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1. Lord Kelvin famouslystated that “when you can measure what you are speaking about,and express it in numbers, you know something about it; but when youcannot measure it, when you cannot express it in numbers, yourknowledge is of a meagre and unsatisfactory kind: it may be thebeginning of knowledge, but you have scarcely, in your thoughts,advanced to the stage of science” (Thomson 1889: 73).

2. In what follows I will usethe word “object” to refer to a system under measurement.This designation is meant to cover processes and events as well. Aswill be clarified shortly, what exactly constitutes a system undermeasurement is a debated topic among measurement theorists.

3. While mathematicalmeasurement theories consider the Celsius zero point to be arbitraryin principle, from a practical perspective the selection of fixedpoints has involved complex empirical considerations. See Chang 2004:Chapter 1.

4. Kyburg takes“direct” measurement to involve only unaided observationand does not require that such measurements be additive. Hence forKyburg the ordering of minerals by hardness constitutes directmeasurement because the scratch test does not involve the measurementof any other quantity, while the use of an equal-arm balance tomeasure weight constitutes indirect measurement because it involvesmeasuring the angle of the arms’ deflection. See Kyburg (1984:Ch. 5–6) for discussion.

5. The Mohs scale is based ona scratch test, i.e., on observing which materials can scratch others.Other methods of measuring hardness, like those proposed by Brinell,Vickers and Knoop, allow representing hardness on ratio scales. SeeTabor (1970) for discussion.

6. As the same number mayrepresent several objects, e.g., different rods of the same length,RTM focuses on many-to-one rather than one-to-one mappings (cf. Krantzet al. 1971: 8 ff. 1).

7. The RepresentationalTheory of Measurement is closely related to the semantic view ofscientific theories, to which Suppes is a key contributor (Suppes1960, 1962, 1967). According to Suppes’ version of the semanticview, a scientific theory can be thought of as a hierarchy ofmodels. Here the term “model” is borrowed frommathematical logic, and denotes a structure that satisfies certainaxioms. RTM analyzes measurement as the construction of morphismsbetween mathematical and empirical models, and in this respect can beviewed as a successful application of the semantic view of theories(Suppes 1967: 59).

8. Bridgman’sempiricist caution was inspired by the success of Einstein’sspecial relativity theory, which exposed the naive assumptions behindclassical, substantivist conceptions of space and time and replacedthem with operational concepts. Bridgman’s operational analysiswas intended to “render unnecessary the services of the unbornEinsteins” (1927: 24).

9. Operationalists did notalways make explicit their metaphysical and semantic commitments,leaving some room for interpretation. In Bridgman’s case, hisinitial strong operationalist views were tempered in his later works.See also fn. 10.

10. See also Gillies(1972). Bridgman later revised his account and no longer claimedoperationalism was a comprehensive theory of meaning (1938; Chang2009: section 2.1). For Bridgman’s later views see also his1945, 1956.

11. For an exception, seeDingle (1950). Recently, Chang (2004: 141–158) has defended amoderate version of operationalism that does not involve the reductionof meaning to operational definition. Instead, Chang recastsoperationalism as a theory of the semantic extension of concepts tonew domains of application.

12. Thatsomeaspects of measurement are conventional is undisputed. Whether wemeasure temperature on the Celsius or Fahrenheit scales, or whether weuse the meter or inch as a unit of length, are choices that ultimatelyhang on consensus among humans rather than facts about nature.Conventionalism about measurement aims to additionally show that somenontrivial aspects of the application of quantity-concepts,previously not thought to rest on human consensus, are in factconventional.

13. An important refinementof operationalist and conventionalist traditions is provided by BrianEllis in hisBasic Concepts of Measurement (1966). Instead ofdefining quantity concepts in terms of particular operations asBridgman did, Ellis views quantity concepts as “clusterconcepts” that may be “identified by any one of a largenumber of ordering relationships” (1966: 35). Differentinstruments and procedures may therefore measure the same quantity.What is common to all and only those procedures that measure the samequantity is that they all produce the same linear order among theobjects being measured:

If two sets of orderingrelationships, logically independent of each other, always generatethe same order under the same conditions, then it seems clear that weshould suppose that they are ordering relationships for the samequantity. (1966: 34)

14. Despite theseobjections, it is worth noting that some realists have made partialconcessions to moderate conventionalist claims. In particular, bothSwoyer (1987: 257–8) and Michell (1994: 398) agree that amultiplicity of relational structures exist on the reals that satisfythe axioms of additivity, and that the choice among them isarbitrary.

15. The realistinterpretation invokes Russell’s (1903: Ch. 19) distinctionbetween “absolute” and “relative” theories ofmagnitude and advocates the absolute conception, as did Russellhimself.

16. See also van Fraassen(1980: 58–59).

17. This notion of“model” is developed by Morrison (1999), Morrison andMorgan (1999) and Cartwright (1999), among others.

18. Frigerio et al. (2010)employ a slightly different terminology and divide the secondsub-condition into two: (ii) objectivity, i.e., the mutual consistencyof measurement outcomes across different environments; and (iii)intersubjectivity, i.e., the mutual consistency of measurementoutcomes across different measuring instruments and metrologicalmodels. They call setups that satisfy objectivity but notintersubjectivity “pre-measurement” setups (2010:136). See also Mari (2000).

19. Models are also used toanalyze data in economics experiments, and to test the robustness ofexperimental results across different circumstances and socialcontexts. Francesco Guala (2008) argues that certain experimentaldesigns, such as the Ultimatum Game, function like standardizedmeasuring instruments in that they allow experimenters to discoverinter-cultural variations among subjects’ behaviors. Bydetecting discrepancies between the predictions of a self-interestedrationality model and actual experimental results, investigators areable to elicit phenomena that suggest differences in norms of fairnessacross different cultures.

20. See also McClimans& Browne (2012) and Angner (2013). For additional challenges tomeasurement in the social sciences see Chang and Cartwright (2008:372–5).

21. For the sake ofstandardization, a quantity concept may be any concept employed inquantitative representation, such as a kind of quantity (e.g.,temperature), a relation among quantities (e.g., equality amongtemperature intervals), a measurement unit, or a measurementscale.

22. Note thatstandardization by itself does not imply any kind of operationalism orconventionalism. The prescription of a determinate mode of applicationfor a concept is required for its consistent application, regardlessof the question whether such prescription determines the meaning ofthe concept or whether it involves nontrivial choices.

23. Shapere (1982)describes the counting of solar neutrinos as an observation of thecenter of the sun, because indications from neutrino receptors can bereliably used to infer properties of the sun’s core. This broadconstrual of the concept of observation is motivated by an analogybetween sensory organs and measuring instruments. However, even thisbroad construal of “observation” assumes a causalconnection between the particular objects being observed and theindications being produced, a connection that is not present in thecases Morrison and Parker describe. See Israel-Jost (2011) for arelated discussion.