Good introduction of the literature review on infinite idealizations

[github-actions]

Specifically, make it clear WHY you are looking at these sources, what you hope to gleam from them, and why doing such a survey is useful to begin with.

https://github.com/ThomasFKJorna/thesis-writing/blob/c2380155eb1597c6aa72ebbf6d62e6eaed0ac2a7/Chapters/II. Idealizations/2. Previous work on infinite idealizations.md#L6

# 2. Previous work on infinite idealizations



{/**
	* TODO: Good introduction of the literature review on infinite idealizations
	* Specifically, make it clear WHY you are looking at these sources, what you hope to gleam from them, and why doing such a survey is useful to begin with.
	* labels: write, II
	*/}

	

{/**
	* TODO: Expand the current literature review on infinite idealizations with at least one more source
	* labels: write, II
	*/}



## Norton

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 one:

>[!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 the limit not being logically possible or conflicting in some other way with other assumptions we hold, ==which is bad==.

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 should however not assume such strong requirements for idealizations, as whether "being consistently describable" is a good feature for a model to have _is_ what is under discussion. For now we will 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.


{/**
	* TODO: Create figure which shows the differences between ellipsoid and cylinder volumes
	* labels: visualization, II
*/}


In short: for Norton infinite idealization simply is the end result of the process of a limiting operation. Furthermore, these idealizations can sometimes be well-behaved, and sometimes ill-behaved.

## Strevens

@Strevens2019a defines infinite idealization (or “asymptotic idealizations” as he calls them, we will stick with infinite here) slightly differently than Norton. Luckily, Strevens does provide a clear definition, which is in contrast to what he calls a “simple” idealization, which "is achieved by the straightforward operation of setting some parameter or parameters in the model to non-actual values, often zero". A clear example is the air-resistance coefficient above. At first, he contrasts this straightforwardly with infinite idealizations in the Nortonian sense, as “in \[infinite\] idealization, by contrast, a fiction is introduced by taking some sort of limit”. We will take this definition to be identical to Norton's.

However, later on in the paper Strevens adds another layer to the definition, 
{/*TODO: find a good quote for Strevens adding another layer to the definition for idealizations*/}
 namely that scientists use infinite idealizations when it is not possible to use a simple idealization to directly set the relevant property to zero (or infinity) 
{/*TODO: Clarify the distinction between infinite and "normal" idealizations for Strevens*/}
. Furthermore, he adds, “\[Infinite\] idealization is an interesting proposition, then, only in those cases where a simple substitution cannot be performed, which is to say only in those cases where a veridical model for mathematical reasons falls apart or otherwise behaves badly at the limiting value.” While Strevens later argues why these interesting cases (Norton's mismatches) _do_ make sense, we do not have to concern us with evaluating their correctness just yet, we simply need to note that Strevens makes the same distinction as Norton here. 

Then, we can define

>[!definition] **Infinite Idealization (Strevens):**
> 
> An infinite idealization is made by performing a limiting operation on a system, taking some "extrapolation" parameter (such as length, number, volume) to either zero or infinity **in order to set some other parameter to zero or infinity.** The infinite idealization is the system with the extrapolation parameter and the relevant paramenter set to either zero or infinity (dont' need to be the same). However, sometimes these systems misbehave, **which is interesting**.
_(bold to highlight differences with Norton)_


{/*Batterman also has some definition but it is rather vague.*/}


## The problems with existing definitions

I am not satisfied with definitions as they stand. They contain too many distinct criteria, such as whether the idealizations misbehave, how the limit is taken, and what kind of subset infinite idealizations form of idealizations in general. All putative infinite idealizations are discussed using the same umbrella term, even when the previously mentioned criteria differ. I believe that the discussion of infinite idealization would be much clearer if we were able to distinguish between the factors that contribute to the idealization braking down.

# Footnotes
[^well-behaved]: Again, Norton actually does not consider ill-behaved idealizations to be idealizations at all, but for now we shall simply pretend he does in order to compare his stance.