Chapter 18.


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Ladyman, "Idealization" and Norton, "Approximation and Idealization"

Nothing is as we say it is

Many of us had to memorise Newton's laws when we were younger. Thankfully, they're so darn memorable! Do you remember the first one? Newton wrote it like this (but in Latin of course):

Newton's Law I. Every body preserves its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed thereon.

This is sometimes called the law of inertia. Being inertial means one of two things: you are either at rest without any forces acting on you, or moving with constant velocity without any forces acting on you.

Newton's Law I applies to all and only inertial systems. That seems helpful at first, until you realise that there are no inertial systems in our world. There are always forces acting on things: gravity, electricity, or magnetism. You could try to escape by heading out into the outer reaches of space, but even then you wouldn't be truly inertial. There would always be a tiny gravitational effect.

(Note for experts: The situation in general relativity, our modern theory of gravity, is no better. There, the analogue of Newton's first law applies only to test particles, meaning massive particles that occupy only a point in space, and do not exert any gravitational influence. But every massive body exerts some gravitational influence! So, there are no true test particles, and the law is strictly vacuous.)

The law of inertia involves an idealisation, a statement that is strictly inapplicable in its description of the real world, but nevertheless thought to capture some important aspect of reality. They are everywhere in science. Almost nothing is exactly as we say.

How is it possible that a correct law of nature can fail to apply in general? How can it be a law when bodies are always changing their state of motion in practice? How should this influence our perspective on science? These are the puzzles that we'll be considering today.

Idealisations are the rule, not the exception

Almost every law or model in science is idealised in one sense or another. In a moment we will discuss some of the different ways that an idealisation can occur. But first, can you think of any examples that you might have encountered before?

Here's one. Idealisations are very common in the social sciences. For example, people participating in economic systems are often assumed to be rational in order to derive their preferences, or in order to predict the behaviour of a market. One often even assumes that human preferences satisfy a set of mathematical statements like the von Neumann-Morganstern axioms, thought to characterise how a rational human should behave. Perhaps we should behave that way, but unfortunately humans are often irrational, perhaps more often than we'd care to admit. Nevertheless, these models serve as an effective guide to describing social groups.

Another example: when biologists model the way that a population of foxes and hares is changing, they will typically treat all animals of the same species as identical. Of course they are not. Some animals are more effective predators than others, some are non-hunting babies or adolescents, and some will stray from the group and die of natural causes. However, the models still provide effective tools to describe the rough behaviour of the group.

Even a description merely representing things using real numbers involves many radical idealisations. Real numbers are perhaps the most familiar manifold of points, which we use all the time to represent our weight, our driving speed, or our location in space. But the real numbers also have many bizarre properties, such as the Banach-Tarski construction. This construction provides an algorithm for decomposing a real-valued sphere into a finite number of disjoint pieces (originally six pieces), and then reassembling it into two spheres of the same size as the original! It is a strange and unrealistic feature of representations using real numbers, that is made possible by considering what is known as an unmeasurable set.

Banach-Tarski paradox

These idealisations are more than merely incorrect descriptions. What's curious about them is that there is something about them that these descriptions get right in spite of their strict falsehood. We would like to understand what exactly that is. But to begin, let's discuss a few particular features that idealisations may have.

Formal vs Empirical Idealisations

To begin to understand idealisations more seriously, the historian and philosopher of science Ernan McMullin identified two broad classes of idealisations, which have sense been influential in their description.

He called the first a formal idealisation, where 'formal' refers to a scientific theory. These idealisations occur when concepts in a scientific description are either simplified or removed.

For example, in many scientific theories involving the earth, such as our theories of gravity, the earth is idealised as a sphere. But you may know that it is more nearly an ellipsoid, which bulges a little bit around the equator. But it gets worse: the ellipsoid description idealises away the description of the large bulging undulations that the oceans known as the geoid. And even that idealises away the irregularities formed by irregularly shaped matter on the surface such as ourselves. Our formal descriptions of the earth are full of idealisations!

The second class of idealisation is called a causal or empirical idealisation. McMullin uses the word "causal," which is unfortunate, because it brings in an unnecessary ambiguity about the meaning of a "cause." To avoid this, let us use the word "empirical" instead. These are idealisations that involve ignoring unnecessary or undesirable parameters in an experiment.

For example, Galileo argued that all bodies near the earth will fall in the same way. In particular, two bodies that fall from rest at the same height will reach the ground at the same time, regardless of their mass, shape or size. You can do this experiment yourself by dropping a book and a heavy pencil from the same height.

But Galileo's experiment involves many experimental idealisations. The color, shape and temperature of the body are all ignored. Some of these have no effect on the experimental outcome, but some of them do. In fact, the shape of a body can have a dramatic impact when it results in significant air resistance.

In order to remove the experimental idealisation, one might try to remove the air from a room, as Brian Cox does in this NASA spacecraft testing chamber.

After the test-chamber is evacuated there remain 2 grams of air in there. That may not seem significant, and is a fairly accurate way to describe low-orbit space environments near the earth. But it's still enough to ruffle the feather a little bit, it's a lot more atmosphere than the moon, where in a comparable room there would be about a hundredth of a microgram of atmosphere. A less-idealised test was actually given several decades ago during the 1971 Apollo 15 mission.

There are many further kinds of idealisations that could be mentioned. McMullin suggests that a further "material idealisation" occurs when one leaves out irrelevant parameters in a description, such as in the theoretical statement of Galileo's law without a description of colour or shape. He further suggests that empirical idealisations can apply to both real experiments and to thought experiments. We will set this latter possibility aside for now, and come back to it in a later lecture. You should feel free to explore the taxonomy of what it means to be an idealisation further on your own.

For now, let us turn to a helpful recent distinction that was recently introduced, which expresses a view about what an idealisation is not.

Approximation and Idealisation

John Norton (2011) has argued that there are important philosophical consequences in distinguishing what an idealisation is not. He begins by distinguishing the concept of an idealisation from an approximation. For Norton, that distinction is the following.

An approximation is an inexact description of a target system.

An idealisation is a real or fictitious system, distinct from the target system, some of whose properties provide an inexact description of some aspects of the target system.

Norton suggests we begin by picking out a target system that we wish to describe, and then ask whether our description refers to that system (in which case it's an approximation), or to some other system (in which case it's an idealisation). For example, suppose our target system is a falling skydiver.

A pretty accurate way to describe the skydiver's motion would be to say that his velocity is effected by gravity plus air resistance. The velocity increases with the acceleration due to gravity (g), but is resisted by the presence of air pushing back. In case you're interested (not that you need to remember it), the equation that describes that situation when the skydiver begins at rest at time t=0 is,

where v(t) is the velocity at a given time, and g and k are constants. All those extra terms to the right of the "gt" are capturing the effect of air resistance. But they are a bit hard to deal with, and in first few seconds of freefall they have a small effect anyway. So, we can approximate the skydiver's velocity by dropping those terms, and using the equation,

Our description still refers to the same skydiver. It's just less accurate than it was before. In that sense, it is an approximation.

To make it an idealisation in the Norton sense, we would have to consider a totally different system, which we recognise is not the same as our original. For example, we could consider the skydiver in a complete vacuum, like the NASA test chamber above without any air whatsoever. That system is correctly described by the equation without any air-resistance terms.

In referring to a vacuum, which is different than the original skydiver system, we are describing an idealisation. However, it is by analogy similar to the original target system in relevant ways. So, it may still serve as a useful way to describe the system.

What Norton argues is philosophically important is that we recognise that an analogy may break down. This happens in particular when an idealisation refers to systems involving the infinite.

The danger in idealisations

One kind of approximation and idealisation that often happens in science is the infinite limit. For many reasons, it is simply easier to consider an infinite number of objects instead of a finite number.

For example, we often model population changes as a continuous curve. Of course we recognise that there are always a finite and discrete number of people. But going to the infinite limit of continuous numbers of people provides a nice way to approximate real-world systems, and allows us to work with nicer continuous curves that we can apply formal methods to like differential equations.

population as a continuous curve

However, infinity is very strange. It has many strange properties that you may have heard of. For example, there are the set of even numbers and the set of even+odd numbers have the same size, even though one is a proper subset of the other. And there are other oddities as well. Infinite systems can often fail to be deterministic. In physics they can sometimes lead to divergences like infinite energy, which aren't observed in the real world.

So, there are dangers in going to the infinite limit. An infinite limit often doesn't provide a true approximation, instead confusing us by suggesting strange infinitary properties that the real world doesn't actually share.


An example of an innocuous limit is the following capsule. We're interested in its surface area and volume as it gets very large. As we increase its length, the ratio of its surface area to volume turns out to get closer and closer to the number 2. And so, in the infinite limit, it makes sense to say that the ("infinitely long") capsule has a surface area to volume ratio of 2. That is, the infinite capsule provides a good approximation of a finite but very large capsule, at least with regard to its surface area to volume ratio.


An example of a dangerous limit is the same kind of system, but in which the capsule has been replaced with a sphere. Here the ratio of surface area to volume is actually approaching zero. And so one might say that the ratio in the infinite limit actually is zero. However, this would be a mistake, because the infinite sphere has no surface area at all. In particular, there are no points in space where the infinite sphere has surface area; it has simply been reduced to all of space. So, the infinite sphere is an idealisation that shares some but not all properties of a large but finite sphere, and in particular fails to correctly approximate its surface area to volume ratio.

These dangers raise real questions for science. For example, are our infinite descriptions approximations or idealisations? And if they are idealisations, are they legitimate? This question has been raised in particular in physics in the context of the so-called thermodynamic limit, in which physicists consider properties of a system as the number of particles and volume in a system both go to infinity. The debate over the character and legitimacy of such systems are not yet settled.

Do the laws of nature lie?

Nancy Cartwright

The philosopher of physics Nancy Cartwright has argued that the non-approximating idealisations are dangerous for a different reason. That is that they ignore relevant causes in the description of a system. So, idealising descriptions are not even true descriptions, because they leave out essential stuff.

How the laws of physics lie

Cartwright's main concern is that dangerous idealisations are not just present in a few isolated instances. They run rampant in what we call the "fundamental" laws of nature. In ignoring essential causes, these laws incorrectly describe the world, and therefore should be taken so seriously. However, less fundamental laws may still be ok so long as they provide good approximations of the actual world.

Thus, dangerous idealisations may lead one to an argument for antirealism if one decides that they are endemic in science.

And realism as well

A different perspective on idealisations is that they actually provide some support for realism! In particular, idealisations allow one to identify a part of reality is that correct and success-producing, while allowing a different part be incorrect.

Think of the example caloric, the substance discussed last week that was once proposed to account for changes in temperature, but later discovered not to exist.

The 19th century proponents of caloric considered caloric to be a factual description that approximated reality. It was thought to be a real substance. So, when one described the behaviour of caloric in a system in which heat was involved, this was thought to be a good approximation of reality.

Today we view the situation differently. The caloric theory can now be viewed as an analogy: correct in some senses, but not in others. In particular, when we a description involving caloric is a fictional description of the world, and not one that correctly approximates any particular target system that we may wish to study. In other words, caloric is an idealisation.

However, like analogies, idealisations may describe some aspects of the world correctly and other aspects incorrectly. This provides an opportunity for the realist: by describing the caloric theory as an idealisation we are saying that some parts are correct, while other parts may be incorrect. And it is only the correct parts that, according to the realist, are responsible for the successful predictions of the theory.

Thus, both realists and antirealists may take the concept of idealisations to support their view. Which one is correct? I leave that consideration to you.

What you should know

  • Examples of idealisations in the natural and social sciences
  • How to distinguish formal and empirical idealisations
  • The difference between approximations and idealisations
  • How idealisations can lead to dangerous or incorrect inferences
  • The possible implications of idealisations for scientific realism and for antirealism.

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