Separation Theorems in Econometrics and Psychometrics: Rasch,Frisch,Two Fishers and Implications for Measurement

Frisch’s Decisive and Direct Influence on Rasch

In a 1979 interview (Andrich, 1997, 2005) conducted a year before his death, Rasch recounted three important events in his life. First, in the Spring of 1935, he studied with Ragnar Frisch in Oslo and, second, he went to London to study with Ronald Fisher for the 1935–1936 academic year. In the interview, Rasch mentions Frisch’s confluence analysis and his occasional use of it, but stressed repeatedly the central importance that Fisher’s concept of the sufficient statistic had in the development of his approach to measurement.

The third key event occurred later, in 1959, when Rasch visited Frisch and outlined some of his recent work for him, including the development of the model for which Rasch is now well-known. As Rasch (1977, p. 63; also see Andersen & Olsen, 2001; Andrich, 1997, 2005; Olsen, 2003; Wright, 1980, pp. xvi–xviii) remembered it, his

understanding of what the model entails tarried several years. At the 1959 anniversary of the University of Copenhagen, the highly esteemed Norwegian economist Ragnar Frisch—later Nobel Prize winner—was to receive an honorary doctorate. I visited him by appointment the next day, and when our business was finished, he asked me what I had been doing in the 25 years since I stayed at his institute in Oslo for a couple of months to study a new technique of statistical analysis (confluence analysis) that he had developed.

Rasch then proceeds to recount for Frisch his development of a model for measuring students’ reading speeds, writing out a series of equations (II:1–II:7 in Rasch, 1977):

On seeing (II:7) Frisch opened his eyes widely and exclaimed: ‘It (the person parameter) was eliminated, that is most interesting!’ And this he repeated several times during our further conversation. To which I of course agreed every time—while I continued reporting the main results of the investigation and some of my other work.

Only some days later I all of a sudden realised what in my exposition had caused this reaction from Ragnar Frisch. And immediately I saw the importance of finding an answer to the following question: ‘Which class of probability models has the property in common with the Multiplicative Poisson Model, that one set of parameters can be eliminated by means of conditional probabilities while attention is concentrated on the other set, and vice versa?’

What Frisch’s astonishment had done was to point out to me that the possibility of separating two sets of parameters must be a fundamental property of a very important class of models.

Rasch only then, in light of the insight prompted by Frisch’s astonishment, generalised his results into a formal model supported by what he called a separability theorem and the concept of specific objectivity (Rasch, 1960, 1961, 1966a, 1966b, 1977). Building on the acquaintance established in 1947, Rasch contacted L. J. Savage at the University of Chicago about a visiting appointment. Savage asked Benjamin Wright, his neighbour and friend, a young assistant professor of education and psychology, to act as Rasch’s host. The future of Rasch’s ideas was to be extensively influenced by Wright.

Wright was profoundly affected by what Rasch had to say, eventually making several trips to Denmark to study with him (Andrich, 1995; Wright, 1988, 1996). Wright brought Rasch’s ideas to bear in measurement research and practice on a broad scale in education, psychology and healthcare. Wright was recognised by Rasch in a 1972 letter (Rasch, 1988) as the foremost representative of his work, having ‘practiced an almost unbelievable activity in this field’ in the period from Rasch’s 1960 visit to Chicago, through Wright’s visits to Rasch in Denmark in 1964, 1965, 1967 and 1970, when Rasch was a visiting professor at Chicago in the 1967–1968 academic year, and in the week-long, first-ever software training workshop given in conjunction with an American Educational Research Association meeting, held in 1969 in Los Angeles, California.

Wright (2009/1993) offers further documentation of the work he did in this period in a 1993 letter to Eddie Roskam. Wright took up many challenges that Rasch himself did not. Wright’s energetic dissemination of Rasch’s measurement ideas combined with the work of his students and many others to create a global movement in research methods that continues to grow to this day. It is significant that Rasch emphasises an analogy from Maxwell’s treatment of Newton’s second law in the formulation of his model, and that Wright’s background was in physics (Linacre, 1998a; Wright, 1988). Wright had worked with Nobel laureates Mulliken and Townes, absorbing methodological principles and research values from them. He used yardsticks and rulers throughout his career as simple and intuitive illustrations of the principles of structural invariance embodied in Rasch models.

Rasch and Wright independently insisted on thinking for themselves and building models from the ground up (Wright, 1988) on their own terms. Andrich (2016) documents their struggle to formulate a model of measurement for ordered categorical data illustrated by rating scales that would satisfy the demand for sufficient statistics. As acknowledged by Wright (1988), Andrich (1978, 2010) resolved the struggle by taking a somewhat different approach but building directly on Rasch (1961) and Andersen (1977a) to set the form of the model now used routinely in applications. The effects of these efforts on measurement practice globally (Wilson & Fisher, 2017) can be understood only in the context of previously unacknowledged influences. Pursuit of the questions that arise from Frisch’s astonishment at the ‘disappearing parameter’ and Rasch’s response to it, lead in some interesting directions.

Frisch’s Decisive and Indirect Social Influence on Rasch

An obvious question that arises has to do with why Rasch contacted Savage in Chicago as soon as he realised the importance of the disappearing parameter. How did it happen that Savage was inclined to extend an invitation to him? What prior history did they share that would have given Rasch such credibility with Savage? Wright (1988), Andrich’s (1979) interview with Rasch and the Cowles Commission Report for 1947 (Cowles Commission for Research in Economics, 1947; Linacre, 1998b) provide some answers, as they document Rasch’s social connections and the role Rasch played in the Commission’s investigations into economic problems.

Rasch is identified twice in the 1947 Cowles report as a Fellow of the University of Chicago. He gave two papers late in the year that served as the starting points for staff discussions (Rasch, 1947a, 1947b). One was entitled, ‘A Biometric Multidimensional Model’, and the other, ‘Remarks on Estimation’.

On 20 November 1947, Rasch gave a Cowles seminar in Chicago under the heading, ‘Statistical Analysis of Growth Curves’ (Rasch, 1947d). His was the last of the year. It was preceded by seminars given by later Nobel Laureates Herbert Simon (1978), Milton Friedman (1976) and Ragnar Frisch (1969) (see https://cowles.yale.edu/commission-seminars).

Others giving seminars in the same series that year included Rensis Likert, Sewall Wright and Harold Hotelling. The papers given by Friedman and Rasch were presented at joint meetings of the Cowles Commission and the Statistical Techniques Group of the Chicago Chapter of the American Statistical Association, which had Kenneth Arrow, a 1972 Nobelist, as its chairman. L. J. Savage gave a paper at the Cowles Commission in 1947, as well. He spoke about estimating parameters for a continuous stochastic process, a topic closely related to the work Rasch would come to present at Chicago in 1960.

This would seem, then, to be when Rasch and Savage came to be acquainted. But how did Rasch come to be at the Cowles Commission in the first place? It would seem natural that his presence there would have come about through his association with Frisch. And indeed, as is summarised by Wright (1980, pp. xii–xiii) from Andrich’s (1979) interview with Rasch,

… a friendship with Chester Bliss made in London in 1935 brought Rasch to the United States in 1947 to participate in the founding of the Biometrics Society (Rasch, 1947c) and in the postwar reorganisation of the International Statistical Institute in Washington. In the course of these meetings Tjalling Koopmans, a fellow student of Ragnar Frisch’s confluence analysis and Fisher’s sufficient statistics, invited Rasch to spend two months with the Cowles Commission on Economic Research at the University of Chicago, where he met L. J. Savage. This set the stage for Savage to bring Rasch back to Chicago in 1960, to finish writing Probabilistic Models and to give the series of lectures that introduced this writer [Wright] to Rasch’s new psychometrics.

Koopmans had also studied with Fisher in London, and Rasch had attended Koopmans’ lectures while studying with Frisch in Norway in early 1935 (Bjerkholt, 2001, p. 9). Then, while in Chicago in 1947, Rasch collaborated with Koopmans (1975 Nobelist) and another of Frisch’s students, a Norwegian Rockefeller Fellow, Olav Reiersøl, in

a comparative study of problems arising from the specification of models, in particular problems of identification, in three fields: A multivariate model designed for the study of human metabolism; a model designed for the analysis of mental aptitudes; and an econometric model of the type employed by the Cowles Commission in its economic research. The similarities and differences encountered help to understand the nature of scientific induction (Cowles Commission for Research in Economics, 1947, p. 6).

In a 1979 letter to Wright, Rasch (1979) recounts other work he did in 1947 in Denmark that resulted in distributions independent of demographic criteria. This work, combined with Rasch’s studies under Frisch in Oslo and Ronald Fisher in London 12 years earlier, was vitally important preparation for the development of Rasch’s models for measurement.

In sum, Rasch’s presence in Chicago in 1947 was in large part due to his relationships with Frisch and Ronald Fisher. Rasch’s 1960 invitation from Savage stemmed from the acquaintance they formed when they were both working at the Cowles Commission and that they cultivated through their participation in Biometric Society meetings. It was at one of these meetings, in 1959, that Rasch impressed Savage with his need to communicate his measurement work in Chicago (Andrich, 1995, p. 1).

Having established the network of social connections leading to Rasch’s presence in Chicago, what insight into the development of the intellectual substance of his work can be gleaned?

Frisch’s Decisive and Indirect Intellectual Influence on Rasch

By Rasch’s own account (Rasch, 1979), he did not follow any kind of logical, smooth, simple or direct process in formulating the models for measurement. Andrich (2016) recounts the course of that struggle in relation to the sufficiency of rating scale model parameter estimates as this develops in letters between Rasch and Wright. In one of these letters (Rasch, 1979), Rasch urged Wright to ‘consider how erratically the human mind may work!’ Rasch (in Andrich, 1979, 1997; Rasch, 1960, p. 21; Rasch, 1977; Wright, 1980) repeatedly and strongly emphasises the vital importance of Ronald Fisher’s concept of the sufficient statistic, recast by Rasch from the group-level mean and standard deviation to the individual-level score, as an invaluable factor in the work informing his models, separability theorem and epistemological focus on specific objectivity. As noted by van der Linden (1992), Rasch makes only infrequent mention (Rasch, 1977, 1968, p. 21), on the other hand, of the Darmois–Koopman–Pittman type of exponential distributions that support statistical sufficiency (Barndorff-Nielsen, 1978; Fraser, 1963; Koopman, 1936), though this was developed at length by his student, Andersen (1970, 1977b, 1999).

But further insight into Rasch’s models, and a feeling for some of the issues at stake in economics at the time Rasch was with the Cowles Commission, emerges in a review of Frisch’s work and influences (Arrow, 1960; Edvardsen, 1970, 2001). Frisch was the editor of Econometrica for over 20 years, 1933–1954, and coined many words now taken for granted, such as ‘macroeconomics’, ‘econometrics’ and others. Frisch was heavily influenced by the work of the Yale economist, Irving Fisher, one of the most important economists of the first half of the twentieth century (Dimand, 1997; Dimand & Geanakaplos, 2005; Fisher, 2005; Frisch, 1947; Tobin, 2005). Frisch and I. Fisher were among the cofounders of both the Cowles Commission and the Econometric Society (of which I. Fisher was the first President).

One of I. Fisher’s primary areas of interest was in the mathematical structure of value and the indices used to measure it (Fisher, 1921, 1930, 1967/1922). Frisch (1930) was among the first and most effective to engage constructively with I. Fisher on these issues, basing his work in his own highly original axiomatic formulation of measurable utility (Frisch, 1926), which ‘introduced the axiomatic approach into the theory of economic choice’ (Boumans, 2001a, p. 333, 2005; Chipman et al., 1971, p. 326). So, ‘while Fisher’s approach was to select among a number of index formulae by testing them, Frisch’s method was to derive mathematically the appropriate form of these tests’ (Boumans, 2001a, p. 333). This aptly describes not only Frisch’s approach but Rasch’s as well.

Irving Fisher could not, however, have worked from an infinite number of all possible index formulae, even if he was not as disciplined as Frisch in his selection of them. Insight into his principles is gained from consideration of his incurable tinkering; he was always working problems out in relation to some form of technical instrumentation or another. His most famous invention is now known as the rolodex. Significantly, one of his earliest papers was on Kant’s theory of geometrical axioms, and was entitled, ‘Mathematical Contribution to Philosophy’. This early involvement with geometry influenced I. Fisher’s use of mathematics in that he tended to define problems as exercises in instrument design (Boumans, 2001a, p. 316; 2005, p. 157).

I. Fisher’s 1891 Yale PhD dissertation, ‘Mathematical Investigations in the Theory of Value and Prices’, was written under the direction of J. W. Gibbs, who was widely recognised for his work in statistical mechanics (Gibbs, 1902). Fisher’s dissertation was followed by his promotion to full professor at Yale less than seven years later. In 1947, the same year Rasch was with the Cowles Commission, Frisch celebrated I. Fisher’s 80th birthday in print, saying ‘…it will be hard to find any single work that has been more influential than Fisher’s dissertation’ (Frisch, 1947). Economists still refer to the invariance of the real interest rate relative to equal and opposite changes in the expected inflation rate and the nominal interest rate as the ‘Fisher Effect’ (for instance, see Berument & Froyen, 2021; Koustas & Serletis, 1999; Rubinstein, 2003; Shobande & Shodipe, 2021), or in terms of the ‘Fisher hypothesis’ and the ‘Fisher equation’, both of which are found in Google Scholar searches to be mentioned in hundreds of articles published just within the last four years.

I. Fisher’s instrumental approach contrasts strongly with Frisch’s axiomatic approach. Fisher was concerned that the formulae from which indexes were derived should allow for ‘qualitative differences’ that would be ‘a bridge to reality’ (Fisher, 1967/1922, p. 274; quoted in Boumans, 2001a, p. 340). I. Fisher’s ‘rule that an instrument’s performance should approximate a standard within a satisfactory margin’ (Boumans, 2001a, p. 341) gives a practical and applied perspective that can provide an important and often-needed contrast with overly pure mathematical approaches. Though there is no evidence of direct influence on Rasch from I. Fisher, there is striking similarity between this practical orientation and Rasch’s (1960, pp. 37–38) sense that models are not meant to be true, but should be useful, and that they always remain on trial.

Irving Fisher’s separation theorem (Fisher, 1930, Chapters 6–8) advances the same basic principle as Rasch’s separability theorem. I. Fisher’s theorem separates managerial opportunities for productivity from entrepreneurial market opportunities. The point is that a firm’s basic objective is the maximisation of its current value, no matter what the investment preferences and financing sources of the owners happen to be. The Fisher Separation Theorem posits that investment budgeting decisions are made in a two-stage process. First, entrepreneurial capital investment decisions are held to be independent of the preferences of the owner, and second, the investment decision is independent of the financing decision.

The ongoing relevance of this separation theorem is remarkable. Hirshleifer (1970) discussed, and Smith and Nau (1995) extended, the theorem to complete markets under uncertainty. Hochstein (2001) provides a geometric proof and Keynesian view on I. Fisher’s separation theorem, noting its continuous presence in standard econometrics textbooks. Löfgren (1995) relates Wicksell’s work on economically optimal forest management to I. Fisher’s separation theorem and also provides insight into Frisch’s perspective on that work. Smith (1996) extends the Fisher separation theorem to partially complete markets. Rubinstein (2003) refers to Fisher (1930) as providing the first formal econometric equilibrium model, and as ‘the seminal work for most of the financial theory of investments during the 20th century’.

The story told by these relations of managerial and market opportunities thus became the basis of neoclassical macroeconomic theory (Dimand, 1997, p. 442; Tobin, 2005), and could be written as a multifaceted Rasch model (Linacre, 1989),

ln ( P n i j / P n i j - 1 ) = D i − B n − K j

such that the log of the odds of observed investment probabilities P should be equal to the difference between pro and con decisions D as to investments i, owner n’s preferences B, and the source j of the financing K. The unidimensionality, additivity and separability of each of these parameters would then be independently evaluated, in the same manner as more conventional dichotomous or rating scale Rasch model applications (Andrich, 2010; Andrich & Marais, 2019; Bond & Fox, 2015; Fisher & Wright, 1994; Wilson, 2005).

The implications of I. Fisher’s separation theorem prompted Frisch (1930) to formulate a paper on the necessary and sufficient conditions under which an index number shall satisfy the requirements of I. Fisher’s tests. Frisch does not cite R. Fisher (1922) on the sufficient statistic, but Rasch makes a point of saying that he ‘soon got hold’ of Fisher’s paper after arriving in London in 1935 (Olsen, 2003, p. 28; Wright, 1980, p. xi; Wright and Olsen conflict on whether it was 1934 or 1935, but 1935 seems correct).

In Rasch’s models, counts of correct responses function as minimally sufficient statistics (Andersen, 1970, 1977b, 1999; Andrich, 2010, 2016; Fischer, 1981, 1995; van der Linden, 1992). These counts, or the analogous sums of ratings in the context of surveys or assessments, are minimally sufficient, and so, are necessary, since they are functions of all the other statistics that are sufficient in the sense of summarising data with no loss of information. What this means is that, ‘if there exists a minimal sufficient statistic for the individual parameter Theta which is independent of the item parameters, then the raw score is the minimal sufficient statistic and the model is the Rasch model’ (Andersen, 1977b, p. 72). As Wright (1977b, p. 114) points out, it then follows that,

Unweighted scores are appropriate for person measurement if and only if what happens when a person responds to an item can be usefully approximated by a Rasch model …. Ironically, for anyone who claims scepticism about ‘the assumptions’ of the Rasch model, those who use unweighted scores are, however unwittingly, counting on the Rasch model to see them through. Whether this is useful in practice is a question not for more theorising, but for empirical study.

That is, if a count of correct answers or a sum of ratings can provide a meaningful basis for invariant, additive quantification in terms of an interval unit, then Rasch’s separability theorem must be satisfied and the model holds. Even when data are not evaluated for fit to a Rasch model, even when the invariance and additivity properties of quantitative measurement are ignored, use of test, survey or assessment scores as measures inherently implies acceptance of Rasch’s separability theorem.

Rasch’s parameter separation theorem is a formal representation of the rigorous independence of figure and meaning, or of name and concept, posited at the birth of philosophy (Fisher, 2003, 2004), which classically used geometric demonstrations to illustrate the relevant processes and principles (Fisher & Stenner, 2013). Tests of the qualitative hypothesis of quantitative meaningfulness (Luce, 1978; Narens, 2002) are made more accessible and practical by Rasch than is most usually the case for work in this area. And in doing so, Rasch model applications tap deeply into the history of measurement and deploy rich possibilities for mathematical thinking (Engelhard, 2008, 2012; Fisher & Stenner, 2013; Wright, 1988, 1997). Many of these possibilities reside in the field of econometrics (Fisher, 2002, 2009, 2011, 2012a, 2012b), but are deeply rooted in concepts historically associated with measurement and mathematical models in physics (Fisher, 2010; Fisher & Stenner, 2013; Narens & Luce, 1986; Rasch, 1960, pp. 110–115). The value and validity of these connections and general claims about the scientific status of Rasch’s measurement theory (Fisher, 2009; Stenner et al., 2013; Wilson, 2013a, 2013b) and applications of it (Fisher & Stenner, 2016; Williamson, 2018) have been borne out in recent collaborations of psychometricians and metrology engineers and physicists (Cano et al., 2019; Mari & Wilson, 2014; Mari et al., 2021; Melin et al., 2021; Pendrill, 2019; Pendrill & Fisher, 2015).

This trajectory begins in the late eighteenth and early nineteenth centuries, when scientists took Newton’s successful study of gravitation and the laws of motion as a model for the conduct of any other field of investigation that would purport to be a science. Heilbron (1993) documents how the ‘Standard Model’ evolved and eventually informed the quantitative study of areas of physical nature that had previously been studied only qualitatively, such as cohesion, affinity, heat, light, electricity and magnetism. Referred to as the ‘six imponderables’, scientists were widely influenced in experimental practice by the idea that satisfactory understandings of these fundamental forces would be obtained only when they could be treated mathematically in a manner analogous, for instance, with the relations of force, mass and acceleration in Newton’s Second Law of Motion.

The basic concept is that each parameter in the model has to be measurable independently of the other two, and that any combination of two parameters has to predict the third. These relationships are demonstrably causal, not just unexplained associations (Stenner et al., 2013). So force has to be the product of mass and acceleration; mass has to be force divided by acceleration; and acceleration has to be force divided by mass.

The ideal of inversely proportional relations not only guided much of nineteenth century science, the effects of acceleration and force on mass were also a vital consideration for Einstein in his formulation of the relation of mass and energy relative to the speed of light, with the result that energy is now separated from mass in the context of relativity theory (Jammer, 1999, pp. 41–42). Einstein realised that, in the same way humans experience nothing unpleasant or destructive as body mass (or, as is now held, its energy) increases when accelerated to the relatively high speeds of trains, so, too, we might experience similar changes in the relation of mass and energy relative to the speed of light. The basic intellectual accomplishment, however, was one in a still-growing history of analogies from the Standard Model.

Working on an independent line of research, historians of economics and econometrics have documented another extension of the Standard Model. Analogies to the new field of energetics made in the period 1850–1880, and the use of the balance scale as a model by early economists, such as Stanley Jevons and Irving Fisher, are widespread (Boumans, 2005; Maas, 2001). For instance, in Walras’ first effort at formulating a mathematical expression of economic relations, he ‘attempted to implement a Newtonian model of market relations, postulating that “the price of things is in inverse ratio to the quantity offered and in direct ratio to the quantity demanded”’ (Mirowski, 1988, p. 2).

Jevons similarly studied energetics, in his case, with Michael Faraday, in the 1850s. Pareto also trained as an engineer, and made ‘a direct extrapolation of the path-independence of equilibrium energy states in rational mechanics and thermodynamics’ to ‘the path-independence of the realisation of utility’ (Mirowski, 1988, p. 21).

The concept of econometric equilibrium models stems from this work and was also extensively elaborated in the analogies Jan Tinbergen was well known for drawing between economic phenomena and James Clerk Maxwell’s encapsulation of Newton’s second law. In making these analogies, Tinbergen was deliberately employing Maxwell’s own method of analogy to guide his thinking (Boumans, 2005, p. 24).

In his studies with Frisch in Oslo and with Ronald Fisher in London in the 1930s, Rasch (Andrich, 1997; Wright, 1980) made the acquaintance of a number of Tinbergen’s students, such as Tjalling Koopmans (Bjerkholt, 2001, p. 9), from whom he may have learned of Tinbergen’s use of Maxwell’s method of analogy. Rasch employs such an analogy in the presentation of his measurement model (1960, p. 115), pointing out that,

… the acceleration of a body cannot be determined; the observation of it is admittedly liable to … ‘errors of measurement’, but … this admittance is paramount to defining the acceleration per se as a parameter in a probability distribution—e.g., the mean value of a Gaussian distribution—and it is such parameters, not the observed estimates, which are assumed to follow the multiplicative law [acceleration = force/mass].

Thus, in any case an actual observation can be taken as nothing more than an accidental response, as it were, of an object—a person, a solid body, etc.—to a stimulus—a test, an item, a push, etc.—taking place in accordance with a potential distribution of responses—the qualification ‘potential’ referring to experimental situations which cannot possibly be [exactly] reproduced.

In the cases considered [earlier in the book] this distribution depended on one relevant parameter only, which could be chosen such as to follow the multiplicative law.

Where this law can be applied it provides a principle of measurement on a ratio scale of both stimulus parameters and object parameters, the conceptual status of which is comparable to that of measuring mass and force. Thus … the reading accuracy of a child … can be measured with the same kind of objectivity as we may tell its weight ….

What Rasch provides in the models that incorporate this structure is a portable way of applying Maxwell’s method of analogy from the Standard Model. Data fitting a Rasch model show a pattern of associations suggesting that richer causal explanatory processes may be at work. Model fit alone cannot, of course, demonstrate causality or provide a construct theory (Burdick et al., 2006; Wright, 1994), as was emphasised repeatedly by I. Fisher and by Tinbergen in their remarks on the difference between the mathematical model and the substantive meaning of the relationships it represents.

This distinction also informs the reason why Ludwig Boltzmann was so enamoured of Maxwell’s method of analogy (Boumans, 1993, p. 136; also see Boumans, 2005, p. 28):

it allowed him to continue to develop mechanical explanations without having to assert, for example, that a gas ‘really’ consists of molecules that ‘really’ interact with one another according to a particular force law. If a scientific theory is only an image or a picture of nature, one need not worry about developing ‘the only true theory,’ and one can be content to portray the phenomena as simply and clearly as possible.

Rasch (1960, pp. 37–38) similarly held that a model is meant to be useful, not true. The point of using a model is not to describe reality so much as it is to prescribe the structures necessary and sufficient to the task of constructing sustainable relations between ideas and things. Models function as heuristic fictions in much the same way great art does, by telling a story in which we can all find ourselves, even though the specific details of the story may not be true of any of us in particular. The challenge is one of figuring out how to portray human, social and natural phenomena as simply and clearly as possible while also paying close attention to those inevitable failures of invariance that make us each unique individuals (Fisher, 2004).

Frisch Between Fisher and Fisher, Relative to Rasch

Rasch makes no mention at all of any influence of Frisch’s work on himself apart from confluence analysis. Nor does he mention Frisch’s close association with Irving Fisher, I. Fisher’s work, or Tinbergen’s frequent use of Maxwell’s method of analogy. Instead, he repeatedly stresses the crucial importance of Ronald Fisher, to the point of having the reputation of being the ‘great communicator’ of R. Fisher’s ideas in Denmark (Olsen, 2003, p. 42). Rasch’s student, Erling Andersen, remarked that he recalled Rasch quoting R. Fisher and using his examples well into the 1960s (Olsen, 2003, p. 28).

In contrast with this ongoing enthusiasm for R. Fisher’s ideas, Rasch confessed to using Frisch’s confluence analysis less as time passed because he did not ‘think there is anything in social sciences that is linear’ (Olsen, 2003, p. 28). The latent influence Frisch exerted, however, may have been even more profound than the overt influence exerted by R. Fisher.

Issues circulating at a time in history sometimes crystallise simultaneously in multiple works as independent developments. It is probably not just a coincidence that Rasch develops his separability theorem and bases it in statistical sufficiency in a context of proximity to Frisch, who thought through the conditions in which a minimally sufficient statistic would be possible for an economic index number, and whose close colleague I. Fisher had developed a separation theorem for the creation of rigorously constituted index numbers.

But the question remains as to how, when and why Frisch might have prepared Rasch for what R. Fisher had to teach him in London about sufficiency. Rasch was with Frisch only months before meeting Fisher, and Frisch had published his paper on minimally sufficient index statistics only five years before.

The details of Rasch’s relationship with Frisch that prepared Rasch for what R. Fisher had to teach him in London about sufficiency may never be known. Many implications for the conduct of the social sciences, however, are hidden within the methodological overlaps existing between Rasch models and models commonly employed in econometrics, such as matched case-control conditional logistic regression models (Gautschi, 2001; McFadden, 1974; Rice, 2004). Kærgård (2003), for instance, suggests that Rasch’s lack of impact on econometrics in his own day contrasts with the situation today in (a) the similarity between Rasch’s concept of specific objectivity and today’s emphasis on data mining methods (also see Linacre, 2001), (b) the emergence of robust estimation methods less dependent on the normal distribution, which accords with Rasch’s focus on distribution-free estimation (Kærgård, 1987) and (c) the established use of hazard functions and Weibull distributions involving the duration of waiting times, the length of periods of unemployment increases or the durability of various goods (Kærgård, 1970; Kiefer, 1988).

These overlaps might have been expected had there been some awareness of, or way of recognising, Rasch’s implicit incorporation into his models of the principles first elucidated by I. Fisher and Frisch, principles now built into the foundations of econometrics. The conceptual identity of statistical sufficiency with mathematical invariance (Hall et al., 1965) has led to a general consensus among statisticians ‘that statistical analysis should depend only on a sufficient statistic’ (Arnold, 1985, 1988, p. 79). Indeed, not only is the meaningfulness of quantitative scales contingent on demonstrable invariance but meaningful communication itself requires implicit qualitatively mathematical considerations of invariant reference (Falmagne & Narens, 1983; Mundy, 1986; Narens, 1981, 2002; Roberts, 1985, 1994).

Piecemeal implementations of seemingly unrelated principles and methods introduced by I. Fisher, R. Fisher and Frisch are effectively and compactly integrated in Rasch’s models. Consider, for instance, Frisch’s (1930, pp. 402–403) observation that,

The [previously referred to] formulae (8) and (9) must hold good with respect to any of the commodities entering into the index. In order that the index number Pts (with the continuity properties here assumed) shall fulfill the base test it is, therefore, necessary and sufficient that it be of the form

P t s = P t Q s ( 10 )

This plainly builds out from I. Fisher’s Separation Theorem and reads like a statement of a Rasch model. The difference is that equations of this kind are conceptualised statistically as models of group-level, intervariable relations, and not as measurement models of individual-level, intra-variable relations. The result is that Frisch’s concept of autonomy is realised only in the context of specific data sets, and is assumed to apply only in terms of broad scale top-down policy implementations. This contrasts markedly with the generalised autonomy realised via Rasch models, when locally applicable instruments are calibrated and disseminated for point-of-use interpretation, informing bottom-up co-ordinations, alignments and harmonisations of individual behaviours and decisions (Bezruczko, 2005; Chien et al., 2018; Masters et al., 1999; Wilson, 2018; Wright et al., 1980).

Though not distinguishing between group-level statistical modelling and individual-level measurement modelling, recent conceptual analyses of economic modelling note a significant difference between descriptive and prescriptive measurement models (Boumans, 2001b). The generality of models depends more on being able to predict and explain facts about phenomena than on being able to account for variation in observed data. Boumans presents economic models in terms of instruments that could be calibrated so as to provide the invariant profiles needed for generality. He points out that ‘since its introduction to economics, calibration has been controversial’, echoing long-standing debates and controversies in education, psychology and healthcare concerning the epistemological status of Rasch’s models (Andrich, 1988, 2002, 2004; Fisher, 1994; Lumsden, 1978; Maul, 2017; Wright, 1977a, 1984), and, more generally, concerning differences between the statistical and measurement paradigms (Embretson, 1996; Lumsden, 1978; Meehl, 1967; Michell, 1986). Boumans (2001b, p. 429) points out that study results should not be dependent on the particular instrument or data-gathering method used. Rasch models similarly have long been held to facilitate the isolation of contextualized invariances across respondent samples, instruments, time and space.

Making no mention of Rasch, Boumans directly addresses the same issues of mathematical modelling, instrument calibration, reliability, invariance and parameter estimation taken up routinely in these contexts. He (Boumans, 2001b, p. 430) also situates these issues in relation to metrological traceability to universally uniform reference standards, as has been pursued at length elsewhere (Fisher, 1997, 2000, 2009; Fisher & Stenner, 2016; Mari & Wilson, 2014; Mari et al., 2021; Pendrill, 2019; Pendrill & Fisher, 2015) relative to Rasch’s models. Boumans also contends that these issues are also of central concern in Haavelmo’s (1944) probability approach in econometrics. Most important, perhaps, is the fact that Haavelmo’s probability approach was highly influenced by Frisch and his method of confluence analysis (Haavelmo, 1948), as is also stated by Bjerkholt (2001, p. 8).

Frisch’s concern for the stability of evidence across the ephemeral vagaries of different sets of observations led him to the concept of autonomy as a matter of identifying stable, structurally invariant economic relations (Aldrich, 1989; Boumans, 2001b, pp. 430, 440; Haavelmo, 1948). Similarly, Rasch-based item banking, adaptive instrument administration and judge/rater calibration methods (Barney & Fisher, 2016; Choppin, 1968; Linacre, et al., 1994; Wright & Bell, 1980; see chapters in Bond & Fox, 2015; Masters & Keeves, 1999) are designed for situations in which the stability of the observational and inferential frames of reference is of paramount importance.

Frisch explicitly mentions a need for a bridge between his confluence analysis and R. Fisher’s work. In an April, 1935, letter to Koopmans, Frisch refers directly to Ronald Fisher as having preceded Koopmans in the definition and solution of a problem in which the probability distribution of the second term in an observational series is not independent of the first term (Bjerkholt, 2001, p. 9). Both Haavelmo and Rasch attended lectures given by Koopmans at Frisch’s invitation, in Norway in 1935 (Bjerkholt, 2001, p. 9), one of which involved R. Fisher’s theory of estimation. Bjerkholt says that ‘an important part of Haavelmo’s original inspiration towards the probability approach came from Frisch’s ideas and from his work with Frisch on applying confluence analysis’ (p. 8).

One wonders to what extent Rasch also might have found inspiration for his probability approach in his studies with Frisch concerning autonomy and confluence analysis, and in listening to Koopmans speak of R. Fisher’s estimation theory. As Rasch later recounted (Wright, 1980, pp. xi–xii), soon after arriving in London, he,

got hold of his [R. Fisher’s] 1922 paper where he developed his theory of maximum likelihood, because I was especially interested in that matter. What caught my interest most was his idea that this is a form of generalisation of just the same kind as Gauss attempted when he invented the method of least squares. The meaning of least squares is not, in Fisher’s interpretation, just a minimisation of a sum of squares. It is a maximisation of the probability of the observations, choosing such values as estimates of the parameters as will maximise the probability of the set of observations at your disposal…. This philosophy went further when Fisher got to his concept of sufficiency…..

To purely mathematical minds sufficiency may appeal as nothing more than a surprising and singularly nice property, extremely handy when accessible, but, if not, then you just do without it. But to me sufficiency means much more than that. When a sufficient estimate exists, it extracts every bit of knowledge about a specified feature of the situation made available by the data as formalised by the chosen model. ‘Sufficient’ stands for ‘exhaustive’ as regards the feature in question….

What is left over when a sufficient estimate has been extracted from the data is independent of the trait in question and may therefore be used for a control of the model that does not depend on how the actual estimates happen to reproduce the original data.

The realisation of the concept of sufficiency, I think, is a substantial contribution to the theory of knowledge and the high mark of what Fisher did…. His formalisation of sufficiency nails down the… conditions that a model must fulfill in order for it to yield an objective basis for inference.

Koopmans returned to London in November of 1935 to study further with R. Fisher, by which time Rasch was also there. To what extent did their conversations set the stage for Rasch to build a bridge between Fisher’s estimation theory and concept of sufficiency, on the one hand, and Frisch’s concept of autonomy and method of confluence analysis, on the other?

At about the same time, in October, 1935 (Bjerkholt, 2001, p. 10), Frisch wrote a letter to an American colleague explaining the need for a means of determining the extent to which the differences between his confluence analysis and R. Fisher’s sampling theory could be bridged. Frisch despairs in the letter of being able to find such a bridge in economics because of the difficulty of obtaining data that can be adequately controlled. Some years later, he advocated expanding the use of interviews with the aim of obtaining more control over the kinds of data gathered (Boumans, 2001b, p. 447; Frisch, 1948, p. 370). That kind of control, and the kind of sampling from the universes of possible examinees and test items that would provide sufficient statistics, are precisely what Rasch focused on in the development of his probabilistic models.

Unfortunately, in contrast with Frisch, I. Fisher’s place in this story has been obscured by accidents of history. He is remembered most notably ‘for his spectacular misprediction of stock prices in October 1929 and for eccentric crusades’ even though he ‘emerges in retrospect as a major figure in the development of economics’ (Dimand, 1997, p. 442; also see Cowles Foundation for Research in Economics, 1967; Dimand & Geanakoplos, 2005; Fisher, 2005). Recent celebrations of Fisher’s legacy have held that ‘Fisher is widely regarded as the greatest economist America has produced’ and have elaborated on the large extent to which ‘much of standard neoclassical theory today is Fisherian in origin, style, spirit and substance’ (Tobin, 2005, p. 19). And so it happens that

Fisher vanished from citation lists by the 1940’s as John Maynard Keynes captured the profession’s attention, yet contemporary macroeconomics builds largely upon Fisher’s foundations. … Fisher’s contributions closely parallel much of modern macroeconomics, yet his role was long neglected. … The ‘Mark I monetarism’ of Friedman and his students had many Fisherian features …. Nonetheless, Friedman placed less emphasis on links with Fisher than with Chicago oral tradition (Dimand, 1997, pp. 442, 443).

But what issues and themes were likely at the heart of the Chicago oral tradition? Both Frisch and I. Fisher were instrumental in the establishment of the Cowles Commission (Christ, 1952), to the extent that the Chicago oral tradition was in many ways an extension of the conversation they started. The Cowles Foundation, as it is now called, itself holds that ‘The most important link [between it and I. Fisher] is intellectual. It is the great influence of Irving Fisher’s economic thought in the entire range of topics of research activity at the Cowles Foundation’ (Cowles Foundation for Research in Economics, 1967).

I. Fisher and Frisch’s influences converged in the context of the concept of identified models. Separable parameters are evidence of the autonomy—the structural invariance—of the data relative to the model (Aldrich, 1989, p. 41) and so are essential to future practical policy implementations.

The Cowles Commission Report for 1947 mentions Rasch’s work with Koopmans and Reiersøl on a ‘comparative study of problems arising from the specification of models, in particular problems of identification’. Models are identified when Frisch’s sense of autonomy is obtained.

Koopmans and Reiersøl (1950) formalise Frisch’s concept of autonomy, developing the necessary and sufficient conditions for identifiability by specifying the independence data ought to have relative to the econometric model (Aldrich, 1989). They (p. 165) explicitly thank Rasch, as well as Thurstone, for ‘fruitful’ discussions at the 1947 Cowles Commission meeting in Chicago. Rasch (1953, p. 65; 1960, p. xii) similarly thanks Koopmans and Reiersøl, also referring to the Chicago discussions as ‘fruitful’.

Though neither Rasch, Wright, Andersen, Fischer, nor others among Rasch’s close colleagues referred to Rasch’s models as identified, this term has emerged in recent years as a model choice criterion of note (San Martin & Rolin, 2013; San Martin et al., 2009, 2015; Verhelst & Glas, 1995, p. 235). Given documentation showing Rasch was present alongside Reiersøl at Koopmans’ lectures in Oslo in 1935 (Bjerkholt, 2001, p. 9; 2005, p. 525), and that he was with both of them in Chicago in 1947, further possible conceptual correspondences may stand to be uncovered in documents dating to the time Rasch was working on these problems (for instance, Fisher, 1959, 1961; Fraser, 1963).

Rasch made little or no effort to communicate or collaborate with economists, and roughly criticised their methods. Much of what he criticised, such as reliance on normal distributions, regression and structural equation models, and factor analysis, continue to dominate. Aldrich (1989, p. 34) remarks on the fact that the philosophical issues raised before 1960 were unresolved in 1989, and the evidence today is that the same problems still remain unaddressed (Biørn, 2017). No changes may be possible as long as the statistical approach to measurement dominates and the common but mistaken presumption that quantification is inherently a matter of data analysis—and not a matter of reading calibrated instruments at the point of use—persists.

Rasch’s work suggests another path, one in accord with the results of studies in the history of science that point towards the calibration of instruments measuring in meaningful, interval, quality-assured, standard units and the distribution of those instruments throughout multilevel social ecosystems. The task may well be one of learning the lessons taught by Hayek (1988), another Nobel economist whose core ideas are disregarded: The fatal conceit is assuming that the Cartesian approach to centralised control and reductionist decision-making is the only possible way to proceed.

Rasch’s models for measurement, in contrast, provide methods of gathering together collectively determined stochastic expressions of invariant constructs that can meaningfully be brought into language as common metrics distributed to the point of use throughout metrological networks (Fisher, 2007, 2009, 2011, 2012a, 2019, 2020a). The ultimate proof of Rasch’s separability theorem may reside, however, only in the pudding; persuasion and education may continue to be futile until successful new economies of thought and language are created as consequences of innovative socio-cognitive infrastructures (Fisher & Stenner, 2018; Fisher & Wilson, 2015, 2020; Mislevy, 2018). Evidence suggests momentum in this regard is growing (Williamson, 2018).

Subtler links can also be discerned. Because of the death of Henry Schultz, the leading econometrician at the University of Chicago, in late 1938, ‘the University was in a position from which the possibility of adopting a group such as the Cowles Commission appeared particularly attractive. Likewise, the University was an ideal environment for the Cowles Commission’ (Christ, 1952). The move of the Cowles Commission from Colorado in 1939 to Chicago thus took place in a context in which the Chicago oral tradition had been interrupted. Macroeconomics’ general Fisherian themes would have been even more pronounced in an econometrics organisation that I. Fisher had helped launch, and in a context in which new leadership was imperative.

Chicago economics department faculty held positions with the Cowles Commission from 1939 on, and the role of the Cowles Commission in the intellectual vibrancy of the Chicago oral tradition is also evident in the number of future Nobel Prize winners who first came to Chicago as Cowles research associates, such as Koopmans, Arrow and Simon.

When the Cowles Commission for Economic Research left the University of Chicago for a new home as the Cowles Foundation, it moved to Yale University, with which I. Fisher was associated for most of his career, 1890–1935. The 1967 Cowles Report of Research Activities states that, ‘the Cowles Foundation is now contributing to the continuation at Yale of the approach to economic theory and measurement so brilliantly initiated and represented by Irving Fisher’. The continuation of I. Fisher’s approach to economic theory and measurement is also evident at Chicago, as well as in the field as a whole, and in the social sciences at large.

Rasch in Relation to I. Fisher, Frisch and R. Fisher

This history suggests Rasch’s Separability Theorem emerged as an implicit synthesis of Irving Fisher’s Separation Theorem, Ragnar Frisch’s proof concerning the necessary and sufficient conditions for satisfying I. Fisher’s theorem, and Ronald Fisher’s formulation of sufficiency. Three lessons can be drawn from these historical considerations.

First, Frisch may have capitalised on the opportunity created by Rasch’s supervisors in Copenhagen, Nørlund and Madsen, when they sent Rasch to London to study with R. Fisher (Andrich, 1997, p. 542; Wright, 1980), by suggesting that R. Fisher’s work on statistical sufficiency could be particularly important. Rasch spent time with Frisch in Oslo for three months before joining Fisher in London, and Frisch’s 1930 paper is strong evidence of the value he placed on the concept of sufficiency. Far from discovering sufficiency as a concept directly from contact with R. Fisher, Rasch likely arrived in London primed by Frisch to be looking for it.

Second, given these developments, we should be less surprised to find econometric models being applied to ‘everything from tax evasion to teenage pregnancy’ (Hayes, 2007), since the mathematics used in these models is basically the same mathematics as that incorporated in the structure of natural laws, and as used in a wide variety of measurement models for test, survey, assessment, questionnaire and rating scale data across the social sciences.

Third, the connection between measurement as defined by Rasch and others (Wright, 1997) and the definition of capital as requiring additive, divisible, and transferable representations (Brown, 1980; Denny, 1980; De Soto, 2000; Diewert, 1980; Fisher, 2002, 2009, 2011, 2012a, 2012b; Latour, 1986, pp. 12, 31, 1987, p. 223) now has a more fully elaborated context. Echoing North (1981, pp. 26, 36–43) and others, Palmer (2008, p. 337) recognised, in his contrast of De Soto’s and Latour’s perspectives, ‘It is law that detaches and fixes the economic power of assets as a value separate from the material assets themselves and allows humans to discover and realise that potential’. Practical metrological approximations of I. Fisher’s econometric separation theorem and Rasch’s separability theorem open onto new possibilities for laws detaching and fixing the economic power of intangible assets. The cost of transacting trades of educational, healthcare, social and environmental service outcomes could be lowered by metrological standards separating the substantive value of human, social and environmental assets from people, communities and ecosystems themselves; this would set the stage for a new class of property rights, legal title to shares of human, social and natural capital stocks, and innovative entrepreneurial approaches to discovering and realising the full potential of those outcomes for individuals (Fisher, 2011, 2012a, 2012b, 2020a). Further research will be required to explore the similarities and differences between Irving Fisher’s sense of capital and more recent work relating capital to social networks and metrological infrastructure (Barzel, 1982; Benham & Benham, 2000; De Soto, 2000; Latour, 1987, pp. 223, 247–257; Miller & O’Leary, 2007; National Institute for Standards and Technology, 1996, 2009).

Interestingly, I. Fisher and R. Fisher may have had an opportunity to meet under the auspices of the Cowles Commission, as they both spoke at a conference sponsored by the Cowles Commission at its first home, in Colorado Springs, in the summer of 1936 (Christ, 1952), the year after Rasch had been in London studying with R. Fisher and in Olso with Frisch.

How might the history of economics, psychology, education and social science been different if Rasch had been able to get the two Fishers and Frisch to focus on the importance of statistical sufficiency and maximum likelihood estimation in the context of blended instrumental and axiomatic approaches to tests of a separation theorem? We might have today fewer meaningless but statistically significant results in research in psychology (Bakker et al., 2012) and finance (Kim & Ji, 2015), and less of a crisis of reproducibility in both psychology (Pashler & Wagenmakers, 2012) and a wide range of other disciplines (Baker, 2016), if researchers had measuring tools calibrated to common units, with known precision, expressing substantively significant and quantitatively meaningful established scientific laws? Even more unfortunately, had meaningful social and psychological measurement been deployed in practical information infrastructures throughout business, government, education and healthcare, could not we have avoided many of the system failures experienced in recent years?

This odd coincidence—Rasch’s relationship with Ronald Fisher and Rasch’s formulation of a separability theorem from Fisher’s concept of the sufficient statistic, alongside Frisch’s relationship with Irving Fisher and Frisch’s construal of necessary and sufficient statistics for Fisher’s separation theorem—is made even more intriguing by the fact that Irving Fisher’s biography at http://www.swlearning.com/quant/kohler/stat/biographical_sketches/bio20.1.html is accompanied by a photo of Ronald Fisher. The same photo is identified correctly at https://www2.stetson.edu/~efriedma/periodictable/html/Fl.html, and many other websites. It is an understandable error to make, given that both were white-haired men with glasses, moustaches and goatees. For a photo of Irving Fisher, go to the Cowles Foundation website at https://cowles.yale.edu/archives/irving-fisher.

Concluding Notes

A popular commentator on econometrics recently made some telling remarks about the increasing application of economic models in a growing number of areas of the social sciences, saying that,

the neoclassical model didn’t leave its mark only on economics. In an audacious burst of methodological imperialism, Chicago Schoolers like Gary Becker used the framework of rational individuals seeking to maximise their utility to analyze and explain everything from tax evasion to teen pregnancy (Hayes, 2007).

What Hayes refers to as econometrics’ methodological imperialism could arguably be seen as covering more than the hegemony of neoclassical economics. Alternative ‘heterodox’ approaches emerging in economics might be even more relevant to the analysis and explanation of everything from tax evasion to teen pregnancy than the neoclassical framework.

As Rasch implicitly understood, though he applied the principle at the individual, instead of the population, level, the basic structure of the relations built into the neoclassical model assumptions concerning parameter separation and statistical sufficiency can be observed to hold across many different kinds of situations. The basic structure of the three-part relationships posited in economic models proposed by Irving Fisher and many others has been adopted in a wide range of other fields. As is pointed out by Rasch (1960, p. 115), this basic structure does not originate in economics, but in physics. Some (Ramsay et al., 1975, p. 258) have speculated whether there is perhaps good reason why mathematical laws must take the particular form of three parameters, with one equal to the product or the quotient of the other two. Others (Burdick et al., 2006; Fisher, 2009, 2010, 2012a; Fisher & Stenner, 2013; Stenner et al., 2013) have demonstrated the structural analogy that holds between the parameters for pressure, temperature and volume in the combined gas law, on the one hand, also holds for the parameters for reader ability, text difficulty and comprehension rate in a Rasch reading law, on the other.

A variation on the question Burdick and colleagues ask might eventually inform research in a new paradigm: How many such variable triplets structured by strict quantitative causal laws exist in the human, social and environmental sciences? Models of this kind provide a basis for fundamental measurement (Narens & Luce, 1986), the development of individually customised, universally uniform units of measurement (Cano et al., 2019; Fisher & Stenner, 2016; Mari & Wilson, 2014; Mari et al., 2021; Pendrill, 2014, 2019; Pendrill & Fisher, 2015), and information infrastructures incorporating qualitatively discontinuous levels of complexity (Fisher, 2017; Fisher et al., 2018, 2021; Fisher & Stenner, 2018). Rasch’s separability theorem and its consequences in both probabilistic and deterministic contexts provide a basis for an explanatory, and not merely descriptive, theory of measurement (Andrich, 2016; De Boeck & Wilson, 2004; Melin et al., 2021; Stenner et al., 2013). A wide range of questions like these follow from the ideas of Rasch, Frisch, I. Fisher and R. Fisher. Though they cut against the grain of contemporary assumptions about modelling and measurement in the psycho-social and economic sciences, the possibilities they represent for meaningful productivity demand exploration.

Motivations for pursuing lines of inquiry in this direction are not restricted to the logic of more efficient communications based in lower transaction costs and shared standards tapping collectively projected structural invariances. Just as Hyde (1979, pp. 273–274) was surprised to find the logos of rational markets complemented by the eros of gift exchanges, so, too, must we also be prepared to see nonbinary integrations of seeming opposites in the implications of measurement modelling. The arts, after all, are infused with the technical effects of sciences ranging from acoustics to electronics to colour theory, while scientists themselves must inevitably make use of metaphor in the derivation of new conceptualisations of the products of their creative innovations (McLeish, 2019).

The mathematics of the separation and separability theorems advanced by I. Fisher and Rasch, and of the identified models and minimally sufficient statistics advocated by Frisch, Koopmans and Rasch, speak not to a reductionist logic homogenising human qualities to monotonous sameness but to an emergent and playful blending of logos and eros. This blending may be best understood via the imagery of music’s capacities for captivating and enthralling sensations. No other art has a greater capacity for possessing the spirit, and no other art more thoroughly combines the logic of mathematical theory with calibrated instrument standards in support of local improvisations. The simultaneously formal, abstract and concrete nature of music is also characteristic of language in general and of the objects of scientific investigation, as has been documented in the history and social studies of science (Bowker et al., 2015; Galison & Stump, 1996) over the last 30 years and more. These seemingly opposed capacities for locally situated openness and malleability, on the one hand, and generally communicable navigability and structural invariance, on the other hand, ‘create a fascinating design challenge–even a new science’ (Fisher et al., 2021; Star & Ruhleder, 1996, p. 132). Frisch, Rasch their colleagues, and their intellectual heirs provide a wealth of tools relevant to addressing this challenge. We would be remiss in not putting their contributions to the test.