Abstract
In this note, I apply Norton’s (Philos Sci 79(2):207–232, 2012) distinction between idealizations and approximations to argue that the epistemic and inferential advantages often taken to accrue to minimal models (Batterman in Br J Philos Sci 53:21–38, 2002) could apply equally to approximations, including “infinite” ones for which there is no consistent model. This shows that the strategy of capturing essential features through minimality extends beyond models, even though the techniques for justifying this extended strategy remain similar. As an application I consider the justification and advantages of the approximation of a inertial reference frame in Norton’s dome scenario (Philos Sci 75(5):786–798, 2008), thereby answering a question raised by Laraudogoitia (Synthese 190(14):2925–2941, 2013).
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Notes
This assumes some delicate details about how classical mechanics ought to treat systems of infinite mass. However, even if one takes infinitely massive systems as falling outside the scope of classical mechanics, the conclusion of the argument would still hold, for in that case no reference frame of an object experiencing a net force can be inertial.
Laraudogoitia (2013, p. 2930) draws the further conclusion that the “infinite mass must be distributed across an infinite region of space because infinite mass distributed in a finite region of space has no physical meaning, at least if one does not ‘decouple’ the gravitational interaction,” i.e., no longer require that gravitational and inertial mass be equal to each other. This conclusion is correct, but not for quite the reasons Laraudogoitia gives. There are perfectly mundane distributions of total infinite mass in a finite region of space—consider, for example, the mass density \(\rho (r)\propto 1/r\) for \(0< r < 1\), and \(\rho (r) = 0\) otherwise—but the mass concerned must nevertheless be distributed in an infinite region of space in order to generate a uniform gravitational attraction. In any case, this conclusion is not essential for the present discussion.
In the terminology of Norton (2008), this is called a strong failure of internal idealization. The dome scenario has also been criticized as involving illegitimate idealizations, e.g. of infinite rigidity (Korolev 2007a, b), but Norton (2008, p. 795) correctly points out that there can be no failure of idealization—what he calls in that paper “external idealization”—if the model in question is not intended to represent an independently specified system or phenomenon.
In fact, one can show that \(|\varvec{\gamma }(r)|\) grows no faster than Ar for some \(A>0\) when r is small, as its derivative at \(r=0\) is positive and finite. To show this, without loss of generality select units in which \(b^2 = g\). Then \(|\varvec{\gamma }(r)| = \frac{2}{3}[(1-(1-r)^{3/2})^2 + r^3]^{1/2} = \frac{2}{3}[2-3r-2(1-r)^{3/2} + 3r^2]^{1/2}\). Direct calculation shows that
$$\begin{aligned} \frac{d|\varvec{\gamma }(r)|}{dr} = \frac{(1-r)^{1/2} + 2r - 1}{[2-3r-2(1-r)^{3/2} + 3r^2]^{1/2}}. \end{aligned}$$Although the limit \(r \rightarrow 0\) yields an indeterminate form, one can perform the substitution \(z = (1-r)^{1/2}\), rewrite the resulting fraction as a square root with which one commutes the limit \(z \rightarrow 1\), then apply l’Hôpital’s rule.
This requirement implies, of course, that the particle must not lose contact with the dome, at least initially. See Malament (2008) for a discussion of this issue in the original dome scenario.
In this sense, minimal approximations fit with the perspective of Knuuttila and Boon (2011), who wish to reduce the representational role for models in favor of their use as tools to forge more general representational relationships.
See also Tamir (2012).
References
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Thanks to two anonymous referees and to Otávio Bueno for suggestions to clarify my argument, especially in Sect. 3.3.
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Fletcher, S.C. Minimal approximations and Norton’s dome. Synthese 196, 1749–1760 (2019). https://doi.org/10.1007/s11229-018-1676-0
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DOI: https://doi.org/10.1007/s11229-018-1676-0