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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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17/17
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18