SHELDON SMITH
VIOLATED LAWS, CETERIS PARIBUS CLAUSES, AND CAPACITIES
ABSTRACT. It is often claimed that the bulk of the laws of physics – including such
venerable laws as Universal Gravitation – are violated in many (or even all) circumstances
because they have counter-instances that result when a system is not isolated from other
systems. Various accounts of how one should interpret these (apparently) violated laws
have been provided. In this paper, I examine two accounts of (apparently) violated laws,
that they are merely ceteris paribus laws and that they are manifestations of capacities.
Through an examination of the primary example that motivated these views, I show that
given a proper understanding of the situation, neither view is optimal because the law is not
even apparently violated. Along the way, I am able to diagnose what has led to the mistaken
belief: I show that it originates from an element of the standard empiricist conception of
laws. I then evaluate the suggestions of how to interpret violated laws with respect to other
examples and find them wanting there too.
1. INTRODUCTION
Where it was once commonly believed by philosophers of an empiricist
stripe that the laws of physics report universal, temporal regularities, it is
now often claimed that most, some claim all, laws of physics suffer from
exceptions, or violations, sometimes gross ones. There have been several
reactions to this (alleged1 ) state of affairs, all of which cause problems
of interpretation that have been addressed in the literature. A somewhat
typical reaction has been to take the laws of physics as ceteris paribus laws
of some sort. As such, a law of physics contains an implicit ceteris paribus
clause that indicates what the conditions have to be for the law to govern
a system’s behavior. Other things being equal, the system governed by the
law will obey it. In the cases where the law is not obeyed, other things were
obviously not equal. Thus, the notion that laws contain implicit clauses is
meant to solve what one might call the “problem of falsification.” The
law is saved from (apparent) violation by the ceteris paribus conditional
clauses. An apparent falsification just means that the antecedent conditions
of the law – when fully spelled out – were not satisfied.
Among other places, in her recent The Dappled World, Nancy Cartwright has suggested this view: “Laws, where they do apply, hold only
ceteris paribus”2 (Cartwright 1999, p. 4). Cartwright’s account grew out of
Synthese 130: 235–264, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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an explicit worry about the apparent falsification of certain physical laws
(Cartwright 1983). Although they do not explicitly say that all laws of
physics should be read in this way, in “When Other Things Aren’t Equal:
Saving Ceteris Paribus Laws from Vacuity,” I believe that Paul Pietroski
and George Rey say much that implies the view.3 A related sort of account of the break-down of laws is due to Geoffrey Joseph in “The Many
Sciences and the One World” where he claims that violated laws include
(implicit) ceteris absentibus clauses, the law applies other things being
absent.4 In either case, the additional clauses are meant to keep the law
from being falsified, and thus, to solve the problem of falsification.
Even if it apparently solves the problem of falsification, the addition of
escape clauses has been thought to push rapidly towards a whole family
of complexly interrelated problems that have been widely and diversely
addressed by those noted above. Along with the problem of falsification,
this paper will focus predominantly on a discussion of these problems as
they apply to different examples. First, there is what I will call, following Peter Lipton (Lipton 1999, p. 157), the “problem of instantiation”. It
seems to many that it is the case that other things are never – or only very
rarely – equal; likewise that other things are very rarely absent. Thus, the
additional clauses (of whatever type) would normally fail to be satisfied
and the law would fail to be instanced by any physical system. The laws in
question, then, are not only not universal, contrary to standard empiricist
expectations, but they apply to almost nothing. The question also arises
as to why the claim that a law is ceteris paribus or ceteris absentibus is
not just vacuous because it amounts to the claim that a system will obey
the law except in cases that it doesn’t. I will call this the “the problem of
vacuity.” This vagueness is especially pressing because many believe that
one could never spell out the clauses explicitly.5 If either (1) things are
never equal – whatever that exactly means – or (2) the attempt to shore up a
violation of a law leads one to a vacuous statement, then it is not clear what
entitles physicists to continue applying the law. For, what could justify the
application of the law to cases where other things are not equal or where
other things are not absent? Also, what could be the point of applying a
vacuous law? I will call this problem the “problem of application” where I
mean by “application” both the use of an equation in physical models and
in explanations of the behavior of a system. At its most problematic, the
problem of application is the problem of how to justify the application of
an equation to situations that it does not even approximately describe, if
such there be.
There have been many attempts to handle these problems within the
literature. Because the bulk of this paper will focus upon examples from
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237
physics that are thought to bolster the claim that as normally stated – that
is, without ceteris paribus clauses – the laws of physics are false, I have
space here only to discuss two in depth. Pietroski and Rey attempt to solve
these problems with an account of when ceteris paribus clauses are warranted. This account is supposed to distinguish between genuinely vacuous
ceteris paribus claims and those that have some substance, and thus, can
be legitimately applied, thus solving the application problem. A different
sort of approach due to Cartwright attempts to solve these problems with
an account of “capacities” and the role they play within physics.
Within the philosophy of physics, many of the worries surrounding
these various problems first arose from an examination of Newton’s laws
along with the Law of Universal Gravitation.6 Thus, in this paper, I shall
discuss various proposed solutions to the problems listed above – particularly the problem of falsification that is thought to push us rapidly towards
the others – first as they arise with respect to a certain case, the motion
of particles subject to Newton’s laws of motion and Universal Gravitation.
Roughly, I think that one can show that contrary to certain arguments, one
should not think of Newton’s laws nor the law of Universal Gravitation
as ceteris paribus because they are not even apparently falsified by the
examples given by those who want to read them as ceteris paribus laws.7
Thus, one needs no solution to the problem of falsification in this case,
particularly not a vague appeal to either ceteris paribus clauses or “capacities.” Because of this, one is not driven to the other problems at all. Later,
I shall discuss other apparently violated laws of physics along with the
suggestions of how to read them. I think that none of the suggestions fairs
very well because, it seems to me, they were generally tailor made to fit the
case of Universal Gravitation. Surprisingly, it will turn out that many laws
are semantically vacuous and are taken to be vacuous by the very people
(engineers) who apply them. Obviously, then, the problem of vacuity does
not lead directly to any problem with application. Thus, the impression
that there is a family of problems that need to be addressed en bloc is illfounded. Much of the lesson is that the exceptions to different laws (even
when they are genuine exceptions) have different sources and it is unlikely
that anything unified can be said about them.8 Among other things, in this
paper, I shall attempt to get clearer about the varying status of different
equations of physics that are commonly thought of as laws and also to
delineate a class of equations that are, in my view, generally mistaken for
laws, all of which tend to be lumped together as ceteris paribus laws or as
capacity claims.
Many of the conclusions I come to here are common to the recent
“Ceteris Paribus, There is no Problem of Provisos” (Earman and Roberts
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1999) by John Earman and John Roberts who claim – plausibly, I shall
argue – that “. . . it isn’t ceteris paribus all the way down – ceteris paribus
stops at the level of fundamental physics” (Earman and Roberts 1999, p.
470). In this paper, I shall show much the same about the status of many9
physical laws – and not just about the fundamental ones, whatever exactly
those are,10 but, as we shall see, about many not so fundamental ones as
well. (I shall also argue that an understanding of the laws of physics as
capacity claims does not help matters any.) Even though I find myself in
agreement with many of the specific points made by Earman and Roberts,
I think that one can provide a more detailed argument than what they
provide against the specific examples from physics that Cartwright and
Pietroski and Rey believe force upon us the claim that the laws of physics are merely ceteris paribus. Many of those who think that the laws of
physics are ceteris paribus are led to this view because they think that they
can find specific examples – and many of them – in physics that defy a
strict reading. Thus, such people will probably be unmoved by Earman
and Roberts’ claim that certain laws are “intended as entirely strict by
physicists.”11 If the arguments that the laws of physics are false when
interpreted as strict are sound, then this response does not help. Thus, I
think the detailed examination and criticism of the discussions of specific
examples that led to the view is in order. Moreover, once we have done
this, I think that we can diagnose fairly precisely what exactly led to the
erroneous belief that certain laws of physics are merely ceteris paribus. In
fact, I think that we can show that it is the empiricist presumption that laws
express constant temporal conjunctions. I take the diagnosis that the paper
provides to be as important as the claim that the laws of physics are not
ceteris paribus laws.
2. NEWTON ’ S LAWS , UNIVERSAL GRAVITATION , AND KEPLER ’ S LAWS
Although many discussions of ceteris paribus laws have focused upon the
(supposed) ubiquity of such laws in the “special sciences,” the primary
example of the failure of physical laws that recurs in the literature comes
from classical, point-particle mechanics.12 Much of the evidence – more
will be supplied in later sections – that the laws of physics are merely
ceteris paribus laws comes from an examination of Newton’s laws of motion along with Universal Gravitation and the Kepler problem, the problem
of two bodies under the influence of mutual gravitation alone. It is wellknown that among other things, Newton showed that Kepler’s laws are, at
best, approximately true of planetary behavior. No planet travels around
the sun precisely in an elliptical orbit because of perturbations from the
VIOLATED LAWS, CETERIS PARIBUS CLAUSES, AND CAPACITIES
239
other planets and the asteroids and passing comets, etc. Moreover, the
setup of the “Kepler problem” assumes not only that there are no other
gravitational influences on each planet but that of the sun, it also assumes
that there are no other types of forces than gravitation. For the planets,
this is approximately true because they are essentially electrically neutral.
For other sorts of systems, such as an electron orbiting an atomic nucleus,
there are other forces that, in this case, swamp the gravitational force.13
So far, each of these facts is well-known, but the various interpretations of
them that appear in the philosophical literature are quite controversial. Part
of what is puzzling to me about the discussions of Universal Gravitation
and the Kepler problem is that there is nothing obscure about the physical
situation. That is, there is nothing that would make a physicist think that we
do not understand the problem, but many of the discussions of it strike me
as hopelessly obscure and misguided. Part of what I hope to do here is to
show that the situation does not call for any sort of obscure interpretation.
In this section, however, I will describe – without thereby endorsing the
problematic nature of this example – various reactions to it.
The most well-known claim of the falsity of universal gravitation that
leads directly to the other problems with violated laws noted above is
Nancy Cartwright’s. She claims,
For bodies which are both massive and charged, the law of universal gravitation and Coulomb’s law (the law that determines the force between two [static] charges) interact to
determine the final force. But neither law by itself truly describes how the bodies behave.
No charged objects will behave just as the law of universal gravitation says; and any
massive objects will constitute a counterexample to Coulomb’s law. These two laws are not
true; worse, they are not even approximately true. In the interaction between the electrons
and the protons of an atom, for example, the Coulomb effect swamps the gravitational one,
and the force that actually occurs is very different from that described by the law of gravity
(Cartwright 1983, p. 57).
Cartwright claims that the existence of other forces shows that the law
of Universal Gravitation is not true,14 not even approximately true.15 Thus,
as indicated in the introduction, the problem of (apparent) gross falsity
calls for some sort of interpretation of the laws involved – in this case,
Universal Gravitation – that justifies, among other things, their application
by physicists.
2.1. Universal Gravitation as a Ceteris Paribus Law
Initially, (i.e., in (Cartwright 1983)), in response to the problem of
falsity, Cartwright pushes,16 for a ceteris paribus reading of Universal
Gravitation.17 Cartwright claims that Universal Gravitation can be made
true if it is read as follows: “If there are no forces other than gravitational
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forces at work, then two bodies exert a force [*] between each other which
varies inversely as the square of the distance between them, and varies
directly as the product of their masses” (Cartwright 1983, p. 58). But, it is
hard to know exactly how to read this ceteris paribus law so as to make
it true, for if one reads the word ‘force’ that I have marked with ‘*’ as
the total force, then it will still not be true if there are, say, three bodies
all mutually influenced by gravitation; the total force on a body will not be
determined by looking at two bodies alone. However, it seems more natural
to assume that Cartwright intends this to be a special force (e.g., the Gravitational pull from a single body), not necessarily the total force, because
she thinks of it as arising from one body acting on another whereas the total
force acting on a certain body generally will be a sum of forces originating
from more than one body. But, if one reads it in this way, then the law tells
us nothing about whether there is a gravitational force at work when there
is some other type of special force involved. (As Cartwright notes, “Once
the ceteris paribus modifier has been attached, the law of gravity is irrelevant to the more complex and interesting situations” (Cartwright 1983, p.
58)). Thus, the application problem threatens: Physicists apply this law in
cases where the antecedent is not satisfied because there usually are other
forces at work; What would entitle them to do that? Also, the problem of
instantiation threatens: Aren’t there no instances of this law because every
body has other forces acting on it?18 Thus, giving a ceteris paribus reading
of Universal Gravitation virtually forces upon us the set of problems noted
in the introduction. Granted, one might wonder whether we are forced here
into the problem of vacuity: That problem was supposed to be generated
by the fact that one cannot spell out in detail the ceteris paribus clauses of
a law. Thus, the claim that a law is ceteris paribus amounts to “the system
obeys the law unless it doesn’t” which is a semantically empty tautology.
But, because Cartwright spells out the antecedent clause explicitly, her law
is, apparently at least, subject to empirical test, and, thus, refutation.
Although they agree with Cartwright that Universal Gravitation is
merely ceteris paribus, Pietroski and Rey are more pessimistic about the
possibility of explicitly filling in the required clauses. In general, they
think that Universal Gravitation requires ceteris paribus clauses for the
very reasons Cartwright gives, and they think that ceteris paribus clauses
are needed precisely when it isn’t clear what the other things that need to
be equal are (Pietroski and Rey 1995, p. 87). Otherwise, one would just
explicitly add the clauses to the falsified law and have a new, modified,
unfalsified strict law. But, physicists do not add the required clauses; they
leave the expression of the law as is. Thus, one might wonder whether it
is as simple to add explicit conditional clauses as some claim. But, if one
VIOLATED LAWS, CETERIS PARIBUS CLAUSES, AND CAPACITIES
241
cannot add them, the threat is that Universal Gravitation is vacuous, that
is, it amounts to “Other (not completely specifiable) things being equal,
there will be a gravitational force between two bodies”. But, then it is the
vacuity of this law that leads directly to the problem of application: Why
would any physicist want to apply a law that is entirely empty to a certain
modeling? To what cases is she entitled to apply it? Moreover, how can
an empty statement function in an explanation? Motivated by such questions, Pietroski and Rey give the following substantive account of ceteris
paribus laws that is supposed to distinguish Universal Gravitation which
one ought to think of as a true and non-vacuous ceteris paribus law from a
bogus law like “Ceteris paribus all bodies go straight towards Cleveland”:
“Briefly, we claim that such [ceteris paribus] clauses are ‘cheques’ written
on the banks of independent theories, their substance and warrant derive
from the substance and warrant of those theories, which determine whether
the cheque can be cashed” (Pietroski and Rey 1995, p. 82). We ought to
expect, then, that a law like Universal Gravitation does not hold in every
instance, but we ought to be able to explain its failure by appeal to an
independent theory: “Thus C-abnormal instances [instances where the law
is not obeyed] are to be expected. But such instances must be explicable
by citing the factors ignored, else the putative cp-law is either vacuous or
false” (Pietroski and Rey 1995, pp. 91–2). For Pietroski and Rey, one does
not need to actually have the explanation. The mere existence of factors
that explain why the law does not hold in a certain case is sufficient to
keep the exception to the law from guaranteeing vacuity or falsity. On
the other hand, the failure of the existence of “interfering factors” that
explain the exceptions to the law will render the law false: “If there are
no such interfering factors, then the apparent counter-example [to the law]
is a genuine one, and the putative law is false. Thus, cp-laws are far from
tautologous on our proposal” (Pietroski and Rey 1995, p 93). Philosophically this certainly appears to be a virtue. If laws are tautologous, then the
application problem looms large.19
To muddy the waters even more, it has also been suggested that rather,
than Universal Gravitation and Coulomb’s law, it is Kepler’s laws that are
merely ceteris paribus laws. Peter Lipton claims, “Many cp laws appear to
have no instances at all, because things are never ‘equal’ in the requisite
respect. The planets may move in ellipses, cp, but no planet actually does
move in an ellipse, because of the influence of other planets and of nongravitational forces”20 (Lipton 1999, p. 157). Although he ultimately rejects the view, Lipton, like Pietroski and Rey, also initially suggests that
ceteris paribus clauses should be read as implying the possible existence
of an interfering force: “Intuitively, the idea is that a cp law typically
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has the form ‘All Fs are G, unless there is some interfering force’, and
the privileged completion would be a C that listed all possible forms of
interference and maintained that none of them is in play” (Lipton 1999, p.
160). This convergence of suggestion – even if not of assertion – shows
that this view is, initially at least, attractive as a way to account for the
counter-instances of laws.
2.2. Universal Gravitation as a Capacity Claim
For now, enough about ceteris paribus interpretations of Universal Gravitation. In work more recent than that quoted above, Nancy Cartwright
has moved towards an interpretation of Universal Gravitation in terms of
“capacities” that bodies have.21 The proper reading of Universal Gravitation on this view is not that it has an implicit ceteris paribus antecedent,
but rather one should read it as “Bodies have the capacity to attract each
other according to G md1 m2 2 .” But, these capacities or tendencies to behave
in a certain way are often thwarted, as is the case for Gravitation when
there is a Coulomb force present. Cartwright describes this the other way
around, but the point is the same:
Coulomb’s law tells not what force charged particles experience but rather what it is in
their nature, qua charged, to experience. To say it is in their nature to experience a force of
q1 q2
is to say at least that they would experience this force if only the right conditions
4π0 r 2
occur for the power to exercise itself ‘on its own’, for instance, if they have very small
masses so that gravitational effects are negligible. But it is to say more: it is to say that
their tendency to experience it persists even when conditions are not right; for instance,
when gravity becomes important (Cartwright 1999, p. 82).
The advantage of this sort of approach is supposed to be that it handles the
application problem in a way that ceteris paribus interpretations cannot
easily do. If a law is of the form “Other things being equal, the system will
obey . . . ” then, as we have seen, the law seems to tell us nothing about what
happens when other things are not equal, and would appear to be useless
in explanations as well when other things aren’t equal. But, physicists do
apply Universal Gravitation to cases where it is not the only force present.
Thus, we need to think of the force of gravity as something that is always
there even in non-ideal, non-isolated situations; it is there as a “capacity”
even if it does not get manifested in behavior.
Although we have seen two reactions to the problem of apparently
falsified laws, there is a commonality between them. That is, whether the
approach is in terms of ceteris paribus clauses or “capacities” of bodies,
many assume that what has gone wrong with certain laws is the result of
looking at a closed system when, in reality, there are no closed systems.
Thus, it is typical to think that laws involve idealizations of some sort.
VIOLATED LAWS, CETERIS PARIBUS CLAUSES, AND CAPACITIES
243
Pietroski and Rey claim, “[A] cp law holds only in a ‘closed’ system,
i.e., a system considered in abstraction from other, independently existing
factors” (Pietroski and Rey 1995, p. 89). Cartwright makes the analogous
claim that laws only work “inside of walls.” The claim that laws involve
idealizations is nearly ubiquitous: “Laws describe the behavior of physical
systems under very special conditions that are hardly ever realized, namely,
in isolation. But they can be applied to non-isolated systems as well” (Hüttemann 1998, p. 129). Moreover, Lipton claims, “Most laws are cp laws
because the world is a messy place, and we need to invoke idealizations
and approximations in order to describe it” (Lipton 1999, p. 155). Part of
what I want to show in the next section, however, is that laws like Universal
Gravitation do not assume anything about “isolated systems.”
3. UNIVERSAL GRAVITATION IN THE CONTEXT OF THE EULER RECIPE
In this section, I want to suggest that the initial step that got this entire problematic started was misguided. That is, I shall argue that, when understood
properly, there is no reason to think that Universal Gravitation is false in
the way generally supposed.22 That is, the law is not shown to be violated
by the sort of behavior that Cartwright’s argument describes. Thus, one
does not need to interpret Universal Gravitation as either a ceteris paribus
law or “capacity” or even a “tendency” of bodies. All of these ways of
speaking were driven by what I take to be a pseudo-problem generated
by a false assumption about laws, that they tie more closely to temporal
behaviors than they actually do. Thus, I think that when discussing Universal Gravitation one should stick to the more physically acceptable and
more intelligible language of forces because, when properly understood,
this vocabulary is not as likely to confuse. More on this point later.
It is widely recognized that a physical modeling will require more than
just what generally go by the name ‘laws’ to describe any sort of concrete temporal behavior. However, such recognition generally takes the
form of pointing towards initial and boundary conditions so that an exhaustive classification of the elements of a modeling would be something
like the following: Laws, initial conditions, and boundary conditions. But,
this taxonomy misses many features of a modeling which are vital to the
understanding of the role laws play in constructing concrete descriptions of
motion. What is generally missed is the distinction between a differential
equation, the solution of which describes the concrete temporal behavior
of a system, and the laws (e.g. Newton’s second law or Universal Gravitation) used to derive the differential equation. What missing this distinction
typically leads to is the appellation of these two distinct elements of the
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modeling as “laws.” Moreover, having made this conflation, one generally fails to properly analyze the logic of the derivation of the differential
equation.
Much of the utility of the notion of law in physics derives from the
ability to model different situations using the same set of laws according to
a “modeling recipe” which allows for the derivation of concrete differential
equations of motion. The general ingredients – we will look at specific
examples below – of such a recipe involve:
1. General dynamical principles which are independent of the type of
body in question.
2. Special “constitutive equations” which describe properties of different
types of material.
It is, perhaps, unfortunate that members of both of these elements which
play distinct roles in the dynamics go by the name ‘law’ in the literature,
for they play different roles in the modeling and, moreover, they have
different degrees of defeasibility should one’s model go awry. I will argue
later that it is even more unfortunate that the resulting differential equation
of motion is often taken to be a law. For, this is, I believe, what leads to
much of the confusion.
For point particle mechanics, the recipe, which is due to Leonhard Euler
even though the laws involved generally go by the name “Newton’s laws,”
is as follows:23
1. Specify the class of bodies (let them be point particles) whose behavior
one is concerned with.
2. Specify what types of special force laws hold between these particles
(e.g. Gravitation and the Coulomb force).
3. Choose a set of Cartesian coordinates along which to decompose the
special forces. That is, specify, say, the force of gravity acting on the
first particle from the second particle acting along the x-axis.
4. For each particle, sum the special forces acting on that particle along
a certain coordinate axis.
2
5. Set the sum for each particle equal to m ddt x2 .
The “constitutive principles”24 in this case are the special force laws such
as Universal Gravitation or Coulomb’s Law, and the general principle is
embodied by the last instruction in the recipe which has us plug the sum
2
of the special forces into Newton’s second law, Ftotal = m ddt x2 .25 This
recipe has been used (often implicitly) since the time of Euler to derive
differential equations using “Newton’s Laws.”
In order to get clearer about the content of Universal Gravitation, it is
important to note what the recipe does not imply: The recipe does not im-
VIOLATED LAWS, CETERIS PARIBUS CLAUSES, AND CAPACITIES
245
ply that the only force operating on any body is the force of gravity from a
single other body. In fact, the last step which has us sum the various forces
on the body suggests that there will, in general, be more than one force
acting on a body. The Universal Gravitation law, itself, does not imply
anything about the total force either or about what other forces might be
operating on a body. All that Universal Gravitation asserts is that between
any two bodies there is a force (not necessarily the total force) that points
2 26
There is
in the line joining the bodies and has the magnitude, G m1r m
2 .
no reason to think that this claim is at all problematic just because some
other force is also pulling on a body. It does not even imply that the body
will have an (instantaneous) acceleration in that direction. For, talk of the
acceleration of the body occurs only at the last step where the special forces
have been summed and one has a concrete differential equation (or coupled
set of them) that, when integrated, describes a motion. Thus, to find out that
there is also a Coulomb force acting on the body does not contradict any of
the prescriptions of the recipe or the law of Universal Gravitation; neither
of these asserts otherwise. What is particularly important to note here is
that Universal Gravitation is extremely modest as to temporal content: It
does not even purport to describe the motion of any body. Thus, it cannot
be accused of lying about any “behavior.”27
When we derive concrete differential equations with this recipe so as
to describe actual temporal behavior, there is generally some falsity introduced. I believe that it is this falsity, the falsity of the resulting concrete
differential equation of evolution type, that those who speak of ceteris
paribus laws in this context are sensing. For example, the Euler recipe can
be used to derive the differential equation for the “Kepler problem” which
describes the motion of a planetary body of mass m1 revolving around
another of mass m2 under the influence of gravity alone.28 For instance,
we invoke Newton’s law of universal gravitation
(1)
m1 m2 d
Fgravity = −G
2 |d|
|d|
as our lone constitutive equation (where d points from the origin to our
particle), and we plug it into Newton’s second law of motion as required
by the last step of the recipe to arrive at the following concrete differential
equation:
(2)
−G
m1 m2 d = m1 d¨
3
|d|
This differential equation can be integrated so as to arrive at a solution
which describes the temporal trajectory of the planet in the virtually empty
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universe that we have imagined. As Newton showed, in accordance with
Kepler’s Laws, the planet will trace out an ellipse as it moves through
space. Of course, no real planet traces out an ellipse because any planet
feels the influence from several bodies which perturb it out of its basic
elliptical orbit. Thus, there is falsity somewhere in the model (although as
noted above not in the individual laws because the laws alone do not even
purport to describe any temporal trajectories).
The equation for the Kepler problem is a sort of “minimally working
model” in the sense that it includes the minimum number of laws needed
to arrive at a correctly set problem of evolution type, a correctly set differential equation with time as the independent variable. Any fewer laws and
we would not have enough information to describe any temporal trajectory,
let alone an inaccurate trajectory. In this context, the minimum set of laws
is Newton’s second law and a single special force law. Without at least
one special force law, we could not know what total force would be acting
on a particle for any given configuration of the system. Without Newton’s
second law, we could not know how a particle reacts to a total force. We
should not expect that such minimally working models for describing a
process will typically be the ones we want to use, for we will typically
have to take into account other bodies and, perhaps, other special forces
arising from already accounted for bodies. The recipe does not limit us
in this way, however. We can always derive a new differential equation to
include the other factors that we left out of the present model.
According to Carl Hempel, it is only the application of a theory that
contains escape provisos, not the laws, themselves:
. . . the quantity f in the second law is understood to be the total force acting on the given
body, and the envisaged application of the theory therefore presupposes a proviso to the
effect that the constituent bodies of the system are subject to no forces other than their
mutual gravitational attraction (Hempel 1988, p. 23).
Thus, he seems to think that the differential equation involves provisos
because it is the differential equation that represents an “application” of the
laws to a specific case. This seems closer to the truth than to think that it is
Universal Gravitation that contains provisos, but if Hempel’s proviso were
literally included, the Kepler differential equation would be applicable to
no system in the world because there are always other forces. The real
proviso when using it to model a situation is that whatever has been left
out of the model – though it is, in fact, there – is a negligible influence on
the motion (at least over some time period of interest). If it turns out not to
be, we have to derive a new, usually more complex equation. But, Hempel
is correct that it is at the level of theory application (i.e. differential equation) that the provisos enter, not at the level of the laws. But, at this level
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the existence of provisos (of this sort) does not have the same disturbing
quality because, for reasons to be given, a differential equation is not a law.
That is, explaining the failure of a body to trace out an ellipse, we might
appeal to “interfering factors” in Pietroski and Rey fashion. In fact, talk
of interfering factors or Lipton’s “other force at work formulation” seem
well suited to this task. (It is in this sense that I think that their solutions
were tailor made to fit this case.) But, we are not explaining the failure of
any law here. What we will soon see is that once one arrives at temporal
claims within physics (i.e., differential equations of evolution type), one is
generally trafficking in something other than a law.
In claiming it to be Kepler’s laws that should be given a ceteris paribus
reading, Peter Lipton suggests, like Hempel, that it is the product of the
Euler recipe – not the nomic ingredients – that is at issue. This seems
natural because for those who think of laws as expressing temporal regularities, Kepler’s laws (unlike Universal Gravitation or even Newton’s
second law) have the proper temporal content. But, among what Newton
did was that he showed that Kepler’s “laws” are not laws. This is not because, as Duhem pointed out long ago, they turn out not to be true,29 but
because they involve additional non-nomic modeling assumptions, such
as the assumption that there are only two bodies and the assumption that
the bodies only interact gravitationally. No one would think that it is a
law that there are only two bodies in the universe, not even a ceteris
paribus law. Since the derivation of the differential equation giving the
Kepler ellipses involves non-nomic elements, it cannot be considered in
any straightforward sense a law; it involves non-law elements as well. The
derivation also generally contains anti-nomic elements, assumptions that
violate laws. For instance, the action of the planet on the dominant body,
the sun, is generally ignored in the derivation in violation of Newton’s
third (action=reaction) law. (The derivation of the “law” of the pendulum
also contains this same anti-nomic assumption.)30 Moreover, the Kepler
ellipses only result for certain initial conditions, and initial conditions are
not generally considered to be nomic. For other initial conditions, one
does not get elliptical orbits (even given the other assumption of only two
bodies under the influence of gravitation alone), but one gets parabolic or
hyperbolic scattering. Thus, Kepler’s “law” that planets travel in ellipses
is (at least) triply not a law. It seems to me that the appellation ‘law’ to it
is largely a matter of historical accident and should not be taken seriously.
It will quite generally be the case that differential equations of evolution
type involve non-nomic (and often false or even anti-nomic) modeling assumptions, and, thus, are not laws. To give another example, the standard
wave equation for the vibrating string generally involves the assumption
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that the amplitude of vibration will be small. This makes the equation
easier to solve than the equation that does not involve this assumption
but at the expense of absorbing something that should have been left to
the initial conditions into the differential equation (Antman 1995, p. 11).
Alternatively, it is derived by assuming that there are only transverse vibrations and no longitudinal vibrations in the string. There might be some
types of materials that have approximately this type of behavior (i.e., constitutive equation) (Antman 1995, p. 32). But, the assumption that any
given string is of that type of material seems clearly non-nomic. Either
way, the involvement of non-nomic modeling assumptions in one’s differential equation disbar it from being a law. Thus, it is not profitably read as
a ceteris paribus law either.
One can multiply these examples indefinitely. In fact, most of the alleged ceteris paribus laws in physics are actually differential equations
whose derivations have the same features as those above or are consequences of such differential equations. Thus, they are not laws at all.
For instance, Pietroski and Rey give the “law of free fall” as an additional
example of a ceteris paribus law. But, this is a consequence of a differential
equation derived under the assumption, for instance, that there is no air
resistance. But this involves either (1) an assumption that no one would
think is a law of nature, even a ceteris paribus one, (i.e., that there is no air
surrounding the Earth.) or (2) an anti-nomic assumption (i.e., that the air
surrounding the Earth provides no resistance even though it is a viscous
fluid.) Either way, the intrusion of non-nomic assumptions disbars this
from being a law. The famous “law of the pendulum” relating the length
of the pendulum arm to the period (given as an example by Giere in (Giere
1988)) is a consequence of a differential equation that is derived under the
assumption that the pendulum faces no external forcing of any kind. But,
that is either (1) a non- or anti-nomic assumption because there will always
be forcing either from the damping of the air (see the comments about the
law of free fall) or (2) it involves the non-nomic assumption that there are
no bodies in the universe (other than the Earth) which force the pendulum
which no one would think is a law. As one last example, Marc Lange’s
example of “the law of heat expansion of a metal” (in (Lange 1993)) which
is violated when, for instance, there are external pressures on the bar which
keep it from expanding, is derived from laws under the assumption that
the bar is subject to no boundary pressures. But, there is absolutely no
reason to think that assumption is a law. This is a boundary condition
which seems clearly non-nomic.31 Thus, the “law” of heat expansion is
not a law because it involves non-law assumptions in its derivation.
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All of this ought to be sufficient evidence to reject the standard
presumption of empiricists following Hume that laws express constant
conjunctions of temporally contiguous events. Laws like Universal Gravitation say nothing about temporal successions by themselves. When we
do have a differential equation that does tell us something about temporal
succession, it is generally not a law because it is derived from laws and
non-law ingredients. But, the standard empiricist presumption seems to be
nearly ubiquitous, particularly in the literature on ceteris paribus laws.32
For instance, in their official schema of laws, Pietroski and Rey tie laws
explicitly to temporal behavior: “Such laws [as “Boyle’s Law of gases”]
have traditionally been schematized as universal generalizations of the
form: (x)(F xt → (∃y)Gyt + )”.33 Moreover, they claim, “Laws say
that whenever some initial condition obtains, some other condition obtains
as well” (Pietroski and Rey 1995, p. 83). I assume that by “initial condition” they mean something temporal.34 Of course, they think that such
claims need a ceteris paribus claim out front, but other than that, the basic
format is not tested against examples. Cartwright also claims, “By ‘laws’
I mean descriptions of what regularly happens” (Cartwright 1999, p. 4).
Strictly, the claim of something “happening” does not need to be given a
temporal reading,35 but such a temporal reading seems quite natural. But,
it seems to me that it is this Humean belief that laws tie closely to concrete
temporal behavior that has led to the erroneous belief that Cartwright’s
original argument (given above) shows that Universal Gravitation is false,
and thus, that it requires a ceteris paribus or “capacity” interpretation. For,
when she claims that “No charged objects will behave just as the law
of Universal Gravitation says; and any massive objects will constitute a
counterexample to Coulomb’s law,” she is suggesting that it is the temporal
behavior implied by Universal Gravitation that has gone wrong. But, as I
have noted above, Universal Gravitation implies no temporal behavior at
all. Moreover, no law is violated when a body does not trace out a Kepler
ellipse. Rather, it seems to me that what has been done in Cartwright’s
original argument is that the content of Universal Gravitation has been
confused with that of a minimally working model, in this case, that of the
two-body problem with only Universal Gravitation; and, the failure of that
model in certain more complex cases is blamed on Universal Gravitation.
But, the only reason to equate the content of Universal Gravitation with
such a model is the presumption that since it is a physical law, it must report
a temporal regularity as expected by Humeans. Thus, it is the empiricist
presumption that laws tie very tightly to behavior that is at fault because
it forces us to look for some temporal content in Universal Gravitation
that just is not there. But, once properly understood, the only descriptive
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inadequacy in the vicinity of Cartwright’s argument is that of a certain
differential equation, the one for the Kepler problem, which is, I have
argued, not a law. Moreover, because they have content about temporal
succession, the empiricist assumption tempts us into mistaking differential
equations of evolution type for laws. But, as I have argued, this would be
a mistake.
If the problem of falsification is spurious because Kepler’s laws are not
laws and Universal Gravitation is not false, or, at least, has not been shown
to be false by the argument above, it would seem that the other problems
noted in the introduction are also spurious. I think that this is, in fact, the
case. Ronald Giere expresses that the problem of instantiation is pressing
with respect to Newton’s laws and Universal Gravitation:
Could one find, for example, any two bodies, anywhere in the universe, whose motions
exactly satisfied these laws [i.e., Newton’s laws of motion and Universal Gravitation]?
The most likely answer is “no.” The only possibility of Newton’s Laws being precisely
exemplified by our two bodies would be either if they were alone in the universe with no
other bodies whose gravitational force would effect their motions, or if they existed in a
perfectly uniform gravitational field (Giere 1999, p. 90).
Cartwright also seems to think this in her recent work. She claims, “. . .
we have the small mass m located a distance r from the larger mass M.
.
Now we can look to see if the small mass moves with an acceleration GM
r2
If it does, we have a model for Newton’s laws” (Cartwright 1999, p. 44).
(Presumably it won’t because there are always other forces, even other
instances of the gravitational force.) But, even if it doesn’t move with the
acceleration Cartwright gives, we still might have a model of Newton’s
laws. There are many – infinitely many – motions that satisfy Newton’s
laws of motion and Universal Gravitation. A motion that satisfies any differential equation derivable from the Euler recipe will satisfy those laws as
long as one adds Universal Gravitation between all bodies whatever other
forces one adds between the bodies. That is, as long as Ftotal = ma for
each particle throughout the motion and there is a gravitational force of
the proper magnitude – even if it isn’t the total force acting on a particle –
between all massive particles, we have a model of F = ma and Universal
Gravitation. Only the erroneous belief that gravitation would have to be the
total force between particles would make one believe that we do not. (More
on the status of non-total forces later.) Contrary to Giere’s and Cartwright’s
expectations, Newton’s laws and Universal Gravitation are instantiated in
many, many systems, not just simplified or isolated ones. Although no
system exactly instances Kepler’s formulae, there is no “problem of instantiation” that needs to be explained there. We know antecedently what
we have ignored in that model.
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What about the application problem? If Universal Gravitation is not
shown to be false, then there is no question of what entitles us to apply it
for that question was forced upon us by the alleged falsity of that law. But,
maybe it is with the Kepler equation that the application problem resides.
The equation for the Kepler problem does not apply to the three body problem, but there is no problem of application here. No one would think that it
does apply to this case in general. The application problem was supposed
to be the problem of how we apply equations to situations that they do
not (even approximately) fit. It seems that the correct answer is that we
don’t. We only apply the Kepler differential equation in cases that it does
describe approximately. In other cases, we will need to use the Euler recipe
to derive a more complex differential equation. It can sometimes look like
we apply the Kepler differential equation to, say, certain three body cases.
For in certain special cases of the three body problem (for instance when
one of the bodies has an almost negligible mass), we treat the problem as
a perturbation on the Kepler orbits. However, in doing this we are merely
trying to get information out of the correct differential equation that would
otherwise be hard to get. We are not in any real sense “applying” the Kepler
equation; we are using it in the service of getting information out of some
other differential equation.
Finally, the problem of vacuity is no problem at all in this case. Universal Gravitation is not vacuous – it asserts the existence of a gravitational
force (again, not necessarily the total force) between all massive bodies
– even though its content is much more minimal than people generally
expect.
4. DO SPECIAL FORCES REPRESENT CAPACITIES ?
One might still think that even though the initial problematic, that led
to Cartwright’s talk of capacities is unsound (i.e., the alleged falsity of
Universal Gravitation), there is still something useful in thinking of special forces like gravitation as capacities to behave in a certain way. For
instance, inter alia, the move towards “capacity” talk decouples Universal
Gravitation from any actual temporal behavior – in precisely the way the
Euler recipe shows that it should be – because rendered as a capacity claim
Universal Gravitation does not state what temporal regularity can be expected, but only what there is a “capacity” or “tendency” to have happen. The
behavior otherwise implied by Universal Gravitation is usually thwarted by
some other force.36 What capacity talk reminds us, then, is that contrary
to empiricist expectations, Universal Gravitation does not state a temporal
regularity.37
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But, talk of forces as “capacities” strikes me as hopelessly vague. For
one, the capacities of a body are truly unlimited. For instance a massive
body like a rock has the capacity to attract another, but it also has the
capacity to be used as a murder weapon or to serve as a chair. There are
a million capacities that bodies have that are not forces in the mechanical
sense. (Cartwright gives the example “Aspirin has the capacity to relieve
headaches.”) Thus, even if there were something right about thinking of
forces as capacities, when someone states that the force of gravity represents a capacity of bodies, this is more vague by far than the claim that
gravity is one of the mechanical forces acting on a body. Until one has
a recipe telling us what to do with the various capacities of a body so as
to derive a differential equation, it seems to me that nothing is gained by
this talk that cannot be had from talk of forces. The Euler recipe tells us
how to use the notion of force so as to get something specific and useful, a
differential equation of evolution type. The same cannot be said of talk of
capacities. There is nothing particularly unclear about the notion of force
in mechanics (particularly as compared with the notion of capacities). In
fact, there are precise, mathematical axioms for forces given by Walter
Noll. (See, for instance, (Truesdell 1991)). Since force talk is more specific
and there are clear axioms governing its use, we should, I maintain stick
to it when discussing Gravitation. When understood properly as we have
done above, talk of forces does not commit us to the belief that Universal
Gravitation represents a temporal regularity. It is the empiricist expectations that lead one astray, not any particular obscurity in the notion of
force that needs to be removed by calling them ‘capacities.’ One can show
the erroneous nature of those empiricist assumptions without appeal to
“capacities” but only by getting clearer about the nature of forces as we
have done here.
But, at this point, it will probably be objected that I should not be so
sanguine about forces, for their existence is often thought to be somehow
problematic. Thus, there is another, related role for capacity talk in that
it supposedly reminds us that forces are not “occurrent” properties; they
are something more akin to dispositions to behave in certain ways. For instance, Cartwright claims, “The term ‘force’ in the equation of gravity does
not refer to yet another occurrent property like mass or distance that could
appear in a typical philosopher’s list of occurrent properties” (Cartwright
1999, p. 52). One might note that in much of what I have said above, I
have assumed that they are. For instance, when I say above that Universal
Gravitation should be read as “between any two bodies there is a force
(not necessarily the total force) that points in the line joining the bodies
2
and has the magnitude, G m1r m
2 ,” I have assumed that there is an occurrent
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force there even if it does not issue in any sort of characteristic behavioral
manifestation. But, if this is false and capacity talk is supposed to highlight
that falsity then the interpretation of gravity as a capacity claim might be
thought to be preferable to my interpretation of the law at face-value, as
asserting the existence of a force, not necessarily the total force.
Unfortunately, it is never made clear why we should not think of the
special forces as occurrent. Cartwright claims, “The relevant vocabulary
of occurrent or measureable properties in this case is the vocabulary of
motions – positions, speeds, accelerations, directions and the like” (Cartwright 1999, p. 65). But, the motivation from measurability seems entirely
insufficient to prove the point, for we do measure forces, and not just total
forces that issue in accelerations, but special forces as well. For instance,
when I am standing on a scale, that is a measure of the impressed gravitational force on me. But, it is not the total force, for the total force on me is a
net zero; I am not accelerating at all. Moreover, some of the other items in
her list seem to me to be less straightforwardly measurable. For instance, it
is not obvious that we can measure instantaneous accelerations.38 We can
only measure velocity differences at two different times and try to get the
times as close to each other as possible. Furthermore, none of the items
in her list are always measureable. For instance, we cannot measure the
position of bodies that are outside of our light cone. If it is often the case
that we cannot measure forces, this should not indicate that they have a
special status as properties because no property can always be measured.
Thus, it seems that if the problematic nature of force was supposed to
be that it is not measurable, this worry is misguided. Again, we have another empiricist presumption that is best scrutinized, and, I believe, found
wanting.
Moreover, aside from the vagueness of capacity talk, it seems to me
that much of the talk of forces as “capacities” (rather than as occurrent
properties) still ties the special forces too closely to actual behaviors, albeit
behaviors that are only manifested in certain ideal situations. Thus, this
way of speaking is likely to mislead. Cartwright claims,
My use of the terms capacity and nature are closely related. When we ascribe to a feature
(like charge) a generic capacity (like the Coulomb capacity) by mentioning some canonical
behavior that systems with that capacity would display in ideal circumstances, then I say
that that behaviour is in the nature of that feature. Most of my arguments about capacities could have been put in terms of natures had I recognized soon enough how similar
capacities, as I see them, are to Aristotelean natures (Cartwright 1999, pp. 84–5).
But, if one has understood the Euler recipe, there is no single, canonical
behavior that is “in the nature of a charged particle” in the same way that
there is no single behavior that is in the nature of a massive particle.39 The
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content of laws like Coulomb’s law or Universal Gravitation should not
be equated with any characteristic behavior. (To be sure, given a physical setup, we can derive a behavior, but there is no reason to equate
this behavior with the content of Coulomb’s law even as a capacity that
is somehow thwarted or that Coulomb’s law still “tries (unsuccessfully)
to bring about.”40 ) Much of this confusion is engendered by the belief
that forces are not occurrent and that they are more akin to dispositions.
Disposition talk, for instance, clearly implies that there is some specific
behavior that constitutes manifesting the disposition. For example, fragility is manifested when something breaks when it is struck. But, there is no
single behavior that manifests being under the influence of gravity. Any behavior consistent with a differential equation derived from the Euler recipe
counts as a manifestation of gravity as long as one added a force of gravity
between all bodies. I do not see why the content of Universal Gravitation
needs to be equated with any “natural” behavior at all or even a capacity to
behave in some specific way. For, as shown, Universal Gravitation does not
enter into physical modelings at the level of temporal behavior; it comes in
before any talk of behavior. To equate its content with some behavior is to
illicitly equate it with some minimally working model, or what Cartwright
calls behavior in “ideal circumstances.” (We have already seen that this
is the error that led to the claims of the falsity of Universal Gravitation.)
In reality, the law does not tie to behaviors at all, not even to capacities
to behave in a certain way. All of this is clear from the actual practice of
physical modeling as displayed by the Euler recipe.
In light of the considerations of this section, I would suggest that nothing is gained in terms of our understanding of Universal Gravitation by
referring to it as a capacity or disposition of bodies. Rather, the gain comes
from understanding the Euler recipe and the role that forces play in it.
5. CONSTITUTIVE EQUATIONS IN CONTINUUM MECHANICS
To take stock, Universal Gravitation is not a ceteris paribus law because
although a law, there is no reason to think that it is not quite strict. It
does hold between any two bodies in any situation that the bodies might
find themselves in.41 It just does not say as much about behavior as some
feel that it should. Kepler’s laws are not laws at all because, among other
reasons, they involve non-nomic modeling assumptions. These are just a
few examples, however, and the merits of any account of ceteris paribus
laws or of laws as capacity claims should be judged by a wider range of
examples, even if they fail for the specific case that they were designed
to handle.42 Thus, even though the primary example that is used in this
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255
literature involves a pseudo-problem, how do the accounts of violated laws
given above fair in other contexts?
Among the other examples that they give, Pietroski and Rey give the
ideal gas law as a law that does not apply to a large class of gases. Thus,
on their view, it expresses a ceteris paribus law, and one needs to explain
its legitimacy – i.e., its non-vacuity – in the fashion described above:
On our view, then, a chemist holding that cp(P V = nRT ), is committed to the following:
if a gas sample G is such that P V = nRT , there are independent factors (e.g., electrical
attraction) that explain why P V = nRT with respect to G. In general, these independent
factors [that explain why a certain cp-law does not hold in a certain instance] count as
‘noise’ or ‘interference’ with respect to the cp-law in question (Pietroski and Rey 1995, p.
91).
But, the ideal gas law is one of a large class of laws that are known
not to be universal that are called “constitutive equations” within classical
continuum mechanics. Within continuum mechanics, there are essentially
two types of laws: Cauchy’s laws which apply to all materials (and, thus,
are not ceteris paribus laws) and constitutive equations which describe the
behavior of specific materials. In addition to the ideal gas law, examples
of this latter sort of law are the equations of state of a Van der Waals gas,
Hooke’s law relating stress and strain in an elastic medium, and the constitutive equation for an ideal fluid. None of these laws – unlike Cauchy’s
laws – is universal. That is, not all materials are governed by Hooke’s law.
In fact, most are not. For instance, ordinary, liquid water is not a Hookean
elastic solid. Even many materials that under small values of strain act like
Hookean elastic solids undergo plastic deformation at higher strains. One
might, then, want to render Hooke’s law as “ceteris paribus the stress in a
material is linearly proportional to the strain.” Since Hooke’s law and the
ideal gas law are just two examples among many of constitutive equations,
it looks as if we have found a large class of ceteris paribus laws against
which we can judge Pietroski and Rey’s account.
But, what is interesting is that, surprisingly, the very engineers who
apply these laws take them to be definitions of materials. For instance,
in his Mechanics of Continua, A. Cemal Eringen claims, “A constitutive
equation defines an ideal material” (Eringen 1967, p. 144). Thus, satisfaction of Hooke’s linear relation between stress and strain counts as the
lone criterion for being a Hookean elastic solid.43 Hooke’s law should
be read, then, as “if the material is a Hookean elastic solid, then it will
have a linear relationship between its stress and its strain.” How do we
know if something is a Hookean elastic solid? We check to see if it has a
linear relation between its stress and its strain. What does one say, then,
about materials that do not obey the relation implied by Hooke’s law? One
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merely says that they are not Hookean elastic solids and tries to figure out
what the heck they are for the sake of modeling them.44 Thus, constitutive
equations are generally taken by the very people who apply them to be
empty tautologies. But, what does this show? Are we led down the garden
path towards the problem of application? It seems to me that we clearly
are not in this case. The laws are tautologous – at least the constitutive
equations are – but, the application problem is not made unsolvable thereby
because one knows that certain materials are approximately, say, Hookean
elastic solids. Contrary to Pietroski and Rey’s assumption, to recognize
Hooke’s law as not being applicable to a material, one does not even need
to be committed to some “interfering factors” which explain why the law
failed to hold of that new material. One needs only to point out that the
material is not a Hookean elastic solid. In some moods, Pietroski and Rey
claim, “We ourselves are inclined to a fairly liberal attitude about what
can explain what” (Pietroski and Rey 1995, p. 92). Thus, this might be the
sort of explanation that would fit their account of the failure of a certain
law to hold, but it certainly doesn’t seem to be one in terms of “interfering
factors”.45
Hooke’s law has not been applied to liquid water, say, not because,
scientists were (implicitly) committed to some underlying theory that explained why it did not hold, but because the behavior of water in bulk
showed the untenability of applying Hooke’s law. Moreover, it does not
seem to me that the decision not to apply Hooke’s law to a certain material
or to apply it to a certain material involves one – even implicitly – in
the claim that there are factors which explain why not all materials are
Hookean. Apparently to the contrary, Pietroski and Rey claim, “Note that,
just as the truth of a cp-law depends on their being independent factors that
explain its C-abnormal instances [exceptions], so the warranted assertability of a cp-law depends on the available evidence for such interfering
factors” (Pietroski and Rey 1995, p. 94). It is not clear what “warranted
assertability” amounts to here, but I would think that it would mean (at a
minimum) that to be warranted in applying a certain law to a certain physical situation, a physicist would have to have evidence for such “interfering
factors” as explain why it does not hold of certain materials. But, belief that
use of Hooke’s law was warranted in certain modeling situations predated the independent theory that Pietroski and Rey see as the only thing
that supports its use. James Bernoulli knew that not all materials obeyed
Hooke’s law but it is not clear why this would commit him to the existence
of independent factors that explain why. For all he knew, it was just a brute,
inexplicable matter that some materials are Hookean and others are not. We
feel committed to the existence of explanations for the failure of Hooke’s
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257
law on independent grounds that Bernoulli could not have had. That is,
we understand the microscopic structure of different materials and how
this gives rise to various bulk responses. But, Bernoulli’s failure to know
this should not impugn his using Hooke’s law to model some material and
not others. The only thing that matters for application here is behavior in
bulk.46
Given my general remarks about capacities above, it will be clear that
I am unsympathetic to the claim that constitutive equations describe capacities of bodies. But, it seems to me that capacity talk fits even less well
here. For instance, how should we read Hooke’s law? Should we say that
Hooke’s law represents a capacity that bodies have to respond to a strain by
a certain stress? In the plastic range, the range where bodies do not have a
Hookean elastic response, bodies fail to respond in that way. I can make no
sense of the claim that nonetheless they still have the capacity to respond
in that way, for it seems to me – to the extent that I understand the capacity
talk – they have lost the capacity (sometimes permanently!) to respond
according to Hooke’s law. At any rate, talk of capacities in these contexts
has yet to show its worth. What we have seen, then, is that constitutive
equations are not understood as ceteris paribus laws – nor as claims of
capacities either – but as definitions of materials. But, this semantic vacuity
does not lead to any problems of application that need to be explained by
appeal to underlying “interfering factors”.
6. CONCLUSION
It seems to me that talk of the breakdown of laws as resulting from “interfering factors” or “competing capacities” or the “idealization to closed
systems” was driven by a desire to explain the breakdown of Kepler’s laws
or the alleged breakdown of Universal Gravitation. But, we have seen that
there is nothing essentially problematic in this example. There is only a
non-nomic differential equation where what has gone wrong are the nonnomic assumptions, not any of the laws, used in its derivation. We have
seen that the empiricist assumption that laws express temporal regularities
has driven much of the confusion here. When we move to other examples,
such as Hooke’s law, the accounts examined seem even less at home.
Certainly, to handle each of the examples discussed in this paper with a
unified “they are all merely ceteris paribus” or their success in applications requires “capacities” is overly optimistic. I hope as an alternative
approach to have examined the differing character of each different type
of law and to have said something reasonable about how the law should
be understood. Once we have done this, we can see that contrary to the
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impression often given, there is no family of problems that pushes one
from worries about vacuity to worries about what warrants application.
Engineers feel right at home treating Hooke’s law as an empty definition
and still applying the relation between stress and strain that it involves to
certain (approximately) Hookean materials. If I were to summarize grossly
what I take to be one of the implications of this paper, I would say that
contrary to the popular belief that all laws of physics are merely ceteris
paribus there are no obviously good examples from physics. Moreover,
there are no examples from physics that can profitably be read as capacity
claims. Certainly, we have failed to find any here.
ACKNOWLEDGEMENTS
I would like to thank Robert Batterman, and, especially, Mark Wilson for
generous help on this paper. All errors that remain are, of course, my own.
NOTES
1 Much of the point of this paper is to argue that this is not the actual state of affairs and
to diagnose why it has seemed to many that it is the actual state of affairs.
2 Although, I will note below that she draws a distinction between ceteris paribus laws
and capacity claims and she generally sides with the latter interpretation of “violated” laws
(Cartwright 1999, p. 28).
3 In fact, much of their paper is worried about laws in the “special sciences” rather than
in physics. However, many of their examples (see below) come from physics, and after
defining a strict law as “one that contains no cp-clause [that is, ceteris paribus clause],
even implicitly” they worry “given current science, the appropriate question would seem
to be whether any laws are strict” (Pietroski and Rey 1995, p. 88).
4 I do not mean to imply that this brief summary captures the primary concern of Joseph’s
paper which has to do with how well the different parts of physics – particularly General
Relativity and quantum field theory – fit together. Moreover, Joseph seems to abandon the
suggestion that the laws of physics are ceteris absentibus. Nonetheless, the suggestion can
be assessed on its own merits, and, at a minimum, shows the initial attraction of thinking
that the laws of physics need to be supplemented with escape clauses of some sort.
5 In their, (Earman and Roberts 1999), Earman and Roberts note that Carl Hempel, whose
(Hempel 1988) is sometimes taken as the inspiration for this line (e.g., by (Lange 1993)
and Fodor 1991)), is not one of these people.
6 In this paper, I will assume unapologetically that Universal Gravitation is a law, for
to the extent that we have any handle on the referent of the word ‘law’ this seems to
be a paradigmatic example. I am aware that there are some that will take this to beg
certain important questions. For instance, about Newton’s laws of motion and gravitation,
Ronald Giere claims, “Everyone uses these equations. The issue is how to interpret them,
whether as ‘laws’ or as something else” (Giere 1999, pp. 91–2). But, here I will assume
VIOLATED LAWS, CETERIS PARIBUS CLAUSES, AND CAPACITIES
259
no substantive philosophical account of what it is to be a law. As such, I will be begging
no questions about the content of Universal Gravitation. Rather, I want to get clear about
what role Universal Gravitation has in modelings, for I think that this has led to much
confusion in this vicinity. I will especially not assume as Cartwright does that “a law of
nature is a necessary regular association between properties antecedently regarded as OK”
(Cartwright 199, p. 49). If I assume anything about laws here, it is merely the commonly
accepted claim that initial conditions – at least of ordinary differential equations – are
not laws. This seems to be widely agreed upon. Part of the point of this paper is that
antecedent philosophical accounts of what content a law is supposed to have – such as
that Cartwright adopts from empiricists – have forced upon us various confusions. From
one’s philosophical account one expects certain content to be found in a law that is not
present in Universal Gravitation. This, then, generates confusion. But, rather than denying
to Universal Gravitation the name ‘law’ it seems to me that we should jettison the bogus
account of laws that forces the problems upon us. Since I am more neutral than other
authors on the issue of laws, I am less likely to beg important questions.
7 I do not mean to imply that they are straightforwardly true because we know from both
Relativity and Quantum Mechanics that they are not.
8 It must be said that none of the authors mentioned above takes him or herself to have
given a general solution to the problem of violation of physical law. However, I think that
since their solutions are tailor made to fit what is ultimately an unproblematic case, that of
Universal Gravitation, their solutions fit no examples in physics.
9 Of course, no paper could argue this about all physical laws. Nor do I think that it is true
for all of them.
10 Sometimes to call a law ‘fundamental’ implies that it governs the most basic entities
in our currently accepted physical ontology. So, on this account, no part of classical
mechanics can be considered fundamental since the objects in our most basic ontology
do not behave classically. But, sometimes, fundamental just means “most basic within
a certain theory.” On this reading, Newton’s Second Law is one of the most basic laws
of classical mechanics and is, therefore, fundamental. Sometimes ‘fundamental’ means
having to do with the microscopic details. On this reading, classical statistical mechanics
that (approximately) handles the microscopic details – even though it does not use quantum
mechanics which is in one sense (the first) more fundamental – counts as fundamental, at
least it is more fundamental than, say, thermodynamics that ignores the details. Given these
ambiguities, I think that it is best to avoid the unexplicated use of the word.
11 They claim, for instance, “. . . the putative laws at issue in Hempel’s example – Newton’s
second law of motion and his law of gravitation – are intended as strict laws which require
no provisos” (Earman and Roberts 1999, p. 445).
12 As Earman and Roberts note, much of the stature that they have in the special sciences
derives from their supposed existence in physics as well.
13 Of course, in the latter case the problem needs to be solved with quantum mechanics
rather than classical mechanics. Nonetheless, an exactly analogous situation occurs there.
14 Joseph suggests roughly the same thing with respect to the Coulomb force (Joseph 1980,
p. 777).
15 This latter claim is of utmost importance because it is well-known (for other reasons)
that Universal Gravitation is not true – because General Relativity is more accurate – but it
is generally thought to be approximately true.
16 See below for her more recent attitude and its supposed advantages over this interpretation.
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17 Apparently motivated by similar considerations, Pietroski and Rey also push for a
ceteris paribus reading of Universal Gravitation:
According to this simple proposal, bodies obey the law of gravity and Coulomb’s law,
cp [ceteris paribus] but cetera just aren’t paria with respect to the former law when the
bodies in question have significant charge; and other things are not equal with respect
to the latter law when the quanta in question are affected by nearby dense masses
(Pietroski and Rey 1995, p. 86).
18 We will see below that this is precisely the position taken by Ronald Giere.
19 I will not here enter any debates within the philosophy of language about whether e.g.,
tautologies exist at all or whether being tautologous implies being vacuous or non-refutable
by empirical data, etc. Rather, I will just assume all of the above. If one or more of these
assumptions is false, it will not alter the overall position of this paper.
20 Lipton does not ultimately endorse the account of laws implied.
21 Some authors describe this as just an interpretation of what it is to be a ceteris paribus
law rather than a conflicting view. It will not matter to the overall view of this paper whether
the capacity view conflicts with the ceteris paribus view or is merely an interpretational
variant of it.
22 It is, of course, false because of General Relativity. I will say little about this falsity
in the remainder of this paper. Suffice it to note that gravitation is not treated as a force
in General Relativity but is absorbed into the (non-Euclidean) structure of space-time.
Moreover, the classical trajectories implied by F = ma are not followed by microscopic
bodies like electrons. However, the advance that General Relativity has over Newton’s
Universal Gravitation – or that Quantum Mechanics has over classical mechanics – does
not seem to me to be properly described as the discovery of a new “capacity” or of a new
“interfering factor.” If one wants to describe the non-Euclidean nature of space-time, for
instance, as an “interfering factor” or as representative of a certain “capacity,” it shows only
the looseness of that vocabulary.
23 The details of this recipe are taken from the excellent (Wilson 1998).
24 Strictly, these are the point-particle analogs of what in continuum mechanics are called
‘constitutive equations.’
25 Lest one think that the existence of such recipes is merely a relic of classical mechanics,
there is a similar procedure for building up the Hamiltonian operator in Quantum Mechanics. Like in the Hamiltonian formulation of classical mechanics, the Hamiltonian is a
sum of the kinetic energy, T , and the potential energy U . Most of the work goes into the
construction of the potential energy operator. (Although, depending upon the coordinate
system being used hassles can attach to the kinetic energy operator as well.) This operator
is formulated by summing the various potentials which act on the particles.
26 Later in this paper, I will address the often made claim that the special forces such as the
pull of gravity are not “occurrent properties”, and therefore, a claim like this one that treats
them as if they are occurrent is still false. Here I only mean that nothing issuing from the
Euler recipe or from the behavior of any body shows Universal Gravitation to be false.
27 To be sure, there have been many revised versions of Universal Gravitation that have
been put forward even within the context of classical mechanics, mostly because of the
problem of the advance of the perihelion of Mercury but also because of certain lunar
anomolies (Roseveare 1982). But, the fact that the same argument used by Cartwright and
others could be used to show that these revised laws are false or merely ceteris paribus
ought to be a sign that something has gone amiss.
VIOLATED LAWS, CETERIS PARIBUS CLAUSES, AND CAPACITIES
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28 I assume below that the body of mass m is fixed at the origin which is, itself, fixed in
2
an inertial frame. This reduces the number of equations needed for the illustration, but at
the cost of introducing the assumption that m2 is much greater than m1 .
29 Much – though not all – of Earman and Roberts’ claims that certain equations are
not laws issue from their falsity. (For instance, their example of Boyle’s law described
in Footnote 46 below.) But, this invites the claim that it is, therefore, merely a ceteris
paribus law. This is, in fact, the typical reaction of the ceteris paribus crowd to the alleged
falsity of a law. It seems to me that in many cases, it is not the falsity of the equation, but
the non-nomic assumptions – to be described – used in the derivation of the equation that
disbar it from being a law. (Although this remark does not, I think, apply to Boyles’ law
because it has a special status described in Section 5 below.) Universal Gravitation still, I
would think, ought to count as a law (of classical mechanics) even though it is false – due
to General Relativity, not due to Cartwright’s argument. But, little I say here depends upon
the possibility of there being false laws. If Universal Gravitation is not a law because it is
false, then it is not a ceteris paribus law either. Moreover, as noted in Footnote 22 above,
its falsity (shown by General Relativity) does not seem to be well described in terms of
capacities. Also, my claims about the content of Universal Gravitation stand.
30 I do not mean to imply that there is anything amiss here with the physics. There are
perfectly principled reasons why these factors are irrelevant and can be ignored.
31 It would be wrong to think that all boundary conditions are non-nomic. See, for instance,
(Wilson 1990) who argues otherwise. But, in this case, there isn’t much to recommend the
idea that “There are no pressures on the boundary of the bar” is nomic rather than merely
de facto.
32 In (Smith 2000), I have argued that Russell’s failure to find causation in the equations
of physics issued, in part, from this expectation.
33 Boyle’s law particularly does not fit this evolutionary bill because it is generally –
although not always – used in an equilibrium context.
34 This is, of course, not always the case in that one might refer to the initial condition of
a differential equation whose independent variable is not the time.
35 For instance, one can talk about what happens when a system is in equilibrium (e.g., the
pendulum just hangs there vertically).
36 In fact, I have heard it said that Cartwright could happily acquiesce to everything I say
above. Even stronger, I have heard it said that these just are her points. But, I find this hard
to believe, for these considerations render the example of Universal Gravitation essentially
unproblematic, as we have seen above. For instance, her claim that there are no exact
models of Universal Gravitation is still driven by an erroneous picture – the belief that for
a motion to be a model of Universal Gravitation, Universal Gravitation has to be the total
force. This is not true as described above. In this section, I hope to show further that many
of the presumptions I debunk above are still lurking.
37 Sometimes it is thought that this decoupling of laws and behavior implies a nonregularity view of laws that is, in some sense, metaphysically robust. But, to my mind, it
only shows that laws do not express regularities of temporal succession. Universal Gravitation still expresses a regularity (i.e., between any two bodies there is a gravitational force.)
A Humean who is willing to give up the idea that laws have to be temporal regularities
can still think that they are just brute regularities not underwritten by any sort of “relation
between universals” or anything more metaphysically robust. I will not lean on this claim
heavily here, but it seems to me to be correct.
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38 This is sometimes given as a motivation for replacing instantaneous functions of time
with spread out Schwartz distributions which do not even have values at an instantaneous
point (Zemanian 1987).
39 Sometimes Cartwright is more careful than this. For instance, she claims, “. . . capacities, as I use the term, are not restricted to any single manifestation” (Cartwright 1999,
p. 59) But, in my mind, this only shows the instability of capacity talk. Sometimes it
gives rise to misleading associations as in the quote in the body of the text because it
is natural to assume that capacities are capacities to behave in a certain way and that there
is some specific behavior attached. Given that Cartwright, herself, often succumbs to this
temptation, it seems to be worth abandoning capacity talk. For instance, in her previous
(Cartwright 1989), Cartwright claims, “ ‘capacity’ is reserved for a special subset of these
[tendencies] – those tendencies which are tendencies to cause or bring something about”
(Cartwright 1989, p. 226). If these are supposed to be the tendency to cause something
specific it sounds as if there is a canonical behavior attached, the specific thing that they
bring about. If, on the other hand, they are supposed to be the tendency to cause something
or other, then they are too vague to possibly be useful for helping us to understand forces.
40 Cartwright often talks in this way (e.g., “it is enough for the system to exercise its capacity regardless of what results, i.e. for it to try to produce the associated effect” (Cartwright
1999, p. 66). But, again, there is no temporal effect associated with Coulomb’s law.
41 This is probably too strict, for if the bodies are point-particles and they collide, the gravitational force will be undefined (infinite). However, it is true of any other case, especially
the cases that those who suppose it to be false envision.
42 No author claims universality for his or her account. Nonetheless, I think that these
accounts apply to no examples from physics. Nothing but an exhaustion of examples, a
task that I cannot carry out here, could prove this. Nonetheless, a few more examples
might help make it more persuasive.
43 Lest one think that this is just an idiosyncratic feature of Eringen’s text, Clifford Truesdell claims, “. . . in continuum mechanics ideal materials are defined by particular relations
between the stress tensor and the motion of the body” (Truesdell 1991, p. 159). Also,
Malvern claims, “The ideal elastic solid, or Hookean elastic solid, is the ideal material
most commonly assumed for stress analysis in structures and machine parts. It is assumed
to obey Hooke’s law . . . ” (Malvern 1969, p. 274).
44 The latter part is usually the hard part.
45 To endorse the tautology view with respect to constitutive equations is not, however,
to assent to discussion of laws as holding “only in models” which they exactly describe.
(This is another popular reaction to the break-down of physical laws.) For instance, Giere is
often tempted to claim this sort of thing: “By stipulation, the equations of motion describe
the behavior of the model with perfect accuracy” (Giere 1999, p. 92). But, physics has
no interest whatsoever in describing non-existent things even if it describes them totally
accurately. Scientists would never care to describe such a model unless there were systems
that approximately behaved in the manner described by the model.
46 It is for similar reasons that I do not agree with Earman and Roberts’ claim that the
ideal gas law is not a law because it is false. (The standard mechanical practice of taking
the ideal gas law to be a definition of an ideal gas, for one, ought to speak against this
claim.) After having claimed that “the ideal gas law assumes that the gas molecules have
no volume and interact only by contact,” they claim,
We also think that it is improper to classify Boyle’s law as a ceteris paribus law. The
obvious things to say here are that it is not a law because it is false; that it is false
VIOLATED LAWS, CETERIS PARIBUS CLAUSES, AND CAPACITIES
263
because it is based on unrealistic idealizations (e.g., that the gas molecules have no
volume); but that, nevertheless, for some gases and some pressure-temperature ranges
the idealization provides an approximation that is good enough for most applications
(Earman and Roberts 1999, pp. 461–462).
I agree that Boyle’s law should not be read as a ceteris paribus law, but I disagree with the
reason – I take the standard reading from continuum mechanics of the law as a definition of
an ideal gas seriously. It seems to me that Boyle’s law is not based upon any assumption as
to the nature of the constituents of the gas. In an epistemological sense, it was based upon
certain experimental evidence before it was known that there were molecules. In a semantic
sense, it is entirely consistent with there not even being molecules; it says nothing about the
existence of molecules. So, nothing about molecular constitution can show it to be false.
Certainly, early appliers of the law would not have known – and many energeticists would
not have even believed – that there were molecules. The relative autonomy of modeling
in continuum mechanics from any hypotheses as to microscopic constitution is the reason
equations like Hooke’s or Boyle’s could remain intact through various changes in view
about atomic constitution. It is, of course, true that it can be derived from “fundamental
physics” using the idealized assumptions that Earman and Roberts note. But, it does not
follow from this that the ideal gas law contains any of those assumptions as part of its
content. Moreover, the justification for applying it does not depend upon the truth of the
assumptions as to the constitution of the gas. If it did, then, one would never be justified
in applying it since those assumptions are always false. And, if its warranted use depends
upon the derivation from micro-physics, then no one would have been justified in using it
in a model before the derivation of which Earman and Roberts speak which, it seems to
me, is an overly harsh judgement. (I do not mean to imply that Earman and Roberts claim
otherwise.) Again, the only thing that matters for successful application here is behavior
in bulk, not microscopic constitution. When the behavior in bulk no longer matches the
equation, we obviously do not have an ideal gas. Even if one does not favor the claim
that constitutive equations are definitions, it seems to me that it would not be the falsity of
Boyle’s law, but the non-nomic assumptions – the ones pointed out by Earman and Roberts
– used in its derivation that disbar it. If it is derived from non-law assumptions, then it is
not a law but contains a blend of law and non-law elements. To claim directly that it is its
falsity that keeps it from being a law is to invite the claim that it must, therefore, be shored
up with a ceteris paribus clause. This, after all, is the standard move.
REFERENCES
Antman, S.: 1995, Nonlinear Problems of Elasticity, Springer-Verlag, New York.
Cartwright, N.: 1983, How the Laws of Physics Lie, Clarendon Press, Oxford.
Cartwright, N.: 1989, Nature’s Capacities and Their Measurement, Clarendon Press,
Oxford.
Cartwright, N.: 1999, The Dappled World: A Study of the Boundaries of Science,
Cambridge University Press, Cambridge, U.K.
Earman, J. and J. Roberts: 1999, ‘Ceteris Paribus, There is no Problem of Provisos’,
Synthese 118, 439–478.
Eringen, A. C.: 1967, Mechanics of Continua, John Wiley, New York.
Fodor, J.: 1991, ‘You Can Fool Some of the People all of the Time, Everything Else Being
Equal’, Mind 100, 20–34.
264
SHELDON SMITH
Giere, R.: 1988, ‘Laws, Theories, and Generalizations’, in Grünbaum and Salmon (eds),
pp. 37–46.
Giere, R.: 1999, Science Without Laws, University of Chicago Press, Chicago.
Grünbaum, A. and W. Salmon (eds): 1988, The Limitations of Deductivism, University of
California Press, Berkeley.
Hempel, C.: 1988, ‘Provisos: A Problem Concerning the Inferential Function of Scientific
Theories’, in Grünbaum and Salmon (eds), pp. 19–36.
Hüttemann, A.: 1998, “Laws and Dispositions’, Philosophy of Science 65, 121–135.
Joseph, G.: 1980, ‘The Many Sciences and the One World’, Journal of Philosophy 77,
773–791.
Lange, M.: 1993, ‘Natural Laws and the Problem of Provisos’, Erkenntnis 38, 233–248.
Lipton, P.: 1999, ‘All Else Being Equal’, Philosophy 74, 155–168.
Malvern, L.: 1969, Introduction to the Mechanics of a Continuous Media, Prentice-Hall,
Englewood Cliffs, NJ.
Pietroski, P. and G. Rey: 1995, ‘When Other Things Aren’t Equal: Saving Ceteris Paribus
Laws from Vacuity’, British Journal of the Philosophy of Science 46, 81–110.
Roseveare, N. T.: 1982, Mercury’s Perihelion from Le Verrier to Einstein, Clarendon Press,
Oxford.
Smith, S.: 2000, ‘Resolving Russell’s Anti-Realism about Causation: The Connection
between Functional Dependencies and Causation’, The Monist 83, 274–295.
Truesdell, C.: 1991, A First Course in Rational Continuum Mechanics, Academic Press,
Boston.
Wilson, M.: 1990, ‘Law Along the Frontier: Differential Equations and their Boundary
Conditions’, PSA 2, 565–575.
Wilson, M.: 1998, ‘Mechanics, Classical’, Routledge Encyclopedia of Philosophy.
Zemanian, A. H.: 1987, Distribution Theory and Transform Analysis, Dover, New York.
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