Abstract
I argue that an adequate understanding of the practice of constructing models in physics requires a distinction between two strategies that are commonly both labeled ‘idealization’. The formal characteristic of both methods is to let a parameter in the equations for a target system go to zero. But the discussion of examples from various applications of perturbation theory shows that there is in general a difference with respect to the aims such limiting procedures are supposed to serve; and with different aims comes the need to characterize the means (the interpretation of the limits) differently. I therefore suggest that we distinguish ‘idealizations’ from ‘perspectives’ or perspectival representations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
There are other views on idealization in which this problem is addressed. Cartwright (1989), for instance, has insisted on a distinction of ‘idealizations’ and ‘abstractions’. While idealized models have a place for a factor that in principle influences the behavior of the system but set this factor equal to zero (and hence make a literally false claim about the system), abstract models omit the factor altogether and are therefore, in some sense, saved from claiming what is not true.
These labels are not optimal. I do not want to deny that all scientific representation may be, in some sense, perspectival (cf. Fraassen 2008). Giere, for instance, uses such a wide sense of perspective in his “perspectival realism”, in which one can speak of a “Newtonian perspective”, a “Maxwellian perspective”, etc. (2006, p.14) As we’ll see in Sect. 2, the sense I give to perspective is much more restricted, more literal. Nothing hangs on these labels; what counts is that a distinction be made.
See for instance Holmes (1995, Chap. 4).
We’ll see in Sect. 2 that this constraint is important.
See Holmes (1995, p. 106).
For further defense of this view see, e.g., Batterman (2009).
Pincock uses a different example; I have therefore adapted the symbols in the quotations.
Pincock himself has discussed this example in some detail in Pincock (2009).
See, for instance, Chorin and Marsden (1990, pp. 69).
Pincock does not seem to acknowledge this problem. He writes: “Maybe an account of idealization which is developed for the regular case will fail when it is extended to the singular case. On a first pass, it is hard to see why this mathematical difference would correspond to any difference in the interpretative significance of the representations which result. In both ... cases a mathematical transformation decouples part of the representation from its original interpretation.” (Pincock, 22).
The ordinary derivative formally has to be replaced in Eq. (5) by a partial derivative: d/dx \(\rightarrow \partial / \partial x +(1/ \upvarepsilon ) \partial / \partial y\).
This new approximation of the target is not good when we move away from \(\mathrm{x}=0\); the old approximation—u\((\mathrm{x}) =\mathrm{x}\)—, which failed for \(\mathrm {x} \rightarrow 0\), is much better in that regime. One can however construct a universally valid approximation by ‘matching’ the two approximations and combining them. See, e.g., (Holmes (1995), Chap. 2).
See Rueger (2005).
Comparison with the exact solution, Eq. (4), shows that the new model captures only the change in amplitude but not the shift of frequency \(\upomega _{0}^{2} \rightarrow \upomega _{0}^{2}-\upvarepsilon ^{2}\)/4. To account for both modifications, one has to re-scale the frequency itself, \(\upomega (\upvarepsilon )=\upomega _{0}+\upvarepsilon \upomega _{1}+\upvarepsilon ^{2}\upomega _{2} + {\cdots }\) and the time as \(\mathrm {t}^{*} = \mathrm {t} \upomega (\upvarepsilon )\). The new constants \(\upomega _\mathrm{i}\) can then be determined so as to cancel secular terms in the expansion for x(t). Such re-scaling of the frequency is analogous to a renormalization of \(\upomega \) which is now distinguished from the ‘bare’ frequency \(\upomega _{0}\) in the initial statement of the problem. Cf. (Castiglione et al. (2008), p. 221) and below, footnote 23.
See, for instance, Holmes (1995, Chap. 5).
For D think, for example, of a function like D(x, x/\(\upvarepsilon )=\mathrm{sin} (\frac{2\pi x}{\upvarepsilon }).\)
The x-dependence is integrated over and the \(\upvarepsilon \)-dependence vanishes in the limit.
The arithmetic mean would be \(\mathrm {lim}_{\upvarepsilon \rightarrow 0} \int \nolimits _0^1 D\left( {x,\frac{x}{\varepsilon }} \right) dx.\)
Batterman (2013, p. 267) has also emphasized this point.
See Weisberg (2007) for discussion.
For examples of this, see, e.g., (Holmes (1995), p. 12) or (Hinch (1991), p. 19) Hinch remarks that “worrying about the higher terms [of an asymptotic expansion] as N \(\rightarrow \infty \) runs counter to the philosophy of asymptoticness in which the first term is virtually correct as \(\upvarepsilon \rightarrow 0\).” (1991, p. 21)
Such absorption of the details into effective coefficients is also a characteristic of renormalization group approaches to systems that exhibit behavior at different scales. For a discussion of the close relations between the multi-scale techniques to singular perturbation problems and the renormalization group approach, see Castiglione et al. (2008, Chap. 7, 8) and Batterman (2013).
References
Barenblatt, G. (1996). Scaling, self-similarity, and intermediate asymptotics. Cambridge: Cambridge University Press.
Batterman, R. (2009). Idealization and modelling. Synthese, 169, 427–446.
Batterman, R. (2013). The tyranny of scales. In R. Batterman (Ed.), Oxford handbook of the philosophy of physics (pp. 256–286). Oxford: Oxford University Press.
Cartwright, N. (1989). Nature’s capacities. Oxford: Oxford University Press.
Castiglione, P., Falcioni, M., Lesne, A., & Vulpiani, A. (2008). Chaos and coarse graining in statistical mechanics. Cambridge: Cambridge University Press.
Chorin, A., & Marsden, J. (1990). A mathematical introduction to fluid mechanics (2nd ed.). New York: Springer.
Giere, R. (2006). Scientific perspectivism. Chicago: University of Chicago Press.
Hinch, E. J. (1991). Perturbation methods. Cambridge: Cambridge University Press.
Holmes, M. (1995). Introduction to perturbation methods. New York: Springer.
Pincock, C. (2013). How to avoid inconsistent idealizations. Synthese (in press).
Pincock, C., et al. (2009). Towards a philosophy of applied mathematics. In O. Bueno (Ed.), New waves in the philosophy of mathematics (pp. 173–194). London: Palgrave.
Rueger, A. (2005). Perspectival models and theory unification. British Journal for the Philosophy of Science, 56, 579–594.
Torquato, S. (2002). Random heterogeneous materials. New York: Springer.
Van Fraassen, B. (2008). Scientific representation. Oxford: Oxford University Press.
Weisberg, M. (2007). Three kinds of idealization. Journal of Philosophy, 104, 639–659.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rueger, A. Idealized and perspectival representations: some reasons for making a distinction. Synthese 191, 1831–1845 (2014). https://doi.org/10.1007/s11229-013-0371-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-013-0371-4