2.1. Epistemic Iteration

Chang (Reference Chang2004) defines epistemic iteration as “a process in which successive stages of knowledge, each building on the preceding one, are created in order to enhance the achievement of certain epistemic goals” (45). He takes this process to support a “progressive coherentism”: on the one hand, the criteria for measurement success are internal to a practice, so scientific knowledge does not rest on an independent foundation; on the other hand, these internal criteria may be used to evaluate new practices as improvements or refinements on their predecessors, thereby allowing for scientific progress (in contrast to traditional coherentism; Chang Reference Chang2007). In the context of measurement, this means that later measurement practices may be understood as in some sense “better” than earlier ones, yet the “epistemic achievement” involved should not be understood in terms of greater degree of correspondence to quantities in the world.

For instance, thermometry as a practice begins with subjective assignments of relative heat on the basis of our bodily experiences. Noticing that fluids appear to change volume in rough correspondence with these subjective sensations, one may construct a thermoscope, or device allowing comparison of relative fluid volumes in different circumstances. Already a theoretical leap is required to identify the cause of these changes in relative volume with the cause of our differing subjective sensations, especially given the discrepancies between these sensations and our thermoscopic readings (e.g., contrary to experience, caves are warmer in summer than they are in winter). Nevertheless, the move to the thermoscope constitutes an epistemic achievement, in the sense that it allows for greater regularity in the assignment of relative temperatures, both across contexts and across observers. A similar pattern is seen in the move from thermoscope to thermometer, which enables assignment of numbers to temperatures. Numerical representation constitutes yet a greater epistemic achievement, insofar as it allows comparison of temperature assignments across devices. Nevertheless, this practice does not itself guarantee greater veracity of temperature assignments, since it rests on the assumption that temperature varies linearly with changes in the height of thermometric fluid. But this assumption cannot itself be verified, as that would require access to temperature in the world by some means independent of thermometry. Similar achievements, (seemingly) inextricably entangled with theory, may be seen at each further stage in the development of thermometric practice.

The avowed moral of this case study is the historical contingency of thermometry and, thus, also of its results. At each stage in the development of thermometry, an advance in theory was required to extend measurement practice. Internal criteria of consistency and increased precision in the assignment of numerical values establish the new practice as an advance over the previous one. Yet, the application of these criteria is not empirically constrained. When one assumes that “temperature” (whatever it may be) varies linearly with changes in the height of the indicator column in an air thermometer, one is making an assumption both necessary for measurement progress and in principle nonempirical, since no independent access to “temperature,” outside the behavior of the very devices and procedures under investigation, is possible: “Prior to the construction of a thermometer, there is no thermometer to settle that question” (van Fraassen Reference van Fraassen2008, 126). Chang (Reference Chang2004) argues that, in order to make sense of the “progress” exemplified by cases like these, we have to “look away from truth” and appeal only to historically contingent criteria for success (227)—“scientific progress … cannot mean closer approach to the truth” (228). “Truth, in the sense of correspondence to reality, is beyond our reach” (Chang Reference Chang2007, 20). The delusion that one may evaluate the correspondence between assignment of temperatures and the objective state of the world rests on the mistaken and “impossible god-like view in which nature and theory and measurement practice are all accessed independently of each other” (van Fraassen Reference van Fraassen2008, 139). Rather, the only relevant notion of “truth” for assessing the success of thermometry “rests first and foremost on coherence with the rest of the system” (Chang Reference Chang2012, 242).

2.2. Models and Calibration

Another kind of epistemic loop is found in synchronic measurement practice in which models play a constitutive role in determining measurement outcomes. The crucial concept here is calibration, the process of correcting a measurement device for inferred discrepancies between its readout and the target value. Calibration is a necessary feature of all sophisticated measurement, yet the process of calibration illustrates the ineliminable role of theoretical posits in the very assignment of quantity values in an act of measurement. When measuring, scientists do not (as one might naively suppose) read values directly from nature; rather, they employ models of the interaction between the measurement device and target system in order to “correct” the readout value to a final assigned value (Mari and Giordani Reference Mari, Giordani, Boumans, Hon and Petersen2014).

Tal (Reference Tal2014) illustrates this point through the example of the measurement of time, in particular coordinated universal time (UTC).Footnote ¹ The second is currently defined as 9,192,631,770 periods of the hyperfine transition between the two ground states of a caesium-133 atom at zero degrees Kelvin. Models feature at every step of the process leading from devices that interact directly with caesium atoms to the UTC. First, it is impossible to probe caesium atoms at absolute zero, so the enumeration of hyperfine transitions output by a caesium clock must be corrected for this discrepancy. This as well as other corrections rely on models of the physical interaction between the device and the atom in order to infer the discrepancy between the actual state of the system and the idealized state referred to in the definition. Caesium clocks are too complex to run continuously, so their output is used to calibrate more mundane atomic clocks (301). Furthermore, the UTC itself is not identified with the output of any one clock; rather, it is calculated retrospectively by a weighted average over all participating atomic clocks, with weights determined by the degree of past fit between each clock and previous calculations of UTC (302–3).

The lessons of this example are analogous to those of epistemic iteration: measurement improvement appears to rest on internal standards of coherence rather than on correspondence with external quantities. The weighting procedure that leads to UTC, for instance, “promotes clocks that are stable relative to each other” (Tal Reference Tal2014, 304). Success at achieving this stability indeed demonstrates “genuine empirical knowledge”—not knowledge in the first instance about a regularity in the objective world but rather a regularity “in the behaviour of instruments” (327). Consequently, it is a “conceptual mistake” to think that “the stability of measurement standards can be analysed into distinct contributions by humans and nature” (328). On an extreme interpretation of this view, even computer simulation constitutes a form of measurement (Morrison Reference Morrison2009). The basic idea is that, once we grant the ineliminable role of models in measurement, it is a small conceptual step to accept that the aspect of measurement involving empirical contact with the world may be arbitrarily distant from that involving modeling (Parker Reference Parker2017).

3.1. Convergence

Coherentists have emphasized the theory-ladenness of both diachronic and synchronic aspects of measurement refinement. However, a hallmark of sophisticated scientific measurement is its efforts to factor out the contributions of theory by developing diverse procedures for measuring a single quantity. A measurement practice converges when procedures employing different theoretical commitments arrive at the same outcome.

For instance, in the early twentieth century, a wide variety of phenomena were investigated, employing distinct methods and theoretical commitments, in the attempt to measure Avogadro’s constant N _A, the number of particles in a mole of substance. Perhaps most well known are Perrin’s experiments on Brownian motion, which, in combination with Einstein’s theoretical analysis, allowed an assignment of value to N _A. However, similar values were achieved by radically different means. For instance, Millikan was able to determine N _A by measuring the charge of the electron through his oil drop experiments and dividing the Faraday constant (charge of a mole of electrons) by his result. Millikan’s measurement relied on Stokes’s theoretical analysis of the movement of spheres through a viscous fluid—insofar as Brownian motion was a factor, it was as a source of noise, not (as for Perrin) a source of evidence. Black body radiation and the blue of the sky are examples of other phenomena that, when combined with theoretical models of photon emission and diffraction respectively, allow alternate means of measuring N _A. Insofar as these procedures assign the same value to N _A, they converge.

I want to stress that the point being made here is not the traditional realist one that these practices provide converging evidence for the particulate nature of matter, whether as a “common cause” (Salmon Reference Salmon1984) or most likely hypothesis (Psillos Reference Psillos2011). Those arguments are instances of abduction, while I am interested in whether a stronger, nonabductive conclusion may be drawn from convergence. A better analogy is with the discussion of robustness in the modeling literature: a result is robust if it is obtained by a plurality of models that each make different simplifying assumptions (Weisberg Reference Weisberg2006). The particulate nature of matter is not robust in this sense across different measurement practices, since it is assumed by all of them. However, the value of N _A is robust, since that value is not itself assumed and is obtained with a great degree of agreement despite differences in the assumptions made by each measurement practice (and its supporting models). I claim that convergence toward this value provides robust, nonabductive evidence for an objective feature of the world.

This example is in no way exceptional: convergent measurement practices are rife across the sciences. Smith and Miyake, for instance, have investigated a number of examples. Thomson’s convergent measurements of the charge of the electron employed a variety of different methods and assumptions (Smith Reference Smith, Buchwald and Warwick2001). Early attempts to measure the density of the interior of the earth likewise assumed a variety of different theoretical models (Miyake Reference Miyake and Saatsi2018). In more recent research, measurements of the constants that govern molecular vibration converge across spectroscopy, chemistry, thermodynamics, and femtochemistry (Smith and Miyake, Reference Smith, Miyake, Vickers and Lyonsforthcoming). To pick an example from an entirely different area of science, measurements of the spectral sensitivity of mammalian retinal receptors employing psychophysical methods (extracting sensitivity curves from behavioral color-matching experiments, as performed by Helmholtz in the late nineteenth century) converge closely with twentieth-century physiological methods (detecting the rate of nerve firing in, e.g., cow retinal tissue in response to single wavelength lights; Wandell Reference Wandell1995). In all of these cases, “What is being shown through the convergence of these measurements is that the discrepancies between the different measurements … are due to the particularities of the models being used” (Miyake Reference Miyake and Saatsi2018, 336). In other words, convergence factors out model-sensitive features of measurement; in order for it to occur, “the empirical world has to cooperate” (Smith Reference Smith, Buchwald and Warwick2001, 26).

3.2. Precision

Traditionally, measurement success was evaluated with respect to two features: accuracy and precision. Accuracy was the degree of approach to the true value, while precision was the degree of specificity in the value provided. The considerations in section 2 undermine the criterion of accuracy, since they show we have no independent access to “true values” and thus cannot use them as standards for evaluating measurement (Mari Reference Mari2003). Nevertheless, we can still assess measurements for precision, since it may be defined operationally: a measurement is precise to the extent that it returns the same result when performed repeatedly. The number of significant figures in a numerical assignment indicates the degree of measurement precision, since these characterize the size of the region within which repeated measurements fall.

Coherentists stress the fact that increased precision is a purely internal criterion for improving measurement. Here, however, I want to stress the way in which increased precision constitutes a qualitatively different, and more impressive, epistemic achievement than other forms of empirical success, such as qualitative prediction or improved coherence of classification. These qualitative achievements are subject to worries about semantic and theoretical holism: one may always succeed in classification, or correct qualitative prediction, by suitably redrawing the boundaries of one’s theoretical concepts. As LaPorte (Reference LaPorte2004) argues, when faced with anomalies in the relationship between guinea pigs and prototypical rodents, or birds and dinosaurs, scientists face a choice whether to expand or contract their previous categories to include or exclude perceived outliers (a similar case is made by Slater [Reference Slater2017] for Pluto and planethood). Nothing about the prior conceptual framework itself forces this choice one way or another, nor do demands for internal consistency.

Measurement is different from mere categorization, precisely because it maps states into a metric space. The crucial point to note here is that a metric space has antecedent structure: the distances between points on the real line and the algebraic relationships between them are fixed before we employ them to represent height or temperature or electric charge. This antecedent structure constrains the relationship between measurement outcomes, independently restricting our assessment of them as the same or different, or converging or not, in a manner impervious to ad hoc revision. Increase in precision occurs when successive measurement practices are able to shrink distances (between repeated measurements within each practice) determined by the metric of the representing space. Thus, the metric of this space serves two functions: (i) it represents the distances between different measured quantities, but (ii) it also provides a directed metric for improving measurement of a single quantity, since it determines the distances between repeated measurements that characterize their precision. Consequently, pace van Fraassen, attempts to increase precision are empirically constrained, since this directed metric for improvement can only be satisfied through the cooperation of nature: if nature is not sufficiently stable when we probe it, no choice, convention, or increased coherence can reduce the distances between our repeated attempts to measure it. Some examples will illustrate this point.

Consider, for instance, determinations of the boiling point of water. Chang (Reference Chang2004, chap. 1) surveys the sequence of choice points in the early practice of thermometry leading to relative stability in the measurement of this temperature: What are the visual indicators of boiling, where should the thermometer be positioned, what should be the shape of the vessel holding the water, its material, and so on?Footnote ³ Decisions on each of these points affect the relative stability in the thermometric reading, illustrating the naivety of a view on which “boiling point” is a simple phenomenon merely waiting to be observed. Nevertheless, in committing to represent the boiling point numerically, investigators subjected themselves to a criterion for success distinct from coherence. If the numbers assigned by thermometers within this-shaped vessels and that-shaped ones differ during phenomenologically similar bubblings, then the distance between those numbers provides a criterion of difference that must be respected if thermometric practice is to count as measurement. Restricting attention to those vessels that minimize distances between numerical outcomes is thus not a mere choice or gerrymandering of the category “boiling,” since it is forced on the investigator by an antecedent metric for success.

Likewise, consider again the determination of UTC through the retrospective weighting of the comparison set of atomic clocks. For Tal, the success of this procedure is evidence for stability in our clocks but not for any human-independent feature of the world. Nevertheless, UTC is constrained by the world in two distinct ways. First, through empirical contact with caesium atoms. While this contact is mediated by models, these models themselves are the result of convergent measurements of atomic phenomena through a wide variety of means, employing distinct theoretical assumptions. Second, the distance metric of the real line constrains the assessment of fit between clocks in the set. While the algorithm that weights them takes degree of internal agreement as the standard for higher weighting, the metrical structure of the space in which relative rates of the clocks are assessed ensures relative agreement cannot be stipulated, fudged, or gerrymandered. The clocks need to cooperate by performing stably enough that they may be compared with a high degree of precision, and this region of modal stability remains tethered to a robust regularity in the world through checks with the convergent behavior of caesium.

While UTC is in some respects atypical (n. 1), these three features—internal coordination of outcomes, empirical checks, and directed improvement constrained by the real line—are features of scientific measurement in general. What Tal’s discussion of the UTC obscures is the sheer number of empirical checks typically involved and the strictness of the demands placed by conformity to the metric of improvement the measuring space provides. In official determinations of fundamental physical constants, convergence is demanded across all measurement procedures, as assessed by the law-governed interrelationship between physical quantities, and the degree of precision achieved illustrates the strictness of this demand. For instance, in late nineteenth-century measurements of N _A by Perrin and e (charge of electron) by Thomson, only two to three significant figures were typically obtained within method, and convergence across methods often only agreed as to order of magnitude. By 1911, Millikan was measuring both e and N _A to four significant figures and demonstrating that the models employed to calibrate the oil drop method converged closely with other aspects of physical theory (Reference Millikan1911). As of 2014, N _A was being measured at upward of nine significant figures, and e upward of 11 (Mohr, Newell, and Taylor Reference Mohr, Newell and Taylor2016).Footnote ⁴ In each case, the increase in precision has been constrained by the antecedent structure of the real line and, thus, is not itself a matter of mere convention or coherence. Rather, the world must cooperate by remaining sufficiently stable if such precision is to be possible; consequently, precise values constitute robust evidence for points of objective fixity in the world revealed through measurement.