Does Time Flow, at any Rate?

Abstract

The so-called no-rate argument argues that time cannot flow or

pass in the literal sense of that term, because its motion can

be assigned no meaningful rate. This paper examines a yet

unexplored objection to the no-rate argument, which consists

in showing that the argument itself is based on an extended

conception of motion, according to which it is meaningful and

consistent to say that time flows at no well-defined rate.

Keywords: time, rate, speed, flow

1. Introduction

Philosophically unprejudiced language abounds with

expressions such as ‘the flow of time’ or ‘the passage of

time’, which apparently confer objective dynamical properties

on time. Few philosophers are willing to take expressions of

this kind literally: they are sometimes called dynamists, and

their view dynamism. The vast majority of philosophers, on the

contrary, is reluctant to consider similar expressions

anything more than metaphorical characterizations either of

the way we experience temporality or of the irreducibility of

tensed predication. We may collectively refer to them as the

anti-dynamists.1 Several philosophers in the analytic tradition,

in particular, have argued that phrases such as ‘time flows’

or ‘time passes’ are the result of a category mistake or a

semantic shift, to the effect that time, instead of being

conceived as a conceptual or physical precondition of motion,

is treated as the subject of motion itself. This claim is

often supported by what Maudlin (2002, 2007) calls logical

objections: philosophical arguments that purport to demonstrate

that taking expressions such as ‘time flows’ or ‘time passes’

literally would generate either logical fallacies or bare

tautologies.

Certainly the best-known among the logical objections is the

so-called no-rate argument, according to which time cannot

literally flow or pass because it cannot be assigned any

meaningful rate of flow or passage.

[I]f it made sense to say that time flows then it would

make sense to ask how fast it flows, which does not seem

to be a sensible question. Some people reply that time

flows at one second per second, but even if we could live1 The distinction between dynamists and anti-dynamists is thereforetransversal to the more notorious distinction between A-theoristsand B-theorists, who respectively assert and deny that tenses areobjective, irreducible, and non-relational features of time. Movingspotlight theorists are examples of dynamists subscribing to the A-theory, while Prior (1958, 1968) and Zeilicovici (1989) are examplesof non-dynamist A-theorists. The overwhelming majority of B-theorists are non-dynamists, but Zwart (1976) and Maudlin (2002,2007) are exceptions. Similarly, the question whether or not timeliterally flows is logically independent of the debate concerningMcTaggart’s argument against the reality of time, even if the twoissues have been traditionally conflated. See, for instance, Craig(2000: 219 – 222).

with the lack of other possibilities, this aspect misses

the more basic aspect of the objection. A rate of seconds

per second is not a rate at all in physical terms. It is

an adimensional quantity, rather than a rate of any sort

(Price 1996: 13).

This paper examines a yet unexplored objection to the

argument, which consists in showing that the argument itself

is based on an extended conception of motion or flow,

according to which it is meaningful and consistent to say that

time flows at no well-defined rate.

2. What rate?

To begin with let us notice that, for the sake of the

following discussion, terms such as ‘passage’, ‘flow’ or

‘motion’ are perfectly interchangeable with each other. The

no-rate argument, in fact, equally concerns the expressions

‘time passes’, ‘time flows’, and ‘time moves’, as well as all

similar ones. Thereof, we shall indifferently employ all such

terms. For the same reason, we shall consider immaterial

whether time should flow or pass as a whole, or if rather only

the present moment or transient now should glide or shift along

the temporal axis. Similar considerations finally apply to the

terms ‘rate’ and ‘speed’. The premises of the no-rate argument

can accordingly be phrased as follows:

(1) Time flows literally.

(2) If time flows literally, it flows at a well-defined rate.2

2 The phrase ‘well-defined’ should be understood here as ‘compatiblewith the accepted definition or understanding of speed’, and not as

(3) If time flows at a well-defined rate, that rate is one

second per second.

(4) One second per second is one.

(5) One is an adimensional number.

(6) No adimensional number is a well-defined rate.

Premises (1)-(6) jointly entail the contradictory claim that

time flows at a well-defined rate, namely one, that is not a

well-defined rate. This means that at least one premise must

be rejected; but which one?

Those who subscribe to the no-rate argument argue that the

above premises generate a reductio ad absurdum of (1), thus

leading to the conclusion that time does not flow in the

literal sense3. Let us call them the no-raters. They include

Smart (1949, 1954), Price (1996), Olson (2009) and van Inwagen

(2009), just to cite few. Because statements (5) and (6) look

uncontroversial, on the other hand, it appears that dynamists

have only three choices to resist the no-raters’ conclusion.

The first option is to reject premise (4). This is the

currently most debated strategy and it has been chosen by

dynamists such as Maudlin (2002, 2007), Phillips (2009), Raven

(2010), and Skow (2011, 2012a). Despite their different

approaches, all of them share the basic intuition that

temporal dimensions cannot be cancelled from a rate of one

second per second in the same way as two identical numbers can

be cancelled, respectively, from the numerator and the

‘having a definite or precise value’. This means that the no-rateargument is concerned with the form of the purported speed of time,rather than with its value.3 The no-rate argument can also be phrased in modal terms, so as tolead to the conclusion that time cannot possibly flow.

denominator of a fraction. This contention is at the centre of

an on-going debate, and it has been most notably criticised by

Price (2011).

The second option consists in denying premise (3), thereby

arguing that one second per second is not the rate of the flow

of time. This strategy has been chosen by Webb (1960),

Schlesinger (1969, 1982) and Zwart (1972, 1976), and examined

by Markosian (1993), though they disagree as of what the

correct rate should be. Webb and Markosian propose that the

rate of the flow of time should be identical to the

multiplicative inverse of the rate of ordinary changes. This

means that, if a runner moves at the constant rate of twelve

miles per hour, then time must pass at the rate of one hour

per twelve miles covered by that runner. Zwart, instead,

identifies the rate of the flow of time with the number of

events occurring in the universe per standard temporal unit.

Finally, Schlesinger argues that the rate of the flow of time

is measured in units of time per units of a second-order

temporal dimension, called hyper-time or super-time.4 None of the

above proposals, however, has sensibly undermined the no-rate

argument.

The final option is to deny premise (2), arguing that it is

possible for time to flow at no well-defined rate. This

strategy has been never systematically considered. Zwart

(1976) seems to embrace it while arguing, en passant, that the

flow of time is not the kind of flow about which it is

4 The hypothesis of the hyper-time, in particular, has been recentlyrevived by Skow (2012b), though it should be noticed that heconsiders the hyper-time only a heuristic fiction.

meaningful to ask how fast it flows. Markosian (1993) makes

the same claim, arguing that demanding that the flow of time

should possess a well-defined rate would be committing a

category mistake.

Neither Zwart nor Markosian, however, offer compelling

arguments in support of their claims. This potentially makes

their strategy a boomerang for the dynamist. This will become

apparent after having reconsidered the objective of the no-

rate argument. The no-rate argument is most frequently

employed as a reductio ad absurdum of the claim (1) that time

flows in the literal sense. However, it is easy to see that

the same argumentation could be used, modulo minor

corrections, to reject the quite different claim that it is

meaningful to say that time flows in the literal sense. Price

himself, for instance, introduces the no-rate argument as a

refutation of the claim that it could make sense to say that time

flows (Price 1996: 13; see quotation above). On the light of

this, the no-raters could easily reply to Zwart and Markosian

that, if it is meaningless to ask how fast time flows, that is

just because it is meaningless to claim that time flows, in

the first instance.

Still, the path initiated by Zwart and Markosian is worth

exploring. The following sections will be dedicated to show a

different and more effective way of rejecting premise (2).

This strategy will consist in establishing the following two

claims: (i) the no-raters must concede, for the sake of their

reductio, an extended conception of motion, according to which

it is possible that time is both the independent and the

dependent variable of motion; (ii) this conception of motion

is not incompatible with the possibility that time flows or

move without any well-defined rate.

3. Literally, but in what sense?

The no-rate argument, as we have seen, can alternatively be

directed against the claim that time flows literally, or

against the claim that it is meaningful to say that time flows

in the literal sense. Much of its logical strength, in either

case, accordingly hangs on what the literal meaning of that

term is supposed to be. So, what is the literal content of the

idea of motion?

Excluding few notable exceptions (e.g. Tooley 1988) it is

generally agreed, at least within the analytic tradition, that

the concept of motion is best analysed as

the fact that, by the occupation of a place at a time, a

correlation is established between places and times; when

different times, throughout any period however short, are

correlated with different places, there is motion; when

different times, throughout some period however short, are

all correlated with the same place, there is rest.

(Russell 1938: 473)

This view is often referred to as the at-at theory of motion: in

brief, it claims that being in motion consists precisely in

being at different places at different times.

The at-at theory analyses motion into three major ingredients

and a triadic relation. The three ingredients are the set X of

the possible movers, the physical space S, and the time T. The

triadic relation, the relation being-at-at, is a subset of the

Cartesian product XST, namely a set of ordered triples whose

elements are, in order, a mover, a spatial position, and a

time. Given the assumption that nothing can be in two

different places at one time, the further constraint is

imposed that, for every mover x and any time t, there can exist

at most one spatial position s such that x, s and t stand in

the relation being-at-at. For every mover x, we then say that x

moves if and only if there exist two distinct times t1 and t2

and two distinct spatial positions s1 and s2 such that the

ordered triples (x, s1, t1) and (x, s2, t2) are both elements of

the relation being-at-at. For this reason, we may alternatively

think of motions as ordered pairs of triples of the form ((x,

s1, t1), (x, s2, t2)) where s1 and s2 and t1 and t2 are pair-wise

different. In plain words, this means that x moves from s1 to s2

in the temporal interval between t1 and t2 just in case x is at

s1 at t1 and x is at s2 at t2.

Is this, then, the concept of motion that the no-raters have

in mind when they claim that time cannot flow in the literal

sense of that term? This might surprise you, but the answer is

no. Let us see why.

By definition, the relation being-at-at holds between a mover, a

spatial position, and a time, where by a ‘spatial position’ we

mean a position in physical space. This means that the at-at

theory applies only to movers that can be spatially located.

However, time seems not to be that kind of thing: sophistries

aside, it is simply meaningless to say that time is here,

rather than there. Hence, the at-at theory of motion turns out

to be overtly incompatible with dynamism: according to it, it

is meaningless to claim that time moves, simply because it is

meaningless to claim that time is at different spatial

locations at different times.

Thus it seems clear that, when the dynamists claim that time

literally moves, they do not mean by this that time moves in

the sense prescribed by the at-at theory of motion: for

instance, when they claim that time literally flows, they

evidently do not mean that time flows in exactly the same way as a

fluid does. Rather, they mean that time moves according to a

conception of motion that encompasses, but which is not

restricted to, the conception of motion and flow that is

proper of the at-at theory.5

Conversely, it is evident that the no-rate argument cannot be

effectively employed as a reductio of the claim that time flows

or moves in the sense prescribed by the at-at theory. Firstly,

because that is not what the dynamists claim, hence employing

the no-rate argument in such a way would amount to attacking a

straw-man. Secondly, because in that case the no-rate argument

would be entirely superfluous: for, as we have just seen, one

5 Notice that this remains true independently of which strategy thedynamists employ to resist the no-rate argument, and in particulareven if they choose to deny premise (3), which will play a majorrole in the discussion to follow. For instance, the hyper-timestrategy requires that time should move with respect to a temporaldimension that is not the one with respect to which all ordinarymovers flow or pass. Similarly, to argue that the speed of the flowof time should be the multiplicative inverse of ordinary speedsevidently implies that the roles of time and speed should beexchanged in either case.

does not need the no-rate argument in order to show that the

at-at theory of motion is incompatible with dynamism.

This means that, at least for the sake of the no-rate argument, the no-

raters have to admit the possibility that the literal meaning

of ‘motion’ is not the one explicated by the at-at theory, but

rather the one that the dynamists have in mind when they claim

that time flows or move in the literal sense. So, our next

question becomes: what could that meaning be?

To establish this, it will be useful to have a further look at

the premises of the no-rate argument, and in particular at

premise (3). Premise (3) claims that, if time moves at a well-

defined rate, that rate must be one second per second. Rates

of passage or motion are ordinarily measured in units of the

dependent variable of motion per units of its independent

variable. So, according to premise (3), the dependent variable

of the motion of time must be measured in seconds. Since

seconds are units of duration, it follows that the dependent

variable of the motion of time must coincide with time itself.

This means that, according to premise (3), time moves by being

at different times at different times.6

Premise (3) thus requires that time could take over the role

of space in the at-at theory of motion. More exactly, the

6 The adequacy of this analysis is confirmed by the fact that one ofthe logical objections to the flow of time, due to Grünbaum (1973:315 - 316), is specifically directed against the meaningfulness ofthis claim. Grünbaum argues that this claim is in fact a plaintautology, which, as such, can have no empirical content. Toproperly address Grünbaum’s objection would go beyond the scope ofthis paper; should it suffice to mention that at least somedynamicists consider the flow of time empirically evident (cfr. forinstance Maudlin 2007: 114).

physical space S should be replaced by a wider space

consisting in the union of S and T, where T is supposed to

denote time, exactly as before. Ordinary motions are

accordingly analysed in the usual way, while the motion of

time is analysed in terms of (pairs of) ordered triples whose

elements are, in order, time (taken as a mover), a time (taken

as a position), and a time (taken as the time at which time,

understood as the mover, is in that position).

Let us call this the extended at-at theory of motion. Contrary to the

at-at theory, this analysis of motion is prima-facie compatible

with dynamism. As a further proof of this, consider the claim,

frequently made by the dynamists, that time flows from past to

future: according to this claim, time traces a trajectory

within itself, as it were, precisely as requested by the

extended at-at theory.

Of course, the fact that the supporters of the logical

objections concede that motion can be conceived in accordance

with the extended at-at theory does not mean that they

subscribe to that conception. The moral of the no-rate

argument, in fact, can be equivalently rephrased as follows:

even provisionally conceding that the flow of time could be

conceived in the way just outlined, that conception will

eventually be proved to be unsatisfactory, because it fails to

grasp an essential feature of motion, namely speed.

But is speed really necessary to motion? This problem will be

discussed in the following sections.

4. Can there be motion without speed?

Denying premise (2) would present us with a rather unusual

possibility, namely that something could literally flow or

pass at no well-defined rate. No-raters clearly reject this

possibility, but the question we shall address is whether such

a possibility is really logically incompatible with the

conception of motion offered by the extended at-at theory: if

it is, then dynamists will have to accept premise (2),

accordingly being forced to reject either premise (3) or (4);

but if it is not, then dynamists will be entitled to reject

premise (2) as a way to escape the conclusion of the no-rate

argument.

To being with, we can safely argue that speed-less movement is

not prima facie incompatible with the extended at-at analysis of

motion: after all, speed is not expressly mentioned anywhere

in that analysis. This provisional conclusion seems to be

confirmed by the fact that premise (3) entails the extended

theory of motion, yet no overt contradiction can be apparently

derived from premises (1), (3)-(6) alone. These premises

jointly imply that time moves at no meaningful rate, but this

claim, as it stands, does not seem to entail any statement of

the form ‘p and not-p’. To obtain a contradiction out of (1),

(3)-(6), premise (2) seems to be needed; but premise (2) is

precisely the object of our present inquiry, so it cannot

simply be taken for granted.

It seems, then, that premise (2) cannot be logically derived

from the extended at-at theory of motion alone. So, what could

compel the dynamists to accept it? Namely, why cannot they

just subscribe to extended at-at theory, while arguing that

speed-less motion is possible in at least some exceptional

cases, notably including the flow of time?

No-raters, as we have seen, typically consider premise (2)

uncontroversial. However, they cannot simply insist that such

a claim is intuitively compelling, without begging the above

questions. Therefore, they will have to offer some independent

argument in support of it, where ‘independent’ means that such

an argument should not deny ex hypothesi the adequacy of the

extended at-at theory of motion, e.g. by being based on some

alternative theory.

Now, there seem to be only two ways they could do that. On the

one hand, they may contend that the idea of speed analytically

follows from the very idea of motion, so that it is not even

possible to conceive or meaningfully talk about speed-less

motion. On the other hand, they may argue that possessing a

well-defined speed logically follows from being in motion. The

difference between the two cases is subtle, but they deserve

independent scrutiny.

5. Does speed-less motion make sense?

The no-raters might contend that premise (2) is self-evident

because the concept of motion analytically presupposes the

concept of speed. This would imply that it is not even

conceivable that something could move at no well-defined

speed, nor it could even make sense to talk about speed-less

motion. The following discussion will be specifically

dedicated to rebut this contention, and to show that the

concept of speed is demonstrably not a constitutive analytical

ingredient of the idea of motion.

To begin with, let us first determine what we should

understand by speed. To keep our discussion as simple as

possible, let us restrict our attention to the physical

understanding of that term. Showing that the physical

conception of speed is not analytically presupposed by the

physical idea of motion will be enough to prove that the idea

of speed is not analytically presupposed by the idea of motion

in general.

Speed, we are told by the physicists, is the first-order

derivative of space with respect to time. This definition is

intended, in particular, to capture the concept of speed at a

time, or instantaneous speed. For the sake of the following

discussion, however, we shall concentrate on the idea of

average speed, namely the displacement or increment of the

spatial variable per unit of time. This choice is motivated by

three reasons. Firstly, it will allow us not to rule out a

priori non-continuous models of time, because the idea of

instantaneous speed is inextricably associated with the

assumption that time is continuous. Secondly, it will help us

avoid bothersome and irrelevant complications concerning the

infinite derivability of the equations of motion. Finally, it

will allow us not to get involved in the tangled issue whether

or not moving objects should necessarily possess a well-

defined speed at each time.

Now is the idea speed, so understood, analytically presupposed

by the concept of motion? My answer is negative. For it is

evident from what we have just said that speed is defined in

terms of motion or displacement, not vice-versa. However, the

die-hard no-rater may insist that the concepts of motion and

speed are in reality co-defined, in the sense that they

analytically presuppose each other, and that it is not

possible to make sense of either of them without the other.

The fact that we ordinarily define speed in terms of motion or

displacement, rather than otherwise, would then be a purely

historical accident. This would suffice to establish that it

is not possible to conceive of or meaningfully talk about

speed-less motion anyway, and the no-rater could still claim

that premise (2) is self-evident on this basis. To this

objection, let me offer the following reply.

If the concept of speed was analytically presupposed by the

concept of motion, then one could not in principle grasp the

concept of motion in a full and adequate way, without also

understanding the notion of speed, namely the measure of

displacement per unit of time. But then it would follow that,

by the same token, one could not in principle fully understand

the idea of speed without also understanding the idea of

acceleration, namely the increment of speed per unit of time.

Similarly, if that was the case, then the concept of

acceleration would similarly presuppose the concept of jerk,

i.e. the increment of acceleration per unit of time.

Evidently, one could proceed this way without end; but then,

due to the chain of implications, the concept of motion would

turn out to presuppose infinitely many concepts, thereby

becoming indeterminate.7 To preserve the idea of motion from

such an unjust fate, we must conclude that the idea of motion

does not analytically presuppose the concept of speed in the

first place.

Let us call this the conceptual regression argument, and let us

thereafter refer to the infinite regress of analytic

entailments that it depicts as the conceptual regression. The most

promising way to avoid the conceptual regression argument is

to contend that the conceptual regression should not even

start, because, even if the concept of speed is analytically

presupposed by the concept of motion, the concept of

acceleration is not similarly presupposed by the concept of

speed. To support this claim, however, one should point to

some asymmetry between the way speed is conceptually related

to motion on the one hand, and the way acceleration is

conceptually related to speed on the other hand. Let us

therefore examine if any such asymmetry can be found; as we

shall see, two different possibilities are in sight.

Firstly, one can point to the fact that, even though it is not

possible for ordinary movers to move without a rate, they can

nonetheless move without accelerating. This should be

considered a clue, if not a proof, that acceleration is not

necessarily required by speed. This objection to the

conceptual regression, however, would hang on an equivocation7 Evidently, in this context we are referring to a conceptual orlogical possibility. Therefore, to argue that it is notepistemically or practically possible for anyone to go through aninfinite chain of conceptual implications is no objection to thepresent argument.

between moving without accelerating and moving without

acceleration. To move without accelerating is to move at a

constant speed; but to move at a constant speed is not to move

without acceleration: rather, it is to move with a well-

defined acceleration, whose value is zero. So, ordinary movers

cannot move without any well-defined acceleration any more

than they can move without a well-defined speed, and the

concepts of speed and accelerations are exactly symmetrical in

this respect.

Secondly, one may point to the fact that rates are the

increments of some quantity or variable per unit of time, and that,

as such, they are not increments in their turn. Hence, speed

is not an increment. Motion, instead, is an increment: notably,

the increment of space, or better the increment of the

variable determining the spatial position of a mover in some

chosen frame of reference. Speed and motion, therefore, play

different roles in the corresponding rates, and this is why

speed and acceleration are not conceptually related in the

same way as motion and speed are.

To make the same point somewhat crudely, but perhaps more

vividly: acceleration is the increment of speed per unit of

time, but what is speed the increment per unit of time of? Not

of motion, but of space. The relation that acceleration bears

to speed is thus the relation that speed bears to space, not

the relation that speed bears to motion. Thereof, the required

conceptual asymmetry.

Despite its apparent force, this objection is in reality no

more effective than the previous one. Let us recall, in fact,

that the upshot of the conceptual regression argument is to

show that if the concept of speed was analytically presupposed

by the idea of motion, then the concepts of acceleration,

jerk, etc. would be as well, thereby making the meaning of

‘motion’ indeterminate. What matters to the argument, then, is

not the conceptual regression as such, but rather the fact

that the conceptual regression implies that, if the idea of

speed was analytically contained in the idea of motion, then

all the remaining notions involved in the conceptual

regression would be analytically presupposed by that concept

in their turn. The objection that we are now examining can

arguably rule out the conceptual regression itself, but can it

similarly exclude this implication?

Let us begin by considering what relation holds between motion

and speed. Speed, we have said, is the displacement or the

increment of space per unit of time; and, as such, it is

mathematically determined as the ratio between the distance

covered by the mover and the temporal interval required by the

mover to cover that distance. Speed, therefore, is a

quantitative relation between the two basic conceptual

ingredients of the ordinary understanding of motion, namely

space and time – and this is, presumably, the reason why the

no-raters seem to consider the idea of speed a necessary

ingredient of that concept.

Now let us turn to acceleration: acceleration, we have said,

is the increment of speed per unit of time. However, this is

not the sole possible way to conceive it. In fact,

acceleration can be equivalently understood as the

displacement or increment of space per unit of time square (or

the increment of space per unit of time per unit of time), as

it is testified by the standard unit according to which

acceleration is measured, namely meters per square seconds.

But, under this light, acceleration reveals to be no less a

quantitative relation between space and time than speed is:

admittedly, a more complex one, but a quantitative relation

nonetheless. The same is similarly true for jerk: jerk is not

only the increment of acceleration per unit of time, but also

the increment of space per cube unit of time (or the increment

of space per unit of time per unit of time per unit of time).

Speed, acceleration, jerk and all the subsequent concepts in

the conceptual regression accordingly bear similar relations

to motion: they are all increments of space per one power of

the unit of time. The fact that, in particular, speed is the

increment of space per the first power of the unit of time

does not seem to make any substantial difference in this

respect. So, if the idea of speed is analytically presupposed

by the concept of motion qua quantitative relation between

space and time, then by the same token, the notions of

acceleration, jerk etc. must be presupposed by the concept

motion as well.

This means that the conclusion of the argument by conceptual

regression can be maintained even if the conceptual regression

itself is rejected along the lines of the objection just

considered.

6. Does motion entail speed?

The other strategy that the no-raters may choose in support of

(2) is to contend that being in motion logically entails

possessing a well-defined speed. Let us now examine what kind

of argument could support that contention.

Here is one possibility. Motion is a form of quantitative

change, namely the variation of a scalar quantity with respect

to another scalar quantity, notably time. On the other hand,

every such kind of variation is arguably a rate. Hence, being

in motion entails possessing a rate of change, which in the

specified case is in fact a rate of passage or speed.

Let us call this the variation-rate argument, and the hypothesis

that every variation of a scalar quantity with respect to time

is a rate the variation-rate hypothesis. The problem with the

variation-rate argument is that its crucial assumption, namely

the variation-rate hypothesis, is logically incompatible with

the premises of the no-rate argument. Hence, the no-raters

cannot consistently rely on it in order to defend premise (2).

To understand why, let us first recall that the no-raters must

provisionally concede the adequacy of the extended at-at

theory, which is implied by premise (3). Thus they must

concede, albeit only for the sake of their reductio, that time

can literally move by being at different times at different

times, and notably that time covers a distance of one second

in an interval of one second. Hence, they must agree that time

undergoes a genuine form of quantitative change, that such a

change consists in the variation of a temporal quantity with

respect to a temporal quantity, and that the value of that

variation is one second per second. Now, given the variation-

rate hypothesis, this variation should necessarily constitute

a rate. However, this result would openly contradict premises

(4)-(6), according to which one second per second is one and

hence not a rate at all. To summarise: given premise (3), and

hence given the extended at-at theory of motion, premises (4)-

(6) become incompatible with the variation-rate hypothesis.

The reader should resist the temptation to understand this

contradiction as a further demonstration that the claim that

time literally flows is false or meaningless. Rather, what

this contradiction shows is that the no-raters cannot

simultaneously assume premises (1), (3)-(6), and the variation-

rate hypothesis. Thereof, as long as they concede (1), (3)-

(6), they cannot rely on the variation-rate argument in order

to justify premise (2).8

Still, one problem remains. To move in the way envisaged by

the extended at-at theory, time would have to cover a distance

of one second per each second of time elapsed. Hence, in order

to reject premise (2), the dynamists would have to admit that

covering a distance of one second in an interval of one second

does not imply moving at a rate of one second per second. But

how could they make such an intuitively implausible claim? The

answer to this question was already contained, albeit somewhat8 This, of course, does not exclude that the anti-dynamist couldconstruct a logical argument along the lines of the variation-rateargument or on the basis of some of its premises. For instance, theymay argue that no quantity can vary with respect to itself,rejecting the extended at-at theory of motion and hence the dynamistconception of the flow of time on that basis. However, what mattersto our present inquiry is that no such argument could be adduced inorder to establish premise (2).

implicitly, in our previous discussion. However, it is now

worth formulating it more explicitly.

Premise (3) prescribes that, if time flowed at some rate, then

that rate would be equal to one second per second. This means

that, if time satisfied the necessary and sufficient conditions

required to have some speed or other, it would do so in such a

way that its rate would be one second per second. What are,

then, those conditions?

Certainly one condition is that, in order for something to

possess a well-defined non-zero speed, its dependent variable

should vary with respect to its independent variable. This

requirement is not substantially different form the

hypothesis, which we have already met while discussing the

variation-rate argument, that motion should consist in the

variation of a scalar quantity with respect to another scalar

quantity; and we have already seen that, given premise (3),

time does satisfy this condition: this is, in fact, what the

claim that time covers a distance of one second in an interval

of one second strictly implies.

There is another requirement, however, that a mover needs to

satisfy in order to possess a rate: namely, that the two

scalar quantities whose relative variation constitutes motion

should not cancel in the same way as the two identical numbers

would if they appeared, respectively, as the numerator and the

denominator of a ratio. This condition is a straightforward

consequence of premise (6), and it amounts to requiring that

the dependent variable and the independent variable of motion

should be quantities of different types. On the other hand it

is evident that, given premises (3)-(5), the flow of time

cannot satisfy it.

To claim that time flows in the way prescribed by (3), yet

without any well-defined rate, is therefore equivalent (given

(4)-(5)) to claim that time flows in such a way as to satisfy

the first but not the second of the above two conditions. What

makes this claim so implausible is that, in any ordinary

instance of motion, this would not be possible. Hence, we are

intuitively inclined to think that the first of the above two

condition logically implies the second. This is, in the last

instance, what seems to motivate the variation-rate

hypothesis, and more generally what could support the claim

that motion logically entails speed. However, as we are going

to see, this intuition cannot be relied on in the case at

hand, and hence it can offer no argument in support of (2).

Ordinary movers, as we have noticed, are located in space and

move through space, in the sense that space is the dependent

variable of their motion. This is the reason why all ordinary

motions can satisfactorily be analysed in terms of the at-at

theory, but is also the reason why they are necessarily endowed

with a well-defined speed. Ordinary motions, in fact, consist

in the variation of a scalar quantity, notably space, with

respect to time. Hence, all ordinary motions satisfy the first

of the above two requirements. Moreover, space and time are

different types of quantities, so there is no reason why they

should be cancelled from a rate in the same way as identical

numbers can be cancelled, respectively, from the numerator and

the denominator of a ratio. Thus, ordinary motions invariably

satisfy the second of the above requirement, too.

Within the at-at theory of motion, then, it is perfectly safe

to assume that no mover could possibly satisfy the first of

the above two requirements without also satisfying the second,

and therefore it is perfectly legitimate to infer that nothing

could move without a well-defined speed. But, it may be

recalled, time cannot flow in the sense dictated by the at-at

theory. Contrary to ordinary movers, in fact, time is not

located in space, nor it would make sense to say that time

flows from here to there. Rather, dynamists claim that time

moves from past to future, which entails that time, not space, is

the dependent variable of the motion of time. This is the

reason why the extended at-at theory is required to make sense

of the flow of time, and this is also the reason why (given

(4)-(5)) the flow of time can have no speed. For, as we have

already noticed, it is exactly because of this reason that the

flow of time can satisfy the first but not the second of the

two conditions above.

Within the extended at-at theory of motion, then, it is

perfectly legitimate to admit that some mover, notably all the

movers whose dependent variable is time, can move without

speed. The reason why we find this claim perplexing is that

our intuitions about motion are shaped on the observation and

analysis of the motion of material objects, i.e. of ordinary

movers, which can be adequately analysed according to the at-

at theory. However, it would be improper to deny a logically

consistent feature of the extended at-at theory on the basis

of an intuition that is shaped on the implications of another

theory, which has been provisionally abandoned.

To sum up. The no-raters must concede the adequacy of the

extended at-at theory for the sake of their own argumentation.

But this means that, for consistency, they must provisionally

concede all of its logical implications, unless they are

contradictory or false – for that would suffice to their

purpose of getting a reductio. The possibility of speed-less

motion is among these logical consequences, and as we have

seen it is not overtly contradictory. The fact that this

consequence clashes with our intuition about motion, however,

is no reason to dismiss it as false, because such intuition is

based on another theory of motion, namely the at-at theory,

which the no-raters have provisionally conceded not to be the

correct one.9

7. Conclusion

Let us finally take stock. Firstly, we have seen that the no-

rate argument, and in particular premise (3), implicitly

obliges the no-raters to concede, albeit provisionally, that

‘motion’ should be understood in the sense prescribed by the

extended at-at theory. Secondly, we have shown that neither

the extended at-at theory nor any independent argument seem to

9 Incidentally, this should prevent any van Inwagen-style repliessuch as: ‘I cannot make sense of the claim that time flows bycovering one second at each second without moving at a rate of onesecond per second’. To those who are tempted to advance similarobjections, I simply reply that they are relying on the wrongintuition.

oblige the dynamists to accept premise (2). Therefore, there

is apparently no reason why the dynamists could not sensibly

claim that time moves literally at no speed.

This, in turn, entails that the dynamists can consistently and

effectively argue that the flow of time is not a kind of

motion about which it makes sense to ask how fast it flows.

This is not because the claim that time flows is itself

meaningless but because, contrary to ordinary movers, time is

not located in physical space, but in time, in the sense that

time is the dependent variable of its own motion.

This, of course, does not mean that the anti-dynamists are

forced to accept this claim, and to subscribe to the

conception of motion that underlies it. However, our

discussion has demonstrated that they seem to have no means to

reject it, once they concede all the premises required to

formulate the no-rate argument. The no-rate argument can thus

be resisted.

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