomenon using a numerical representation is warranted. (ibid., 102/103)

If I report that today’s high and low temperatures are respectively 70^oF and 35^oF, how much information I have given you?  Not the non-invariant judgment that the high temperature is twice the low.  The point applies, in different ways, to non-numerically presented outcomes.  A photo of a dog is two-dimensional, flatter than a pancake, but the dog is not to be inferred to be anything like as flat as a pancake.  We understand a measurement outcome properly only if we infer from it only what we could infer from it after any of the relevant transformations: that is to say, only if we grasp the structure it reveals.  

REFERENCES

Krantz, David H., R. Duncan Luce, Patrick Suppes, and Amos Tversky (1971) Foundations of Measurement. New York: Academic Press.

Mach, Ernst. (1883/1974)  The Science of Mechanics: A Critical and Historical Account of its Development. New York: Dover.

Wolff, J. E. (2023) “The Philosophical Significance of the Representational Theory of Measurement – RTM as semantic foundations”. Critica 55: 81-107.