Abstract
Idealized scientific representations result from employing claims that we take to be false. It is not surprising, then, that idealizations are a prime example of allegedly inconsistent scientific representations. I argue that the claim that an idealization requires inconsistent beliefs is often incorrect and that it turns out that a more mathematical perspective allows us to understand how the idealization can be interpreted consistently. The main example discussed is the claim that models of ocean waves typically involve the false assumption that the ocean is infinitely deep. While it is true that the variable associated with depth is often taken to infinity in the representation of ocean waves, I explain how this mathematical transformation of the original equations does not require the belief that the ocean being modeled is infinitely deep. More generally, as a mathematical representation is manipulated, some of its components are decoupled from their original physical interpretation.
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Notes
The wave case is again noted at Maddy (2007, p. 316).
See also Colyvan (2009).
In my reconstruction I am ignoring the subtleties needed to find the pressure throughout the fluid.
There are other features of water wave dispersion that I am ignoring here for simplicity of presentation.
A somewhat surprising fact is that it is actually the shallow-water idealization that is used to understand the ocean in geophysical applications. See, for example, Kundu and Cohen (2008, ch. 14). This is because there is a natural boundary in the ocean known as the thermocline. It is the region of greatest temperature gradient as we travel downwards. The bottom of the thermocline can then be treated as the lower boundary of a relatively thin layer of fluid which affects the long wavelength waves relevant to geophysics.
The interpretive issues are discussed briefly at Segel (2007), pp. 340–341.
Indeed, determining the roots directly, say using the quadratic formula, yields values for \(m\) in terms of \(\epsilon \) that are undefined when \(\epsilon = 0\).
See also Batterman (2002, p. 96).
See Kundu and Cohen (2008, pp. 4–5) for an informal justification of this “continuum hypothesis”.
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Acknowledgments
Earlier versions of this paper were presented to the Society for Realist/Antirealist Discussion (Group Session), Central Division American Philosophical Association Meeting, Chicago, IL, USA, February 2009 and the Department of History and Philosophy of Science, University of Athens, Athens, Greece, March 2009. I am grateful to both audiences for their useful suggestions. In addition, I would like to thank the issue editors and the anonymous referees for their help in improving this paper.
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Pincock, C. How to avoid inconsistent idealizations. Synthese 191, 2957–2972 (2014). https://doi.org/10.1007/s11229-014-0467-5
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DOI: https://doi.org/10.1007/s11229-014-0467-5