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Individuating quantities

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Abstract

When discrepancies are discovered between the outcomes of different measurement procedures, two sorts of explanation are open to scientists. Either (i) some of the outcomes are inaccurate or (ii) the procedures are not measuring the same quantity. I argue that, due to the possibility of systematic error, the choice between (i) and (ii) is underdetermined in principle by any possible evidence. Consequently, foundationalist criteria of quantity individuation are either empty or circular. I propose a coherentist, model-based account of measurement that avoids the underdetermination problem, and use this account to explain how scientists individuate quantities in practice.

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Notes

  1. A canonical example is Perrin’s different measurements of Avogadro’s number. See Hudson (2018) and references therein.

  2. Dingle (1950) provides an account of quantity individuation very similar to Bridgman.

  3. The notion of value region is to be construed broadly to include e.g. unique values, value intervals, and probability density functions over a set of values.

  4. The type-token distinction among quantities is not completely sharp. Whether a quantity is considered a type or a token depends on the level of detail in the specification of the quantity and on how finely one individuates the objects or events to be measured.

  5. In some cases quantified instrument indications are obtained by a secondary measurement of a part of the instrument itself, e.g. when measuring the height of the mercury column of a thermometer. Such recursions are allowed as long as they bottom out, i.e. as long as they culminate in indications that can be obtained without further measurement.

  6. My terminology follows the official vocabulary of the International Bureau of Weights and Measures as published by the Joint Committee for Guides in Metrology in two guides: the International Vocabulary of Metrology (VIM) (JCGM 2012) and the Guide to the Expression of Uncertainty in Measurement (GUM) (JCGM 2008).

  7. For discussion of type-A and type-B uncertainties see JCGM (2008). Note that the distinction between type-A and type-B uncertainty is unrelated to that of type I versus type II error. Nor should it be confused with the distinction between random and systematic error.

  8. This conception of models follows closely the views expressed in Morrison and Morgan (1999), Morrison (1999) and Cartwright (1999).

  9. Some systematic errors can be evaluated purely statistically, such as random walk noise (aka Brownian noise) in the frequency of electric signals.

  10. See Mach (1966 [1896]), Ellis (1966), Carnap (1995 [1966], Ch. 6), Chang (2004, Ch. 2) and van Fraassen (2008).

  11. This is a consequence of what Chang (2004, 57) calls the ‘Problem of Nomic Measurement’.

  12. The specific quantity in question may be a specific quantity token, like the temperature of a particular object, a specific quantity type, like the temperature of a type of object, or both. A direct consequence of assuming that the procedures measure the same specific quantity is that they also measure the same general quantity type, e.g. temperature.

  13. A constant-volume gas thermometer measures temperature through variations in the pressure of the gas under constant volume.

  14. This assumption is distinct from Chang’s principle of single-valuedness (2001, 11, 2004, 90). The common quantity assumption does not require quantities to exist mind-independently, nor quantity values to be completely sharp. Moreover, the common quantity assumption is empirically testable (albeit not in isolation) rather than postulated a priori.

  15. Elsewhere I have called this equation a ‘calibration function’ (Tal 2017a, 35).

  16. For a more detailed description of this method see Tal (2011).

  17. See also the VIM definition of “Compatibility of Measurement Results” (JCGM 2012, 2.47).

  18. For a similar criticism of operationalism see Hempel (1966, Ch. 7).

  19. By ‘intervening’ I do not mean to suggest that measurement necessarily affects the object being measured—a distant star is not affected by measuring its location—but only that concrete actions are required to obtain and record the indications of a measurement process.

  20. The model-based account of measurement is neutral about the specific ontology of models. As long as models can fulfill certain functions, such as abstraction, idealization and prediction, it is immaterial to my account whether they are linguistic entities, abstract objects, fictions or something else.

  21. The indications are, of course, collected before the outcome is obtained. Hence measurement involves a backward inference from indications to their best predictors.

  22. What makes a parameter quantitative is a question that cannot be addressed in sufficient depth in the present work. In the following I will assume that quantities have sufficient structure to be represented on interval scales.

  23. Realists about measurable quantities would argue that the best explanation for the success of measuring some quantities is their mind-independent existence. For realist accounts of quantity see Byerly and Lazara (1973) and Swoyer (1987).

  24. Interestingly, the rich literature on construct validation in psychometrics arrives at a similar condition of quantity individuation. See Cronbach and Meehl (1955).

  25. Diez (2002) discusses this sort of weak individuation of concepts under the heading ‘formal content’.

  26. The geoid is an imaginary surface of equal gravitational potential that roughly coincides with the earth’s sea level.

  27. The core idea behind this method is known as Galilean idealization (McMullin 1985).

  28. I am grateful to Wayne Itano of NIST for discussing this episode with me.

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Acknowledgements

The author would like to thank Margaret Morrison, Ian Hacking, Anjan Chakravartty, Mary Morgan, Jacob Stegenga, Allan Franklin and Aaron Zimmerman for helpful comments on drafts of this article. I am thankful to Richard Healy for inviting me to speak at the 2018 meeting of the Pacific Division of the American Philosophical Association and for nominating my paper for publication. I am also grateful for the feedback I received from audiences at the University of Hannover, Bielefeld University, University of South Carolina, University of Cambridge, and the conference “Error in the Sciences” held in Leiden in 2011. This work was supported by a Research Grant for New Academics from the Fonds de Recherche du Québec-Société et Culture (FRQSC) (Grant No. 2018-NP-205463).

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  1. Department of Philosophy, McGill University, Leacock Building, 855 Sherbrooke St. West, Montreal, QC, H3A 2T7, Canada

    Eran Tal

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Tal, E. Individuating quantities. Philos Stud 176, 853–878 (2019). https://doi.org/10.1007/s11098-018-1216-2

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