The Rotating Discs Argument Defeated

J. Butter�eld1

All Souls College, Oxford OX1 4AL

Tuesday 25 May 2004

Abstract

The rotating discs argument (RDA) against perdurantism has been mostly

discussed by metaphysicians, though the argument of course appeals to ideas

from classical mechanics, especially about rotation. In contrast, I assess the

RDA from the perspective of the philosophy of physics. I argue for three main

conclusions.

The �rst conclusion is that the RDA can be formulated more strongly than is

usually recognized: it is not necessary to \imagine away" the dynamical e�ects

of rotation. The second is that in general relativity, the RDA fails because of

frame-dragging.

The third conclusion is that even setting aside general relativity, the strong

formulation of the RDA can after all be defeated. Namely, by the perdurantist

taking objects in classical mechanics (whether point-particles or continuous bod-

ies) to have only temporally extended, i.e. non-instantaneous, temporal parts:

which immediately blocks the RDA. Admittedly, this version of perdurantism

de�nes persistence in a weaker sense of `de�nition' than pointilliste versions that

aim to de�ne persistence assuming only instantaneous temporal parts. But I

argue that temporally extended temporal parts: (i) can do the jobs within the

endurantism-perdurantism debate that the perdurantist wants temporal parts to

do; and (ii) are supported by both classical and quantum mechanics.

This is an extract from a much longer paper, which is available at:

http://philsci-archive.pitt.edu/archive/00001760. The main di�erences are that

the longer paper: (i) gives much more detail about the form and scope of the

RDA, the interpretative subtleties of classical mechanics, and the physics of

rotation; and (ii) reports and assesses several other replies to the RDA, especially

those by Callender, Lewis, Robinson and Sider.

1
email: jb56@cus.cam.ac.uk; jeremy.butter�eld@all-souls.oxford.ac.uk



1 Introduction

This paper is an attempt to connect what modern physics says about matter, with

the debate in analytic metaphysics about whether an object persists over time by the

selfsame object existing at di�erent times (nowadays called `endurance'), or by di�erent

temporal parts, or stages, existing at di�erent times (called `perdurance'). Though this

is a multi-faceted debate, with various connections to the philosophy of physics, I will

focus on just one connection: a metaphysical argument against perdurantism, based on

the idea of rotating homogeneous matter, and nowadays often called the `rotating discs

argument' (RDA). I will argue, against much of the literature, that the argument fails,

because of some features of classical mechanics (including how it should be interpreted

in the light of quantum mechanics).

I begin by reporting the RDA and some of the literature's discussion of it (Section

2). Then I propose an improved version of the argument; but point out that even

this version fails in general relativity (Section 3). In the last two Sections, I set aside

general relativity and propose a new perdurantist line of reply to the argument (even

my improved version of it). The idea is that the perdurantist should accept only

non-instantaneous temporal parts. This idea can be supported both by considerations

drawn from classical mechanics (Section 4), and by considerations drawn from quantum

mechanics (Section 5).

This paper is a fragment of a larger project. My (2004, 2004a) describe some other

connections between the endurantism-perdurantism debate and aspects of physics and

its philosophy. My (2004) also presents the debate in much more detail than I can

here. SuÆce it to say that I conceive the debate in much the same terms as Sider's and

Hawley's �ne recent monographs (2001). As to the RDA itself, my (2004b) expands

on the material to follow. (Hawley and Sider also both discuss the RDA, at pp. 72-90,

and 224-236 (respectively), of their monographs (2001).)

2 The story so far

2.1 The RDA

The argument envisages that the perdurantist with her ontology of temporal parts

faces the project of de�ning persistence: since persistence is not identity, she needs

to tell us what it is. (This project is also called `analyzing persistence', and `analyz-

ing or de�ning the genidentity relation between temporal parts'.) In particular, she

needs to de�ne persistence so as to distinguish \ordinary persisting objects" (i.e. the

referents of ordinary terms, and elements of ordinary domains of quanti�cation) from

the countless other \spacetime worms", i.e. mereological fusions of temporal parts.

(Most perdurantists accept unrestricted mereological composition, so that they also

accept these worms as genuine objects.) On pain of circularity, the de�niens is not to

1



presuppose the notion of persistence.

The argument urges that the perdurantist cannot succeed in this project.2 It is

based on two ideas:

(i) Homogeneous: In a continuum (i.e. a continuous body whose composing matter

entirely �lls its volume) that is utterly homogeneous throughout a time-interval con-

taining two times t0; t1, a spatial part at the time t0 is equally qualitatively similar to

any spatial part congruent to itself (i.e. of the same size and shape) at the later time

t1. (The properties of the continuum can change over time, but must not vary over

space, even on arbitrarily small length scales; e.g. the continuum could cool down, but

must at each time have the same temperature everywhere.)

(ii) Follow: The perdurantist will presumably try to de�ne persistence in terms of

suitable relations of qualitative similarity between temporal parts. The obvious tactic

is to have the de�niens \follow" the curves in spacetime that are timelike and track

maximum qualitative similarity.

The tactic of Follow seems to work well when applied to point-particles moving in

a void each with a continuous spacetime trajectory (worldline). For however exactly

we de�ne `maximum qualitative similarity', there will no doubt be, starting at a point-

particle at t0, a unique timelike curve of qualitative similarity (or occupation-by-matter)

passing through it: the worldline of the particle. Similarly for point-particles moving,

not in a void, but in a continuous 
uid with suitably di�erent properties|a di�erent

\colour", or made of di�erent \stu�", than the point-particle.3

But Homogeneous implies that Follow's strategy stumbles when applied to an ut-

terly homogeneous continuum. There are altogether too many spatial parts at t1 that

are tied-�rst-equal as regards qualitative similarity to the given spatial part at t0: any

congruent spatial part will do. In other words: the curves of qualitative similarity run

\every which way".

This problem is made vivid by urging that the perdurantist cannot distinguish two

possibilities that, the argument alleges, must be distinguished. One main example,

on which I will focus, is the case of a rigid disc made of homogeneous matter that is

stationary; and a duplicate disc (rigid and congruent to, made of the same homogeneous

material as, at the same temperature as etc. the �rst) that is rotating about the axis

through its centre. It will be convenient to have labels for two such possibilities: call

them `(Stat)' and `(Rot)'.

Hence the argument is nowadays often called the `rotating discs argument' (RDA).

(In some discussions, both discs are rotating, but with di�erent velocities, maybe even

in di�erent senses.) But all agree that countless other examples would serve just as well

2
The RDA arose in recent philosophy in Kripke (unpublished lectures) and Armstrong (1980).

Zimmerman (1998) reports how the argument goes back at least to Broad in 1925. Sider (2001, p.

226) notes that Leibniz (1698, sect. 13) deploys essentially the same argument: but Leibniz's target

is Descartes' doctrines about matter and motion.
3
Agreed, one can object to this tactic. I address these objections in Sections 4.1-4.2.1 of my (2004a).

But the details of my replies are not needed for this paper: for they are no help to the perdurantist

in facing the trouble made by the RDA.

2



as a disc: e.g. a sphere; or a body of 
uid, like a river, that can be either stationary

or 
owing (or 
owing with di�erent speeds, or in di�erent directions).

It seems that the endurantist can easily distinguish the two possibilities, according

to whether the very same non-circularly-symmetric part, e.g. a segment, is in the same

place at two times. My (2004b) pursues the question whether this is really so: can

the endurantist legitimately use the notion of being in the same place at two times,

i.e. the notion of persisting spatial points? (This question is almost entirely ignored

in the metaphysical literature: authors often appeal without further discussion to the

idea of \the same place" (e.g. Hawley 2001, p. 85).) Here I just assume, in order to

give the RDA as good a run as possible, that the answer is Yes; though I will say more

in Section 3 about what this notion involves.

On the other hand, it seems the perdurantist has a problem. Surely she must say

that all the relations (and therefore, all her pro�ered \suitable relations" for analysing

persistence) between two temporal parts of the disc (say, second-long parts at noon and

12.01) are the same|whether the disc is rotating or not? And similarly for temporal

parts at the two times of any spatial part of the disc, such as a segment.

The rest of this Section clari�es the scope of this argument, and the kinds of reply

the perdurantist can give to it. This will yield a statement of a consensus which is

widespread in the literature|and an announcement of how the remainder of the paper

will argue against that consensus.

2.2 Intrinsic properties and velocities

So far I have expressed the RDA's main idea as the inadequacy, for de�ning persistence,

of qualitative similarity. But in some versions of the argument, the emphasis is instead

on the inadequacy of intrinsic properties. The intrinsic-extrinsic distinction among

properties is controversial, but the rough idea is that possession of an intrinsic property

implies nothing about the possessor's environment, i.e. about matters of fact beyond

the instance. So in these versions, the target is a perdurantist who seeks a de�niens

using intrinsic properties of temporal parts. And in some versions, the target is a

yet stronger neo-Humean doctrine to the e�ect that (roughly speaking) all facts|not

just facts of persistence|are determined by all the various intrinsic properties of all

the points of spacetime. The most in
uential version of this sort of extreme pointilliste

doctrine is Humean supervenience, as formulated and defended by Lewis (1986, p. ix-x;

1994, p. 474; 1999).

This emphasis on intrinsic properties goes along with an objection to the obvious

suggestion that what distinguishes the two cases is the direction of the instantaneous

velocity of the disc's (or sphere's, or river's) constituent parts. Thus for the stationary

disc, all the disc's parts have zero velocity; while for the rotating disc, the parts have

various velocities (and for a perfectly rigid disc, a common angular velocity); and

similarly for the sphere or river. But, says the RDA, the notion of velocity presupposes

the persistence of the object concerned. For average velocity is a quotient, whose

3



numerator must be the distance traversed by the given persisting object: otherwise

you could give me a superluminal velocity by dividing the distance between me and

the Sun by a time less than eight minutes. So average velocity's limit, instantaneous

velocity, surely also presupposes the notion of persistence. Accordingly, says the RDA,

the perdurantist cannot adopt the obvious suggestion, of distinguishing the cases in

terms of instantaneous velocity (or angular velocity)|on pain of circularity.

The notion of presupposition, like the intrinsic-extrinsic distinction, is controversial.

Besides, Tooley (1988) proposes a heterodox account of instantaneous velocity as an

intrinsic property of an object at a time|though he is not concerned with the debate

over persistence; and Bigelow and Pargetter (1989, 1990 Section 2.6) propose a similar

account of velocity. But in this paper, I will set aside these controversies and concede

the above objection. That is, I will assume that both average and instantaneous

velocity presuppose the notion of persistence, and are extrinsic properties. But I shall

nevertheless argue in Section 4 that a certain kind of perdurantist can endorse the

obvious suggestion, i.e. can appeal to velocities to distinguish the two cases.

2.3 \Naturalism"

So far, my description of the perdurantist project of de�ning persistence, and of the

RDA against it, might well be read within the tradition of conceptual analysis. By this

I mean that the perdurantist's de�nition would be both �nite in length, and formulated

using everyday concepts. But nowadays, the literature also considers a \naturalized"

perdurantist project of

(a): providing only a supervenience basis for persistence (i.e. allowing in�nitely

long de�nitions), rather than a �nite de�nition or analysis of it; and-or

(b): appealing to technical notions, and contingent bodies of doctrine, in partic-

ular the laws of dynamics. (Some authors combine (b) with use of the Ramsey-Lewis

technique for simultaneous functional de�nition; e.g. Sider (2001, 230-236).)

Accordingly, the RDA is nowadays sometimes formulated as targeting even: (a) the

supervenience of persistence on qualitative similarity among, and-or intrinsic proper-

ties of, the perdurantist's temporal parts; where (b) such supervenience may even be

contingent, say relative to the laws of a dynamical theory.

This situation prompts two comments: the �rst relates mostly to (a), the second

mostly to (b).

(1): Non-reductive perdurantism:| There is also an even more naturalistic con-

ception of perdurantism, which might well avoid the RDA. On this conception, the

perdurantist seeks a theory of perdurance and related concepts, that can appeal to

scienti�c technicalities, that can revise rather than describe our concepts|and that

does not have to de�ne persistence in terms that do not presuppose it. Of course,

analogous \non-reductive" conceptions are nowadays commonplace in the philosoph-

ical study of many concepts, such as causation, perception and action. So just as a

philosophical theory of causation might decline to de�ne causation (even in�nitarily,

4



even by Ramsey-Lewis functional de�nition), a perdurantist might decline to de�ne

persistence (even in these liberalized senses), on the grounds that she nevertheless says

enough to adequately distinguish \ordinary persisting objects" from other \spacetime

worms". I shall return to this modest (because non-reductive) perdurantism in Section

4. But until then I shall consider the more ambitious perdurantist, who aspires to

de�ne persistence, and so faces the RDA.

(2): How many worlds?:| Once we allow that a perdurantist theory of persistence

might appeal to a contingent body of doctrine, such as a physical theory, the discussion

of the RDA (or even the whole endurantism-perdurantism debate) is liable to become

relative to a theory. There are two aspects to this; the �rst leads on to the second.

First: the RDA might hold good in one theory, and fail in another. Thus it is a

familiar thought that any consistent theory lays out a set of possibilities: in philo-

sophical jargon, possible worlds according to the theory; in physics jargon, a space of

solutions. So relative to any consistent theory about matter and rotation (describing

them no doubt partially rather than completely|and perhaps falsely), the two cases

(Stat) and (Rot) are either two distinct possibilities: or they are not, either because

at least one is not possible (since e.g. the theory denies that matter is homogeneous),

or because they are the same possibility.

This point is independent of \scienti�c realism", which concerns whether we should

believe the theoretical claims of our best theory to be at least approximately true. But

of course, any \naturalist" will be especially interested in whether the RDA holds good

in our best theory of matter and rotation.

Second: But we should beware of just dismissing the RDA on the ground that

according to our best theories, matter is in fact made of atoms and so not homogeneous.

For presumably:

(i): A continuous, rigid and utterly homogeneous form of matter could exist and

be formed into a disc that either rotates or is stationary. And:

(ii): No philosopher of persistence is \so far gone" in naturalism as to be interested

only in how objects persist, given all the contingencies of the actual world.

In what follows, I will agree with these presumptions, so as to give the RDA against

perdurantism as good a run as possible. But it is worth drawing attention to them

since, as we shall see:

(i'): The sort of continuous and homogeneous matter the RDA needs is a much

subtler and more problematic a�air than the RDA literature typically recognizes.4

This leads in to (ii'):|

(ii'): Some perdurantists reply to the RDA by saying that for the possibilities (Stat)

and (Rot) to exhibit no di�erence to which the perdurantist can appeal, the advocate

of the RDA needs to \imagine away" so many actual laws, technical and-or everyday,

which describe various causes and e�ects of rotation, that the RDA's possibilities (Stat)

and (Rot) are, though logically or metaphysically possible, very arcane. Indeed, they

are so arcane that a naturalist perdurantist need feel no shame in being unable to

4
My 2004, 2004b develop this theme.

5



accommodate them.

To put the reply (ii') in the jargon of possible worlds: the perdurantist claims their

theory of persistence, though contingent and unable to discriminate the possibilities

(Stat) and (Rot), is true in so broad a class of possible worlds that excluding (at least

one of) (Stat) and (Rot) is a small price|and worth paying. (Examples of this reply

include: Lewis (1986, p.xiii, 1994, p. 475), Callender (2001), and (less explicitly) Sider

(2001, 230-236).) This leads to the next Subsection.

2.4 The accompaniments of rotation

Rotation has countless typical causes and e�ects; or if one is wary of causal talk: count-

less typical accompaniments. Typically, a rotating object was previously set in motion,

say by being pushed by someone, and exhibits distinctive dynamical e�ects: for ex-

ample, a solid object tends to become oblate, and a 
uid, like water in a whirlpool,

develops a concave surface. These accompaniments do not depend on matter being in

fact atomistic (or on the laws of physics being relativistic and quantum). So in a possi-

ble world that contained continuous and homogeneous matter but was otherwise \like

the actual world", these accompaniments|even the \technical" ones, like oblateness

and concavity|would occur. In which case, the RDA needs to block the perdurantist

appealing to them so as to distinguish the cases.

True to the tradition of conceptual analysis, the literature on the RDA almost

entirely sets aside the technical accompaniments, and concentrates on the everyday

ones, like having been pushed in the past; and on related everyday counterfactuals,

such as `were I to spray a spot of paint on the disc, I would see it move', or `were

I to grasp the disc, I would feel friction'. More speci�cally, the literature tends to

assume that the RDA can legitimately set aside all the technical accompaniments by

just stipulating that the rotating disc is not only solid but perfectly rigid, so that it

does not become oblate; (hence Section 2.1's mention of rigidity). The philosophical

battle can then be joined on two battle�elds familiar to metaphysicians; as follows.

First, there is debate about whether the RDA can legitimately \imagine away"

the everyday accompaniments of rotation, so that the perdurantist cannot appeal to

them. In particular: if (as usual) the RDA stipulates that the present and \occurrent"

everyday accompaniments are absent, can the perdurantist appeal to past or future

accompaniments, or perhaps to counterfactuals about them? For example:

(i) Can the perdurantist make the distinction by appealing to a past cause, such as

a push, or to a present counterfactual about seeing a paint-spot move?

(ii) Or would appealing to a past cause amount to postulating an unacceptable

\temporal action-at-a-distance" (e.g. Robinson 1989 p. 405-406; Hawley 2001 p. 81)?

(iii) And would appealing to a present counterfactual amount to postulating unac-

ceptably \ungrounded" counterfactual truths (Robinson 1989 p. 403; Hawley 2001 p.

74-75)?

Second, there is debate about whether the perdurantist can appeal to di�erences

6



between (Stat) and (Rot) that are distinctively metaphysical (neither everyday nor

technical-physical). For example: Can the perdurantist appeal to:

(i) a special (non-Humean) relation of immanent causation between temporal parts

that subvenes (or even yields an analysis of) persistence (Armstrong 1980, 1997, pp.

73-74); or

(ii) special vectorial properties that are numerically equal to, yet di�erent from,

velocities (Robinson 1989 pp. 406-408, Lewis 1999: incidentally, this idea echoes Leib-

niz's proposal against Descartes (1698, sect. 13)); or

(iii) non-causal relations between temporal parts that are not supervenient on the

intrinsic natures of the parts that are the relata, and yet are not just spatiotemporal

relations (Hawley: 1999, p. 63-66; 2001, p. 85-90)?

I do not need to enter either of these battle�elds; (fortunately, since they remain

well-populated, despite the cross�re!). As to the �rst, I can set aside the \everyday

accompaniments". For I shall argue in Section 3 that the RDA should not just set

aside technicalities, in particular the technical accompaniments of rotation; and that

in any case, it cannot do so just by stipulating perfect rigidity. As to the second, I can

set aside controversial notions like immanent causation etc. (of which I am in any case

wary). For I shall argue in Sections 4 and 5 that technicalities from physics suggest

another way the perdurantist can reply to the RDA.

2.5 Two kinds of reply: Against the consensus

We can sum up \the story so far" in two stages. First, there are two main ways

perdurantists can reply (and have replied) to the RDA. They can either:

(`Appealing Di�erences'): argue that there are di�erences between the discs to

which they can appeal; whether everyday (e.g. `someone pushed it'), technical (e.g.

`it's oblate') or metaphysical (e.g. `the timelike curves of immanent causation are

helical, not straight'); or

(`No Di�erence'): argue that possible worlds in which the discs show no such

di�erence are too arcane to matter: i.e. they do not fall within the scope of their

\naturalist" account of persistence.

Of course, these replies can in a sense be combined. For in the back-and-forth of

debate, a perdurantist might move from one reply to the other: `well, if you imagine

away all of those accompaniments of rotation, I will then reply that there is after all

no di�erence'.

Second: In the literature on the RDA, considerations of metaphysics, and in par-

ticular conceptual analysis, tend to dominate. This dominance has led to a widespread

consensus on four points: two in support of the RDA, and two against the perdurantist.

Namely:

(I): The RDA can legitimately

(a) imagine away the usual accompaniments of rotation: both the everyday

ones; and the technical ones such as discs tending to be become oblate (in the latter

7



case, by requiring the discs to be rigid);

(b) assume the intuitive notion of rotation, with its idea of persisting spatial

points.

On the other hand:

(II): the perdurantist cannot legitimately

(c) appeal to di�erences of velocity, since velocity presupposes persistence; nor

can they

(d) appeal to the atomic, indeed quantum-theoretic, nature of matter, since the

topic of debate is our common-sense conception of persistence|which surely allows

continuous matter.

Turning to this paper: I shall now argue against the consensus (a)-(d). Section 3

argues against (a) and (b); Sections 4 and 5 against (c) and (d). The overall e�ect

will be twofold. As to (a) and (b): I will concede that there are sound versions of

the RDA. Indeed, the RDA can be formulated more strongly than in Section 2.1:

for it need not imagine away the usual accompaniments of rotation. But as to (c)

and (d): a certain sort of perdurantist|roughly speaking, one who accepts only non-

instantaneous temporal parts|can both appeal to di�erences of velocity, and garner

support for their position from quantum theory.

3 Describing rotation

The main theme of this Section is the practice by the RDA's advocates, of imagin-

ing away the technical accompaniments of rotation (especially dynamical e�ects, like

oblateness) that the perdurantist might latch on to as marking the distinction between

the discs, (Stat) and (Rot). I will argue that:

(i): One main aspect of physics' description of rotation supports this practice (Sec-

tion 3.1).

(ii): But the RDA's advocates are lucky|it is an undeserved victory|since they

are largely unaware of (i); and their actual reasons for imagining away such accompa-

niments are worse (Section 3.2).

(iii): In any case, the RDA can be formulated more strongly, i.e. without imagining

away such accompaniments. The idea is that the perdurantist cannot distinguish two

possibilities that di�er about the sense of a disc's rotation; (Section 3.3).

(iv): Both the original version of the RDA and this stronger version fail in general

relativity (Section 3.4).

3.1 Rotation is kinematic

The description of motion and in particular of rotation, in the various spacetime theo-

ries (non-relativistic and relativistic, special and general), is a large and subtle subject.

In particular, general relativity holds considerable surprises (Malament 2002, 2003).

8



But in this Subsection, I only need the following points:

(i): The various theories analyse motion using a four-dimensional connection (not

necessarily supplied by an absolute space), and spatial and temporal metrical struc-

ture; and they model continuous matter with a congruence of timelike curves: i.e. a

continuously in�nite collection of worldlines whose points of intersection with a (pos-

sibly �nite) spacelike slice completely �ll the slice.

(ii): Given such a connection, metric(s) and congruence, the theories de�ne (in

much the same way) at each point in the congruence a rotation tensor, usually sym-

bolized as !, which gives a quantitative measure of the speed and direction of rotation

of the congruence in an arbitrarily small neighbourhood of that point.

(iii): This construction is a \robust local" limit of de�nitions of rotation for ex-

tended objects; in the following sense. There are various intuitively compelling (and

experimentally realizable) criteria for whether an extended object, such as a disc, is

rotating; but as one considers smaller and smaller discs, the verdicts of these various

criteria as to whether a given disc is rotating converge on the verdict given by the

rotation tensor, i.e. by whether or not ! = 0. (The surprises about rotation in general

relativity (Malament ibid.) concern extended objects. For more details about (i)-(iii),

cf. e.g. Misner et al. 1973, p. 566; Dixon 1978, p. 121-128, 140-145, 163-166; Wald

1984, p. 216-218.)

There is also a large and subtle philosophical subject hereabouts: can the per-

durantist and endurantist equally well accommodate the contents of these spacetime

theories, or does one have an advantage over the other? Thus the rise of relativity

theory prompted philosophers such as Russell and Whitehead to claim that relativity

theory favoured an ontology of events, not substances|corresponding, at least roughly,

to today's perdurantist ontology. But on the other hand: nowadays, many enduran-

tists are \scienti�c realists", and even \substantivalists" about spacetime so that they

advocate a mixed ontology: though matter endures, spacetime regions are also genuine

objects and they perdure. For the sake of brevity, I must set aside this debate: (for

some discussion and references, for both sides of this debate, cf. Sider (2001, pp. 75-76,

110-119) and my 2004b.) As mentioned at the end of Section 2.1, I will just assume,

so as to give the RDA as good a run as possible, that the endurantist has at least as

much right as the perdurantist to these theories.

Points (i)-(iii) imply that ! is de�nable in kinematic terms, without regard to

dynamics: rotation makes sense without reference to forces, energies etc. I claim this

supports the RDA's tradition of imagining away rotation's usual accompaniments. To

put it in terms of possible worlds: (i)-(iii) suggest there are possible worlds with a

spacetime manifold, spatial and temporal metrics and compatible connection, and a

congruence of timelike curves representing continuous matter|and no dynamics. A

pair of these worlds can match in countless ways and yet di�er as to whether the matter

is rotating at a point in spacetime. For we can imagine that the worlds match so well

as to justify talking of a point common to the two spacetimes, and yet in one world

the rotation tensor is zero at the point, while in the other it is non-zero. I take this

as evidence that perdurantism should strive to accommodate the distinction between

9



these possibilities.5

3.2 Beware of rigidity

But advocates of the RDA have usually had di�erent|and I submit, worse|reasons

than that just given, for insisting that the perdurantist should distinguish (Stat) and

(Rot) even without any of the usual accompaniments of rotation. I will not try to

catalogue people's errors, but will focus on one prevalent reason: that the RDA can

just stipulate that the discs are perfectly rigid. (Parts of this Subsection's critique will

carry over to versions of the RDA that use a homogeneous 
uid, rather than a rigid

solid.)

This reason is defective in two ways. First: to say these two words `perfectly rigid',

so \trippingly o� the tongue", is to forget that within the theories of classical continuum

physics, perfect rigidity is a very strong idealization|it violates central principles of

these theories.

To take the example of a disc: what in fact would happen when a (classical, con-

tinuous, homogeneous) stationary disc is pushed at its edge to make it rotate, is very

complicated. A disturbance would travel outward (at the speed of sound for the disc's

material) from the place where the push is applied, leading to a complex process that

settled down so that the whole disc rotated approximately uniformly, with internal

cohesive forces exerting the required centripetal forces on parts of the disc.6 So assum-

ing perfect rigidity requires that the disc's cohesive forces should respond \in�nitely

quickly" to distorting in
uences. More precisely, it amounts to vetoing any account of

how the whole disc is set in motion as a consequence of the motions of the parts. In

physics jargon: it vetoes any constitutive theory.7

Second: it is not true that perfect rigidity gets rid of all the actual technical accom-

paniments of rotation. For not all such accompaniments are kinematically manifested,

i.e. associated with changes in shape or size. In particular, dynamical e�ects involv-

ing forces and energies would be present in a perfectly rigid rotating disc. (In physics

5
I do not claim that it is conclusive evidence. Some perdurantists will still prefer the `No Di�erence'

reply to the RDA. That is: they will say that worlds with no dynamics are so unlike the actual

world, that perdurantists have no responsibility to distinguish rotation and non-rotation within them

(cf. Callender 2001, p. 38). But I do not need to resolve this dispute between myself and fellow-

perdurantists. For all perdurantists can agree to the more important conclusions in the next three

Subsections.
6
In the actual quantum world, this description is of course an excellent approximation for solid

discs, the cohesive forces being electromagnetic forces between atoms.
7
Two incidental remarks about rigidity. (1): There is also the worry that perfect rigidity violates

relativity's prohibition on faster-than-light signals. But in fact, relativistic theories allow generalized

notions of rigidity: for a philosopher's introduction, cf. Earman (1989, Chapter 5.5, pp. 98-101).

(2): Among Bigelow and Pargetter's arguments for their heterodox account of instantaneous velocity,

as not always a limit of average velocities, is a thought-experiment involving perfectly rigid spheres

(1989, pp. 292-293, 1990, pp. 67-68). As it happens, I disagree with their argument, but I will not go

into details: in Section 2.2, I already set aside such heterodox accounts of velocity.

10



jargon: some e�ects involve stress rather than strain.) There will be cohesive forces

throughout the disc's interior which would be absent if the disc were stationary; and

the disc's energy is greater (which in relativity means that its mass is greater). Though

such e�ects are more technical than being oblate, that is no reason to think the per-

durantist cannot appeal to them.8

3.3 An improved RDA: allowing the actual accompaniments

The previous Subsection prompts the suspicion that the RDA needs to imagine away all

the actual accompaniments of rotation, in something like the way suggested in Section

3.1. But in fact, not so! Just by altering appropriately the possibilities considered, we

can get a formulation of the RDA which keeps all the actual accompaniments.9 Thus

the endurantist challenges the perdurantist to distinguish the possibilities:|

(Same): Two perfectly circular discs, d1 and d2, both made of continuous homoge-

neous matter and lying in the same spatial plane|but otherwise as di�erent as you

please from one another|spin in the same sense (i.e. both clockwise as seen from one

side of the plane, and so anti-clockwise as seen from the other side).

(Di�erent): Two discs, d01 and d
0

2, match d1 and d2 respectively in all respects (at

all times); except that d01 and d
0

2 spin in opposite senses relative to one another.

The idea is that all the usual accompaniments (stress as well as strain: forces and

energies as well as distortion) match between d1 and d
0

1; and similarly between d2 and

d02. So there is no need to imagine them away.

Nor is there any need for discs within one of the possible worlds to match in any

respect, except being perfectly circular, made of continuous homogeneous matter, lying

in the same spatial plane|and for (Same), spinning with the same sense.

Three comments, in descending order of importance, by way of clarifying this for-

mulation of the RDA:|

(i): Intuitively, (Di�erent) describes equally well two distinct possibilities: one in

which d01 spins in the same sense as both d1 and d2; and the other in which instead,

d02 shares their common sense of rotation. This contrast of course depends on there

being a �ducial spatial direction in common between the possibilities. I agree that this

idea is perfectly coherent. But I emphasise that the direction needs to be speci�ed by

something salient in the environment, such as a local gravitational �eld giving one an

8
At least: it is only a reason if we take the endurantism-perdurantism debate as entirely a matter

of analyzing everyday concepts. In particular, the RDA cannot just consider an oblate rotating disc

and a non-rotating one moulded so as to be congruent to it (as proposed by Hawley 2001, p. 83-84).
9
Paul Mainwood and David Wallace devised the following formulation in a seminar in autumn

2003. The idea of exploiting the distinction between two senses of rotation, so as to avoid having to

imagine away the usual accompaniments, had already been brie
y advocated by Dean Zimmerman

(1998, p. 268-269), crediting an anonymous referee. But beware: Zimmerman's discussion can be

read as placing each disc in a separate possible world|in which case it fails, as explained in (ii)

below. Zimmerman kindly points out (personal communication) that this was not his intention; so

that his formulation of the RDA is essentially the same as that invented by Mainwood and Wallace.

For novelty and precision, I present theirs. My thanks to them and Zimmerman.

11



up-down distinction, on pain of its being a distinction without a di�erence, i.e. a spuri-

ous distinction|an artefact of a diagram, or of our visual imagination. So: given such

a speci�cation, (Di�erent) indeed represents two possibilities. No matter: to challenge

the perdurantist, the RDA can simply consider either one of them.

(ii): (This follows on from (i).) The danger of making a distinction without a dif-

ference also crops up in another way. As mentioned in footnote 9, a formulation of the

RDA in terms of distinguishing two senses of rotation (and thereby keeping the usual

accompaniments) has been urged before, by Zimmerman (1998, p. 268-9). But Zim-

merman's brief discussion can be read as challenging the perdurantist to distinguish

between a disc rotating clockwise, alone in its possible world, and a duplicate disc ro-

tating counterclockwise at the same rate, alone in its world. And this formulation fails

for the reason emphasised in (i): the clockwise-counterclockwise distinction assumes a

�ducial spatial direction in common between the possibilities, which for these \lonely"

discs is a spurious distinction. (Callender (2001, p. 32, 36-7) seems to read, and object

to, Zimmerman in this way.) Our formulation above avoids this diÆculty by consider-

ing two discs in each possible world, so that we need only intra-world comparisons of

the sense of rotation.

(iii): The possibilities can be modi�ed in various ways. In particular, to secure

the needed intra-world comparisons of sense of rotation, we do not need two discs.

(Same) could instead contain just one disc, rotating in the same sense as a curved

arrow drawn on a sheet of paper lying beside it; (Di�erent) would then similarly con-

tain a single disc rotating contrary to the sense of another curved arrow drawn on an

adjacent sheet of paper.

3.4 The RDA fails in general relativity

But in general relativity, the trajectory of a test-particle falling towards a massive body

depends on whether (and how) the body is rotating: the rotating mass \drags", albeit

very slightly, the inertial frames in its vicinity (Misner et al. 1973 pp. 699, 879, 1117).

This frame-dragging means that the RDA fails in the sense that, in the usual version,

the inertial frames (the worldlines of test particles) are dragged around the rotating

disc (Rot), but not around (Stat); and in Section 3.3's version, there cannot be the

perfect match in rotation's accompaniments both between d1 and d
0

1 and between d2
and d02, since the dragging of inertial frames around a rotating body is di�erent, for

di�erent senses of rotation. In short: the RDA fails because there is a dynamical e�ect

of rotation, to which a \suÆciently naturalist" perdurantist can appeal so as to answer

the challenge of distinguishing the possibilities.10

Could the endurantist improve the thought-experiment so as to allow for frame-

dragging, in the kind of way that (Di�erent) and (Same) improve on (Stat) and (Rot)

by allowing for the usual accompaniments of rotation? Perhaps, but I do not see how.

10
This argument against the RDA, in its usual version, is due to Callender (2001, p. 38); it is part

of his `No Di�erence' reply.

12



On the other hand, the endurantist has two lines of reply, even if she cannot thus

improve the thought-experiment. Both return us to some questions raised before.

First, she might emphasise that in developing the RDA for general relativity (in the

usual, or Section 3.3's, version) she can stipulate that the discs are \lonely", i.e. that

there are to be no test-particles travelling the dragged worldlines. Does this stipulation

make the di�erence to which the perdurantist appeals|viz. whether the frames are

dragged, and if so, how|counterfactual? The answer depends on the interpretation of

general relativity. Roughly speaking, a substantivalist will answer `No', since they take

the metrical structure of spacetime to be real and occurrent: it is not just an encoding

of how suitable bodies would behave. But the endurantist may argue that she can ac-

cept general relativity, and so develop the RDA for it, without being a substantivalist

in this sense; (cf. Section 3.1). On the other hand, even if we accept that the di�erence

is counterfactual, perhaps the perdurantist can still appeal to it: (cf. Section 2.4).

The second reply is the obvious one about philosophical method. Surely no philo-

sophical account of persistence should be \so far gone" in naturalism as to depend on

general relativity: it should be able to accommodate continuous matter in classical and

special relativistic spacetimes (cf. (2)(ii) in Section 2.3). And for these cases, the RDA

remains unrefuted, at least in Section 3.3's improved version.

I think the second reply has force. But in the next two Sections, I will argue that

the perdurantist can meet the challenge of defeating the RDA even outside general

relativity: in short, by accepting only non-instantaneous temporal parts. Besides, this

version of perdurantism is supported by some heterodox proposals about the intrinsic-

extrinsic distinction among properties: proposals which are themselves supported by

some features of classical and quantum physics.

4 Perdurantism without tears: the classical case

4.1 Rejecting instantaneous temporal parts

At the end of Section 2.2, I said I would take it that both average and instantaneous

velocity presuppose the notion of persistence, and are extrinsic properties. But when

we consider a non-instantaneous temporal part, the second point needs to be quali�ed.

For one of the part's constituent pieces of matter having a certain worldline segment

within the part is surely an intrinsic property of the part. And similarly for lesser, i.e.

logically weaker, information than the entire worldline segment. For example, that a

constituent piece of matter has a certain average velocity over a time-interval \within"

the temporal part is intrinsic to the part: notwithstanding the fact that average velocity

presupposes the notion of persistence. Similarly for instantaneous velocity at a time

\within" the temporal part.

At least, these properties are intrinsic to the part, modulo the topic I set aside at

the end of Section 2.1, viz. how to justify the appeal to persisting spatial points, and

a spatial metric, that is needed for the idea of the distance traversed by the persisting

13



object.

This situation prompts some terminology. Since in this Section (and the next one),

I will be focussing on whether the possession of a property P by an object o at a time

implies propositions concerning matters of fact, especially about o, at other times, it

will be convenient to use the phrase `temporally intrinsic property'. By this I mean

\intrinsic as regards time": i.e. roughly, a property whose possession by o at a time

implies nothing about matters of fact (especially about o) at other times (though it may

imply propositions about other places). Similarly, I shall talk of temporally extrinsic

properties; and of spatially intrinsic and extrinsic properties.

Two warnings about this terminology. (1): I agree that my explanation is vague,

not least because the general intrinsic-extrinsic distinction on which it rides is itself

vague; (indeed probably ambiguous|cf. Humberstone 1996, Weatherson 2002). But

my vague explanation will be enough for this paper. (2): Note that a property could

be temporally extrinsic for one instance and not for another. Velocity itself provides

an example: it is temporally extrinsic for an object o at an instant t; but, as just

emphasised, temporally intrinsic to a non-instantaneous temporal part (say, a stage of

just o's \career") that includes o at t. Humberstone (1996, p. 206, 227) notes that a

similar phenomenon|extrinsic for one instance, but intrinsic for another|occurs for

extrinsicality and intrinsicality simpliciter.11

These points are of course independent of whether matter is atomic or continuous.

The piece of matter can be a point-particle or a point-sized bit of matter in a continuum.

(Indeed, the quali�cation could be stated in the very same words for an extended piece

of matter, provided it was small enough for us to model it as point-like, i.e. having a

worldline, and a single velocity: but I can focus on unextended pieces of matter.)

To sum up: a non-instantaneous temporal part has a rich set of intrinsic, or at

least temporally intrinsic, properties concerning the worldline-segments and average

and instantaneous velocities, during the part, of its constituent pieces of matter.

Now consider a version of perdurantism that accepts only non-instantaneous tempo-

ral parts. (I will not discuss the pre-history of this proposal in authors like Whitehead:

for details cf. Grattan-Guinness (2002). But I will soon discuss whether it should

accept all such parts, i.e. parts with an arbitrarily short, though non-zero, temporal

extent.)

Since such parts have a rich set of intrinsic properties, the prospects for the per-

durantist project of de�ning persistence (or providing a supervenience-basis for it, or

at least some non-reductive account of it) look a great deal better than for a pointil-

liste version of perdurantism accepting only instantaneous parts (or accepting also

extended parts, yet requiring persistence to supervene on the intrinsic properties of

instantaneous parts, as in Lewis' Humean supervenience). For with these rich sets of

properties, there are so many more ingredients which one could use in the de�niens

of persistence (or more generally, in the account of persistence). More precisely: the

perdurantist's prospects are a great deal better, provided their de�nition or account

11
My (2004, 2004b) further discuss temporal and spatial extrinsicality.

14



of persistence can legitimately refer to these intrinsic properties of non-instantaneous

temporal parts.

In the rest of this paper, I will endorse this version of perdurantism, both in general

and as a reply to the RDA (both the usual formulation and Section 3.3's stronger one).

The reply it a�ords to the RDA is as follows. The worldline segments, average

velocities and instantaneous velocities of point-sized bits of matter within a homoge-

neous disc provide intrinsic properties of the disc's temporal parts. Assuming that the

perdurantist can appeal to these intrinsic properties|an assumption I will discuss in

Section 4.2|she can certainly distinguish the discs. Indeed, with these intrinsic prop-

erties to hand, she may well have no more of a problem about her project of de�ning

persistence, for the parts of a perfectly circular homogeneous disc, than for the parts of

an inhomogeneous one. There are two aspects to this, which we can call `kinematical'

and `dynamical'.

4.1.1 \Kinematics" First, the perdurantist can appeal to the mathematical fact

that every suitably smooth vector �eld U de�ned on a open region R of spacetime

has integral curves throughout R: curves which are timelike, by de�nition, if U is

timelike. (`Suitably smooth' requires only that U be C1, i.e. the partial derivatives of

its components exist and are continuous.) So the idea is that the intrinsic properties of a

non-instantaneous temporal part of a classical continuous body (whether homogeneous

or not) specify the vector �eld U, of instantaneous velocities (to be precise: 4-velocities)

of the body's point-sized bits of matter, on the spacetime region R of the part. U then

speci�es integral curves, i.e. the worldlines within R of the bits of matter. Besides,

by considering a set of such non-instantaneous parts that \cover" the entire period for

which a given bit of matter exists, its entire worldline can be reconstructed.

There are two points to make about this proposal; of which the second will lead us

to \dynamics".

(1): Agreed, this proposal seems at �rst sight a cheat, a case of theft over honest

toil. But I am for the moment just assuming that the perdurantist can appeal to

intrinsic properties of non-instantaneous parts, even though some of them involve the

notion of persistence: postponing discussion to Section 4.2. And rest assured, I will

there admit that this assumption makes this kind of perdurantism \non-reductive".

(2): The idea of reconstructing an object's entire worldline by concatenating seg-

ments (each lying in one of a \covering" set of non-instantaneous temporal parts) is

closely related to a formal equivalence between the ways that endurantism and perdu-

rantism describe the motions of both point-particles and continuous bodies (in both

non-relativistic and relativistic spacetimes). I develop this equivalence in Butter�eld

(2004, 2004a). The idea, for the simplest case viz. point-particles or point-sized bits

of matter in a continuous body, is that:

(a): an endurantist will represent the motion by a single function q : t 7! q(t) 2

M, mapping times at which the point-particle, or point-sized bit of matter in a con-

tinuous body, exists, to locations in a manifold M (either space or spacetime); while

15



(b): the perdurantist will use a collection of functions, labelled by time-intervals

that together cover the object's lifetime; for example, if it exists throughout the closed

time-interval [a; b], there might be a function q[a;b] : t 2 [a; b] 7! q[a;b](t) 2 M.

In this paper, I do not need more details of this equivalence (e.g. about extending

it to spatially extended objects). It suÆces to say that one can reconstruct worldlines

from such functions, even for a point-sized bit of matter in an utterly homogeneous

continuum, provided the functions' domains are non-degenerate time-intervals, i.e. not

singleton sets of times; (cf. Butter�eld 2004a, Section 3).

This reconstruction of worldlines from a collection of functions raises two points.

First, I admit again that it seems at �rst sight a case of theft over honest toil: the per-

durantist reconstructs worldlines from functions that involve the notion of persistence.

Here I again refer to Section 4.2's discussion.

Second, this reconstruction of worldlines is \kinematical". It uses no information

about the properties of the moving matter, in particular the causes of its motion

(\dynamics"): it simply invokes a set of functions that immediately specify worldline-

segments. So it is natural to ask whether our kind of perdurantist can give an account

of persistence that in some way appeals to (i) the properties of the moving matter, or

(ii) the causes of its motion.

The RDA argues that appealing to (i) cannot work for perfectly homogeneous mat-

ter, though (i) is of course the bread-and-butter of the philosophical search for criteria

of identity for actual objects of various sorts, such as persons. I believe that nothing in

our perdurantist's rejection of instantaneous temporal parts undermines her appealing

to (i) for that kind of search; I will support this in Section 4.2.2 (cf. also Butter�eld

2004b, Section 6.2.2.(2)). On the other hand, (ii) leads to \dynamics".12

4.1.2 \Dynamics" Our perdurantist can indeed appeal to dynamics. That is: if she

is suÆciently \naturalist" that she is willing to appeal to the laws of motion (Section

2.3), then in a classical mechanical world, the de�nition of persistence can \piggy-back"

on the determinism of those laws.

That is: in common cases, the classical laws (above all, Newton's second law, that

Force = mass � acceleration) fully determine the motion of a point-particle, or a point-

sized bit of matter in a continuum, over an interval of time [t1; t2], in terms of its initial

position and velocity at t1 and the regime of forces on it during [t1; t2]: all of which

the perdurantist can take to be given by intrinsic properties of a temporal part that

begins before t1 and ends after t2.

Agreed, that is rough speaking: hence my `in common cases'. For accuracy, I should

note some of the subtleties, in particular the threat to determinism from solutions in

which some quantities become in�nite within a �nite period of time after the initial

12
This \kinematics-dynamics" contrast exempli�es two more general contrasts in the philosophy of

identity (discussed in in my 2004a, Section 4.1) which I call (a) `ontic-epistemic' and (b) `conceptual-

empirical'. (a) concerns whether the criterion or account of identity speci�es the \constitutive facts"

of persistence, or our grounds|everyday or technical, occasional or systematic|for judgments of

persistence. (b) concerns whether the criterion or account eschews the concepts and results of empirical

theories, e.g. physical theories, or is willing to invoke them.

16



time t1. For point-particles, such solutions are known to exist even if we veto collisions;

for a popular account of this, cf. Diacu and Holmes (1996, Chapter 3). For continua,

whether there are such solutions is a deep open question: witness the fact that one

of the Clay Institute's million-dollar Millennium Prizes is for a proof or disproof of

the rigorous existence for all times of solutions of the equations that govern a classical


uid, i.e. the Navier-Stokes equations.13

But for present purposes, I can discount these subtleties: here it is enough to suggest

that a naturalist perdurantist can go about de�ning persistence in terms of integrating

the equations of motion.

So much by way of replying to the RDA. But I need to defend this version of perdu-

rantism, especially the assumption that the perdurantist's account of persistence can

appeal to the non-instantaneous parts' intrinsic properties. I will defend this perdu-

rantism in four stages. The �rst two stages are metaphysical; I expound them in the

next two Subsections. The third and fourth stages will return us to the philosophy

of physics. The third stage, in Section 4.4, concerns the classical mechanical descrip-

tion of motion; the fourth stage, in Section 5, concerns quantum theory. The third

and fourth stages will also each involve a novel proposal about the intrinsic-extrinsic

distinction among properties.

4.2 Intrinsic properties of non-instantaneous temporal parts

Intrinsic properties of non-instantaneous temporal parts raise three issues; which I

address in three Subsections.

4.2.1 Can the perdurantist appeal to them? I claim that the perdurantist can

legitimately appeal to these parts' intrinsic properties, even though some of them

involve the notion of persistence. Does this mean that my sort of perdurantist just

gives up on the project of de�ning persistence (or at least providing a supervenience

basis for it) in terms that do not presuppose it? Agreed, giving up need not spell

defeat for perdurantism. For a non-reductive perdurantism of the sort mentioned in

(1) of Section 2.3 might have various merits|and merits that are not undermined by

accepting only non-instantaneous temporal parts. (I will support this in Section 4.3.)

But does my sort of perdurantist give up?

Yes and No! Yes, in that she aims to give some account of persistence, yet is

willing to have the account invoke notions that presuppose persistence; in particular,

instantaneous velocity.

But also, No: for reasons hinted at in Section 4.1.B's discussion of persistence

\piggy-backing" on the laws of motion. That is: my sort of perdurantist need not

assume persistence as a primitive|or that persistence is somehow satisfactorily de�ned

13
For a popular account, cf. Devlin (2002, Chapter 4); for a monograph discussion of what is known

about the simpler case of a perfect 
uid (Euler's equations), cf. Section 4.4 and Example 5.5.8 of

Abraham and Marsden (1978)|thanks to Gordon Belot for this reference.

17



(or accounted for, say with a supervenience thesis)|for some speci�c set of parts: say,

a set that covers the lifetime of the persisting object in question, or a set containing

all those temporal parts with a temporal extent (lifetime) less than some bound. She

can perfectly well pursue the project of de�ning, or accounting for, persistence as a

relation between any two non-instantaneous parts (including any two sub-parts of any

given non-instantaneous part).

And even if the perdurantist accepts all such parts, so that there are parts with

arbitrarily short, though non-zero, temporal extents, I maintain that this need not

involve a vicious regress of endlessly deferred de�nitions or accounts of persistence. For

the account may, for time-intervals less than some amount, become suitably \uniform",

i.e. with no substantive variations for shorter times. In short: it can be \turtles all

the way down", provided that below a certain level, the turtles are all the same. Of

course, this is in e�ect what happens in an account of persistence that piggy-backs

on the classical deterministic laws of motion, determining future and past positions in

terms of present position and instantaneous velocity (or momentum).

4.2.2 Temporal intrinsicality at an instant is rare I turn to a general point

about the sorts of property invoked in an account of persistence: a point that applies

to both endurantist and perdurantist, and to accounts of criteria of identity for speci�c

kinds of object, e.g. persons, where there are issues, e.g. about the weighing of diverse

factors such as bodily and psychological similarity, absent from the highly general

endurantism-perdurantism debate.

The point is simply that almost no properties are temporally intrinsic to their

instance at an instant. That is: almost all properties require features of their instance

not only at a single instant, but also at other (albeit perhaps close) times. So an

account of persistence, or a criterion of identity for a speci�c kind of object, needs

must appeal to temporally extrinsic properties; (though the other times involved may

be close to the given one).

Unfortunately, this fact is obscured in most philosophical discussion of persistence

(at least in the tradition of conceptual analysis). This discussion focusses on the idea

of giving an account of, or criterion for, o at time t being the same persisting object

(maybe of a speci�c kind, e.g. person) as o0 at t0, that invokes everyday properties.

Almost always, the idea is that the object(s) (in perdurantist terms: the two temporal

parts) need to be:

(i) suitably similar as to these properties: where `suitably similar' allows con-

siderable change provided there is some kind of chain of small changes; (cf. Follow in

Section 2.1); and-or

(ii) suitably causally related, with the properties being the causally relevant

ones (in other jargon: part of the speci�cation of the object's causal state); where

again there can be a suitable chain of stages or states linked by causation.

So far, so good: I have no objection to searching for this sort of account or criterion,

nor to its invoking everyday properties in ways (i) and-or (ii). But the locution `at

time t', and the focus on everyday properties, makes philosophers often choose as their

18



examples observational properties, i.e. properties which can be ascribed \at a glance":

be they \everyday-taxonomic" like `is a rock/leaf/chair' or \purely sensory" like `is

red/hot'. And since they can be ascribed at a glance, philosophers are tempted to

think they are temporally intrinsic in the strong sense of requiring something of their

instance only for a instant.14

And that is false. We are very gross creatures: our perceptual apparatus is insen-

sitive to such properties. Rather, the process of perception \averages", in myriadly

complex (and often adaptive) ways, over the instant-by-instant properties of not only

the object but also the medium, and our perceptual apparatus itself. So any observa-

tional property is temporally extrinsic at an instant: it demands features of its instance

over a time-interval of at least about one twentieth of a second|and in general a very

complex, open-ended and vague array of features, to boot.

When we set aside conceptual analysis and everyday properties, and consider the

properties of technical science, in particular physical theories, the same conclusion

holds good: most properties are temporally extrinsic at an instant (though as empha-

sised, they may well be intrinsic to a non-instantaneous temporal part). Thus most

of the hundred-odd physical quantities that get an entry in a physics dictionary are

clearly temporally extrinsic at an instant. I have already mentioned velocity: obvi-

ously momentum, angular momentum and kinetic energy are temporally extrinsic for

the same reason. Many other quantities, such as temperature, conductivity (thermal

and electrical), permeability and permittivity, depend for their de�nition (as well as

their value) on collective phenomena that require a process or situation to last longer

than an instant (though perhaps much less than a second).

I admit that within classical physics, three familiar quantities are good candidates

for being temporally intrinsic even to an instant: viz. position, mass and electric

charge. Besides, for a point-particle: these also seem to be spatially intrinsic at a

spatial point, not just for an extended spatial region. At least, this is so modulo the

topic I set aside at the end of Section 2.1, about the basis of spatial geometry: that is

to say, a \relationist" about spatial geometry would no doubt object to the claim that

the position of a point-particle is spatially intrinsic to a point.15

I also admit that this trio seeming to be intrinsic|taken together with the great

success of classical physics in reducing much of the behaviour of large complex objects

to the classical mechanics and electrodynamics of postulated tiny components, whether

point-particles or point-sized bits of matter in a continuum (\micro-reductionism")|

has undoubtedly been one strong reason, perhaps the main reason, for the prevalence

14
All parties can agree that among non-observational everyday properties, most are temporally

extrinsic; indeed they often require features at other times of objects other than their instance: for

example, being married requires a spouse at a past wedding, and no intervening divorce or death.
15
Beware! For a point-sized bit of matter in a continuum, the trio of position, mass-density and

charge-density seem to be not only temporally intrinsic, but also spatially intrinsic|provided we can

interpret the densities (i) as de�ning mass and charge through integration, rather than (ii) being

themselves de�ned from the masses and charges of �nite volumes, by taking the limit of smaller

and smaller volumes. But in fact, we cannot interpret the densities like this: (i) fails, and we need

(ii)|another mark against pointillisme, in my view (2004).

19



in philosophy of pointilliste doctrines like Lewis' Humean supervenience.

Of course, the RDA is precisely an argument that such doctrines come to grief on the

topic of persistence.16 And my present point is that the rarity of temporal intrinsicality

at an instant supports my proposal to be perdurantist without being pointilliste|and

so to block the RDA.

4.2.3 A better reason for temporal intrinsicality Finally, an incidental point.

Philosophers discussing persistence have another reason to focus on temporally intrin-

sic properties, in addition to the erroneous tendency to think observational properties

are temporally intrinsic to an instant. I admit that it is a better reason. But it is

a reason only for properties temporally intrinsic for shortish intervals, up to about a

second: not for the stronger notion of temporal intrinsicality at an instant|which is

the target of my anti-pointilliste campaign. In short, the reason is that a property

that is temporally intrinsic for a longish interval is liable to be useless in a criterion of

identity.

In detail: All parties (both endurantists and perdurantists) can agree that an ac-

count of persistence, or a criterion of identity, had better not invoke a property that

requires some feature of its instance within a period of time similar to the time-scale

over which the account or criterion is to be applied. For doing so is liable to make the

criterion hard or even impossible to apply. Thus suppose an account of the conditions

under which o at time t is the same persisting object (maybe of a speci�c kind, e.g.

person) as o0 at t0, invokes a property P : requiring, say, that o at t must be P and

so must o0 at t0. (The argument works equally well with other requirements, e.g. that

only one of the two need be P , but that change as regards P is suitably continuous,

with some kind of chain of small changes.) Then if being P at t requires a feature � at

a time close to t0, it may well be hard to apply the account: having to ascertain that �

holds close to t0 might entangle one in ascertaining whether the persistence claim for o

and o0 holds.

4.3 Non-instantaneous parts can do the jobs

I turn to the second stage of my defence of perdurantism without instantaneous tempo-

ral parts. I claim that, by and large, non-instantaneous temporal parts do the various

jobs, within the endurantism-perdurantism debate, that the perdurantist demands of

temporal parts, just as well as instantaneous temporal parts. More precisely: this is

so once the perdurantist \just says No" to the siren-calls of pointillisme. Of course,

I cannot here discuss all these jobs: I will make do with three comments. The �rst

comment is general, and will be illustrated by the second and third, which concern

particular jobs temporal parts are invoked to do.

16
Philosophers tend to forget that they also have trouble in physics. The classical mechanics and

electrodynamics of point-particles and continua have considerable conceptual tensions, some of which

are aggravated by a pointilliste picture; cf. (2004, 2004b).

20



4.3.1 Humean supervenience revisited The �rst comment is an o�er of a peace-

pipe to the neo-Humean. She envisages the world as \loose and separate", a succession

of \distinct existences": \just one darned thing after another". My version of perdu-

rantism can agree, in that it might well accept all non-instantaneous temporal parts,

no matter how short-lived: my veto is only against utterly instantaneous parts.

Besides, my perdurantist can echo Lewis' Humean supervenience, by making some

claim along the lines that all the facts supervene on the temporally local facts; i.e. the

facts speci�ed by the intrinsic (if you like: temporally and spatially intrinsic) proper-

ties of all the non-instantaneous temporal parts. To state this echo more precisely: she

can claim that for any covering of spacetime M by a family F of non-instantaneous

temporal parts (no matter how short-lived some or all of the parts may be), all the

facts supervene on the intrinsic properties of elements of F. (Here, `covering' is un-

derstood in mathematicians' usual sense: a set M is covered by a family F of sets i�

M � [F; and similarly if M and the elements of F are treated not as sets, but as say

mereological fusions.)

So the only aspect of Lewis' Humean supervenience that my perdurantist needs to

deny is the pointilliste idea that all the facts supervene on the intrinsic properties of

spacetime points (or of spatially extended instants of time, i.e. spacelike surfaces). I

think neo-Humeans should �nd this a price worth paying: having all the facts super-

vene on the intrinsic properties of all the non-instantaneous temporal parts should be

enough to satisfy a Humean's ambition to have the \global" supervene on the \local".17

4.3.2 The problem of change The second comment concerns the so-called `prob-

lem of change'. Perdurantists argue that o's changing in respect of a property P is

best understood in terms of one temporal part having P and another having :P . In

particular, they argue that the endurantist has to understand P (and :P) as a relation

to a time, and that for the case of an intrinsic property P this is surely wrong. Hence

the problem is also called the `problem of temporary intrinsics'; (cf. e.g. Sider 2001

pp. 92-98, Lewis 2002).

So far as I can tell, almost all the arguments for the perdurantist understanding

of change carry over, so as to support my version, i.e. perdurantism without in-

stantaneous temporal parts. (Admitted: as do almost all the arguments against the

perdurantist understanding of change.) The main reason is of course that if within a

single non-instantaneous part of o there is change in respect of P , the perdurantist will

understand the change in terms of one shorter-lived part of o having P , and another

not|and this need not involve a vicious regress (Section 4.2.1). Besides: since tempo-

ral intrinsicality at an instant is rare (Section 4.2.2), the perdurantist's argument that

endurantism has trouble with temporary intrinsics is more persuasive as an argument

17
Agreed: since these parts in general overlap, the \fundamental description of the world", given

by the in�nite conjunction of all (the ascriptions of) the intrinsic properties of all such parts, is highly

redundant. But I say: no worries. After all, the exact spatial analogue occurs in continuum classical

mechanics. To describe a continuum, this theory needs|not the in�nite point-by-point conjunction

of all the properties of points|but the highly redundant in�nite region-by-region conjunction of all

properties of all regions; cf. footnotes 15 and 16.

21



for non-instantaneous temporal parts.

But there is one objection; (my thanks to Oliver Pooley). Suppose that a tempo-

rary intrinsic property such as shape changes continuously over time, so that an object

o is square for merely an instant: to secure an instance of squareness simpliciter in this

scenario, the perdurantist surely needs an instantaneous temporal part.

Reply: Given the supposition, this is certainly right. Here I can only bite the bullet,

by any or all of:

(i) dropping the problem of change from the list of jobs my non-instantaneous

temporal parts are to do; or

(ii) urging that since temporal intrinsicality at an instant is rare (Section 4.2.2)

my temporal parts can solve the problem of change for the vast majority of temporary

intrinsic properties; and besides, urging that succeeding with this vast majority should

satisfy the neo-Humean (cf. the �rst comment above); or

(iii) adopting a \mixed" view, more congenial to pointillisme, that admits in-

stantaneous parts as well as non-instantaneous ones, but then argues that it is le-

gitimate to account for persistence (and so answer the RDA) by invoking only the

non-instantaneous ones, as I have.

Of these options, I on the whole prefer reply (ii). But I will not in this paper try

to choose between these replies: in particular, I will not refer again to the mixed view,

though I agree it is tenable.

4.3.3 Puzzles of coincidence Thirdly, the situation as regards the debate over

`puzzles of coincidence' is similar to that for the problem of change. The puzzles (re-

viewed by Sider 2001, p. 5-10, 141-152) concern such cases as the statue and the clay,

or the �ssion and fusion of objects such as amoebae|or even persons. For example,

after an artist makes on Tuesday a statue out of a lump of clay, the statue and clay

seem to be the very same object. But they seem to di�er in their temporally extrinsic

properties (often in this debate called `historical properties', e.g. by Sider 2001, p.

5, 142): the statue but not the lump was created on Tuesday, the lump but not the

statue existed on Monday. Perdurantists argue that these puzzles are best understood

in terms of distinct objects sharing temporal parts, just as objects can share spatial

parts (such as two roads having a stretch in common).

Again: so far as I can tell, almost all the arguments for the perdurantist understand-

ing of these puzzles carry over, so as to support my perdurantism without instantaneous

temporal parts. (As do, I admit, the arguments against!). For example, almost all the

arguments in Sider's critique of endurantist approaches (2001, p. 154-188), and in his

advocacy of perdurantism (2001, p. 152-153, p. 188-208), carry over.

I said `almost all the arguments' carry over. For there are two wrinkles. First, Poo-

ley puts the analogue of his objection in Section 4.3.2. Suppose that two objects fuse

for merely an instant: here the perdurantist surely needs an instantaneous temporal

part. I reply: I think this objection is weaker than its analogue in Section 4.3.2, be-

cause its supposition is more of an idealization, more a merely logical or metaphysical

possibility, rather than part of the content of classical mechanics. That is: classi-

22



cal mechanics does describe deformable objects changing shape continuously, as the

objection in Section 4.3.2 requires. But it does not describe instantaneous fusions.

Indeed, classical mechanics �nds collisions, even of point-particles, problematic|let

alone fusions and �ssions. (There is of course no problem about the spatial analogue

of instantaneous fusions, i.e. two 3-dimensional objects sharing a 2-dimensional part:

think of two semi-detached houses!)

The second wrinkle is that the issue whether to accept instantaneous temporal parts

does bear on one signi�cant division within the perdurantist camp. This distinction

concerns how the perdurantist treats temporal language. The traditional perdurantist

view is that an object of ordinary ontology|i.e. a referent of an ordinary term, a

subject of ordinary predications, an element of ordinary domains of quanti�cation|is

the whole four-dimensional object, the \maximal spacetime worm"; (Sider calls this

the `worm view'). But both Sider (2001, p. 188-208) and Hawley (2001, pp. 30-32,

41-64) defend the rival `stage view', that the referents of our ordinary terms, subjects

of ordinary predications etc. are the temporal parts.

This is not the place to assess their arguments for this proposal. They concern, for

example, counting: the stage view says that at each time before an amoeba splits into

two, there is one amoeba (the stage), a verdict which matches everyday thought and

language; but since there are then two maximal spacetime worms, the worm view has

to say that there are stricto sensu two amoebae, and explain away everyday thought

and language by invoking some conventions about counting.18

For my purposes here, it suÆces to comment on Sider's position that the stages he

claims to be the referents of ordinary terms are indeed instantaneous|and so do not

persist: `no person lasts more than an instant' (2001, p. 193)! Sider of course agrees

that everyday thought and language take: (i) ordinary objects to persist, as in `Ted was

once a boy'; and (ii) most of their properties to be temporally extrinsic at an instant,

as in `Ted believes perdurantism is true' (Section 4.2.2). So he goes on to argue that he

can accommodate (i) and (ii) with a temporal analogue of Lewis' counterpart theory

(2001, pp 111-113, 193-198).

My comment on Sider's position is now obvious. While I admit that temporal

counterpart theory is coherent and powerful enough to cope with (i) and (ii)|the

stage view does not have to be so pointilliste as Sider! That is: one could combine my

perdurantism, the rejection of instantaneous temporal parts, with the stage view. Not

only do most arguments for a perdurantist understanding of the puzzles of coincidence

carry over and support my perdurantism (as I said above). Also, one could combine it

with some arguments speci�cally for the stage view: e.g. a version of my perdurantism

that denies overlapping parts could retain Sider's counting argument for favouring the

stage view over the worm view ... But I leave developing this topic for another occasion.

18
For this line of argument, cf. Sider 2001, pp 152-153, 188-193. But Sider has to admit that

sometimes we count by maximal spacetime worms, not by stages, as in `Fewer than two trillion people

have set foot in North America throughout history'. He writes (2001, p. 197): `if `person' refers to

person stages, this sentence will turn out false, since more than two trillion (indeed, in�nitely many

if time is dense) person stages have set foot in North America throughout history'.

23



This concludes my metaphysical defence of this version of perdurantism. I hope to

have made it plausible, quite apart from its blocking the RDA. But the philosophy of

physics has some more support to o�er it. In the next Subsection, the support comes

from the classical description of motion. In Section 5, the support comes from quantum

theory. But this support is not \just technical": each of them involves a novel proposal

about the intrinsic-extrinsic distinction among properties.

4.4 Instantaneous velocity is hardly extrinsic

Extrinsicality is usually considered an all-or-nothing a�air. But it is natural to suggest

that it comes in degrees. Intuitively, a property is more extrinsic, the more that

its ascription implies about the world beyond the property's instance: (compare the

philosophy of mind's jargon of `wide' and `narrow' mental states|some are wider

than others).19 That is rough speaking; and all the rougher because of controversies

about the intrinsic-extrinsic distinction. But I expect that in many suÆciently limited

contexts, the idea could be made precise in a natural way. In any case, I shall in this

Subsection only consider the temporal extrinsicality at an instant of the properties of

position and its time-derivatives (velocity, acceleration etc.), in the classical description

of motion. This is certainly a suÆciently limited context for the idea to be made precise.

After making it precise, I will return to perdurantism without tears, i.e. without

instantaneous temporal parts.20

In (1) and (2) below, I present two closely related ways of making it precise. On

both ways, instantaneous velocity is \hardly extrinsic", i.e. hardly temporally extrin-

sic, since its ascription to an object o at t implies \little" about matters of fact at

other times.21 Both ways are based on the obvious point that the only \categorical"

proposition that an ascription of a velocity (or indeed, of a higher derivative of posi-

tion) to o at t implies about other times is that the o exists for some open interval

(a; b) containing t: all the other implications are hypothetical. This will support my

perdurantism without instantaneous temporal parts.

(The di�erence between the two ways will be that according to the �rst, which is

19
It is all the more natural when we consider how few properties are intrinsic: the extrinsic properties

form so large a class as to merit being sub-divided.
20
Here I should stress a limitation of my discussion, which I have imposed since Section 2.2 and

will maintain until Section 5.1. Namely: in order to give the RDA as good a run as possible, I go

along with the RDA literature's assumption that velocity is de�ned as the time-derivative of position

(so that position is conceptually prior to velocity, and momentum is de�ned as mass times velocity).

But this assumption is questionable, for various reasons. In particular, one can develop classical

mechanics by taking momentum as primitive, together with position and mass, and de�ning velocity

as momentum divided by mass. And in such a presentation, momentum does not need to be \secretly

understood" as mass times velocity: one can introduce it abstractly, and without reference to time,

as the generator of spatial translations. Thanks to Gerard Emch for explaining this point.
21
It is also hardly temporally extrinsic, on a third construal of that notion discussed in my 2004. Of

course none of this is to deny that instantaneous velocity is temporally extrinsic at an instant, since

it presupposes persistence.

24



\read o�" the calculus, successively higher time-derivatives of position are more ex-

trinsic; while on the second way, which is more logical and less mathematical, velocity

acceleration and all higher derivatives are equally|and only mildly|extrinsic.)

In what follows, we can think of o as a point-particle; but it could equally well

be a point-sized piece of matter in a continuum, or an extended body small and rigid

enough to be treated as a point-particle. It will also be clear that the temporal extrin-

sicality of average velocity, acceleration etc. is mild for essentially the same reasons

as for instantaneous velocity, acceleration etc. But to save space, I will focus on the

instantaneous quantities.

(1): The sequence of time-derivatives:| The discussion will be tidier if we consider

ascriptions, not of speci�c values of position, velocity, acceleration etc. to o at time

t, but of some or other value. Then successive ascriptions are of increasing logical

strength: having a velocity implies having a position, having an acceleration implies

having a velocity etc.

So consider a sequence of ascriptions to o at time t: viz.

(Pos): an ascription of a position, i.e. a proposition saying that o has some or

other position at t;

(Vel): an ascription of an (i.e. some or other) instantaneous velocity at t;

(Acc): an ascription of an instantaneous acceleration at t.

These ascriptions are of course the �rst three members of an in�nite sequence of

ascriptions stating the existence of higher time-derivatives of o's position. This gives an

obvious sense in which instantaneous velocity is only mildly extrinsic. Each ascription

is logically stronger than its predecessor; so (Vel), being almost at the start of the

sequence, implies little in comparison with later members.

In more detail: if a real function f has a derivative at a point t 2 IR, it must be

de�ned on a neighbourhood of t and be continuous at t. So the existence of f 00(t)

requires the existence of f 0 in a neighbourhood of t and its (i.e. f 0's) continuity at t;

and this in turn requires the continuity of f in that same neighbourhood of t. And so

on. In short: the existence of the nth derivative gives more information about times

other than t than does the existence of the (n � 1)th derivative.

(2): The \only categorical implication":| But there is also another sense in which

velocity and the higher derivatives of position are only mildly temporally extrinsic.

This sense is more directly tied to the basic idea that the only categorical proposition

that an ascription of such a quantity to o at t implies about other times is that the o

exists for some open interval (a; b) containing t.

In more detail. Let us ask what exactly is implied about other times by the as-

criptions in the sequence; starting with (Pos). The metaphysical literature invariably

assumes position to be temporally intrinsic: why? The answer seems clear: `because

(Pos), or even an ascription of a speci�c value `o is at x at t', implies nothing about

o's position at other times'.22

22
Once again, as mentioned at the end of Section 2.1: this answer ignores possible implications

about other objects' positions, at t or other times, and so sets aside the absolute-relational debate

about space.

25



But to be more precise about `implying nothing' (apart of course from necessary or

analytic propositions), we need:

(a) to decide whether to allow that the object o might exist only for an instant; (as

many metaphysical discussions of persistence do: true to the tradition of conceptual

analysis, they allow all metaphysical or logical possibilities, not just the nomic ones);

and

(b) to distinguish categorical from hypothetical propositions.

Although the categorical-hypothetical distinction is vague and contentious (because

`logical form' is), I will not need to be precise or partisan about this: for it will be

obvious from the calculus' de�nition of a limit which propositions implied by ascriptions

such as (Pos)-(Acc) to count as hypothetical.

If we allow o to exist only for an instant (if we say `Yes' in (a)), then indeed (Pos)

implies no categorical proposition about o's positions at other times: there may be no

such positions! But consider a hypothetical proposition along the lines: `if o exists at

a later time t0, and some value (or upper limit) is assumed about its average speed

(de�ned in the usual way as distance traversed divided by time elapsed) over [t; t0],

then o is at t0 within a sphere of a certain radius, centred on x'. Such a hypothetical

proposition is of course not analytic; but it follows by just de�nitions and logic from

`o is at x at t'.

When we turn to the next member of the sequence, (Vel), we of course get many

more implications. o must exist throughout some open interval, maybe tiny, around

t; and since di�erentiability implies continuity, o's position at a time t0 in the interval

tends, as t0 tends to t, to o's position x at t; and so on. But these implied propositions

are, with one exception, hypothetical. The hypothetical propositions include those

about average velocity discussed in the previous paragraph, and various others one can

spell out by applying the de�nitions of continuity and di�erentiability. The exception

is of course the categorical proposition that o exists throughout some open interval of

times around t; (and, to be precise: its analytic consequences, like o's existing at some

time t0 not equal to t). In particular, (Vel) is compatible with o being anywhere at any

other time t0, no matter how close t0 is to t.23

Similarly again, for (Acc). There are again more implications, but they are almost

all complicated hypotheticals: the only categorical proposition about other times that

the ascription implies is the same one again: that o exists throughout some open

interval of times around t. And so on along the in�nite sequence of ascriptions.

To sum up this discussion of (1) and (2): the temporal extrinsicality of velocity

and higher derivatives of position is mild. For almost all of the implied propositions

are hypothetical; and even the temporally intrinsic ascription (Pos) implies countless

such propositions. Besides, the categorical propositions implied by an ascription of

velocity, or of any higher derivative, are all just consequences of the one proposition

that o exists throughout some open interval of times around t. So as regards categorical

implications about other times, the temporal extrinsicality gets already at stage (Vel)

23
I here assume there is no limiting velocity, as in relativity.

26



as \bad" as it ever gets along the sequence: and that, I submit, hardly deserves the

name `bad'|it is mild.24

Finally, I can connect this discussion with the previous Subsections' advocacy of a

perdurantism accepting only non-instantaneous temporal parts. The main idea here

has been that the \only proposition that goes beyond the instance" implied by an

ascription of velocity (or of other derivatives of position) is that o exists throughout

some open interval, perhaps very short, around the time t. So the \only proposition

that goes beyond the instance" that velocity (or other derivatives of position) implies,

corresponds to an intrinsic property of any (no matter how short!) non-instantaneous

temporal part containing t.

Obviously, this idea meshes with two points in previous Subsections:|

(a): A perdurantist who rejects instantaneous temporal parts can account for persis-

tence: either by fusing segments, perhaps arbitrarily short, of worldlines (\kinematics";

Section 4.1.1); or (more naturalistically) by \piggy-backing" on solving the determin-

istic classical laws of motion, given o's initial position and velocity, and the forces on

it; (\dynamics"; Section 4.1.2).

(b): A perdurantism without instantaneous temporal parts can accept all non-

instantaneous parts, no matter how small their temporal extent. Besides, if one accepts

all such parts one can add a claim that all facts supervene on the \temporally local"

facts, in a strong enough sense to satisfy all but the most pointilliste neo-Humeans; (cf.

Section 4.3.1).

5 Support from decoherence in quantum theory

5.1 Classical and quantum: relativizing the intrinsic-extrinsic

distinction

As I said in Section 2.3, I am not so far gone in naturalism as to just dismiss the

RDA on the grounds that matter is in fact atomic. I agree: a classical mechanical

continuum could exist|prompting the RDA, modulo the above replies. My argument

in this Section will instead be that the way in which classical mechanical objects (both

particles and continua) are in fact emergent from the quantum realm provides further

support for Section 4's perdurantism without tears, i.e. without instantaneous temporal

parts.

This argument will use two new assumptions: one about philosophical method, the

other about the intrinsic-extrinsic distinction.

24
This view is re
ected in the jargon of mathematics and physics. For example, mathematicians

call not only (Pos), but also the ascriptions (Vel) etc., `local'; and physicists call equations of motion

that determine the object's motion at t in terms of its position and some of its derivatives then (but

without reference to facts a �nite temporal interval from t) `local in time'. For more discussion, cf.

Arntzenius 2000 pp. 192-195, Smith 2003 and my 2004b.

27



(1): I will now assume that the interpretation of classical mechanics|in particular,

our conception of how its objects (both particles and continua) persist|should be

sensitive to how classical mechanical objects in fact \emerge from the quantum". I

agree that this assumption is controversial: why not just interpret each theory on its

own, as best you can? After all, there is no lack of work: as I remarked in Section 2.3,

classical mechanics is interpretatively subtle and problematic, even without considering

the dreaded quantum. But I am not alone in endorsing this assumption, even as regards

the interpretation of a classical theory being sensitive to an \adjacent" quantum theory.

Thus for Belot (1998, p. 550-554), it is the main moral of his examination of classical

electromagnetism and the Aharonov-Bohm e�ect.

(2): My second assumption is that it is legitimate to relativize the intrinsic-extrinsic

distinction among properties to a body of doctrine. The distinction is of course usually

discussed in terms of logical or metaphysical possibility: the literature discusses taking

a property P to be intrinsic i� it is logically or metaphysically possible for an object

o to have P \while lonely", or \whatever the rest of the world is like", or ... But I

now assume that it is legitimate to relativize the modality to a body of doctrine, such

as a scienti�c theory T . (I will not need the metaphysically more ambitious idea of

relativizing to the \laws of nature", or to the laws of nature of some possible world.)

Therefore I shall talk, for any such body of doctrine or theory T , of nomic intrinsicality

and extrinsicality.

Unless T is logically or metaphysically necessary|a case I need not consider|the

relativized modality will be a restricted one. That is: not all logically or metaphysically

possible worlds make T true. In general, this will strengthen the notion of intrinsicality,

and correspondingly weaken the notion of extrinsicality|however exactly we under-

stand the original intrinsic-extrinsic distinction. That is: nomic intrinsicality will imply

intrinsicality simpliciter, and extrinsicality simpliciter will imply nomic extrinsicality.

For intrinsicality is a matter of \not implying propositions about the instance's en-

vironment"; and once we assume a theory T is true, any proposition in T can be an

implicit premise in an implication|yielding more implications. So in general, once we

assume T , more properties will be classi�ed as extrinsic. So extrinsicality simpliciter

implies nomic extrinsicality; and vice versa for intrinsicality. (Similar remarks apply to

my notions of temporal, and spatial, intrinsicality and extrinsicality; and to the case

where we consider two theories T1 and T2, one implying the other.)

In fact, this idea of relativized intrinsicality has surfaced in the literature (Hum-

berstone 1996, p. 238); but so far as I know, it has not been pursued. I agree that

many a metaphysician will at �rst sight doubt its value, though they will probably

accept it as coherent. Thus Humberstone writes, after 
oating the idea of relativizing

intrinsicality to a class of possible worlds that match in their laws of nature: `From

a suitably elevated position [i.e. suitably general philosophical stance], this has an

element of arbitrariness about it: why not restrict attention to worlds|not with the

same laws as ours, but|with the same tourism statistics for Naples as ours?' (ibid.).

But I submit that relativization to (our best guess for) the laws of physics has some

interest! In any case, I can at least show that in the present context, it has the interest

28



of being surprising. For in Section 5.2 I will argue that the position, and even the

existence, at a time of an emergent classical object (whether a particle or a point-sized

piece of matter in a continuum) is extrinsic, relative to the laws of quantum theory.25

But before arguing for this, I should brie
y set aside another way in which quantum

theory bears on persistence, and apparently on the RDA.

5.1.1 Unitarity: momentum as temporally intrinsic

Quantum theory violates an assumption that the RDA depends on, viz. that velocity

is not part of the instantaneous state of an object (in particular a point-sized piece of

matter). (I registered this assumption already in Section 2.2; and set aside Tooley's

heterodox proposal that velocity should be thus viewed, on the grounds that Tooley's

proposal in any case assumed the notion of persistence.) This assumption is often

endorsed in the metaphysical literature about space, time and motion, even apart from

the RDA: for example, Sider (2001, p. 39) says `�xing the properties and relations of

present objects will not �x their velocities' (cf. also his p. 34-35).

The assumption tends to be associated with the fact that in classical mechanics, in

order to determine an object's future (and past) motion, you need not only its present

position and the forces acting on it (in the time-interval concerned), but also its present

velocity; i.e. the fact that classical mechanics' equations of motion are second-order

in time. For in a theory in which position and forces were enough to determine the

motion (a theory that is �rst-order in time), it would be more tempting to say that

velocity is part of the present instantaneous state. At least, it would be as tempting to

say this, as that the whole future (and past) history of the system is part of the present

instantaneous state (because of the determinism). Certainly, in such a theory the RDA

itself would have much less sting for a \naturalistic" perdurantist, who is willing to let

her account of perdurance depend on the actual laws. For in such a case, ingredients

that the RDA's advocate presumably agrees to be available to the perdurantist, viz.

position and forces, are enough to determine future positions.

But quantum theory violates this assumption.26 It is �rst-order in time. It combines

the position and velocity (better: momentum) aspects into a single instantaneous state

of a system which, together with the forces acting on the system, determines its future

(and past) states (setting aside controversy about whether there is a \collapse of the

wave-packet" on measurement).27

25
Besides, the extrinsicality has nothing to do with the possible involvement of other objects in

de�ning position, as urged by a relational conception of space: an issue I set aside at the end of

Section 2.1. The extrinsicality is what I have called temporal extrinsicality, rather than spatial; and

it arises from decoherence.
26
As readers who are cognoscenti of quantum theory will have long ago noticed: at least by the

time that Section 4.1 proposed we could have perdurantism without tears, by letting the perdurantist

\have" velocity, and even have their account of persistence \piggy-back" on integrating the classical

equations of motion. Apologies for the delay!
27
Agreed, the Hamiltonian formulation of classical mechanics also combines position and momentum

29



So it is tempting to say that in quantum theory, velocity and momentum are just

as intrinsic (or temporally intrinsic) to the system at a time, as is position; (Arntze-

nius (2003, p. 282) says this). A bit more precisely: once we are willing to relativize

the intrinsic-extrinsic distinction to a physical theory (as proposed in (2) above), it is

tempting to say this.

Furthermore, just as Section 4.1.2 proposed that in a classical setting, a perdu-

rantist accepting only non-instantaneous parts could have their account of persistence

\piggy-back" on integrating the classical equations of motion: so in quantum the-

ory, the perdurantist's account of persistence could appeal to integrating the quantum

equations of motion. (But as the weasel-word `system' hints, it is controversial how to

relate persisting objects to quantum systems, even if you know the systems' complete

histories: cf. the next Subsection).

So be it, say I. But again: I am not so far gone in naturalism about persistence|I

am loath to just dismiss the RDA on the grounds that quantum theory is �rst-order

in time. A theory of persistence should accommodate classical continua, and this

Subsection's points do not bear directly on how it can do so. However, I will now

argue that quantum theory has other light to shed on our topic|once we ask the

interpretation of classical mechanics to take note of how classical mechanical objects

emerge through decoherence.

5.2 Position and existence as nomically extrinsic

So let us adopt the idea in (2) of Section 5.1, of nomic intrinsicality and extrinsicality.

The intrinsic-extrinsic distinction among properties is to be relativized to bodies of

doctrine|in particular, to quantum theory.

Warning: Choosing logically strong bodies of doctrine can yield odd-sounding ver-

dicts of extrinsicality. Given our interest in temporal extrinsicality, the obvious example

of this is provided by a deterministic theory. Thus suppose you choose to relativize,

not just to the deterministic theory itself, but to the conjunction of the theory and

the regime of forces imposed on a system in some time-interval (a; b). This yields the

verdict that every instantaneous state28 is temporally very extrinsic: indeed, about as

extrinsic as it could be. For given the laws of the theory and the forces imposed, any

instantaneous state of a system determines the system's states during (a; b). But it

sounds wrong to say that every instantaneous state is temporally very extrinsic.

The solution of course is to exercise some judgment about what is a natural or use-

ful body of doctrine to which to relativize. In our example, the theory is presumably

in its conception of state, and so is �rst-order in time. But there is a crucial disanalogy: neither of

the pair, position and momentum, determines the other. (Indeed, the formulation is equivalent to

the Lagrangian or Newtonian formulation, under certain conditions, in particular taking the phase

space to be the cotangent bundle of a con�guration space.) But in quantum theory, the position and

momentum representations each determine the other.
28
Since for some philosophers, a state is not a property, it is better to say: every property that

speci�es such a state.

30



such a body of doctrine, but its conjunction with a speci�ed regime of forces is not:

that is too particular (logically strong). More generally, we should allow some distinc-

tion between \central" and \peripheral" statements (or more generally; features) of an

ambient body of doctrine, and relativize the intrinsic-extrinsic distinction only to (the

conjunction of) the central ones. That is, only the central ones are held �xed in all the

nomic possibilities, and so by nomic intrinsicality and extrinsicality. Then you may

say in the example that (maybe part of) the speci�cation of the forces is not central,

so that instantaneous states are not so very temporally extrinsic.

Let us now apply this sort of relativization to how classical mechanical objects

emerge through decoherence. Fortunately for us, although the quantum measurement

problem remains controversial and there remain many open technical questions in the

physics of decoherence, we need not address these controversies and questions. We

can sidestep the measurement problem, and manage with only the most basic and

best-established features of decoherence.29

Classical mechanical objects (both particles and continua) are in fact transient and

approximate patterns in the quantum state of an underlying quantum system. They

are patterns that emerge from an ubiquitous, continuous and very eÆcient process of

decoherence, which continues throughout the lifetime of the classical object. Roughly

speaking, decoherence is di�usion (spreading) in to the quantum system's environment

of coherence, i.e. of the puzzling interference e�ects in the probability distributions

that are the system's state.

To keep things simple, I shall discuss this in terms of the elementary quantum

theory of particles, not quantum �eld theory. But I should note that:

(i): quantum particles are themselves transient and approximate patterns in the

quantum state of an underlying quantum �eld or �elds; for discussion of this, cf. my

(2004b) and Wallace (2004, especially Section 5.2);

(ii): decoherence also happens within quantum �eld theory; for a review, cf.

Guilini et al. (2003, Chapter 4).

Here are some details about a well-studied model of a quantum particle immersed

in an environment (called `quantum Brownian motion'). Take as the initial quantum

state of a tiny dust-particle (radius 10�3 cm) in air, a superposition of two positions

for the centre of mass of the particle, with the two positions just 10�4 cm apart (i.e.

a tenth of the particle's radius), and with (say) the two positions not moving relative

to one another. The bombardment of the particle by air molecules is very eÆcient in

di�using the coherence in to the environment: the superposition's interference e�ects

converge to zero like exp(t=10�36 sec) and remain small for a very long time (1010

years)!

This means that very soon the probabilities for any quantity on the particle you

care to measure are as if there is an even chance of the (centre of mass of the) particle

29
Bacciagaluppi (2003) is an excellent introduction to decoherence for philosophers; for more techni-

cal details, Guilini et al. (2003), Schlosshauer (2003) are also excellent. By the way, all these sources

endorse the consensus that decoherence cannot by itself provide the solution of the measurement

problem, but is an important ingredient in any such solution.

31



being in the two positions; (i.e. probabilities for quantities other than position are

also given by a 50-50 mixture corresponding to the two positions). Similarly for other

initial states: if the initial superposition had the two positions separating from each

other at say x cm sec�1, then after a second, the probabilities would be as if there is an

even chance of the (centre of mass of the) particle being in two positions x + 10�4 cm

apart. Indeed, more generally: it is even possible to deduce the approximate validity

of the deterministic classical mechanical equations of motion of a dust-particle from

the underlying equations for the quantum system, together with a description of the

decoherence process.

So the classical object, \the dust-particle we see", corresponds to one of these

two decohered possibilities (in my example: possibilities for the position of the centre

of mass). It is a pattern in a quantum state, which also contains another pattern

corresponding to the other possibility. If the quantum state were suÆciently di�erent,

not only would the classical object not have the position and momentum we see: it

would not exist. In particular, if the decoherence process did not occur, it would never

exist; and if the decoherence process did not continue, it would cease to exist. That

is: the quantum system would continue to exist, but the classical dust-particle would

not: it would \disappear into a quantum fog".30

I propose that we take these points, about how classical objects are in fact patterns

in a quantum state that are formed because of an ongoing process of decoherence,

as what I called `central'. After all, they are crucial to how such objects are in fact

constituted. That is: I propose they are to be held �xed in assessing whether a property

is nomically intrinsic or extrinsic. So they are to be available as implicit premises for

implications from ascriptions of a property to propositions about the world beyond the

property's instance.

It follows that an ascription to a classical object such as a dust-particle, of a position

at t (to be precise: for its centre of mass, say), is nomically extrinsic. (I would say:

temporally extrinsic, since the implications are about facts at times other than t). For

the ascription (together with the implicit premises) implies the (categorical) proposition

that the object has a position at all other times in a (very short but non-zero!) interval

of times around t. Here, the length of the interval is determined by the decoherence

process' time-scale.

Similarly, a statement that the object exists at a time is nomically extrinsic. For

it implies that the object exists at all other times in an interval about as long as the

decoherence time-scale.31

30
For more discussion of the idea of classical objects as patterns in quantum objects, cf. Wallace

(2003).
31
A point of clari�cation for quantum a�cionados. You might object that since

(i) the reduced state density matrix of the dust-particle (strictly: of its centre of mass degree of

freedom) is nearly diagonal (upto some desired level of approximation) in position, at an instant;

it surely follows that:

(ii) the position and existence of the classical particle is not nomically temporally extrinsic.

I reply: (i) is of course true, but does not imply (ii). For I am taking as \central", not just the

formalism of reduced states etc., but also the physical fact of a decoherence process over time.

32



So far I have only discussed the emergence of a classical particle, such as a dust-

particle. But the discussion just given carries over to continua, as regards both physics

and philosophy.

Admittedly, there are more technical questions about decoherence in quantum 
uids

that are still open than about quantum Brownian motion, which is by now very well-

studied. But there is already a good understanding of decoherence in quantum 
uids,

and so of the emergence of classical continua. In short: recent work shows that even in

a quantum 
uid, where there is no clear distinction between system and environment,

decoherence selects certain quantities (roughly, hydrodynamic variables) as \behaving

classically". Again, one can deduce the approximate validity of the classical equations

of motion for a 
uid. (For details, cf. Halliwell (1999) and references given there.)

As regards philosophy: I said above that the fact that classical mechanical objects

(both particles and continua) are in fact emergent from the quantum realm should be

re
ected in the interpretation of classical mechanics, and so in a naturalistic theory of

persistence. One way to do this is now clear. Namely: take the nomic extrinsicality of

position at a time, and even existence at a time, as favouring the denial of instantaneous

temporal parts. Thus decoherence supports the perdurantism without the tears of

pointillisme which I defended in Section 4: a naturalistic perdurantist can interpret

classical mechanics in terms of temporally extended temporal parts|and thereby block

the RDA.

Acknowledgements:| I thank: G. Belot, C. Callender, W. Demopoulos, G. Emch,

I. Gibson, L. Humberstone, P. Mainwood, A. Oliver, O. Pooley, M. Redhead, T. Sattig,

S. Saunders, N. Shea, D.Wallace, A. Wilson and D. Zimmerman for discussions and

comments on previous versions; audiences in Florence, Granada, Leeds, London Eng-

land, London Ontario, Montreal and Oxford; and participants at a seminar in autumn

2003.

6 References

R. Abraham and J. Marsden (1978), Foundations of Mechanics, 2nd edition, Ben-

jamin/Cummings.

D. Armstrong (1980), `Identity through time', in ed. P. van Inwagen, Time and Cause,

Dordrecht: Reidel.

D. Armstrong (1997), A World of States of A�airs, Cambridge: University Press.

F. Arntzenius (2000), `Are there really instantaneous velocities?', The Monist 83, pp.

187-208.

F. Arntzenius (2003), `An arbitrarily short reply to Sheldon Smith on instantaneous

velocities', Studies in the History and Philosophy of Modern Physics 34B, pp. 281-282.

G. Bacciagaluppi (2003), `The role of decoherence in quantum theory', in ed. E. Zalta,

The Stanford Encyclopedia of Philosophy, available at: http://plato.stanford.edu.

G. Belot (1998), `Understanding electromagnetism', British Journal for the Philosophy

of Science 49, pp 531-555.

33



J. Bigelow and R. Pargetter (1989), `Vectors and Change', British Journal for the Phi-

losophy of Science 40, pp 289-306.

J. Bigelow and R. Pargetter (1990), Science and Necessity, Cambridge University Press.

J. Butter�eld (2004), `Classical mechanics is not pointilliste and can be perdurantist',

in preparation.

J. Butter�eld (2004a), `On the persistence of particles', forthcoming in Foundations of

Physics; available on the Los Alamos arXive at http://xxx.lanl.gov/abs/physics/0401112,

and on the Pittsburgh philosophy of science arXive at: http://philsci-archive.pitt.edu/archive/0000158

J. Butter�eld (2004b), `On the persistence of homogeneous matter', available on the

Pittsburgh philosophy of science arXive at: http://philsci-archive.pitt.edu/archive/00001760/.

C. Callender (2001), `Humean supervenience and rotating homogeneous matter', Mind

110, pp. 25-42.

F. Diacu and P. Holmes (1996), Celestial Encounters: the Origins of Chaos and Sta-

bility, Princeton University Press.

K. Devlin (2002), The Millennium Problems, London: Granta Books.

W. Dixon (1978), Special Relativity, Cambridge: University Press.

J. Earman (1989), World Enough and Space-Time, Cambridge MA: MIT Press.

I. Grattan-Guinness (2002), `Algebras, Projective Geometry, Mathematical Logic and

Constructing the World: intersections in the philosophy of mathematics of A.N. White-

head', Historia Mathematica 29, pp. 427-462.

D. Guilini et al. (2003), Decoherence and the Appearance of a Classical World in Quan-

tum Theory, New York: Springer (2nd edition).

J. Halliwell (1999), `The Emergence of Hydrodynamic Equations from Quantum The-

ory: A Decoherent Histories Analysis', to appear in Proceedings of the 4th Peyresq

Conference, 1999; available at: http://xxx.soton.ac.uk/abs/quant-ph/9912037

K. Hawley (2001), How Things Persist, Oxford University Press.

I. Humberstone (1996), `Intrinsic/Extrinsic, Synthese 108, pp. 205-267.

G. Leibniz (1698), `On Nature itself, Or On the inherent force and actions of created

things'; pp. 498-508 of L.E. Loemker ed. (1989) Gottfried Wilhelm Leibniz: Philosoph-

ical Papers and Letters, Dordrect: Kluwer.

D. Lewis (1986), Philosophical Papers, volume II, New York: Oxford University Press.

D. Lewis (1994), `Humean Supervenience Debugged', Mind 103, p 473-490; reprinted

in his Papers in Metaphysics and Epistemology (1999), Cambridge University Press

pp.224-247.

D Lewis (1999), `Zimmerman and the Spinning sphere', Australasian Journal of Phi-

losophy 77, pp. 209-212.

D Lewis (2002), `Tensing the copula', Mind 111, pp. 1-13.

Malament (2002), `A No-Go theorem about rotation in relativity theory', in D. Mala-

ment ed. Reading Natural Philosophy Chicago: Open Court; p. 267-293.

Malament (2003), `On relative orbital rotation in relativity theory', in A. Ashtekar,

R.Cohen et al. eds. Revisiting the Foundations of Relativistic Physics: Festchrift for

John Stachel, Kluwer Academic; pp. 175-190.

C. Misner et al. (1973), Gravitation, San Francisco: Freeman.

D. Robinson (1989), `Matter, Motion and Humean supervenience', Australasian Jour-

34



nal of Philosophy 67, pp. 394-409.

M. Schlosshauer (2003), `Decoherence, the measurement problem, and interpretations

of quantum mechanics', available at the Los Alamos physics arXive: quant-ph/0312059.

T. Sider (2001), Four-Dimensionalism, Oxford University Press.

S. Smith (2003), `Are instantaneous velocities real and really instantaneous? An an-

swer in the aÆrmative', Studies in the History and Philosophy of Modern Physics 34B,

261-280.

M. Tooley (1988), `In defence of the existence of states of motion', Philosophical Topics

16, pp. 225-254.

D. Wallace (2003), `Everett and structure', Studies in the History and Philosophy of

Modern Physics 34B, 87-105.

D. Wallace (2004), `In defence of naivete: The conceptual status of Lagrangian quan-

tum �eld theory', forthcoming in Synthese;

available at: http://philsci-archive.pitt.edu/archive/00000519.

R. Wald (1984), General Relativity, Chicago: University Press.

B. Weatherson (2002), `Intrinsic and extrinsic properties', in ed. E. Zalta, The Stan-

ford Encyclopedia of Philosophy, available at: http://plato.stanford.edu.

D. Zimmerman (1998), `Temporal parts and supervenient causation: the incompatibil-

ity of two Humean doctrines', Australasian Journal of Philosophy 76, pp. 265-288.

35