This special issue presents a collection of articles that grew out of the “Measurement at the Crossroads” conference held at the University of Paris, June 27–29, 2018.1 It is the third set of publications from an ongoing series of interdisciplinary conferences exploring the history, philosophy and sociology of measurement. It follows from two books, Standardization in Measurement (Schlaudt and Huber 2015) and Reasoning in Measurement (Mössner and Nordmann 2017), based on the “Dimensions of Measurement” conference held in 2013 at Bielefeld University; and from the special issue “The Making of Measurement” published in Studies in the History and Philosophy of Science (Mitchell, Tal, and Chang 2017) which came out of the eponymous conference organized in 2015 at the University of Cambridge. The preparation of a fourth conference in 20222 at the Università Cattolica del Sacro Cuore, Milan, confirms the success of this cycle of conferences and demonstrates that the revival of interest for the topic of measurement, that arose in the 2000s (Chang 2004; van Fraassen 2008), has now firmly settled into a rich, diverse and enduring scholarly movement.The 2018 “Measurement at the Crossroads” conference aimed, like the earlier ones in the series, to present the state of research in the history and philosophy of measurement and proposed, in addition, to put the issue of measurement in a world-wide and a long-term perspective (notably, with contributions on measurement in ancient Mesopotamia). Although this special issue contains only a limited selection of the contributions offered at this event, the collection provides a representative overview of the current direction of measurement research in a variety of disciplines: the philosophy of science, physics, chemistry, medicine and economics. As for the broader historical and international issues of the conference, they will gradually emerge from the discussion of the themes in the collection.The revival of the philosophy of measurement in the early 2000s goes hand in hand with a novel approach to measurement issues that has now reached a certain maturity. Three key features, highlighted by Tal (2013), are especially useful in characterizing this new trend: the contemporary outlook tends to be epistemological, practice-oriented, and model-based. The first two features are clearly in contrast to the foundational and formal outlook of the “representational theory of measurement” (hereafter RTM) which held sway during most of the second half of the twentieth century in the wake of the semantic conception of theories. The issue of models, which is central to many of the articles in this collection, deserves more attention. The RTM essentially focused on the conditions under which a qualitative domain of experience can be treated mathematically and expressed numerically, and it cast these conditions into the notions of model and representation used by the semantic view to account for the way in which theories relate to the world. In this sense, models played a key role in the RTM in the guise of numerical relational structures which, on the one hand, satisfied a set of mathematical axioms and, on the other, provided representations of qualitative domains of experience by being isomorphic to them.3 The model-based character of the contemporary epistemology of measurement shifts significantly away from this view by taking its root in a different account of models, closer to the actual use of models in scientific practice (Achinstein 1965; Cartwright 1983; Morgan and Morrison 1999). Models are no longer seen as providing representations of qualitative fields of experience by virtue of certain intrinsic structural properties they share with their target. The new approach adopts a pragmatic standpoint: although models partly originate from theories, they are constructed and unavoidably informed by the intentions and beliefs of the agents engaged in measurement activities and can vary according to their purposes. In this respect, they are also intrinsically linked to a management of uncertainty. Thus, the role of models in quantitative representation and evaluation no longer relates to descriptive or foundational issues, but rather to understanding how measurements are involved in the production of reliable, yet also inevitably fallible results that can serve either epistemic aims, by supporting knowledge claims, or practical as well as social objectives, by guiding and securing the coordination of human activities at large. These last features indicate that the new approach in philosophy of measurement is naturally linked to other disciplines of research that address measurement from historical, sociological, and political perspectives which fall quite outside the scope of the RTM.The articles presented in this collection clearly encapsulate the break with the RTM’s perspective on measurement that has just been described, and contribute, in addition, to make the new epistemology of measurement a genuine stakeholder in the contemporary debates in the philosophy of science. As the following articles demonstrate, this trend in measurement research not only adds to the critical movement raised in reaction to the semantic conception of theories and the representational view of knowledge (Morgan and Morrison 1999; Suárez 2003; Giere 2004; Knuuttila 2011); it also contributes to the debate on scientific realism and the theory of reference and, further, to issues regarding the construction of the instruments and categories through which we understand and govern ourselves as social beings.The first two articles, by Eran Tal and Roman Morawski, draw on the pragmatic understanding of models endorsed by the new approach to measurement. Tal’s article directly challenges the conceptual core of the RTM, while Morawski deals with the practical problems involved in the process of inverse modelling under uncertainty leading to the estimation of a measurement result. The articles by Alessandro Giordani and Luca Mari, and by Jean-Pierre Llored, shift attention towards the target of measurement and examine more pointedly what measurement results, as outcomes of model-based, fallible inferences, actually succeed in referring to. It will be seen that this issue requires different treatments depending on the complexity of the discipline concerned. Finally, the two last articles, by Rebecca Jackson and Oliver Schlaudt, take the point of view of the agents and users of measurement. Their pragmatic perspective highlights that measurement is not only a matter of representing and knowledge-seeking but also constitutes a means of intervening in the world and engages wider cognitive and social implications.In the following, we have sought to bring out the way in which, despite the partly contingent factors that helped bring them together, the various articles presented in this special issue manage to establish a dialogue with one another; how they can be considered as taking up, reworking, and deepening from various vantage points, always anchored in scientific practice, and core questions of the philosophy of science whose ramifications extend into everyday life. We thus wish to prompt the reader to go beyond the mere disciplinary division displayed at first sight by this collection of papers, and benefit from the inspirations and insights that the articles are able to provide across disciplinary boundaries.To date, the RTM has mostly been the subject of scattered criticism or has been challenged from the perspective of particular rival positions;4 it has accordingly never given rise to a systematic discussion liable to go beyond topics specifically related to quantification and measurement. Eran Tal (2021) offers an in-depth critique of the RTM, and his exposition reflects why this theory has been so difficult to assess: the representational theory leaves room for different interpretations and, what is more, harbours several pending questions–concerning the way to construe the axioms involved, what entities are related to numbers (are they objects or attributes?), or what conception of magnitude is proposed.One of the most influential interpretations identified and discussed by Tal supports the foundationalist claim according to which axioms can truly serve as criterions of measurability. The RTM provides a “library of axioms and theorems” (Tal 2021, pp. 701–741) whose role is, first and foremost, to establish conceptual conditions at which qualitative relational structures can be described as relations among hypothetical magnitudes. But can these axioms also serve as conditions of measurability, that is, as conditions against which the quantitative structure of empirically established relationships between concrete objects (or attributes) can be tested experimentally? Tal gives the name “evidential interpretation” for the view according to which axioms can indeed serve to detect if a given set of relations between objects is an instance of a quantitative structure.5According to Tal, the RTM’s evidential interpretation doesn’t withstand scrutiny when it is seriously confronted with the practicalities of experimental testing. Indeed, the experimental tests required involve subjecting the domain of objects under investigation to processes of ordering and concatenation that bring into play, besides the particular quality of interest, still other properties of these objects which interact with the instrumental setup and the environment. These interactions inevitably blur the empirical relations recorded for the quality under examination. Such relations can therefore be meaningfully confronted to the measurability conditions only provided that certain theoretical assumptions about the physical behavior of the system and the experimental setup are introduced. Now, according to Tal, this involves the construction of a mathematical model of the experiment which corrects the inconsistencies. The need for such a model irretrievably defeats the foundational project of the RTM since it requires that the qualitative relational structure be already given as part of a constituted array of theories, which presupposes the very quantitative structure that is to be tested.6 In this respect, the key criticism behind the break away from the RTM is that the latter overlooks the instrumental, conceptual, and theoretical mediations, always anchored in specific contexts, that are inevitably involved in our access to the experimental fields under investigation and that undermine any claim to be able to directly grasp their actual structure. Quantification can therefore only “(start) off as a hypothesis that is introduced tentatively in order to regulate the analysis of data” (Tal 2021, pp. 701–741). The foundationalist stance of the RTM gives way to the task of showing that such assumptions are not ad hoc but instead fulfil “certain epistemic and pragmatic desiderata” (Tal 2021, pp. 701–741) and gain support through on-going research. The evaluation of quantitative structure is therefore an iterative process whose validity should only be judged in a coherentist manner, by assessing its integration in an empirically successful whole.It is obvious that the kind of model put forward in Tal’s criticism does not square in the least with the RTM’s view of models as abstract relational structures. Moreover, as Tal forcefully argues, there is no perceivable fact concerning the structure of a qualitative domain of objects that would ground the possibility to subject it to a process of number assignment; the “structure” of the qualitative domain can only, at best, be inferred: it is always the outcome of an agent’s analysis aimed at epistemic or practical objectives; it involves uncertainties and may need to be substantially revised. In this respect, Tal’s criticism of RTM is consistent with the extensive discussions raised within the philosophy of science in the context of the criticism of the semantic view of theories about the nature and function of models and the conception of representation. It appears particularly in line with the pragmatic, agent-based approach of representation according to which the symmetric (intrinsic and formal) two-place relationship of “representation” between models and target empirical domains is replaced by a three-place relation in which the connection of model and target is established by a user reasoning on the basis of a body of knowledge and motivated by certain purposes (Giere 2004, Knuuttila 2011, Gelfert 2017). In this respect, the epistemology of measurement could offer valuable insights into issues that are being raised in this wider philosophical debate, especially on error and misrepresentation.In the next paper in the collection, Roman Morawski (2021) takes up the issue of model-dependence in measurement but, this time, at the level of the quantitative evaluation of “measurands”—the “quantities intended to be measured” (JCGM 2012, p.17). For reasons similar to those given in Tal’s analysis, the derivation of quantity values from experimental data is plagued with underdetermination due to measurement errors. Morawski argues that this derivation, leading to the estimation of a measurement result, is tantamount to an inverse inference. He suggests a systematic approach to carrying out this derivation and, by the same token, connects his discussion to the broader philosophical issues of inverse reasoning and inference to the best explanation.By stressing the inverse nature of the core operation driving the evaluation of a measurand, Morawski argues that quantitative abductive reasoning plays a central role in measurement that may yet have been underestimated. Morawski’s argument involves three steps: (i) measurement is an inverse problem; (ii) inverse problems can be handled as cases of abduction; (iii) understanding measurement as abduction opens up various concrete perspectives for dealing with the complexities of measurand evaluation, and thus cope with the underdetermination due to the situation of uncertainty in which the evaluation is carried out.Morawski’s first step is to point out that elaborate measurements often involve the gathering of instrumental data which do not provide direct information about the quantity that is ultimately aimed at being estimated—what metrologists call the “measurand.” Instead, the measurand is only one of the causal inputs of a measurement chain7 which is designed to convert the measurand’s state into an output of measurable signals (e.g., electrical signals, easier to handle). However, to process the data into information about the measurand, the experimenter has to go in the reverse direction. This requires first modelling the system under measurement and the measuring system, in order to get a formal description of the way the state of the measurand is being physically converted into the output data (this is what Morawski calls a “mathematical meta-model of measurement”). By mathematically inverting the physical models, the experimenter is then in a position to proceed with the “reconstruction” of the measurand and to infer its value from the output data.Measurement thus appears to be driven at its core by a reverse inference, meaning that it pertains to an “inverse problem.” Morawski draws here from Niiniluoto’s argument that inverse problems imply a kind of estimation task that “identifies the best approximate explanation of the data” and thus “can be considered as a special case of Peircean abduction” (Niiniluoto 2011, p. 137). Not only is measurand reconstruction a model-based inference, it is a model-based abduction: the value of the quantity is inferred from the output data by looking for the best explanation of the data.One of the critical points of abduction, Morawski emphasizes, is that it does not provide a unique solution. Because of the pervasiveness of measurement errors, measurand reconstruction involves inverse modelling under uncertainty and is therefore underdetermined by the data. Morawski shows how solving this inverse problem can be done by incorporating a priori information about the measurand in order to narrow the range of possible outcomes. Morawski offers a Bayesian example of implementation of such a priori information. While understanding measurement as abduction may appear as a simple, yet abstract proposition, Morawski’s aim is not just to restate Niiniluoto’s claim in the context of measurement, but instead to expand and enrich it, by devising practical applications in actual cases of measurand reconstruction. Although measurand reconstruction “may be trivial in the case of a scalar static measurand,” more complex situations call for more elaborate tools, such as “regularization” that Morawski presents in section 4, before providing a more thorough exemplification in section 5. As such, the paper ambitions to work “as a bridge between philosophy of science and research practice,” as “it enables one to take into account the plurality of the sources of uncertainty of scientific knowledge when designing measurement instrumentation” (Morawski 2021, pp. 742–756).In calling attention to questions of theory-dependence, the model-based account of measurement does not only cast aside the RTM’s foundationalist overtones; it naturally feeds into the debate on realism. It would seem, at first sight, that the model-based account of measurement should unequivocally undermine realism about measurement. Indeed, as Bas van Fraassen (2018, p. 271) pointed out, “ostensibly scientific descriptions refer to concrete entities and quantities in nature, but what are the referents if those descriptions can only be understood within their theoretical context?” Van Fraassen’s own empiricist account of theory assessment removes altogether the metaphysical question of the existence of referents of quantity terms in nature by merely focusing on the empirical adequacy between a model of the theory under test and a model of data obtained from the experiment. A different anti-realist take is found in Paul Teller, who specifically opposes traditional realist conceptions of measurement and accuracy on the account that quantity terms, unit terms and dimensional quantity terms fail to refer. This, he argues, is because, on the one hand, there cannot exist in the real world determinate quantities such as those “characterized in our idealized theories” (Teller 2018, p. 281)—especially since definitions vary according to the theory; and, on the other, it is impossible for linguistic representations, always using general terms, to become specific enough to pick out particulars.In response to these anti-realist positions, Alessandro Giordani and Luca Mari (2021) propose a defence of measurement realism from within the model-based approach. The authors acknowledge that the model-based account of measurement seems, on first sight, to seriously challenge a realist account of measurement. But, according to them, the pervasive role of models in measurement (in the definition of the measurand, the design of the measurement procedures, and the analysis of the measurement process) does not mean that truth in measurement and measurement reports are relative to theoretical assumptions. The model-based approach is compatible with a realist account of truth in measurement, provided the Fregean distinction between sense and reference is taken into account with regard to individual and general quantity terms (as well as units and dimensions).Although our theories and models determine the sense of the quantity terms—that is, the way the reference of quantity terms are presented to us—they do not fix the references of these terms themselves—what they actually relate to. What these terms relate to, argue the authors, is determined by the causal processes underlying the behaviour of the measurement setup which operates (in tune with Morawski’s meta-model of measurement) by transforming, or “transducing”, the physical property of interest (for instance, temperature) into another property of a different kind more easily accessible to the comparison with a standard (the expansion of mercury in a thermometer) in order to deliver a numerical output. The quantities involved in these transduction operations are thus independent of the theories and models that purport to capture them and constitute the facts in the world that account for the truth value of measurement reports, quite independently from the theories we use to interpret them (Geordani and Mari 2021, pp. 757–781). Although knowledge obtained by measurement is model-based and theory-dependent (Geordani and Mari 2021, pp. 757–781), measurement reports nevertheless have both content and truth value, and, duly associated with measurement uncertainties, succeed in laying hold of actual properties characterizing physical systems in the world which stand as genuine truthmakers for the measurement results.To Teller’s semantic anti-realism, the authors thus oppose a semantic realism that resonates with far-reaching problems in the philosophy of science. While their argument summons the Fregean distinction of sense and reference, it no less resolutely ties in with the rejection of the Fregean descriptive theory of reference and addresses, by the same token, the problem of the cognitive individuation of particulars. The classical Fregean idea that it is sense that provides access to the entities denoted by quantity terms lies implicitly at the core of Teller’s failure of reference argument. The arguments advanced by the authors imply on the contrary that references are by no means identified in a descriptive way by the statement of the theoretical or model dependent properties and concepts but are rather determined independently through causal relationships: enlarging on their analysis, one could say that the reference of a quantity term is not picked out by a description, but in a non-conceptual way as that pre-theoretical quantity “which is [causally] responsible for such and such effects” (Putnam 1979, p. 201) in certain experimental configurations (related to what the authors call an “i-quantity”). This externalist account of reference confers a trans-theoretical status to quantity terms and makes their reference independent from theory. It also invalidates Teller’s anti-realist argument based on the ideality of conceptual descriptions: conceptual resources are certainly not enough to cognitively select an individual; the latter can only be achieved by either using a proper name, or an indexical expression “pointing to the object under measurement in a context, as the i-quantity that triggers the dynamics of the measuring instrument in the context” (Geordani and Mari 2021, pp. 757–781).It remains of course a pending question if the quantities involved in the transduction processes actually “have the structure conjectured and incorporated in the model” (Geordani and Mari 2021, pp. 757–781) which serves to present them. This question is forcefully brought to the fore by the issue of measurement in chemistry analysed by Jean-Pierre Llored (2021). Llored’s article on measurement in chemistry brings the issue of reference into a new light which leads us to approach models from a different angle. The concern with the question of “what do the chemists’ measurement results refer to?,” or “what are chemists attempting to quantify?” does not relate so much to issues of theory- and model-dependence of the reference, but rather to the fact that, in chemistry (and also presumably in research areas such as biology and sociology), questions of context-dependence become overriding and create impediments to ceteris paribus reasoning which cannot be eliminated. One moves here away from the debate on realism to address the question of the reliability (and use) of measurement results in cases where the properties causally “responsible for such and such effects”—the references of quantity terms—cannot be disentangled from other causal lines: it is then not possible to safely make inferences or carry out concrete manipulations intended to involve such properties in a given context on the basis of measurement results that were obtained in other contexts. Here influence factors and disturbances from the experimental setup and processes cannot be dealt with by introducing corrections in a measurement model. In this respect, the “gap between concept and object” pointed out by Schummer (quoted by Llored 2021, pp. 782–801) appears to be much deeper in chemistry than in the situations investigated in the two previous articles.8The specific difficulties raised by measurement in chemistry, Llored argues, come from the fact that the behaviour of chemicals is modified by surrounding substances, even by the smallest amounts of impurities. Moreover, when preparing samples to be subjected to measurement, the products of the purification process remain irretrievably dependent on the “network including operations, instruments, transformations, and other purified bodies or mixtures” (Llored 2021, pp. 782–801), as well as on the matrix from which the sample is originated. In this respect, chemical bodies cannot be said to pre-exist in a compound. The reference of a property term is partly defined, constituted by the external conditions and bodies involved in obtaining it (Llored 2021, pp. 782–801) and from which it cannot be abstracted from. It becomes conspicuous that what the measurement result relates to, what is here causally responsible for certain effects cannot be conceptually isolated and safely captured by a description assuming purity which would claim to qualify it intrinsically; the reference has a relational dimension which must include the experimental circumstances. “The measurement is about this complex and not about the substance only. Chemists assign numbers and units to the outcome of experiments related to this complex” (Llored 2021, pp. 782–801).In chemistry, measurement results therefore cannot straightforwardly be made detachable; they are not comparable across different laboratories, since the conditions for obtaining them cannot be controlled so as to remain the same, which hinders their use in the development of reliable predictions. Llored shows how, faced with these predicaments, chemists resort to a methodological pluralism: they study the compatibility of the measurement outcomes and inferences obtained on the basis of heterogeneous, independent, unrelated complexes of methods, instruments, chemical surroundings, know-hows implemented by different teams, and they coordinate these outcomes in such a way as to provide paths of stable, holistic inferences.Llored’s analysis of measurement in chemistry has shown how the answer to the question of “what” is measured, “what” measurement is about, has to be modified in order to secure the intersubjective meaning and shared use of measurement results in complex situations—that is, apply ceteris paribus reasoning and be able to draw inferences from measurements. The inherent “epistemic limitations” (Llored 2021, pp. 782–801) of chemistry thus seemed to introduce strong pragmatic constraints leading the “semantic realism” defensible in measurement in physics to give way to a pluralistic and holistic approach designed to stabilize a network of chemical compounds, instruments and processes that prove sufficiently resilient to match scientific purposes.By focusing on the meandering history of the determination of units for measuring small quantities of chemical drugs to be administered to patients, Rebecca Jackson (2021) develops these pragmatic concerns in yet another direction. In the previous studies, measurement was primarily broached as an epistemic activity—a means to acquire knowledge and formulate knowledge claims. Jackson’s article allows us to underscore another, no less important function of measurement concerned with action and use. The issue of “what do we measure?” then shifts to: “what and who are the measurements for?” Regarding their use, measurement statements are not only to be evaluated in terms of their descriptive function, with respect to how accurately they represent a quantity or a state of affairs. They also have to be judged in terms of their performance, of their suitability for the intended application—their “felicitous” or “infelicitous” character, to borrow John L. Austin’s terminology (Austin 1962). Although correlated, description and use are liable to be in tension with each other to the point that the scientific, epistemic purposes of measurement may seriously come to affect practical ones. Jackson shows indeed how the attempts to define units and develop standards in order to express small amounts of medicinal substances as accurately as possible in the context of general anaesthesia prompted a conflict between descriptive and use-oriented functions of measurement reports that finally led to temper the pursuit for ever greater accuracy and standardization.Jackson studies a case paramount to public health: the administration of specific amounts of drugs across different contexts and uses. At first glance, the definition of stable, uniform units that can be realized in reproducible and shareable standards appeared to be indispensable in conveying objective information about medicine doses and providing clear instructions to guide practice. Thus, early in the nineteenth century, the minim was introduced as a more uniform and reliable unit for medical dosage than the drop, which notoriously depended on multiple, uncontrollable factors (liquids with different surface tension generate drops of different volume). However, the definition of the minim through a calibrated glass tube not only proved to have its own drawbacks in terms of accuracy; it turned out to be utterly inadequate for achieving vitally important pharmaceutical and medical objectives. The minim could indeed serve as a rhetorical tool