The Relation
between Idealisation and Approximation in Science
DEMETRIS PORTIDES
University of Cyprus
Abstract: The notions of ‘idealisation’ and
‘approximation’ are strongly linked to the question of “How our theories
represent the phenomena in their scope?” Although there is no consensus on the
nature of the process of idealisation and how it affects theoretical
representation, at the level of science education we can still gain much from the
insights of existing philosophical analyses. Traditionally, educational
methodology treats the observed divergence between theoretical predictions and
experimental data by appealing to the more commonsensical notion of
‘approximation’. The use of the latter notion, however, to explicate
discrepancies between theory and experiment obscures the theory/experiment
relation. It does so, I argue, because ‘approximation’ either depends upon or
it piggybacks on ‘idealisation’. Thus ‘idealisation’ is a primary aspect of the
theory/experiment relation, whereas ‘approximation’ depends on the former.
1.
Introduction
That theoretical predictions and experimental
measurements in science do not exactly match each other is commonplace. Had
they matched there would not be much room for doubt that our theories gave us
the truth about particular aspects of the world. The fact that they do not has
led to philosophical debate on several related issues concerning our scientific
enquiry. Some examples are: “Is it the approximate truth or the empirical
adequacy of our theories that could be rationally justified?” “Do we have
rational criteria of choice between competing theories?”, “Given that our
theories do not exactly mirror the world, how is it that they represent it?”,
etc. In this paper I shall explore the relation between ‘approximation’ and
‘idealisation’. These two notions are features of the more general process of
theoretical representation of physical systems and hence are strongly tied to
the relation between theoretical statements and experimental reports. In fact,
the discrepancy in the theory/experiment relation could be attributed to these
two characteristics of scientific methodology in theory and model construction.
In
science and to some extent in philosophy, attention is given primarily to the
notion of approximation, possibly because of its more mundane nature but also
because it is widely recognised that it can be explicated and handled
exclusively by the use of mathematical tools. This has led to conceptual
confusions firstly because the concept of idealisation and the process of
idealisation in science by and large have been ignored, despite their
epistemological and methodological significance, but also because it became
customary to use the concept of approximation as a surrogate to idealisation,
thus hindering the recognition of those elements of scientific practice that
are associated with idealisation. That this indiscriminate use of approximation
has created the impression that the two concepts could be used interchangeably
is not an argument for the synonymy of the two concepts. I shall argue that the
two concepts are in fact distinct, and that their interdependence is such that
if clarified it could illuminate the theory/experiment relation.
As a first step in
distinguishing the two concepts we could follow Suppe (1989) and understand idealisation of the features of physical
systems to involve two primary modes: either (a) abstracting relevant features
of the physical systems from the theoretical description, e.g. ignoring the
effects of friction in the description of the motion of a body on an incline
plane, or (b) distorting the characteristics of their relevant features, e.g.
assuming a projectile to be a point-mass in estimating its trajectory.[1]
Approximation of the features of
physical systems could also be divided into two modes; it is achieved either
(a) by simplifying the relevant parts of the descriptions of individual
features and properties of the physical systems in the overall theoretical descriptions,
e.g. assuming the effect of the damping force due to air resistance to the
motion of the pendulum to be a linear or quadratic function of velocity, or (b)
by simplifying the theoretical description of the physical system as a whole in
order to produce a description that is not exact but it is tractable and close
enough, e.g. assuming that the magnitude of all the effects to the motion of a
body are small thus allowing us to ignore their mutual interactions and treat
them as separate contributions that give rise to linearly independent tractable
equations (later I shall demonstrate both of these modes of approximation with
reference to the process of modelling the pendulum).
Aiming to minimise
the epistemic effects of approximation, in science and mathematics much work is
done in an attempt to minimize the theory/experiment discrepancy by the use of
theories of systematic and random error of measurement. This however does not
explicate the relation of approximation between theory and experiment, which
is, it could be argued, a vague concept both in terms of its constitutive
conceptual components and in regard to its impact on other semantic concepts
like truth. Nevertheless, one thing is clear about the approximation relation;
it is a concept that is closely linked to the concept of truth, and thus it is
not surprising that recent philosophical attempts to explicate the discrepancy
have focused, for instance, on a notion of approximation as degree of truth or
truthlikeness (Popper 1979, 1989), despite the shortcomings of such an approach
(see Psillos 1999).
It is a trivial
matter that science is not concerned with the strict arithmetical sense of the
notion of approximation. That is, the statement that the number a
approximates the number b does not have any scientific significance. If
a statement of such form is to have scientific value then the numbers a
and b must be values of physical quantities that refer to the actual
world. Hence the approximation relation between a and b takes on
a different character when stated in a scientific context, it refers to the
closeness of the values of two quantities. Not to any two unrelated quantities,
but to quantities that purportedly refer to the same thing and which are
computed in different ways, the first via the conceptual resources of a theory
and the second via the experimental apparatus and the theories of measurement
and experiment.[2] In short,
the approximation relation refers to the closeness of theoretical predictions
to experimental measurements, and this closeness has been traditionally
interpreted by philosophers and scientists alike as closeness to truth, or
truthlikeness, or verisimilitude of scientific theories. In the philosophical
literature one can find several theories of approximation as theories of
approximate truth (e.g. Niiniluoto 1987, 1999; Laymon, 1980, 1987), all of
which have one common goal, to explicate what it means for a theoretical
statement to be approximately true of the world. This task has not proved to be
easy partly because of the vagueness of the concept of approximation and partly
because of its relation and dependence on the process of idealisation in
scientific methodology.
In science education
it is tradition to ignore the vagueness of the approximation relation and in addition
to take it as a primitive notion that suffices for the –approximate– truth or
adequacy of the theory in deliberating upon the discrepancy in the
theory/experiment relation. This practice not only does not clarify what it
means for a certain theoretical prediction to approximate an experimental
measurement, but it also obscures the theory/experiment relation and hence it
leads to an elliptic understanding of the nature of scientific theories and
models on behalf of the science student. In this paper I shall not attempt to
explicate the concept of approximation but I will focus only on the second
problem and argue that by hooking up approximation to idealisation we manage to
maintain a more lucid view of the theory/experiment relation. In fact, I will motivate
a much stronger thesis that a clear understanding of the ways by which
idealisation and approximation interrelate is necessary for explicating the
theory/experiment relation.
2.
Distinguishing Idealisation from Approximation
By hooking up the two concepts it should not
be understood to mean that they are indistinct. On the contrary, they are
clearly distinct concepts and this can be seen by inspecting their logical
properties. We generally understand idealisation, but not approximation, as a
directional process. This intuition is captured by the logical property of
symmetry. Idealisation is not a symmetric concept (in fact, it could be claimed
that it is asymmetric) whereas approximation is. That is to say, thinking of
idealisation and approximation as relations in which two statements enter (e.g.
one deriving from theory, X, and the other from experimental reports, Y), if “X
is an idealised description of Y” is true then it is not true that “Y is an
idealised description of X”, whilst if “X is an approximate description of Y”
is true then it is also true that “Y is an approximate description of X”. For
example, the simple harmonic oscillator exemplifies what we would consider to
be an idealised description of a target physical system like the motion of the
pendulum in the lab; however, a description of the pendulum that accounts for
all factors influencing the motion of the bob is not an idealised description
of the simple harmonic oscillator. On the other hand, if the simple harmonic
oscillator predicts that the earth’s acceleration g is equal to 9.8m/s2 and this is accepted to
approximate the value of 9.81m/s2 that results from measurements on
the pendulum motion, it makes equally good sense to claim the converse that
9.81m/s2 approximates the theoretical prediction of 9.8m/s2.
In addition to being
asymmetric idealisation is transitive, i.e. if “X is an idealised description
of Y” and “Y is an idealised description of Z” then “X is an idealised
description of Z”. This property captures our intuition that idealisation is a
scalable concept, which is demonstrated by examples like the following: if the
simple harmonic oscillator is an idealised description of the damped harmonic
oscillator and the latter is an idealised description of the pendulum then the
simple harmonic oscillator is also an idealised description of the pendulum.
Approximation on the other hand is not –unconditionally– transitive since it is
a pragmatic –and context-dependent– issue whether, if “X is an approximate
description of Y” and “Y is an approximate description of Z” then “X is an
approximate description of Z”, and hence the conditional is not true for all X,
Y and Z. This logical difference is in a sense indicative of the fact that the
two concepts differ in their pragmatic components (e.g. their role in
heuristics), an issue to which I shall return in the last section of this
paper. Furthermore, another characteristic that distinguishes the two concepts
is that approximation is understood as having end-point limits, whereas idealisation
cannot have clearly-cut limits. This intuition is partially captured by the
property of reflexivity. Idealisation is not reflexive (and it could also be
argued that it is irreflexive), i.e. it makes no sense to say that X is an
idealisation of itself unless the concept is trivialized, whereas approximation
is reflexive, i.e. the statement “X is equal to itself” can be understood to
mean the limiting case of the approximation relation.
With these logical
properties of the two concepts in mind it is clear that aphorisms like “all
idealisations are forms of approximation” or “all approximations are forms of
idealisation”, which one encounters in philosophical as well as scientific
literature, either distort our intuitions of the two notions or require careful
qualification if they are to shed light on the theory/experiment relation. For
the science educator the problem is twofold, on the one hand there are two
concepts that are related in not very obvious ways and whose distinction is
subtle and not easily comprehensible to the science neophyte. On the other
hand, there are two methodological processes in science, that are equally
important for the best possible understanding of science, that are closely
related to each other but which if not discerned properly the theory/experiment
relation is obscured; moreover, despite the fact that the concepts of
idealisation and approximation are logically distinct, mere inspection of
actual science reveals that the scientific processes in which they are employed
are interconnected.
Looking at
idealisation and approximation from a methodological perspective another
difference can be discerned that can be located in how the two concepts are
employed in scientific representation. It is widely admitted that one of the functions
of idealisation is to reach a level of generality in our representations of
phenomena. When it is claimed, “X is an idealised description of Y” it is
implied that Y is not necessarily a description of a token case. The simple
harmonic oscillator, for instance, is an idealised description of the type
‘pendulum’ and not just of particular token pendulums. The same also holds for
different levels (or degrees) of idealisation. The somewhat de-idealised
version of the simple harmonic oscillator, known as the damped harmonic
oscillator, also represents in a general way even though the class of physical
systems it represents is not necessarily the same as that of its more idealised
predecessor model. The rationale behind representation by the use of the idealisation/de-idealisation
process is that the more idealised the representational model is the less
properties and features of the type target systems it represents. By
de-idealising a model we do not restrict the class of representations, but we
add more of the relevant features in the general representation of the type
target system. The process of idealisation is therefore used for general as
well as for particular representation. This characteristic of representation is
not something that is present in approximation claims. We cannot, for instance,
claim that the simple harmonic oscillator represents the pendulum in general
because it approximates the type pendulum, since there are actual
pendulums with very large damping forces that make them anything but
approximate to the model. In short, approximation is a feature of
representation that concerns the specific or the token cases of target systems.
This is another reason why not all idealisations are approximations, although
the converse may be the case. More importantly, however, in order for an
approximation claim, i.e. “X approximates Y”, to be useful in the explication
of the theory/experiment relation, X must be such as to refer to a token
physical system that involves a large enough number of features and properties
that are present in Y. Otherwise X and Y would not be referring to the same
thing. In other words, the statement “X approximates Y” is useful for
explicating the theory/experiment relation if X is sufficiently de-idealised so
that it can be regarded as a genuine representation of Y. Another way of saying
this is that X must be sufficiently de-idealised so that its reference is no
longer a class of ideal type systems, but types that can be actualised in the
world. Because of this I claim that the process of idealisation is primary and
upon it the pragmatics of the process of approximation depend (e.g. the role
and use of approximation in representing physical systems, or the factors that
are responsible for the successful use of approximation in prediction and
explanation, etc.).
My argument is backed
by an analysis of the well-known model of the simple harmonic oscillator (of
classical mechanics) and the process by which it is used in order to construct
derivative models that can be proposed for the representation of target physical systems such as the simple
pendulum or the torsion pendulum. I employ this analysis of the process of
construction of representational models to demonstrate that idealisation, and
its converse process of de-idealisation, is present at every level of
scientific theorizing whereas the concept of approximation becomes
methodologically valuable, and epistemically significant, after a certain point
in the process is reached when a given theoretical construct (i.e. a scientific
model) may be proposed for the representation of a physical system. Thus I
explicate the dependence of approximation on idealisation on pragmatic grounds.
In other words, although both concepts are epistemic in nature scientific
methodology requires that a process of de-idealisation takes place before we
can meaningfully employ the notion of approximation. Hence idealisation is a
primary process in our scientific methodology and approximation piggybacks on
it. Thus, science education must accommodate this result in the analysis of the
theory/experiment relation if the goal is to elucidate the latter.
3.
Linking Approximation to Idealisation
Although it is not the purpose of this paper
to offer a theory of approximation that would clarify the notion, some aspects
of the use of the concept are worth clarifying since my concern here is the use
of the concept in illuminating the theory/experiment relation and hence it is
crucial that its relation with idealisation is understood. The ambiguity
present in the statement that our theories approximate the world partly has to
do with what the statement refers to and at what level of discourse it is used.
Sometimes the notion of approximation is used as part of a meta-meta-scientific
statement. Such is the case when it is claimed that the idealised description
of the simple harmonic oscillator approximates the motion of the torsion
pendulum, or more generally when it is claimed that idealised descriptions of physical systems approximate their actual target physical systems. In philosophical
discussions of idealisation, this use of the notion of approximation is in a
sense present in the view that good idealisations are distinguished from bad
ones if their claims approximate the world. In such uses, the reference of
approximation is either to the degree or to the kind of idealisation and not to
the actual relation between the theoretical and experimental statements. A
meta-meta-scientific use of the concept is employed to qualify
characterisations of scientific statements and their relation to experiment
hence it must be discerned from its meta-scientific use. Meta-scientific use
means that approximation itself is a characterisation of scientific statements
and their relation to experiment. The focus in this paper is to the latter use
of approximation, which I believe to be the epistemically important use of the
concept. At the meta-scientific level of discourse we could make either of two
kinds of approximation claims. We could claim that X approximates –the truth about–
Y, when X and Y describe properties and processes and the descriptions of X
closely resemble those of Y. We could also claim that X approximates Y when X
and Y are real-valued functions and the value of X is close to the value of Y
for particular values of their arguments. Or we could claim that approximation
refers to a combination of both of the above.
The first kind of approximation claim can easily be
changed over to an idealisation claim of either of the two modes mentioned
earlier, i.e. if X approximates Y, then X is an idealised description of the
properties and processes of Y, either because relevant features of Y have been
abstracted from X or because the characteristics of relevant features of Y have
been distorted in X. The connection between approximation in this sense and
idealisation can easily be seen because when we identify approximation with
closeness of resemblance of the properties and processes in two descriptions it
is either because some of the characteristics of Y are absent from X or because
some of the characteristics of Y have been changed or distorted in X or because
of both reasons. In this sense approximation is used as a surrogate to
idealisation (and coincides with the latter’s meaning) and adds nothing more to
the content of the characterisation of the relation between the statements X
and Y that idealisation would not. Because approximation in this sense is
understood as being proportional, so to speak, to the number of features that
have been abstracted or distorted in the theoretical description, often one is
led to the view that a description X approximates Y better than Z does only if
it is less idealised than Z; but this way of linking approximation to
idealisation is unnecessary since it does not add anything instructive to the
relation between the two concepts because in this sense all approximations are
specific forms of idealisation. Nevertheless the distinction between the two
concepts is still useful because idealisation is a much wider concept and not
every idealisation is an approximation even in this sense, as I have argued
above.
The second kind of
approximation claim presents a more complicated problem. Clearly this kind of
approximation claim is distinct from the notion of idealisation and very hard
to relate to the latter. Because it is a direct consequence of representing
theoretical descriptions in mathematical languages approximation in this sense
seems to be a concept that could be explicated exclusively by mathematical
considerations. Because of this some philosophers (e.g. Laymon, 1980, 1985,
1987) have attempted to relate idealisation to approximation by also
explicating the former primarily in terms of mathematical considerations. Such
attempts, however, fail to achieve a full explication of the process of
idealisation in science because as a conceptual process idealisation is not a
characteristic restricted to mathematical languages alone.
I shall herein
concentrate on yet another problem with this view. If we do understand the
statement that “our theories approximate the world” as referring to a relation
that can be explicated exclusively by mathematical considerations then we are
faced with the following two possible points of view of approximation pointed
out by Redhead (1980). The first is approximate solutions to exact equations.
Consider his example: ‘For the equation 
we might expand our
solution as a perturbation series in λ,
the nth order
approximation being just 
, if we consider the boundary condition y=1 at x=0.’(Ibid. p. 150) The second view of
approximation that Redhead calls to mind is when we look for exact solutions to
approximate or simplified equations. In the example above, yn is an exact solution to the equation 
, which for small λ
is approximately the same as the original equation above. It is easy to prove,
as Redhead indicates, that the two views are equivalent since, ‘…if we consider
an approximate solution yn
for an exact [equation] …we can always specify [another equation] …which is
‘approximately’ the same as the first, for which yn is an exact solution.’ (Ibid. p. 150) Now, the
number of logically possible approximate solutions to an exact equation is
infinite and each of these is an exact solution to another equation which is an
approximate or simplified version of the exact equation. Thus by viewing
approximation only in a mathematical sense we run into the problem that
different equally plausible approximating equations that purportedly represent
the same target physical system will yield somewhat different solutions that
will not be experimentally distinguishable. Hence it follows that we would have
no non-arbitrary way of singling out one solution that approximates the data
and represents the corresponding physical system if we focus only on
mathematical considerations. There are various ways to see the consequences of
this problem. One simple way, for instance, would be to suppose that we have a
choice between two theoretical constructs that are meant to represent a particular
physical system (e.g. a pendulum) such as the models of the simple harmonic
oscillator and the damped harmonic oscillator. For the sake of the argument,
let us suppose that the first model predicts a period of oscillation equal to a and the second predicts a value equal
to b; now suppose that the
measurement of the period of oscillation of the pendulum is such that it
approximates both predictions without distinguishing between them (e.g. (a+b)/2). Based merely upon the criterion of
approximation (understood only in mathematical terms) in choosing the correct
representation of the physical system means that we cannot choose between the
two in a non-arbitrary way. This may seem as a very simple example, with which
we are so familiar that we are tempted to say that the damped harmonic
oscillator is a better choice for modelling the pendulum because had
experimental inaccuracies not been present experiment would have distinguished
the two, since we know that there is in fact a damping force acting on the real
pendulum. This response however is not convincing because we can imagine
encountering the same problem in modelling a system we are not familiar with,
in which case we are not familiar with the factors that influence the physical
quantity in question. With this in mind we are led to the conclusion that
approximation of the experimental value by the theoretical prediction is not a
sufficient condition for proximity to truth, but also neither is it the way
scientists go about in choosing their theoretical representations. The way
around this, I suggest, is to link approximation to the process of idealisation
on pragmatic and methodological grounds so that non-mathematical considerations
also become part of our explication of the concept of approximation and
subsequently of the theory/experiment relation.
4. The Interplay between Idealisation and
Approximation in Modelling the Pendulum
The
process of idealisation enters at different levels of scientific theorising.
Two principal levels could be identified that are useful to our understanding
of how theories are formulated and applied. Assuming that we begin with the
universe of discourse, the first level of idealisation that could be
distinguished is that of selecting a small number of variables and parameters
abstracted from the phenomena and used to characterise the general laws of a
theory. For example, in classical mechanics position
and momentum are selected and used to
establish a relation which we call Newton’s 2nd law or Hamilton’s
equations. By abstracting a set of parameters we thus create a sub-domain of
the universe of discourse in which the scope of the theory is confined and
which we call the domain of a scientific theory. Thus, Newton’s laws signify a
conceptual object of study that we may call the domain of classical mechanics;
similarly the Schrödinger equation signifies the domain of quantum theory, and
so forth. Scientific domains, viewed from this perspective, are clearly
distinct from physical domains, which they could represent only if they are
expanded by or integrated with other conceptual resources. For instance, the
dynamics of bodies may be influenced by factors that are related to electrical
or heat phenomena that are not accounted by Newton’s laws. In all the laws
(which we may call idealised, in the sense that they are established by a small
number of abstracted parameters) something is left unspecified: the force
function in Newton’s 2nd law, and the Hamiltonian operator in the
Schrödinger equation. The specification of these is what would establish the
link between the assertions of the theory and physical systems. The description
I propose of this level of idealisation in scientific theorising is similar, if
not identical, to Suppe’s version of the Semantic View (Suppe 1974, 1989),
where he maintains that by abstracting a small number of variables and
parameters in order to characterise the general laws of a theory we thereby
define a class of mathematical structures or models that may be used for the
representation of phenomena.[3]
It is evident that the notion of approximation does not enter at this level of
theorising.
The
second principal level in which the process of idealisation enters in our
scientific theorising is the process of specifying force functions or
Hamiltonian operators etc. and it is effective in allowing us to bridge the
assertions of the theory to physical systems. At this level, the process of
idealisation is intertwined with that of approximation and in what follows I
shall demonstrate this process and attempt to show the pragmatic nature of the
relation between the two by analysing how scientists model the simple pendulum.
Morrison
(1999) has argued that in order for an idealised model, such as the simple
harmonic oscillator, to accurately represent the respective physical system we
cannot rely on theory alone but we must add several correction factors to the
model. In order to analyse the process of constructing a representational model
of the pendulum by blending a theoretical model with the relevant correction
factors, it
will be helpful if we work with a distinction between two kinds of model that I
shall label the ideal model (modelI) and the concrete model (modelC).
Let the class of ideal models be the class of theoretical models (as understood
by the proponents of the Semantic View, e.g. Gierre 1988; van Fraassen 1980,
1989; Suppe 1974, 1989; da Costa, N. C.A. and French, S. 1990, 2003)[4]
augmented with the class of models that –unproblematically– could be understood
as mathematical approximations of the former. Let the class of concrete models
be the class of those models that are proposed by scientists for the
theoretical representation of physical systems. Distinguishing between modelI
and modelC is not meant to mark a separation between theoretical and
a posteriori models. ModelI is the theoretical model that we
initially attempt to fit the physical system into, however its representational
capacity is only –to say the most– suggestive. We could regard modelC,
on the other hand, as the carrier of all the antecedent knowledge and physical
intuitions that direct us to capture in concrete ways the attributes and
features of a particular physical system. My thesis is that to turn a modelI
into a representation of a physical system we must blend it with conceptual
resources that extend beyond the conceptual confines of the theory and in the
process the result is a distinct entity that I call a modelC. The
distinction is therefore not based on mathematical tractability but it is used
primarily to emphasise the fact that the conceptual resources of modelI
are confined to the theory that gives rise to it, whereas those of modelC
extend beyond the theory.
Frequently in
classical particle mechanics the initial stages of modelling a physical system
involves the employment of one of the available modelsI. The process
by which the modelI is chosen and employed has been analysed by
Cartwright (1983) and dubbed as ‘theory entry’. The ‘fitting of facts to
equations’, Cartwright suggests, is a process that can be divided into two
stages. As a first stage we prepare an informal description of the phenomenon
such as to ‘…present the phenomenon in a way that will bring it into the
theory.’ (Cartwright, 1983, p. 133) In this stage we use our background
knowledge and try to confine the description to those elements that will allow
us to match an equation to the behaviour of the physical system. In the second
stage, we look at the description through the prism of the theory and dictate
the necessary equations, boundary conditions and approximation methods. In the
context of my discussion, the process of theory entry is important because in
preparing a description of the phenomenon as to bring it into the theory we
most frequently distort some features of the phenomenon or abstract others.
Theory entry thus opens up the scene for a third stage which runs parallel to
the second and which is operative in theory-application: the informal
descriptions of the phenomena act as guidelines for the corrections that should
follow the process of theory entry. This stage leads to the construction of a
representational model of the target physical system and thus a relation
between theory and experiment is established. The process involves the
‘moulding’ of the equations of the modelI as to capture as many of
the features of the physical system as possible and the result is a modelC.
To achieve theory
entry for the pendulum we begin with a highly idealised description of the
phenomenon that would sanction the use of a modelI. By assuming a
mass-point bob supported by a massless inextensible cord of length l performing infinitesimal oscillations q about an equilibrium point, the equation of
motion of the simple harmonic oscillator (i.e. a modelI) can be used
as the starting point for modelling a real pendulum and thus attempting to
measure the acceleration due to the Earth’s gravitational field:
The solution of this equation yields a
relation among the period To,
the cord length l and the
acceleration g due to the Earth’s
gravity: 
The experimental
problem of determining g therefore
comes down to measuring l and To. However, To is far from an acceptable
level of accuracy to the experimental value of the period T. This is expected, because it is known that the actual pendulum
apparatus is subject to influences that are not accounted for in the idealised
assumptions underlying equation . That is to say, the modelI,
expressed through equation (1), involves many abstractions and idealisations
that minimise its representational capacity. In fact I encourage an even
stronger claim, that the modelI does not refer to the class of
actual pendulums but to a class of ideal types that cannot be actualised, i.e.
the class of mass-point bobs supported by a massless inextensible cord
performing infinitesimal oscillations about an equilibrium point. Hence to
claim that equation (1) describes
approximately the motion of the pendulum is to commit an error in the
reference of the model, since it does not refer to the pendulum but to a class
of ideal-types that may resemble in some respects the characteristics of actual
pendulums. Moreover, the kind of representation we could have in the relation
between modelI and the pendulum is a type-type kind and to attribute
to such a relation the characteristic of approximation is erroneous since, as
explained earlier, the approximation relation is an attribute of token-token
cases. Furthermore, if we were to claim that equation (1) describes
approximately the motion of the pendulum it would be equivalent to claiming
that To approximates T,
but this would lead us to the problem of non-arbitrary criterion for choice,
explained earlier. That is the solution To of equation
(1) is experimentally indistinguishable from many other possible solutions that
are equally good approximations to T, hence the relation of
approximation does not offer a non-arbitrary criterion of choosing the correct
representation. Hence, even if To
and T numerically approximate each other it would not mean that the truth
about the physical system is approximated by the modelI. But we must also recognise that the reason
physicists expect the two values to differ significantly is because they know
that a large number of important influencing factors are not included in the
theoretical description.[5]
My contention is that when the degree of idealisation is high such that the
theoretical construct refers to a class of ideal-types the concept of
approximation cannot be employed in any scientifically instructive way; hence
we must search elsewhere in order to illuminate the theory/experiment relation.
In their attempt to
construct a representational model of the pendulum, Nelson and Olsson (1986)
give the following list of influencing factors, that modelI does not
account for: (i) finite amplitude, (ii) finite radius of bob, (iii) mass of
ring, (iv) mass of cap, (v) mass of cap screw, (vi) mass of wire, (vii)
flexibility of wire, (viii) rotation of bob, (ix) double pendulum, (x)
buoyancy, (xi) linear damping, (xii) quadratic damping, (xiii) decay of finite
amplitude, (xiv) added mass, (xv) stretching of wire, (xvi) motion of support.
They proceed to show how the value To
can be corrected by introducing the different correction factors into the
equation of motion. In effect, they are attempting to show what is involved and
how it is involved in the construction of a modelC that can be used
for the theoretical representation of the actual pendulum apparatus. Consider
some of the examples analysed by Nelson and Olsson (1986):
(1) Since the
pendulum experiment takes place in air, it is expected that by Archimedes’
principle the weight of the bob will be reduced by the weight of the displaced
air. Since under such circumstances the effective gravity is reduced, this
increases the period. The correction factor is determined by accounting for the
mass of the air displaced.
(2) The air
resistance acts on the oscillating system (pendulum bob and wire) to cause the
amplitude to decrease with time and to increase the period. The Reynolds number
for each component of the system determines the law of force for that
component. The drag force is hence expressed in terms of a dimensionless drag
coefficient, which is a function of the Reynolds number. In the pendulum case
it can be shown that a quadratic force law should apply for the pendulum bob, whereas
a linear force law should apply for the pendulum wire. Hence, it makes sense to
establish a damping force which is a combination of linear and quadratic
velocity terms:
. To determine the
physical damping constants b and c the work-energy theorem is
employed, an appropriate velocity function v=f(θo,t) is assumed, and under the assumption
of conservation of energy they are matched to experimental results. They
proceed to solve the resulting equation of motion and determine the correction
factors.
(3) A real pendulum
has a bob of finite size, a suspension wire of finite mass and in addition the
wire connections to the bob and the support have structure. All these factors
have some contribution to the oscillations. Their effects are incorporated into
the physical pendulum equation: 
. Where, I is the
total moment of inertia about the axis of rotation, M is the total mass and h
is the distance between the axis and the centre of mass. Depending on the shape
of the bob we could calculate its moment of inertia and thus compute its contribution
to the period of oscillation. Nelson and Olsson (1986) assume that the bob is a
perfect sphere of radius a and
proceed to compute a correction to the period. In a similar manner the
correction contributions due to the wire connections and the mass and
flexibility of the wire are computed.
(4) The length of the
pendulum is increased by stretching of the wire due to the weight of the bob.
By Hooke’s law, when the pendulum is suspended in a static position the
increase is 
, where S is the
cross-sectional area and E is the
elastic modulus. The dynamic stretching when the pendulum is oscillating is due
to the apparent centrifugal and Coriolis forces acting on the bob during the
motion. This feature is modelled by analogy with the spring-pendulum system to
the near stiff limit. The result is a system of coupled equations of motion,
which when solved yields the correction factor for the period.
These examples
indicate a number of complexities involved in the process of constructing modelC.
The root of these could be traced in the attempt to relax or overcome the
underlying idealisations and abstractions of modelI, which could be
put in the language of physicists: when the goal is to model a physical system
then the initial problem of starting with a law of force (i.e. Newton’s 2nd
law) and using it to find a modelI for the description of the
physical system does not suffice. In this quest, the general problem of finding the law of force that may be
responsible for a particular constituent of the external force function in
Newton’s law, and which would reduce the degree of idealisation, is of equal
importance. In order to determine the various force laws to be used in modelC
we utilise either the antecedently available empirical laws (such as
Archimedes’ principle, the Reynolds number and the drag force expression, and
Hooke’s law, for the case of the pendulum) or postulate novel physical
mechanisms. By employing the various force laws in the construction of the
modelC we are turning the model into a representation of the
respective physical system because when these correction factors are added the
reference of the model is no longer a class of ideal-types but a class of
actualisable systems. To consider that a modelC approximates the physical
system is not only reasonable but also scientifically useful because all these
factors are approximations to particular aspects of the target physical system.
The construction
procedure of modelC is conventional and not peculiar to the
pendulum. The mathematical functions for each influencing factor are determined
by the use of various empirical theories from disparate areas of physics and
are inserted into the equation of motion in a cumulative manner. Because the
influence of each of these factors on the system is small, it is assumed that
the resulting equation of motion approximates a system of linearly independent
differential equations, each involving a different influencing factor. Each of
the equations is solved individually to determine the values of the individual
effects and the total value of the correction is computed by adding all the
effects linearly (see Nelson and Olsson 1986). The methodological process we
are faced with is the blending of experimental parameters and empirically determined
laws together with a theoretical model to produce a modelC. The
theoretical model is a pure derivative of the theory that we can turn into a representation
of a physical system by blending it with these ingredients. This is done in an
effort to extend the scope of application of the theory beyond the class of
ideal-type systems (e.g. isolated point-masses and inelastic cords) to which
the class of theoretical models may be understood to refer. To achieve this we
give a concrete and specific context to the force function (i.e. to the
abstract concept of ‘force’) for each and every different influencing factor.
It is important to note that de-idealisation is the process by which the model
is turned into a representation of the physical system. Approximation is the
process by which the equation of the modelC is made tractable. Both
processes are in a constant interplay in trying to turn a modelI
into a modelC but the epistemic significance of approximation
depends upon the degree of de-idealisation achieved and this is the pragmatic
aspect of the relation between the two concepts.
In the process of
constructing modelC above the primary concern is to discover those
correction factors that would bridge the gap between modelI and the target physical system, at this stage
only the de-idealisation process is operative. The two modes of the
approximation process enter into the picture once the de-idealisation process
begins. When each correction factor, and the force law responsible for its
behaviour, is discovered it is approximated by a mathematical expression that
gives rise to a tractable equation of motion. This part of the process is an
example of the first mode of approximation which clearly piggybacks on the
de-idealisation process. Once all the correction factors are introduced into
the equation of motion, i.e. when the process of de-idealisation is completed
and the modelC is constructed, the second mode of approximation is
used. The assumption that the effects of all correction factors are small hence
we could approximate the equation of motion with a system of linearly
independent tractable equations also piggybacks on the de-idealisation process,
in the sense that the de-idealising assumptions dictate the approximation
techniques to be used.
In modelling physical
systems the starting point is an idealised model, such as the harmonic
oscillator, whose force function could be expressed through a general
functional relation: 
. A first-step de-idealisation would be to expand the functional
relation by accounting for an influencing factor that has been initially
ignored. This results in a new general functional relation which in its most
simplified logical form could be presented as follows: 
. Supplementing the function H(x) with
cumulative correction factors is a process that goes on until our conceptual
resources and background knowledge are exhausted. The idealised model can be
understood to relate to its derivative de-idealised relatives in the following
general way: 
. In other words, on this account ‘idealisation’ is the
process by which we let factors that are influential to the physical system
tend to zero. De-idealisation is the converse process of allowing these factors
to take finite values. That is, idealisation ignores the influence of factors
and de-idealisation reintroduces their effect into the model. Approximation
enters into this picture because each fi
is represented via the mathematical language of the theory in an approximate
way and because the final Hk
is solved by an appropriate approximation technique.
This process could be
misconceived to mean only that modelI is a description with some
unspecified parameters and the modelC is a description with those
parameters specified, thus the latter is a structure-type nested in the former.
In other words, by specifying parameters we effectively create a sequence of
nested mathematical structures. The idealisation/de-idealisation process viewed
from this perspective is no more than a partial ordering of structures. The
criterion (i.e. relation) of this partial ordering is that of the restriction
of the domain, i.e. two models, M1 and M2, are
partially ordered if and only if the domain of M2 is a restriction of the domain of M1. We could think of the criterion for partial ordering as a
transformation rule that requires the specification (or addition) of a
parameter in the above functional relation. In this picture the reference of
the sequence of models remains constant, i.e. the reference of modelI
would not be different from the reference of modelC, other than the
restriction of the domain. Hence in this understanding of idealisation the use
of approximation is meaningful at every level. However, the misconception in
this view of idealisation is that by specifying a parameter we are not simply
correcting our mathematical description H(x) but we are bringing
our theory in touch with the world, i.e. f1 derives from the
theory alone but f2 derives from empirical laws and experimental
parameters. So de-idealisation is not just a process by which we paste together
different descriptions to create a more complex final description, but it is
the process by which we supply a theoretical description, that refers to a
class of ideal-types, with those conceptual ingredients that would make it
refer to actual physical systems (i.e. in the words I have chosen to present it
in this paper, it is the process of turning a modelI into a modelC).
If idealisation were understood in the former way then approximation would be
suitable at every level. But if idealisation is understood as I suggest then
approximation is useful only when we have turned a modelI into a
modelC. This is what I mean when I claim that the pragmatics of
approximation depend upon idealisation. Unless a modelC is
established by means of de-idealising techniques and hence a plausible
representation of a target physical system is constructed we cannot employ the
relation of approximation without obscuring the theory/experiment relation.
5. Conclusion
The idea that theories do not represent the
concrete circumstances in which naturally occurring physical systems are found
was pointed out by several authors (e.g. Cartwright 1983, 1999; Shapere 1984;
McMullin 1985; Laymon 1985; Morrison 1998). Among them proponents of the
Semantic View like Suppe (1989) well-understand that theoretical models are
abstract and idealised descriptions and as such, it could be claimed, they do
not represent physical systems in any direct sense. My argument leads to the
contention that it is only after they give rise to a modelC,
appropriate for the representation of a particular physical system, that they
acquire a certain capacity of representation. We say that the linear harmonic
oscillator approximately represents the simple pendulum system, only because we
have managed to use it successfully to construct a modelC. We would
not claim that all conceivable theoretical models are representations of
physical systems. But what is more important, in the context of my discussion,
is that a modelC is a representation of a target physical system
that involves a theory derived description blended with empirical laws and
other auxiliaries, and this is why it makes sense to call it an approximation
of the corresponding physical system. Theoretical models refer to a class of
ideal types whose empirical content is supplied when they are used in the
construction of a modelC, by de-idealising them we change the
reference class to actualisable physical systems and thus we can meaningfully
employ the notion of approximation. Given the picture of scientific modelling
that I have drawn and given the arguments that I have given of the difference
between the concepts of idealisation and approximation, one consequence is that
science education must give the necessary weight to the processes of
idealisation and approximation if the student is to be come to terms with the
complexities of the theory/experiment relation.
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This article is from the proceedings of the
Eighth International History, Philosophy Sociology and Science Teaching
Conference, University of Leeds, July 2005, IHPST 2005. A fuller and revised version of the paper
will shortly appear in “Science
and Education”.