Phronesis Volume 11 Issue 1 1966 [Doi 10.2307%2F4181773] Gregory Vlastos -- A Note on Zeno's Arrow

7/21/2019 Phronesis Volume 11 Issue 1 1966 [Doi 10.2307%2F4181773] Gregory Vlastos -- A Note on Zeno's Arrow

1/17

A Note on Zeno's ArrowAuthor(s): Gregory VlastosSource: Phronesis, Vol. 11, No. 1 (1966), pp. 3-18Published by: BRILLStable URL: http://www.jstor.org/stable/4181773.

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2/17

A note nZeno'sArrow.

GREGORY VLASTOS

[1]

For if,

he

says,

[a]

everything

is

always

at rest when it is at a place equal

to itself

(6r=v

1f

xa'r&

r

aov),2

[b]

and the moving object

is always [sc. at a place equal

to

itself]3

in the

"now"

(Iatv 8' acxE o

yep6F,evov

&v r7 v5v), then the

arrow in motion

is mo-

tionless. (Aristotle,

Phys.

239B5-7;

for the text see Ross 657-58).

1

I

shall

deal only

with the main

topics, and with these

far from exhaustively.

For

more

thorough treatments of

the subject

the reader may consult

works

listed

in the bibliography at the

end

of this article

(to

which

reference

is made

in

text and notes by the name

of

author

only).

For

more extensive references to

the

scholarly literature

see M. Untersteiner,

142 ff.

Readers who may

be familiar

with the account of the Arrow in the chapter on Zeno I contributed to Philosophic

Classics, Vol. I, edited

by

W. Kaufmann

(Englewood

Cliffs, N.J.,

1961),

27ff.

at

40-41,

are

hereby

advised that the present

Note is meant to supersede

it

completely.

That chapter

had been prepared

on short

notice to fill an

urgent

pedagogical

need

and,

as

I

explained at

the time (27,

n. 1), presented

"purely

provisional

results of work-in-progress."

The

present

Note incorporates

results

I

have

reached in a more

thorough

study of the Arrow

made possible by

a grant

for research on Zeno from the

National Science Foundation,

to which

I

wish to

express

my

thanks.

2

I follow

the

usual translation of

this peculiar

phrase (cf.

e.g. Burnet,

"when

it

occupies a space

equal

to itself.") I assume

that

if Zeno had used

this ex-

pression, his expansion of it would have been xawr Tov laov kaotcu 'r6nov,

since

r6noq

would have been the only

word

he is

likely

to have

used

in

this

connection: cf. '67rov

&XXicramv

n

Parmenides,

frag. 8,

41 (Diels-Kranz).

However,

Zeno is more

likely to have written

&v

cp

law

kavr&

'r6nc

for the

context

suggests strongly

that the construction

with

xawr&

s Aristotelian.

Aristotle

starts

talking of a

mobile being xaoX tL as

far back

as

239A25,

using

the

phrase

again at 30,

34,

35 and (twice) at 239B3.

Uninterested

in

conserving

the mention of

'r670oq

n his summary of the puzzle

(t67roq

plays

no role

in

the

analysis

of

its

reasoning presupposed

by

his

refutation),

it would be

natural

for

him

to change

&v

T' rfa kmuvr

nto xxra. 'r taov

in

conformity

with his 6 uses

of

xacrci

t

in

the

preceding

15

lines.

3

For the expansion cf. the commentators (Simplicius, Philoponus, Themistius:

Lee

#

30 to

#

34. Lee

puts the expansion

into the

text (he writes:

&v

rij

vUV

xm'd&

r6

taov),

but on

frail MS.

authority.

3

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3/17

[2]

Zeno

argues

thus:

[a]

A

moving thing

moves either

[A]

in the place

in which it is or

[B] in

the

place

in which

it is not.

[b] But

it moves

neither [A]

in

the

place

in which

it is,

nor [B] in the

place

in

which it is

not.

Therefore,

nothing

moves. (Epiphanius,

Adv.

Haer. 3,

11; H. Diels, Doxo-

graphi

Graeci, 590).

brieferversion

of [2],' practically

dentical

with its part

[b],

is

ascribed

to

Zeno

by Diogenes

Laertius

(Vitae

Philos. 9,

72) in a

passage in which he appears to be following a good source: all

citations

in 71-73

are

reliable

and most

of them are

letter-perfect.

Of

the two versions

this

is the one

likely

to be the closer

to the

original:

the longer

one in

Epiphanius

ooks like

an expansion

of

this one

to

make

it

"compl[y]

with the

rules and

conventions

of

post-Aristotelian

syllogisms"

(Frankel

7).

In

any

case,

the ascription

of

[2b]

to Zeno

has

the backing

of both

Diogenes

and

Epiphanius,

and

I know of no

good

reason to

doubt

it. It

is true that

Sextus,

reporting

on three

separate

occasions (Pyrrh.

Hyp.

2,

245 and

3,

71;

Adv. Math.

10,

86-89)

the

use of [2] (with variations) by Diodorus Cronus,& ever mentions its

Zenonian

authorship.

But

neither does

he

say

that

Diodorus was its

originator;

he seems

to

imply

the

very

opposite,

as

Frankel (7,

n.

20)

has

observed, by

introducing

the argument

in one

passage

with the

remark

that

Diodorus '6v

7rpLpopij'nx6v uvep'rq

X6oyov

sr s'r

'

xtveLaOoE

C(A

dv. Math.

10,

87;

cf.

Frankel

7,

n.

20).6

Sextus' failure

to cite Diodorus

as the inventor

of this

argument

could hardly

count

against

its

authenticity:

he

is as silent

concerning

the

authorship

of

the

other

argument

whose

use

by

Diodorus

he

reports

in the

same

connection7- that if time consists of indivisibles "has moved" will be

true of things

of which "is

moving"

was never

true - which

we know

as

one

of

Aristotle's

cleverest

and most characteristic

creations (Phys.

231

B

20ff.).

4

Printed

as frag.

4 in

Diels-Kranz,

with

no defense of

the editorial

decision.

5

Also

in Pyrrh.

Hyp.

2, 242,

without

ascription

to Diodorus

or to

anyone

else.

S

Just

before,

at 86,

Sextus

had given

a fuller

account

of the reasoning

by

which

Diodorus

had

supported

[2bA]

and [2bB].

It would

be hard to believe

that,

when

he goes

on to

cite

the

whole of [2]

in the next paragraph,

he

should

refer

to it in the above terms (note specially the force of auvepc.r), if he had thought

of

[2] as

a Diodorean

invention.

7

Adv.

Math. 10,

48 and

85; it

is connected closely

with

[2b]

in

the

latter

passage.

4



4/17

Once

we decide that on the evidence

[2b]

was

Zeno's

(the

sense,

if

not the exact wording), we should face the question: Could it really

have

been

a

self-contained

composition, independent

of

[1],

as

the

editors have

generally

assumed?

Suppose

it were.

What

would

we

then make of Aristotle's remark that "there are four

arguments

by

Zeno that give trouble to those who

try

to refute them"

(239B

9-11)?

On the

hypothesis, [2b]

would have been a fifth.

Why

then

did

Aristotle

ignore

it? Because he

thought

it a

silly

puzzle,

not

worth

solving,

or too

easily

solved? This would

be hard to

square

with

the

fact

that

Diodorus,

connoisseur of fine

arguments,

thought

so well of

it.

Nor is it likely on aesthetic grounds that a construction trenching so

closely

on the Arrowshould

have

appeared

n Zeno's book as a

separate

puzzle.

And there is a

third,

still

stronger,

reason for

rejecting

the

hypothesis:

As

[2b]

now stands it makes

at

[A]

a

challenging

claim

-

that a

thing

cannot move

'V

4)

ar.L

rO'6rca without

putting

up

the

slightest

defense for it.

To realize how

badly

it

does need

defense

one

need

only

recall that it

would

be

quite

consistent with

general

usage

in

Zeno's time

(and

for a

long

time

after)

to

think

of the

r64os

n

which

a

thing

is as the room or

region

within which

it stands or

moves

-

i.e.,

as its locale, rather than its precise location.8 Hence to be told out of

the blue that

a

thing

cannot

move

"in

the

place

in

which it is"

would

only provoke

the

retort,

'And

why

not?

Why

cannot the

dog move in

the

kennel,

the

man

in

the

courtyard,

the

ship

in

the

bay?'

If

we

take

another look at

[1 a]

with

this

in

mind,

its

aptness

as an answer

to

just

this

objection leaps

to the

eye:

Its

strategy

is to cut down the

thing's

"place" to

a

space

fitting

so

tightly

the

thing's

own

dimensions (just

"equal"

to

its

own

bulk)

as to leave

it

no

room

in

which

to move.

Thus

[

a]

locks

neatly

into

[2bA]

in

point

of

logic,

while

if

it belonged

to a separateargumentwe would have to supposethat Zeno had built

into the

original

of

[2]

some other

backing

for

[2bA]

-

and what

would

that be? In

three

of the

passagesg

n which

[2]

turns

up

in

Sextus the

backing

for it

is

just,

"for

if it is

in it

[sc.

the

place

in

which it

is], it

rests there." This would be

very lame

-

a

patent

begging

of the

question

-

unless

here

again

the

equivalent

of

[

a] were

being pre-

supposed.

And

that this is in

fact the

case

appears from

a fourth

passage

in

Sextus

(A

dv.

Math.

10, 86),

where

the

equivalent

of [la]

8 Many

examples of the wider

usage

in

Liddell

&

Scott, s.v.

r67rog.

9

Pyrrh. Hyp. 3, 71 and Adv. Math. 10, 87; also in Pyrrh. Hyp. 2, 242 (cf. n. 4

above);

but not in

Pyrrh.

Hyp. 2, 245,

where the

argument

is tailored to fit the

joke.

5



5/17

is

given

as

Diodorus'

reason for

[2bA]:

"and

that is

why

[the

body]

moves neither in it [i.e. the place which containsit]

-

for it fills this up,

while

it needs

a

largerplace

in

which

to

move -, nor

etc."

The

fact

that

the

cited

argument

is

loaded

with

the

special

assumptions

of

Diodorean

physics (the

body

is

an

indivisible

which,

if

it

could

move,

would

have

to

move

through

a

space

composed

of

indivisibles)

in

no

way

affects

the point

at

issue,

i.e.

that

Diodorus

too,

when

filling

out

the

argument

in

[2bA] to

make

it

fully

convincing,

resorted

to

precisely

the

same

reasoning

as

is

provided,

more

tersely,

at

[1a]

in

the

Aristotelian

sum-

mary

of the Arrow.This

being

the

case,

and

taking

also

into

account

the

first two reasons above against the hypothesis that [la] and [2bA]

belonged

to

separate

puzzles,

we

have

good

warrant for

rejecting

it

in

favor of the

assumption

-

on

which

the

rest of this

Note

will

proceed

-

that both

[1 a]

and

[2

b]

belonged

to

the

Zenonian

original

of

the

Arrow.

Their

subsequent

separation

in

Diogenes

Laertius

and

Epiphanius

(or

their

sources)

could

be

accounted

for

easily:

in

the

former,

a

truncated

version

of the Zenonian

original;

in

the

latter

an

expansion

of that

fragment.

Nor

would

it

be

hard

to

account,

on

the same

assumption,

for

Aristotle's quite differentversionof the Arrow.We need only suppose

that

he

recognized

n

[1a]

and

[1 b]

the

significant

part

of

the

argument,

and

scornfully

ignored

the rest:

witness

his

drastic

abbreviation

of

the

Race

Course

n

the

same

passage (239

B

11-13).10

That

he

should

have

kept

[1 a]

is

understandable:

on

any

reconstruction

of

the

puzzle,

this

is

the

heart

of

its

reasoning.

As for

[1

b],

all

of

this

too

might

well

have

figured

in

the

original

as

the

sequel

to

[2bA] duly

backed

by [1

a]

(see the

conjectural

reconstitution

of

the

argument

below),

except

for

its mention

of

the viv

at

the

end.1' This is

one

of

Aristotle's

favorite

technical terms. He used it commonly as a name for the durationless

instant;12

but

occasionally,

in

controversial

contexts,

he

also

allowed

o

?

Cf. this with

the

considerably fuller,

though

still

abbreviated,

account

of the

same

argument

at

263 A

5-6.

11

Cf.

Calogero 131-38.

la How

much

of

an Aristotelian innovation

it is

one

can

see

by

comparing

Platonic

usage.

For

Plato

the

"now" remains

an

interval; he

uses 'z

viv

as short

for 6

v5v

Xp6voq

Parm.

152

B5).

The

closest he

comes to

Aristotle's

instantaneous

"now"

is

in

something

he

calls the

kocv(v "this

queer

thing situated

between

motion and rest, not itself in any time [Cornford, Comparing Nic. Eth. 1174B8,

'it occupies no

time

at

all,'

i.e.

has

no

temporal

stretch], while to it and

from

it

the

moving changes

to

a state

of

rest

and

the

resting

to a

state

of

motion"

6



6/17

himself to use it to denote the atomic

quantum

of

duration, integral

multiples of which would make up all larger temporal intervals, if

time were

discontinuous.13

Since neither of these two

uses of

viv

have

any known

precedent,'4

t would be most unsafe

to assume

that

Zeno

had

anticipated

one

of them

acrossa

gap

of a

hundred

years

or

more.

Its presence

here is

explicable

as an

Aristotelian

plant: by

sticking

it

into his account of the

puzzle

Aristotle makes it

all

the easier

for his

readers to feel the

appositeness

of his

refutation,

which centers in

the

claim that "it

[Zeno's

argument]

assumes that time

is

composed of

'nows':

if

this were not

granted,

the

argument

would not

be

valid."'5

If it did not have this function, it is doubtful that Aristotle would

have

found a

place for it in his

ultra-compressedaccount

of the

puzzle,

even

if it had

figured

in the Zenonian

original. How

expendable

it is

becomes

apparent

in a

number

of

the summaries of

the argument

in

the Aristotelian

commentators:

though

undoubtedly

dependent on

our

present

Aristotelian

passage,

they drop the

"now", probably

not even

aware that

they have done

so,

and

certainly

not

expecting that

their

(Parm. 157 D6-E

3). Thus

Plato's

k(aqtpsvr

s a

limit of

temporal extension,

itself

extensionless,

and one

might wonder

if

this,

in all

but

the

name,

is the

Aristo-

telian "now."

But there are

great

differences: Plato does not

explicate

in

rigorous

terms the

concept

he has in

mind,

does not

elucidate

it as the

temporal

analogue

of the

geometrical

point (so

fundamental

for

Aristotle's analysis

of the

"now"

Phys. 231 B

6 ff.

et

passim), does not

specify its formal

properties

(especially the

crucial

one, of

non-consecutive

succession, so

clearly identified

by Aristotle

for the

"now", 218

A

18

et

passim).

Aristotle is at pains to

distinguish

his V5V

from the

&

xEpvij,

remarking (perhaps

with

implied

opposition to Plato's

use of

the

term)

that the

latter

refers

to what

happens

in

an

imperceptibly

small

[stretch of] time. Phys. 222B15. Cf. Owen 2, 101 ff.

18

For this

second usage

(united

with the first

by the fact

that the

durationless

"now"

is also

indivisible,

but never

confused with it

by

Aristotle

to

my know-

ledge) see e.g.

the argument

against the

thesis that time

and

motion (no less than

spatial

extension) "are

composed

of

indivisibles,"

231B18ff.:

at

its conclusion

he

represents

the refuted

thesis as

holding that time

"is

composed of

'nows'

which

are indivisible"

(232A19).

14

There is no hint of

the

latter

in

Plato

(Parm. 152

Aff.; 155 D)

and

certainly

none of

the former

(cf. n. 12

above).

"I

239B31-33; cf. B8-9. That "time is

not

composed of indivisible

'nows"'

(loc. cit.) is the

very

foundation of

Aristotle's

theory

of

the

temporal

continuum.

He announces it (218A8) early in the second paragraph of his essay on time

(Phys.

IV,

217B29ff.) and

reintroduces

it

in

the

first chapter of

Physics VI:

cf. n.

13

above.

7



7/17

readers would

miss it.16

It becomes

quite superfluous to the sense

and to the force of the reasoning when [1 a] and [lb] (the latter with

modifications) are spliced

into

[2b],

with a transitional sentence

added

to smooth out the

reasoning:

The arrow could

not move in the place in which it is not

(= [2 b

B]).17

But neither could

it move in the place in which it is (= [2bA]):

For this is a place

equal to itself [supplied];

And

everything

is always at rest when

it is in a place equal to itself

(=

[1 a]).

But the flying arrow is always in the place

in

which it is

([1 b]

with

modifi-

cations).

Therefore,

the flying arrow is always at

rest.

Pending

a

fuller,

or better grounded,

utilization of our textual

data,

it is

reasonable

to assume

that

some such

argument

was

the

origi-

nal of the Arrow.18

II

Why

should Zeno have thought [1 a]

true? There are two

possible

answers, depending

on

how we read its "when"

(Cf.

Black

128

and

144-46). Is Zeno saying

(I) that everything

is at rest

for any temporal

interval during

which it is "at

a

place equal

to

itself",

or

(II)

that

everything

is at rest

for

any (durationless)

instant"I

in which it is

"at

a

place equal

to itself" ?

16

Of the three summaries

of

the Arrow in Simplicius,

the longer one

(Phys. 1011,

19ff.

=

Lee #

31) includes the "nows",

but the

shorter ones

(1015,

19ff.

=

Lee # 30; 1034,

4ff.

=

Lee

# 32) do not. Philoponus

(Phys. 816, 30ff.

=

Lee # 33) conserves the "now". Themistius does not in Phys. 199, 4ff. = Lee

#

34

or in

200,

29ff.

7

It is entirely possible

that a reason

may have

been given for this

proposition,

such

as the one

in the versions of

[2]

which Sextus

ascribes

to Diodorus

in

Pyrrh. Hyp.

3, 71 and 3, 89: "for it can neither do nor

suffer

anything

where it is

not."

IL

A

trivially

different version, sticking

more

closely to Aristotle's

[1 b]

at the

price

of a slight roughness

in

the transition from

[2 bA] to [1 a],

would repeat

the first two premises

as above

and then

proceed

as

follows:

For everything

is always at rest

when it is in

a place equal to itself

(-= [1 a]).

And the flying arrow

is always

in a place equal to

itself (= [I b]

without

I vu3v).

Therefore, the

flying arrow is always

at rest.

19

Hereafter

I shall always mean 'durationless

instant'

when I

say

'instant.'

8



8/17

Could (I)

be the

right

reading?

Its

linguisticplausibility

is

so

great

that

the second sense of

6Tov

might not even occur to one here,20 as it

apparently

did

not to Aristotle: he seems

to have taken it for granted

that

the

68av

in

[la]

would refer

to a stretch of time.20a But there

is

a

strong

objection

to this reading:

it would

require

us to debit Zeno

with

a

gratuitous

fallacy.

For

only

if we know

that an

object

is in the

same

place for

some

stretch of

time,

would we be

entitled

to infer that it is

at rest during

that stretch. And

that is

just

what we

do

not know

in

the

case of the arrow.

To all

appearance

t is never in the same place

in

the

course

of its

flight during any

sizeable

period.

Is

there

then

any

reason

to think that it would be in the same place during smaller periods?

On the

face

of

it,

none whatever: there

is no more reason a

priori

why

the flying

arrow should

be

in the same

place

for a billionth of a second,

than

for a whole second. So if Zeno wanted us to believe the contrary,

he would have to give

us his

reason. Otherwise,

he would be begging

the

very question

to be

proved

-

that the arrow,moving

by hypothesis,

is in fact resting

-

and to beg

it for smaller periods

would not make

the offense to

logic

the smaller.2'Now on

the present reconstruction

of

20Cf. the behaviour of Liddell & Scott. Elucidating the "when, at the time

when" sense of 6'e

(&8av

=

6're

&v),

he

authors say that

it

is

used

in the

indicative

with imperfect or

aorist "to denote

single events or actions in past time," with

present

"of a

thing

always happening

or

now going

on,"

etc.

They

make no

provision here or subsequently in any

of their statements

or examples

for

sense

(II) above, i.e.

for

the

use

of

85e to refer not to an event or a

period

but to

an

instantaneous limit of an event or

period, e.g. to the start or finish of a

race.

This rarer, but

perfectly authentic,

sense of

6Te

has evidently not occurred

to

the

original

authors or later

editors of

the dictionary.

'0a

I

think we may

infer

from

the context (239 B 1-4) that

the "indivisible nows"

which

Aristotle

believes (ibid. 8-9; 31-33) are being

"assumed" by

Zeno's

argument are atomic stretches, not extensionless instants (i.e. that he is using

the

v5v

in the

second of

the two senses in which he

employs the term:

cf. n.

13

above).

For since

he says that there

is neither motion nor rest in a "now" in

239B1-4,

where

he

is

using

"now" in

the sense of

instant,

the

assumption

that

time

consists of

instants would have warranted in his

view the conclusion

that

the

flying arrow

is

neither moving nor

resting. But he says

(ibid. 30-32) that

the

assumption warrants the conclusion

that the arrow is resting. So unless

he is

being very careless,

he must be

thinking of the "nows" of the supposed

assump-

tion not as

instants,

but as atomic durations.

21

I am not

implying

that the hypothesis is that the arrow

is moving over

every

interval, however

small. It would be quite legitimate to

hold that the hypothesis

is, as such, non-committal as to motion over micro-intervals and does not exclude

a

priori the possibility that motion

might be

discontinuous after all. But

to

assert (I) Zeno would have to go

much further than profess agnosticism

as to

9



9/17

Zeno's

argument there

is

absolutely nothing

-

not one word

-

to

say,

imply, or even suggest, that he was offeringus some reasonfor believing

that the

flying arrow stays put for tiny

intervals. Hence the

only

possibleway

of

bringing

any such premise nto the argument

would be

to assume that

this is something which Zeno's reader

already believes,

so

that Zeno does not

have to

argue

for

it,

or even mention

it: he can

just take

it

for

granted.

Under the

influence

of Paul

Tannery

(249ff.)

a

number of

distin-

guished scholars have made

just

this

assumption.

Supposing (T, i)

-

"T"

n

honour of the father of this

hypothesis

-

that Zeno's

arguments

were directed against Pythagoreans whom they supposed (T, ii) to

hold

a

remarkable

doctrine,

called "number-atomism"

by

Cornford,

these

scholars

have

also

supposed (T, iii)

that Zeno's

opponents

believed that

time,

no less than matter and

empty space,

was

made

up

of indivisible

quanta.

Elsewhere22 have

argued against (T, i)

and

(T, ii).

This is

not the

time to resume that argument. But it

might

be

just

as well

to

remind the reader that that

protest (preceded

by

such

fundamental

work

as

that

of

Heidel, 21ff., Calogero

115ff. and

van der Waerden

151ff.)

has been sustained

in

several later

contri-

butions (includingOwen 1, 211ff.; Booth 90ff.; Burkert37ff., 264ff.;

Untersteiner

197ff.)

However, (T, i)

and

(T, ii)

were at least

put

forward

on the basis

of

presumptive

textual evidence. In this

respect

they

are in a

totally

different

category

from

(T, iii),

for

which

not a

single

item of

positive

textual evidence

has ever been offered.

Aristotle's

last-cited remark

does not

constitute such evidence. For neither

here

(i.e.

239B8-9 and 31-33) nor anywhere else does

Aristotle

say,

or even

suggest,

that this

or

any

other

argument

of Zeno's was directed

against

Pythagorean philosophers.23

Nor does Aristotle tell

us

here that Zeno

said (or claimed, maintained, etc.) that time is composed of "in-

divisible nows." His remarkcan be read

perfectly

well

as

only

tracking

down the

assumption

which,

in Aristotle's own

judgment,

was

logically

entailed

by

Zeno's

argument,

and would have to be added to its

premises,

to validate the conclusion.24

f

I

were to

say,

'In

arguing

that

motion or rest over micro-intervals; he would have to assert

rest;

and this would

be begging the question.

22Gnomon 25 (1953), 29-35 at 31ff.; Philos. Review 68 (1959), 531-35 at 532ff.

23

Nor is any such thing said, or suggested, by Eudemus or by any other ancient

authority.

84 This, I trust, is all Lee (78) means

in

saying that Aristotle here "points

out...

the

necessary presupposition

of the

argument."

10



10/17

P entails

Q,you

are

assuming

the truth of

R,'

I would not be

implying,

and might not even wish to suggest, that you already believe R.

All

I

could

be, strictly,

understood to

imply

is

that,

since 'P entails

Q'

entails

R, you

cannot

maintain

the former

unless

you

are also

prepared

to stomach the latter.

Therefore,

if one wished to cite Aristotle as a

witness

of Zeno's

profession

of the

discontinuity

of

time,

one

would

have to produce

other evidence

tending

to show that

Aristotle

wished

us to understand

his

remark as

ascribing

such

a

doctrine

to

Zeno.

Such evidence does not exist. That the

quantization

of

time was

espoused

by

Zeno himself or

by

his

Pythagorean

contemporaries

thus

remainsa pureconjecture,25 nd a most implausibleone: so abstruse a

speculation

as the

replacement

of

the

temporal

continuum

by

an

atomic

conception

of

temporal

passage

could not have

been

seriously

entertained,

let alone

professed,

until

well

after the much

more

concrete

hypothesis

of the atomic constitution of

matter had become

thoroughly

assimilated

by

the

philosophical

imagination,

i.e.

well

after

Zeno's

time.26

So we can be

confident that

if

Zeno had

expected

his

readers

to concur with

(I),

he could not

have

presumed

on

their

doing

so

because of their

antecedent

philosophical

commitments;

he

wouldhave

had to producean argumentfor (I), if that is what he was assertingby

means

of

[la].

Since there is no trace

of

such

an

argument, we

have

good

reason to discount

this

first

reading

of the

"when."

(II), on the other

hand, allows

a

viable,

and

very

simple,

explanation

of

the

fact

that

Zeno

thought

[1

a] true,

and so

plainly true

that he felt

no need to

argue

the

point or

even so much as

mention it as

his reason

for

[

a];

If we think of the arrow as

occupying

a

given

position for

a

time of zero

duration,

it will

be obvious

enough

that

it

cannot

be

moving just

then: it

will

have

no

time

in

which to

move.27

To derive

[la] we will then requireonly the following additionalpremise:

H.

If the arrow is not

moving

when it is

"at a

place

equal to itself,"

it must be

at

rest

at that

place.

26 For the

"suggestion" that this was

"a

Pythagorean

formulation,

arising out

of their

point-atom

theory"

see

Lee

105-06.

26

For this

reason

among

others I

would

reject the

interpretation

of the

fourth

Zenonian

paradox

of motion

as an

argument

against

temporal

indivisibles.

Cf.

p.

43,

n.

31, of

my

essay cited

in

n.

1

above;

and note that

this

interpretation

of

the

paradox is unheard

of

in

antiquity,

and

goes flatly

against

the

one which

all of our ancient authorities take to be the obvious sense of the paradox.

27

Cf.

Black

133-34, the

first

four

paragraphs

of

his

exposition of

what he calls

"a

modem

version of the

paradox."

11



11/17

This hypothetical is the crucial

tacit premiseof the puzzle. If

this were

granted, [1 a] would certainly follow, and the success of the whole

argument

would

be assured28.

How then

would H have struck Zeno?

Would

it

have

looked to him

a hazardous inference in need

of argu-

mentative

support? I shall try to convince the

reader

-

if he

does need

to

be convinced

-

that, on

the

contrary, it would have

seemed to him

trivially

true.

Let

me

begin

by pointing

out that even today most people

(perhaps

even

some

readers of

this

journal )

would think it so.

Here is

one

example

from

a

distinguished

modern

philosopher:

"If

a

flying arrow

occupiesat each point of time a determinate point of space, its motion

becomes

nothing

but

a

sum

of rests"

(James

157)

-

presumably

because

occupying

a

point

of

space

at

a

point

of

time entails

the

antecedent

of

H

and therefore ts

consequent. Evidently

it

has not

occurred

to

James

that

H

involves

a

substantial inference

-

and an

invalid

one.29

Yet

James had at

his

disposal tools

of

analysis by means

of

which he could

easily

have

satisfied

himself

that

H,

so

far

from

being

tautologously true,

is

certainly

false.

Its antecedent

is

indeed true:

the

arrow

does

not

move

when

(i.e.

in

the instant

of

zero duration

at

which) it occupies a space equal to its own bulk. But its consequentis

false (in

the

broader

sense

in

which

"false"covers senseless

statements

no

less

than

significant falsehoods):

to

say

that the

arrow

is

at rest

for

an

instant is, strictly

speaking,

senseless.

This can

be

established,

S

for

example,

by

means

of the familiar

v

formula

(v,

velocity; s,

distance; t,

time).

Since a

body

at rest

has zero

velocity

and covers

no

distance,

the

values

required

for

v

and s

to

represent

the state

of rest

will be

zero. On

the

hypothesis

that

the

body

is

at rest

in an

instant,

0the value of t willalso be

zero,

and we will then

get

v

=

,

i.e. arithmeti-

28

It might be thought

that [1 a] could be granted

and the conclusion escaped by

arguing that even if at

rest in each instant the

arrow need not be at rest during

the whole of its flight. This would be a mistake.

If we conceded sense and truth

to

the

statement that the arrow is at rest

in

each

instant of its flight, we would

be admitting that the

whole period of the flight can be accounted for in terms

of rest, which would

be only another way of saying that the arrow

is at rest

throughout the whole

period. Cf. Owen 1, 216-17.

'9

A good, clear statement of its error in Chappell

203: "What

is at rest is

motionless, but we cannot infer that what is motionless is at rest without the

added premise that it is motionless through

a

period

of

time,

or

at

more than

one

moment [= instant]: for a single moment there

is neither motion nor rest."

12



12/17

cal

nonsense. The

only

way

to

get

the

required

v

=

0

is

to

assign

a

value greaterthan zero to t, i.e. to representthe body as being at rest

during

some

temporal

nterval,

however

short.30

In

Physics

VI

(234A32-B7;

239A11-17)

Aristotle reached

an

equivalent

result

without benefit of

algebra by

a

conceptual analysis

of the "now". He demonstrated

that,

since

"resting

consists in

being

in

the

same

[place]

in

some

[interval

ofl

time"

(239A26), it follows

that

(1)

in

any

"now"

there can be neither motion nor rest: it is

only

true

to

say

(2)

that

[any body,

whether

moving

or

resting]

is not

moving

while

it is

over against

something [i.e.

while

having

a determinate

position]

in a

"now";

but (3) it would not be the case that [any body, whether moving or resting]

could

be

over

against

a

stationary

body

in a

[period

of]

time: for if the

latter

were

the

case [i.e.

if

a

body had a determinate

position

for

some temporal

interval,

however

small],

then a

moving body

would be

at rest (239

B

1-4).

Here at last we do

get

the denial of H we

have been

looking

for:

It

is implied

in

(2) that the antecedent of H is

true

(the arrowwould

not

be moving

in the

"now")

and

in

(1)

that its

consequent is false

(the

arrow would not be at rest

in

the

"now").

But this

comes only

at the

high point of

an

intensive

exploration

of

temporal concepts,

begun in the Academy,31and continued with rare diligence and

penetration

in

the Physics. And even

so,

for all the

brilliantadvance in

insight this treatise

represents, it

does not take

Aristotle far enough to

enable him to understand

clearly

the

fact that the

antecedent

of H,

though certainly true,

is

true not for

physical, but

semantic, reasons 32

The sense in which the arrow is not

moving

in

any instant is

vastly

different from that in which the Rock of

Gibraltar s

not moving in

any

day, hour,

or second. To

say

that

the

Rock is

moving

in

some

period

would

be

merely

false.

To

say

that the arrow is

moving in any

instant

80

A

perfectly good

sense

may

nonetheless be

given

to

'instantaneous

rest' as a

limit: cf. the

explanation

of 'instantaneous

velocity'

in

Section

III

below.

31

Cf. Owen

2,

92ff.

32

Some readers

may

think

I

am

underestimating

Aristotle's

insight

at this

point.

They

may

retort that since Aristotle

has come to

see

that a

body is

neither

moving

nor

resting

in

an instant

(while

knowing

all too well

that

a

body must be

either

moving

or

resting

during

an

interval),

he

must have

grasped

the

semantic

difference

in the two

types

of assertions. I am

not

questioning

that

Aristotle

has

some

understanding

of the difference. All I

am

claiming is that it is not

extensive

enough

to enable him to

(i)

state

formally

the

crucial

point (in

some

such way as I proceed to do above or in some logically equivalent way) and,

therefore

(ii)

see its full

implications

for

questions

of

immediate

concern to him.

The second

deficiency

will be

apparent

in

n.

35

below.

13



13/17

would be

(strictly

speaking)

senseless:33 t is

non-moving

and non-

resting in the same way in which e.g. a point is non-straightand non-

curved,

non-convex

and

non-concave

-

the

predicatesare

not

falsely

applicable, but

inapplicable.

If

this is

not

understood,

the

antecedent

of

H

will

itself

seem

fully as

paradoxical

as

its

consequent, and

will

provoke

the

question,

'But

if

the

arrow is

not

moving in

any

given

instant of its

flight,

when

and how

does it

manage to

move?' To

answer

this

questionone

must

expose

the confusion

lurking behind

the

expec-

tation that,

if the arrow

is to

move at

all,

it

must move in

the

instant.

One

must point out that to

ask,

'How

can

the

arrow

be

moving

during

an interval when it is non-moving in every instant contained by that

interval?'

would

be

like

asking,

'How

can

the

arc

be

curved

when

none

of its points are

curved?':

motion

(or

rest)

apply

to

what

happens

not

in individual

instants

but

in

intervals

(or

ordered

sets

of

instants),

as curvature

is

a

property

not

of

individual

points

but of lines

and

surfaces

(or

orderedsets of

points).

Only

when

this

has

been

understood

will

one be

in

a

position

to see that

the

arrow's

non-moving

in

any

given instant

is

absolutely

irrelevant to

its

moving

or

resting

during

some

interval

containing

the

given

instant,34

and

that

conversely

the

arrow'smoving (orresting) duringa given interval allows no inference

whatever that

it is

moving

(or

resting)

in

any

instant

contained

by

that

interval.35

0

88

As much so as would the

-

result for v in the formula above,

and for parallel

reasons.

84 And, therefore,

that the disjunction in [2a]

is

not exhaustive:

it stops short of

a third possibility

which (stated with pedantic

completeness)

would run:

"or [C] from

the place,

Po,

in

which it

is

at a given instant,

io,

to

a place,

Pi,

in

whicb

it

is not

at

io,

during an interval which contains both

io

and the later

instant, il, at which it is at

pl."

a6 While certainly

allowing the

inference

that

it

is moving (or

resting)

at

any

instant contained

by that interval. This

-

and the implied distinction

between

motion (or rest)

in (or, for) an

instant

(which

is

senseless,

as

has just been

explained) and

motion (or rest)

at

an

instant

(to

which

sense

can

and must be

given: cf. Section

III below)

-

Aristotle totally failed to grasp.

For positive

evidence of this failure

see

Phys.

236

A

15ff.,

where

he

is

dealing

with a

period

of rest, CA,

immediately

followed

by

a

period

of

change, AD,

with

A

as

the

boundary

instant

common to

the two

periods.

He remarks

that

if

something

were

at rest throughout CA,

it would

follow

that xxl

&v r& A

ipeLe,

line 18. This

inference is,

of course, correct.

But it

appears

to be

in

flagrant

contradiction

with Aristotle's formal doctrine that

'pcrLeZv

is

impossible &v

C

vwov

234 A 32-34;

and

cf. the

citation

from 239B

in

the

text above).

To

get

around

the

apparent

inconsistency

Aristotle

would

have

to understand

how

vastly

different is the

14



14/17

We

may

now

return

to Zeno. It

should now be clear that to see

the

falsehoodof H, or even to suspectit, he would have neededto possess a

clear-cut

understanding

of

the

instant/interval

distinction and to

bring

this

to

bear

on H

in

spite

of the

fact that it does

not mention

"in-

stants"),

thereby

realizing that

its

evident truth

for

intervals

is

not

the

slightest reason

why

it

should

be

true

for

instants.

Zeno's

ability

to

meet this

condition

may

be

gauged

from

the

fact

that,

in all

pro-

bability, he did not

even have a term

for "instant"

and

could

only get

at this

concept

indirectly by

thinking

of

what

would

happen

to

the

time

of

the

flight

"when" the arrow was

"at a

place equal

to

itself,"

i.e. by thinking of durationcut to zero as the distance traversed was

cut

to

zero.

Is

it

surprisingif in

such

circumstances he should

have

thought

of the arrow's

being

at

rest

in

such

a

time as

being

no

more

than

verbally different

from

its

not

moving

in

it,

and

thereforefelt

as

certain

of the former

as

he

was

entitled

to

feel of

the

latter?36

III

We

saw

above

what

William

James made

of

the Arrow.

Here is

one

of

Bertrand Russell's glosses on it:

Philosophers often tell us

that

when

a

body is in

motion, it

changes

its

position

within

the

instant. To

this view

Zeno long

ago made

the fatal

retort

that

every

body

always

is

where

it is;

but

a

retort so

simple and so

brief

was not of

the

kind to which

philosophers are

accustomed to

give

weight,

and

they

have

continued

down to

our

own day to

repeat the

same

phrases

which roused

the Eleatic's

destructive

ardour. It was

only recently

that

it

became

possible

to explain

motion

in detail

in

accordance with

Zeno's

platitude,

and in

opposition to the

philosophers'

paradox. 2,

1582.

Here

Zeno

is

given

credit

for

having

grasped the

following

truth:

If

time-specifications

are

made

not in

terms of

temporal

intervals,

but

in

terms

of

instants, then

it

is possible

to say

that a moving

body

sense of

iv in

236A18

(where

Av

-T&

A

=

"at

instant A")

from

e.g.

Iv

'r

v5v

Ogv

7ri9oxe...

QpqLiv,

234A34

(where

&v

=

"in" or

"within"

[i.e.

for] a

given

instant.)

86

It should be

evident on

inspection

that

in the

Zenonian

argument

as

recon-

structed above

(end of

Section

I)

"not

moving" and

"resting"

are

being

used as

logically

equivalent

expressions:

so e.g.

at

the

third step

where the

arguer

says

"cannot move"

though

his

inference,

if

valid, would have

entitled

him

to

say

"must be at rest." But I am not entering this as evidence for the above con-

clusion,

since the

extent

to

which the

relevant

texts

have

preserved Zeno's

original

wording

is

unknown.

15



15/17

is

at one and

only one place at any given instant just as unambigu-

ously as we can say this of a body which is at rest. This insight liberates

the philosopher from the idea of an instantaneous

state of motion37

an

idea

which, though (strictly) nonsense, is not

obvious nonsense.

For

we do

say such things as 'the ship is now moving

at the rate of

ten miles per hour,' meaning by "now" 'at

this instant,' and our

dynamics

cannot

dispense with the notion of "instantaneous

velocity."

How so,

if the instantaneous state of

motion

makes no

sense? How

could motion at an instant

make

sense,

if motion

in

(orfor)an instant

does

not?

It was

only

with the

greatest

labour

and

after

many false

moves that it was found possibleto give a satisfactory answerto this

question

-

"to

explain motion in detail in accordance with Zeno's

platitude"

-

by showing

that

'velocity

at

instant,

i'

can be understood

to

mean

no

more

than

the

limit

of

average

velocities

over

intervals

approaching

zero

and

always containing i,

where "approaching

zero"

can

be defined

without covert

appeal

to an instantaneous state

of

change (or

to

its

mathematical

twin,

the

infinitesimal) by employing

only

variables

quantified

over intervals of finite

length:

the set of

inter-

vals

containing

i

approaches

zero

if,

and

only if,

for

any preassigned,

arbitrarily small, interval e, there is always a memberof the set such

that

the

differencebetween

its

length

and

zero

is less

than

e.

We

may

be

grateful

to Russell

for

helping

us see

how

important

and

87

I.e. of a

motion

which

is accomplished

in,

or within,

an instant

(cf.

the first

sentence

in

the

citation from

Russell).

38

Russell's

most

frequently

cited

gloss on

Zeno

-

1, 347

ff.: the historical

fable

and

further leg-pulling

of the "philosophers"

- is a

brilliant

piece

of

writing

and

would

be wholly

delightful

if it

were not

confusing for

those

who are not fully

up

to Russell's

tricks (see

e.g. what

poor James

made of it,

186, n. 1). Here

Russell

tells us that Zeno (and Weierstrass following unknowingly in Zeno's footsteps)

have proved

"that

we live

in an unchanging

world

and that the

arrow, at

every

moment [

=

instant]

of its

flight, is truly

at rest"

-

this

on the strength

of

the

fact

that

Zeno is supposed

to have

proved that

there is

no instaneous

state

of change

(= "Zeno's

platitude"

above),

as

though this

entailed

the

vastly

different

propositions

that

there are

instantaneous

states

of rest and

that the

moving

arrow is always

in one of these

For still

another

Russellian

analysis

of the

Arrow

see 4, 805:

here

Zeno's denial

of motion

(suppressed

in the citation

from 2,

1582

in the text

above) is

acknowledged

and pegged

on

Zeno's

assumption

that

there

can be

no motion unless

there

are

instantaneous

states

of motion.

Another

assumption

is imputed

to Zeno

in

the

still different analysis

of the

Arrow in

3,

179: "the plausibility of the argument seems to depend upon supposing that

there

are

consecutive

instants."

There

seem to

be almost as many

Zenos

in

Russell

as

there are Russells.

16



16/17

how true is that

insight

he has

called,

with

pugnacious

understatement,

"Zeno's platitude." With our own analysis of the texts behind us,

we need not be disturbed

by

the

accompanying

historical

fable.38

If the foregoing

interpretation is correct,

Zeno has

indeed seen

that

the arrow does not move in a

given instant. But he could

only

have had

a

faint

glimmering of what this

means, else

he

would not have

jumped

to

the

conclusion

that it must be

resting

at that

instant

and

during

all

intervals

containing that

instant and that all bodies

must be

resting

during

all intervals

containing

all

instants.

Recognizing

Zeno's

mistake,

we need not

belittle

his

achievement.

Zeno's

paradox is not a

bad

first

move in the direction of "Zeno'splatitude."39

Princeton University

89

I

am

deeply

indebted to

Professor Carl B.

Boyer,

of

the

department of

mathematics of

Brooklyn

College,

for

a

criticism which has

prompted

some

revisions in the

penultimate

paragraph

of Section II

and

in

note 35. I

am also

indebted

to Professor Gunther

Patzig

for some useful

suggestions.

Bibliography

Black, M. Problems of Analysis (Ithaca, 1954).

Booth,

N.B.

"Were

Zeno's

arguments

directed against the

Pythagoreans?",

Phronesis 2

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Burkert,

W. Weisheit

und Wissenschaft

(Nuirnberg,

1962).

Calogero,

G. Studi Sull' Eleatismo

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Chappell, V. C.,

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Diels, H. and

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(3)

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18

Documents

Phronesis Volume 11 Issue 1 1966 [Doi 10.2307%2F4181773] Gregory Vlastos -- A Note on Zeno's Arrow