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Mathematics as a Tool pp 213–232Cite as

Idealizations in Empirical Modeling

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Part of the book series: Boston Studies in the Philosophy and History of Science ((BSPS,volume 327))

Abstract

In empirical modeling, mathematics has an important utility in transforming descriptive representations of target system(s) into calculation devices, thus creating useful scientific models. The transformation may be considered as the action of tools. In this paper, I assume that model idealizations could be such tools. I then examine whether these idealizations have characteristic properties of tools, i.e., whether they are being adapted to the objects to which they are applied, and whether they are to some extent generic.

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Notes

  1. 1.

    Mutual adaptedness seems to be a specific property of some ordinary tools. Here, not only the tool is adapted to the object on which it is used, but also the object is adapted to the tool. This is a case of screwdrivers and screws, Allen keys and bolts, or hammers and nails. Some ordinary tools do not share this property, however (e.g., rakes, scissors and shovels). Because this property is specific, it will not be considered in this paper.

  2. 2.

    Abstractions differ from idealizations in that they are omissions of some aspects in the target system which are not relevant for the problem being studied (e.g., to neglect gravitational force in subatomic phenomena), whereas idealizations are distortions (Jones 2005; Godfrey-Smith 2009). Idealizations can be omissions but, in this case, these omissions distort the representation in that they are omissions of relevant aspects.

  3. 3.

    Abstracting can sometimes later be part of mathematization. For instance, difficulties in formulating equations might occur and lead to a different abstraction. This is, however, an additional aspect that I do not treat in this paper.

  4. 4.

    What is relevant might actually depend on the final success in constructing a useful model, and therefore be identified as such at later stages of modeling. In such cases, there might be some back and forth in the process of model building.

  5. 5.

    I shall stress that this way of defining idealization differs from the view on which model as a whole is an idealization. It is considered here that idealizations are only parts of a model. Unlike models, idealizations have no inferential power on their own. For instance, the Ising model will not be considered as an idealization but as being composed of idealizations. A mass point is an idealization, but is not a model in that, alone, it has no inferential power.

  6. 6.

    I borrow the distinction mathematical vs. formal idealizations from McMullin (1985).

  7. 7.

    I take this example from Norton (2012) for a different purpose though.

  8. 8.

    Knudsen number is a dimensionless parameter. It indicates the flow regime depending on the fluid continuity (while Reynolds number indicates the flow regime depending on the turbulence).

  9. 9.

    The development of cellular automata is more recent than the development of differential equations since it started in the 1940s with the work of Ulam and von Neumann at Los Alamos. For a general philosophical discussion on cellular automata, see e.g., Fox Keller 2003 and Rohrlich 1990. There are also different hydrodynamics models based on cellular automata. For an exhaustive presentation of these models, see the forthcoming paper of Barberousse, Franceschelli and Imbert entitled “Cellular Automata, Modeling, and Computation” and Barberousse and Imbert (2013).”

  10. 10.

    The fact that idealizations have a certain scope of application may explain why some models are repeatedly used within and across scientific domains, e.g., the harmonic oscillator, the Ising model, a few Hamiltonians in quantum mechanics, the Poisson equation, or the Lokta-Volterra equations (see Barberousse and Imbert (2014) for an analysis of such recurring models).

  11. 11.

    In the case of waves of water, the equation describes the height of the water at the point x, with a being the maximum height or amplitude of the ripples, and f their frequency. In the sound model, it describes the amplitude of a sound wave at the point x, with a being the loudness and f the pitch. In the light model, it describes the amplitude of a light wave, with a being the brightness and f the color.

  12. 12.

    Some idealizations sometimes denote relevant aspects of the target model that de-idealization would fail to capture. These idealizations are called ineliminable (or essential) (Batterman 2005, 2009; Sklar 2000). They cannot be removed without losing the explanation of the phenomenon that is studied. This is the case with the thermodynamic limit, according to which the number of atoms in the system is infinite, which is necessary for explaining phase transitions, and in particular, the phase transition of a magnet at a certain critical temperature (Batterman 2005).

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Authors and Affiliations

  1. IRFU/Service d’Astrophysique, CEA Paris-Saclay, 91191, Gif-sur-Yvette, France

    Julie Jebeile

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  1. Julie Jebeile
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Correspondence to Julie Jebeile .

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  1. Department of Philosophy, Bielefeld University, Bielefeld, Germany

    Johannes Lenhard

  2. Department of Philosophy, Bielefeld University, Bielefeld, Germany

    Martin Carrier

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Jebeile, J. (2017). Idealizations in Empirical Modeling. In: Lenhard, J., Carrier, M. (eds) Mathematics as a Tool. Boston Studies in the Philosophy and History of Science, vol 327. Springer, Cham. https://doi.org/10.1007/978-3-319-54469-4_12

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