Structure the literature review on idealizations chapter in a less "this is what...

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## Norton

### Characterizations of idealization

Characterizations of idealizations are abound in the literature, although few philosophers dare claim to provide an exhaustive one. @Norton2012 's characterization of idealization as contraposed to approximation has been well received in the literature:

- Approximation: proposition which sort of describes the target
- Idealization: model whose properties sort of describe the target

Norton's standard example is a ball falling through the atmosphere. We have an excellent formula for calculating the velocity $v(t)$ starting at $v=t=0$ as

$$v(t)=\frac{g}{k}(1-\exp(-kt))=gt-\frac{gkt^2}{2}+\frac{gk^2t^3}{6}-\ldots$$
where $g$ is the acceleration due to gravity, and $k$ a friction coefficient.

As any physics student knows, it is often convenient to leave out the influence of air resistance. Setting $k=0$ gives us the much simpler $$v(t)=gt$$ 
This new equation is clearly easier to work with than the bulky on we had before, at the cost of a little precision.

Using Norton's distinction, we can view the above formula in two ways: either as an approximation or as an idealization. If we view it as an approximation, i.e., a proposition describing the actual system (a ball falling through the Earth's atmosphere) inexactly, we can say that this is a good approximation for the initial part of the ball's descent, becoming a worse and worse approximation as the ball continues to fall. We could also view it as an idealization, in which case we view the above equation as representing a (false) model of a ball falling in a vacuum. The description of the model is not false, rather the model (idealization) *itself* intentionally misrepresents our actual world, namely by leaving out air.

In this example, the difference between an approximation and an idealization is simply a matter of perspective: we can freely shift between viewing the equation as an approximation or idealization, as a simplified equation for calculation or a model describing an atmosphere-less Earth. This “shifting” Norton calls *[[Promotion|promotion]]* and *[[Demotion|demotion]]* respectively. Promotion consists in taking the approximation and taking it as the basis for describing a new, idealized model. In this case, removing the effect of air resistance is promoted to an idealized situation of an Earth without any atmosphere.  Demotion on the other hand consists in extracting one or more propositions from an existing idealization and applying them as inexact propositions to the actual system, e.g. seeing that in No-Air-Earth the speed of the ball is $v(t)=gt$ and applying it to the ball falling in our actual Earth. 

First in @Norton2012 and more in depth in @Norton2014, Norton describes his unease with so-called “infinite idealizations”. Although never providing a strict definition, we can make an educated guess to what he means by it:

>[!definition] **Infinite Idealization (Norton)**
>
> An infinite idealization is made by performing a limiting operation on an idealized system, taking some parameter (such as length, number, volume) to either zero or infinity. The infinite idealization is a new system, namely the old system with the parameter _set_ to zero or infinity. However, these systems sometimes misbehave: either by the limit system not being logically possible or conflicting in some other way with other assumptions we hold.

This characterization somewhat goes against the spirit of what Norton intends to argue, namely that such infinite idealizations are not idealizations at all, but can only be sensibly understood as _approximations_ as defined ==above==. For "the essential starting point of the notion of idealization is that we have a consistently describable system, even if it is fictitious." [@Norton2014, pp. 200]  We, however, should not assume such stringent requirements for idealizations, as whether “being consistently describable” is a good feature for a model to have is *exactly* what is currently under discussion. For now, we will (contra Norton) refer to these systems as infinite idealizations.

Norton furthermore distinguishes between _well-behaved_ and _ill-behaved idealizations_.[^well-behaved] _Ill-behaved_ idealizations are infinite idealizations whose limit system (the system with the parameter set to zero or infinity) does not match with target system in some way. This mismatch can take two forms [@Norton2012, (3.2, 3.3)]: the limit system might not exist, e.g., an infinite sphere, or the limit system might have a property which conflicts with a property of the target system. For the former, if we define a sphere as all points which are equidistant from some other point, then an infinite sphere does not exist, as there are no points at infinity. ($\mathbb{R}=(-\infty, \infty)$ not $[-\infty,\infty]$) 

For the latter, Norton imagines modeling an arbitrarily long ellipsoid as an infinite cylinder. While they look similar, the ratio of surface to volume for an ellipsoid is different from that for a cylinder, so the idealization has a fundamental mismatch.




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