Zeno and Naagaarjuna on motion

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Zeno and Naagaarjuna on motion

by Mark Siderits and J. Dervin O'Brien

Philosophy East and West 26, no. 3, July 1976.

(c) The University Press fo Hawaii

p.281-299

P.281

Similarities and differences between Zeno's Paradoxes
and Naagaarjuna's arguments against motion in Chapter
II of Muula-maadhyamika-kaarika (MMK II) have already
been remarked by numerous scholars of Indian
philosophy. Thus for instance Kajiyama refers to
certain of Naagaarjuna's arguments as "Zeno--like,
"(1) and Murti seeks to show that Naagaarjuna's
dialectic is innately superior to Zeno's.(2) In both
cases the assumption is made that Zeno's arguments
are specious; the authors seek to dissociate
Naagaarjuna's destructive dialectic from the taint of
the best-known piece of destructive dialectic in the
Western tradition. On Brumbaugh's analysis of the
four Paradoxes, however, Zeno's arguments are seen to
form a coherent whole which, as a whole, constitutes
a valid argument against a certain type of natural
philosophy (valid, that is, so long as one does not
accept Cantorian talk of "higher-order infinities").
The target of the Paradoxes is now seen as
Pythagorean atomism, with its curious-and to the
modern mind incompatible-mixture of the principles of
continuity and discontinuity as applied to the
analysis of space and time. Zeno's genius lies in
separating out of this muddle the four possible
permutations of spatiotemporal analysis, and then
constructing a paradox to show the implausibility of
each account. Only on this interpretation of the
Paradoxes can we account for the renown which they
enjoyed in the ancient world.(3)
As we shall see, however, the atomisms of ancient
India were strikingly similar in several respects to
the doctrines of Pythagoreanism. This and the clear
correspondence of at least one of Naagaarjuna's
arguments against motion to one of Zeno's Paradoxes,
lead us to wonder whether a new look at the
relationship between the two philosophers might not
be in order. In particular, we wonder whether, armed
with the insight into atomistic doctrines and their
refutation which Brumbaugh's analysis affords, we mig
ht be able to give a more plausible interpretation of
at least some of Naagaarjuna's arguments than has
hitherto been possible. There is no question but that
Zeno and Naagaarjuna put their respective refutations
of motion to completely different uses. The question
is whether the two employ similar strategies. On our
understanding of the Paradoxes a sympathetic account
of Naagaarjuna is no longer in danger of
"contamination" from specious Eleatic reasoning. Thus
the principal aim of the following will be to exhibit
what seem to us to be some striking parallels between
certain of Zeno's and Naagaarjuna's arguments, both
in methodologies and in targets.
Eleatic philosophy, of which Parmenides was the
principal exponent and Zeno the staunch defender, was
in part an attack on Pythagorean science, which
explained the world in terms of a multiplicity of
opposing principles. The Eleatics maintained that
Being was fundamentally one and unchanging-and
therefore, of course, immovable. Such a
counterintuitive position required exceptionally
strong arguments to support it, the best of which
were supplied
_____________________________________________________
Mark Siderits and J. Dervin O'Brien are graduate
students in the Dept. of Philosophy, Yale University.

P.282

by Zeno. The rigor of his arguments overwhelmed his
contemporaries, and the most famous of these
arguments, the Paradoxes, continues to fascinate
laymen and philosophers alike.
Many attempts have been made to explain these
Paradoxes. Taken as separate and independent
arguments, they range from the peculiar to the silly;
yet in the ancient world they enjoyed an enormous
reputation. The best resolution of this problem is
that offered by Robert Brumbaugh in The Philosophers
of Greece: The Paradoxes should be viewed. not as
separate arguments, but as four parts of a single
argument, each part designed to refute one possible
interpretation of Pythagorean philosophy of nature.
Because for many years the Pythagorean order
imposed a rigid code of secrecy upon its members, it
is impossible to determine with any certainty
precisely what its official doctrine was at any given
time, However it seems fair to say that Pythagorean
science was basically atomistic, the universe being
conceived of as additive, that is, composed of atoms
or minims, indivisible "smallest-possible'' units of
space and time. This conception must have been dealt
a severe blow by the Pythagorean discovery that the
hypotenuse of a unit right triangle was
incommensurable with its sides, and that therefore
there could be no one unit, however small, of which
both could be composed. Attempts to resolve this
difficulty led to great ambiguity as to the nature of
atoms, which varied according to context from
entities of definite magnitude to dimensionless
points and instants. The Pythagoreans maintained both
that the world was composed of atoms and that any
magnitude was infinitely divisible, No one definition
of the atom would suffice. If it were taken to have
definite magnitude, then there would be lines which
could not be bisected, and no magnitude would be
infinitely divisible; if, on the other hand, the atom
were made dimensionless to give infinite
divisibility, no quantity of such atoms could ever
add up to any magnitude at all. According to
Brumbaugh, Zeno's Paradoxes were designed to bring
out the inherent absurdities of such a world view and
to show that, however one interpreted this position,
whichever of its premises one adopted, no account of
motion could be given which did not end in
absurdity, Whether space and time were atomistic or
infinitely divisible, no intelligible account of
motion through them was possible.
There are four possible combinations here: Space
might be continuous (that is, infinitely divisible)
and time discrete (that is, composed of extended
minims or atoms); or space might be discrete and time
continuous; or both might be continuous; or, again,
both might be discrete. The Bisection Paradox,
Achilles and the Tortoise, the Arrow, and the Stadium
are designed to refute, respectively, each of these
possibilities. Each Paradox depends for its effect
upon its proper suppressed premise concerning the
nature of space and time.
The Bisection Paradox assumes that space is
continuous (infinitely divisible) and time discrete
(atomistic). Zeno presents it as follows:

p.283

... The first asserts the non-existence of motion on
the ground that that which is in locomotion must
arrive at the half-way stage before it arrives at the
goal...(4)

The problem here is that the walker is required to
traverse an infinite series of distances, which is
impossible. Since time is discrete, in order to
traverse each of the distances involved, the walker
requires at least one minim of time. Therefore the
journey requires an infinite number of such minims of
time, that is, an infinite duration, and for this
reason it can never be completed.
The paradox of Achilles and the Tortoise assumes
that space is discrete and time continuous. It goes
as follows:

The second is the so-called Achilles, and it amounts
to this, that in a race the quickest runner can never
overtake the slowest, since the pursuer must first
reach the point whence the pursued started, so that
the slower must always hold a lead. This argument is
the same in principle as that which depends on
bisection, though it differs from it in that the
spaces with which we successively have to deal are
not divided into halves.(5)

In this case, the difficulty arises from the fact
that there is an infinite series of moments in which
the tortoise is running. In each moment, the tortoise
must traverse at least one minim of space. In order
to overtake the tortoise, Achilles must traverse each
spatial minim through which the tortoise has passed.
Therefore, Achilles would have to travel an infinite
distance in order to catch the tortoise. Like the
Bisection Paradox, this problem can be simply stated
thus: one can never complete an infinite series.
The Arrow Paradox, on the other hand, assumes
both space and time to be continuous.

The third is that already given above, to the effect
that the flying arrow is at rest, which result
follows from the assumption that time is composed of
moments: if this assumption is not granted, the
conclusion will not follow.(6)

Because time is infinitely divisible, and because
moments thus have no duration, at any given moment
the arrow is standing still in a space equal to its
length. Therefore, it is at every moment at rest, and
thus it never moves. Once again, the problem can be
simply stated: one cannot add a number of
dimensionless

p.284

instants together to achieve a duration; no matter
how many such instants are added together, their sum
will always be zero.
The paradox of the Stadium is for the modern
reader the most baffling of the four, and our
interpretation, which follows, differs from
Brumbaugh's. We agree with him, however, that this
puzzle assumes both space and time to be
discrete--composed of minims.

The fourth argument is that concerning the two rows
of bodies, each row being composed of an equal number
of bodies of equal size, passing each other on a
race-course as they proceed with equal velocity in
opposite directions, the one row originally occupying
the space between the goal and the middle point of
the course and the other that between the middle
point and the starting-post. This, he thinks,
involves the conclusion that half a given time is
equal to double that time.(7)

Assume for the moment that we are speaking, as Zeno
originally did, of bodies rather than chariots,
Assume a stationary body (A) divided into three
sections, each section being one minim long. Assume
two more such bodies, one (B) traveling past (A) from
left to right at a certain velocity, the other (C)
traveling past (A) in the opposite direction at the
same speed.

Let (B) be passing (A) at a velocity of one minim of
space per minim of time. Then in the time (one
temporal minim) in which the front edge of (B) passes
one minim of (A), the front edge of (C) will pass two
minims of (B), and in doing so the front edge of (C)
will pass one minim of (B) in half a minim of time,
which is impossible, since the minim is by definition
indivisible.
This puzzle would work just as well looked at in
another way. If we say that the second moving body
(B) is passing the first (A) at the slowest possible
speed, that is, one minim of space per minim of time,
then in the same duration in which the front edge of
(B) passes one minim of (C), it (B) will pass only
half a minim of (A), the stationary body, which is
impossible, since, once again, the minim is
indivisible. Either way the key to understanding this
Paradox lies in understanding that Zeno is not here
assuming the atomicity of empty space or empty time;
these concepts were foreign to the ancient Greeks,
who thought

p.285

instead in terms of the
space-which-something-occupies, or the
time-in-which-something-occurs. What is assumed here
is, for example. the atomicity of the
space-which-something-occupies. and therefore the
atomicity of that which occupies the space, as well.
The interpretation of this Paradox turns on the
phrase, "half a given time is equal to double that
time." It should be borne in mind that the wording
here is Aristotle's, not Zeno's, and that Aristotle
clearly misunderstands this Paradox. He thinks that
Zeno reasons fallaciously that a given object
traveling at a given speed will pass two identical
objects, one stationary and one itself in motion, in
the same amount of time. Modern exponents of this
same interpretation express it differently: Zeno,
they say, is misled by his ignorance of the concept
of relative velocity. Whichever way the alleged
fallacy is stated, Zeno is not foolish enough to have
committed it. He is not saying that (B) will pass (A)
(stationary) and (C) (moving at the same speed as (B)
but in the opposite direction) in the same amount of
time; instead he is pointing out that, if (B) is
traveling at, for example, a speed of one minim of
space per minim of time, it will pass one minim of
(A) in one minim of time, but it will pass one minim
of (C) in half a minim of time, thus dividing the
indivisible minim, which is impossible. The issue of
relative velocity is irrelevant and anachronous.
Not one of these Paradoxes is, by itself, a
convincing argument against motion, but each, when
taken to include its proper assumptions about the
nature of space and time, neatly disposes of one
possible account of the universe in which motion
occurs. (Of course, some of these arguments would
serve for more than one case, but it is reasonable to
assume that four were included for the sake of
elegance.) Once the Paradoxes are seen as a
destructive tetralemna, they then form an impressive
demonstration that any additive conception of the
universe renders an intelligible account of motion
impossible.
Furthermore, these puzzles then can be seen as
part of a comprehensive Eleatic argument against the
possibility of motion. Fundamental to Eleatic
philosophy is the premise that what is unintelligible
cannot exist. Therefore, in order to demonstrate the
impossibility of motion, one need only show that no
matter what kind of universe one assumes, no
intelligible account of motion can be given. It will
then follow that motion cannot occur in any possible
universe.
We begin with the assumption that the universe
must be either additive (that is, made up of parts)
or continuous (that is, made up, not of parts, but of
a continuous, unbroken substance). If it is additive,
then there are three possibilities: (I) that the
universe is composed of bodies separated by a void;
or, (2) that the universe is composed of minims; or
(3) that the universe is composed of dimensionless
points and instants. Case (1) is disposed of by
Parmenides himself; he argues that the void is unin
telligible, and therefore cannot exist, thus
rendering (1) impossible. All possible permutations
of (2) and (3) are refuted by Zeno's Paradoxes; no
conceivable assortment of minims and

p.286

dimensionless points and instants makes possible an
intelligible account of motion. Thus, on Eleatic
terms, the universe cannot be additive.
On the other hand, if the universe is continuous,
then motion can only be explained in terms of
compression and rarifaction. However. these are
clearly species of change, and Parmenides argues that
change of any kind is impossible, since it involves
coming-to-be (that is, arising from nothing, which
"nothing," since it is unintelligible, cannot exist)
and passing-out-of-being (which requires that
something which exists commence to not-exist, which
is likewise unintelligible and therefore impossible).
These arguments, it should be noted, all turn on the
confusion of not-being (for example, being not-red)
with nonbeing (nonexistence), However, if we accept
them, as Zeno apparently did, then they do show that
in a continuous universe, motion is impossible.
Thus, on Eleatic terms, no matter what kind of
universe we suppose-continuous or additive no
intelligible account of motion can be given, and
therefore motion is impossible. Although this and
other of their conclusions never achieved wide
acceptance, their arguments had enormous influence,
establishing the rationalist tradition in philosophy
which survives until today.
Before we proceed to a direct examination of
Naagaarjuna's arguments against motion, we should
like to say a few words about the historical
background behind the writing of the
Muula-maadhyamika-kaarika (MMK) , with particular
reference to Indian notions of space and time. While
far less is known about ancient Indian mathematics
and physics than is known about their ancient Greek
counterparts, it is still possible to discern a few
significant tendencies. And these, it turns out, bear
remarkable resemblances to developments in Greece, It
is known, for instance, that the 'Sulba geometers of
perhaps the fifth or sixth century B.C, discovered
the incommensurability of the diagonal of a square
with its sides.(8) Having done so, they then devised
a means for computing an approximate value of ?
Significantly, however, this was perceived as no more
than an approximation, This suggests that they were
aware that ? is irrational, that is, that its
precise value can never be given with a finite string
of numerals; and from here it is but a short step to
the notion of a number continuum. That is, the
mathematician who knows of the existence of
irrationals should soon come to see that there are
infinitely many numbers between any two consecutive
integers, And with this realization comes the notion
of infinite divisibility, While we cannot say for
certain that the 'Sulba geometers were consc iously
aware of infinite divisibility, developments in
Indian physics require some source for the notion,
and the sophistication of the 'Sulba school makes it
seem the likeliest place to look, The developments to
which we refer are the emergence of the curious
atomistic doctrines of space and time. Material
atomism is quite common in classical Indian
philosophy, and it appears to have been maintained by
Saa^mkhya, Nyaaya, and Sarvaastivaada. For these
schools the paramaa.nu is the ultimate atomic
component of all material entities. While it is
itself imperceptible, this paramaa.nu or ultimate
atom is the material

p.287

cause of all sensible objects. It is said to be
dimensionless, partless. and indivisible, so that we
may say that its size constitutes a spatial minim.(9)
In certain respects. however, the paramaa.nu must be
considered infinitesimal, that is, as having some of
the properties of a geometrical point. Thus the
atomic size of the paramaa.nu is not properly
additive: We should expect the size of the simplest
atomic compounds to be a function of two
factors--number of component atoms and atomic
size--but only the first factor, number, is in fact
involved in computing atomic size.(10) This is to say
that the measure of a dyadic compound is not twice
the size of the constituent paramaa.nu, but is rather
a size which is independently assigned to the dyad.
Thus while the idea of an atomic size of the
paramaa.nu suggests a doctrine of spatial minims, the
doctrine that this size is nonadditive suggests a
conception of a truly dimensionless atom, that is, a
point.
Similar tendencies can be seen in some of the
classical Indian theories of time. Certainly the
Saa^mkhya theory of time must be considered at least
quasiatomistic; the duration required for a physical
atom to move its own measure of space is said to be a
k.sana, or atomic unit of time. And in Abhidharma we
find an explicit temporal atomism, based on the
notion of k.sana as the atomic duration of a dharma
or atomic occurrence. Here we also see a concern with
the problem of divisibility and indivisibility. The
k.sana is first defined as being of imperceptibly
short duration. In order to account for the processes
which must occur during the lifetime of a dharma,
however, the k.sana is divided into three constituent
phases: arising, standing, and ceasing-to-be. The
process of subdivision is then repeated, so that each
phase of the k.sana itself consists of three
subphases, giving in all nine subphases. But here the
process of division ends, the subphases being
considered partless and indivisible, that is, tempo
ral minims. Thus the subphase can be considered a
true atom of time, since it exists outside the flow
of time, in the manner of Whitehead's epochs.(ll)
The natures of these atomisms in pre-Maadhyamika
Indian thought have two important implications.
First, they imply acceptance of the principle of
discontinuity as it applies to our notions of space
and time. This is just what it means to speak of
minims of space (paramaa.nu) and time (k.sana
subphase). That there can be a least possible length
and a least po ssible duration means that space and
time are not continuous but rather discontinuous--for
example, time does not flow like an electric clock,
but rather it jumps like a hand-wound clock. This is
an inescapable consequence of saying that the
paramaa.nu is of definite- but indivisible extension,
and that the k.sana subphase is of definite but
indivisible duration.
The second implication of these atomisms is that
their proponents implicitly accepted the notion of
spatiotemporal continuity. It is one thing to say
that the atoms of space or time are indivisible and
partless; it is quite another to say that they are
dimensionless and nonadditive. The former assertion
might be seen as a counter to the argument of the
opponent of atomism that since a

p.288

physical atom is of definite extension. it must
itself be divisible and so consist of parts. To this
the atomist replies by arbitrarily establishing the
measure of the atom as the least possible extension.
But the second assertion. that the atom is
dimensionless and nonadditive. goes too far. It
implicitly accepts the opponent's thesis of infinite
divisibility. The property of nonadditiveness
properly applies only to true geometrical points on a
line. And with this notion comes as well the idea
that between any two points on a line there are an
infinite number of points; that is, the line
consists of an infinite number of infinitesimal
points. This notion is, of course, suggested by the
discovery of the irrationality of ?. Thus we are led
to suppose that as with the Pythagoreans, so also in
India, the discovery of irrationals led to an atomic
doctrine that treated space and time as, in some
respects, discontinuous and, in other respects,
continuous.
Our aim is to show that some of Naagaarjuna's
arguments against motion, like Zeno's Paradoxes,
exploit the atomist's assumptions about continuity
and discontinuity of space and time. Before we turn
to the direct examination of these arguments,
however, we must perform one brief final task--we
must indicate the point of Naagaarjuna's dialectical
refutation of motion. I think we may safely say that
Naagaarjuna's chief task in MMK is to provide a
philosophical rationale for the notion of 'suunyataa
or "emptiness," which is the key term in the
Praj~naapaaramitaa Suutras, the earliest Mahaayaana
literature. What this comes to is that he must show
that all existents are "empty" or devoid of
self-existence. He must perform this task in such a
way, however, as neither to propound nihilism (which
is considered a heresy by Buddhists) nor to generate
class paradoxes. To this end Naagaarjuna constructs a
dialectic which he considers capable of reducing the
metaphysical theories of his opponents (chiefly
Sarvaastivaada, Saa^mkhya, and Nyaaya) either to
contradiction or to a conclusion which is
unacceptable to the opponent. Unlike Zeno, however,
Naagaarjuna is not refuting the theories of his
opponents simply as a negative proof of his own
thesis: Naagaarjuna has no thesis to defend--at least
not at the object-level of analysis where
metaphysical theories compete with one another.
Instead his dialectic constitutes a meta-level
critique of all the metaphysical theses expounded by
his contemporaries. One of Naagaarjuna's chief
techniques is to exploit the hypostatization or
reification which invariably accompanies metaphysical
speculation. This is to say that he is arguing
against a strict correspondence theory of truth and
is in favor of a theory of meaning, which takes into
account such things as coherence and pragmatic and
contextual considerations. We may thus say that
Naagaarjuna seeks to demonstrate the impossibility of
constructing a rational speculative metaphysics.
As one step in this demonstration, MMK II seeks
to show the nonviability of any account of motion
which makes absolute distinctions or which assumes a
correlation between the terms of the analysis and
reals, that is, any analysis which is not tied to a
specific context or purpose but is propounded as
being universally valid. Thus once again Naagaarjuna
differs from Zeno-here, in that

p.289

he is not arguing against the possibility of real
motion (indeed he argues against rest as well), but
only against the possibility of our giving any
coherent, universally valid account of motion. To
this end he employs two different types of argument:
(a) "conceptual" arguments, which exhibit the absurd
consequences of any attempt at mapping meaning
structures onto an extralinguistic reality; these
exploit such things as the substance-attribute
relationship, designation and predication; (b)
"mathematical" arguments, which exploit the anomalies
which arise when we presuppose continuous or
discontinuous time and/or space. Arguments of type
(a) have already received considerable attention from
scholars of Maadhyamika; thus the bulk of the
remainder of this article will focus on arguments
which we feel belong in category (b).
It is MMK II:1 to which Kajiyama refers when he
calls Naagaarjuna's arguments "Zeno-like." And indeed
there is a clear resemblance between this and Zeno's
Arrow Paradox.

Gata^m na gamyate taavadagata^m naiva gamyate
gataagatavinirmukta^m gamyamaana^m na gamyate

The gone-to is not gone to, nor is the not-yet-gone-to;
In the absence of the gone-to and the not-yet -
gone-to, present-being-gone-to is not gone to.

The model which is under scrutiny here is that which
takes both time and space to be continuous, that is,
infinitely divisible. The argument focuses explicitly
on infinitely divisible space, but infinitely
divisible time must be taken as a suppressed premise
if the argument is to succeed. Suppose a point
moving along a line a-c such that at time (t) the
point is at b:
a b c
??? I
(t)
We may then ask, Where does this motion take place?
Now clearly present motion is not taking place in the
segment already traversed, a-b. Equally clearly,
however, present motion is not taking place in the
segment not yet traversed, b-c. Thus the going is not
occurring in either the gone-to or in the
not-yet-gone-to. But for any (t), the length of the
line is exhausted by (a-b) + (b-c). That is, apart
from the gone-to and the not-yet-gone-to, there is no
place where present-being-gone-to occurs. Therefore
nowhere is present motion taking place.
Our interpretation is confirmed by Candrakiirti's
comments in the Prasannapadaa:

[The opponent claims:] The place which is covered by
the foot should be the location of
present-being-gone-to. This is not the case, however,
since the feet are of the nature of an aggregate of
infinitesimal atoms (paramaa.nu). The place before the
infinitesimal atom at the tip of the toe is the locus
of the gone-to. And the place beyond the atom at the
end of the heel is the locus of the not-yet-gone-to.
And apart from this infinitesimal atom there is no
foot.(12)

p.290

There are two problems involved in making sense of
this passage. The first is that we must assume the
goer to be going backwards! This is easily remedied.
however, by the convenient device of scribal error.
Thus if we assume that an -a- has been dropped
between tasya and gate at lines 21-22, and then
inserted between tasya and gate of line 22,(13) our
goer will be moving forward once again. The second
problem stems from the fact that for the argument to
succeed we must assume that a foot consisting of a
single atom is being considered. This does not
constitute a serious objection, however, since the
analysis may then be applied to any geometrical point
along the length of a real foot--it is for this
reason that Candrakiirti begins the argument by
asserting that our feet are just aggregates of
paramaa.nu. Once these two problem are resolved, it
becomes clear that Candrakiirti's interpretation of
MMK II: I involves the explicit assumption of
infinitely divisible space and the implicit
assumption of infinitely divisible time.
In MMK II:2 Naagaarjuna's opponent introduces the
notion of activity or process:

Ce.s.taa yatra gatistatra gamyamaane ca saa yata.h
Na gate naagate ce.s.taa gamyamaane gatistata.h

When there is movement there is the activity of
going, and that is in present-being-gone-to;
The movement not being in the gone-to nor in
the not-yet-gone-to, the activity of going is
in the present-being-gone-to.

This notion of an activity of going, which takes
place in present-being-gone-to, requires minimally
that we posit an extended present. This is required
since only on the supposition of an extended or 'fat'
present can we ascribe activity to a present moment
of going. Thus the opponent is seeking to overcome
the objections against motion which were raised in
II:I, which involved the supposition of infinitely
divisible time. The opponent's thesis appears to be
neutral with respect to space however; it seems to
be reconcilable with either a continuous or a
discontinuous theory of space.
A textual ambiguity in II: 3 has important
consequences. Where Vaidya has dvigamanam(14) (double
going) , Teramoto has hyagamanam(15) (since a
nongoing) , and May has vigamanam(16) (nongoing).
Vaidya's reading seems somewhat more likely, since
"double going" is supported by the argument of
Candrakiirti's commentary. However both readings
yield an interpretation which is consistent with our
assumption that in II:3 Naagaarjuna will seek to
refute the case of motion in discontinuous time. Thus
on Vaidya's reading II:3 is:

Gamyamaanasya gamana^m katha^m naamopapatsyate
gamyamaane dvigamana^m yadaa naivopapadyate

How will there occur a going of present-being-gone-to
When there never obtains a double going of
present-being-gone-to?

p.291

On this reading the argument is against the model of
motion which assumes that both time and space are
discontinuous; thus it parallels in function Zeno's
paradox of the Stadium. Suppose that time is
constituted of indivisible minims of duration d, and
space is constituted of indivisible minims of length
s. Now suppose three adjacent minims of space, A, B,
and C, and suppose that an object of length 1s at
time t[0] occupies A and at time t[1] occupies C.
such that the interval t[0]-t[1] is 1d. Now since the
object has been displaced two minims of space, that
is. 2s, this means that its displacement velocity is
v=2s/d. For the object to go from A to C, however, it
is clearly necessary that it traverse B, and so the
question naturally arises, When did the object occupy
minim B? Since displacement A-B is 1s, by our formula
we conclude that the object occupied B at t[0] +1/2d.
This result is clearly impossible, however, since d
is posited as an indivisible unit of time. And yet
the notion that the object went from A to C without
traversing B is unacceptable. In order to reconcile
theory with fact, we might posit an imaginary going
whereby the object goes from A through B to C,
alongside the orthodox interpretation whereby the
object goes directly from A to C without traversing
B. This model requires two separate goings, however,
and that is clearly absurd. Thus we must conclude
that there is no going of present-being-gone-to,
since the requisite notion of an extended present
leads to absurdity.
If we accept Teramoto's or May's reading, then
II.3 becomes:

Then how will there obtain a going of
present-being-gone-to,
Since there never obtains a nongoing of
present-being-gone-to?

This may be taken as an argument against the model of
motion which presupposes discontinuous time but a
spatial continuum. Suppose that time is constituted
of indivisible minims of duration d, Now suppose that
a point is moving along a line a-c at such a rate
that at t[0] the point is at a, and at t[1]=t[0]+1d,
the point is at c, Now by the same argument which we
used on the first reading of II:3, for any point b
lying between a and c, b is never passed by the
moving point, since motion from a to b would involve
a duration less than d, which is impossible. Thus
what we must suppose is that for some definite
duration d, the point rests at a. and for some
definite duration d, the point rests at c. The whole
point of the supposition at II:2 was to introduce the
notion of activity, however. Now it seems that this
supposition leads to a consequential nongoing, which
is not only counterintuitive but also clearly
contrary to what the opponent sought when he
presupposed an extended present. While the principles
of cinematography afford a good heuristic model of a
world in which time is discontinuous and space
continuous, we do not recommend them to anyone
interested in explaining present motion through a
spatial continuum.
MMK II:4-6 continues the argument against the
opponent of II.2. Verse 4 is a good example of
Naagaarjuna's "conceptual" arguments against motion,

p.292

which frequently exploit the realistic assumptions
behind the Abhidharma lak.sa.na doctrine of
designation:

Gamyamaanasya gamana^m yasya tasya prasajyate
.rte gatergamyamana^m gamyamaana^m hi gamyate.

If there is a going of present-being-gone-to, from
this it follows,
That present-being-gone-to is devoid of the activity
of going (gati). since present-being-gone-to is being
gone to.

Candrakiirti's commentary, with its use of terms
borrowed from the grammarians, brings out the
linguistic nature of the argument:

The thesis is that there is going (gamana) through
the designation of present-being-gone-to; what
obtains the action of going (gamikriyaa), which is an
existent attribute, from present-being-gone-to, which
is a non-existent term devoid of the action of going;
of that there follows the thesis that
present-being-gone-to is without the activity of
going (gati), [since] going (gamana) would be devoid
of the activity of going (gati). Wherefore of this,
"Present-being-gone-to is being gone to" [is said].
The word " hi" means "because." Therefore because of
the saying that present-being-gone-to, though devoid
of the activity of going (gati), is truly gone to,
here the action (kriyaa) [of going] is employed, and
from this it follows that going (gamana) is devoid of
the activity of going (gati).(17)

In order for us to understand this, it is necessary
that we back up for a moment and look at
Candrakiirti's comments on II:2. There he has the
opponent elaborate his supposition with the following
remarks: "Where gati is obtained, that is
present-being-gone-to, and that is known from the
action of going. It is for just this reason that
present-being-gone-to is said to be gone-to. The one
is for the purpose of knowledge (j~naanaartha), and
the other is for the purpose of arriving at another
place (de'saantarasampraaptyartha) ."(18) The
opponent's thesis is that movement or the process of
going is to be found in the moment of
present-being-gone-to; but since the latter is not an
abiding feature of our world, but rather just a
convenient fiction or conceptual fiction, there must
be available some mark or characteristic whereby it
is known or singled out. This mark is the action of
going (gamikriyaa). The referent of this attribute is
the real process of going, namely, gati, the activity
of going. The term gamana, 'going', is now introduced
in order to signify the product of the assertion that
present-being-gone-to is being gone to, namely, the
going whereby present-being-gone-to is supposedly
being gone-to.
Naagaarjuna's argument is that by speaking of a
going of present-being-gone-to, we forfeit the right
to speak of an activity of going of
present-being-gone-to. Candrakiirti's elaboration of
this argument may be put as follows: The object of
the opponent is to locate the activity of going in
present-being-gone-to, but before this can be done he
must first isolate this moment. Since the notion of
present-being-gone-to is abstracted from a complex
historical occurrence, it is necessary that it be
designated through the arbitrary assignment to it of

p.293

the action of going. that is. we locate the moment of
present-being-gone-to by defining it as that wherein
the action of going takes place. So for there is
nothing objectionable in the opponent's procedure. We
run into difficulties, however, when he insists that
through this assignment of the action of going to
present-being-gone-to. this moment has obtained real
going. that is. it is truly gone-to For in this case
gamikriyaa, ostensibly the lak.sa.na or mark of gati.
has in fact become the lak.sa.na of gamana, the
purported going of present-being-gone-to. The
attribute action-of-going cannot be used at once to
refer to the real activity of going and also to
designate the construct present-being-gone-to, if the
result of the latter designation is the attribution
of going-to this present moment. Either of these two
tasks--reference to a real activity of going or
designation of the construct
present-being-gone-to-with-consequent-going--exhausts
the function of the lak.sa.na action-of-going.
Naagaarjuna pushes this point in II.5-6. In verse
5 he notes that the thesis of the opponent leads to
two goings--that by which there is
present-being-gone-to, and that which is the true
going. Since the designation of present-being-gone-to
as truly gone to has led to the exhaustion of the
lak.sa.na action of-going in assigning a going
whereby the present moment is gone-to, the attribute
action-of-going is now incapable of imparting its
purported referent, real activity of going (gati), to
the going (gamana) which is assigned to
present-being-gone-to. We must now imagine two
goings, one by which present-being-gone-to is
purportedly gone-to, and another which obtains the
real attribute of the action of going and which thus
stands for the activity of going. And as Naagaarjuna
points out in verse 6, the consequence of this
supposition is two goers, since without a goer there
can be no going.
To those unfamiliar with Maadhyamika dialectic,
the argument of II:4-6 must seem sheer sophistry.
Here two things must be borne in mind. First,
Naagaarjuna's argument is aimed at a historical
opponent, not at a straw man, seen in the light of
this historical context, the argument seems somewhat
less specious. The thesis that there is a 'fat'
temporal present within which motion to an other
takes place was held by at least one Abhidharma
school, the Pudgalavaadins.(19) And the lak.sa.na
criterion, whereby only that is a real (that is, a
dharma) which bears its own lak.sa.na or defining
characteristic, was held in common by all the
Abhidharma schools. This latter doctrine, when taken
in conjunction with the strict correspondence theory
of truth which was the common position of early
Buddhism, yields precisely the excessively realistic
attitude toward language which Naagaarjuna so
consistently exploits throughout MMK. In particular,
Naagaarjuna is here taking to task the opponent's
assumption of the possibility of real definition--the
proper manipulation of linguistic symbols gives us
insight: into the constitutive structures of
extralinguistic reality--and with it the assumption
of language-reality isomorphism.
Seen in this light, however, the opponent's
presuppositions are neither as

p.294

farfetched nor as alien to our own philosophical
concerns as they might have seemed. And this brings
us to the second point we should like to make about
Naagaarjuna's line of argument in II:4-6: The attack
is not against motion per se but against a certain
attitude toward language, and so its basic point will
have effect wherever noncritical metaphysics is
practiced. The argument relies on the fact that the
outcome of an analysis depends, among other things.
on the purpose behind doing the analysis. Thus the
notion of a definitive analysis of motion is
inherently self-contradictory. Any account which
purports to be such an analysis can be shown to be
guilty of hypostatization. When the terms of the
analysis--here, in particular gati and
gamyamaana--are taken to refer to reals, they
immediately become reified, frozen out of the series
of systematic interrelationships which originally
gave them, as linguistic items, meaningfulness. This
necessitates the notion of a separate apellate
'going' whereby the real going or the real
present-being-gone-to is known. This, in turn, gives
rise to the problem of the logical interrelations
among these various terms. The result is
Naagaarjuna's demonstration that the supposition of
motion in an extended present leads to paradoxical
consequences. The point we wish to make about this
demonstration is that its efficacy extends far beyond
the limited scope of Pudgalavaadin presuppositions.
Even more than his and Zeno's "mathematical"
arguments, Naagaarjuna's "conceptual" arguments
against motion are of greater than merely historical
interest.
MMK II:7-11 seeks to further demonstrate the
impossibility of motion by focusing on the notion of
a goer. In verse 7 Naagaarjuna states the obvious
point that there is a goer if there is a going.
Verses 8 and 9 then convert this, by means of the
conclusion of II: 1-6 that no going occurs in the
three times, to the consequence that there can be no
goer. MMK II:10-II then utilize essentially the same
argument as verses 4-5, but here apply it to the
notion of a goer:

Pak.so gantaa gacchatiiti yasya tasya prasajyate
gamanena vinaa gantaa gantur-gamanamicchata.h.
Gamane dve prasajyate gantaa yadyuta gacchati
ganteti cocyate yena gantaa san yacca gacchati

The thesis is that the goer goes:from this it follows
That there is a goer without a going, having obtained
a going from a goer.

Two goings follow if the goer goes:
That by which "the goer" is designated, and the real
goer who goes.

Here again we see that the assumption of
language-reality isomorphism leads to paradoxical
consequences; in this case the analysis of the notion
of a goer leads to two goings, one on the side of
language, the other on the side of reality.
MMK II:12-13 allows two divergent
interpretations: one takes it to be an argument of
the "mathematical" type, the other to be an argument
of the "conceptual" type. The verses are as follows:

p.295

Gate naarabhyate gantu^m ganta^m naarabhyate 'gate
Naarabhyate gamyamaane gantumaarabhyate kuha.

Na puurva^m gamanaarambhaad gamyamaana^m na vaa gata^m
yatraarabhyeta gamanamagate gamana^m kuta.h.

Going is not commenced at the gone-to, nor is going
commenced in the not-yet-gone-to;
It is not begun in present-being-gone-to; where,
then. is going commenced?

Present-being-gone-to does not exist prior to the
commencement of going, nor is there a gone-to
Where going should begin; how can there be a going in
the not-yet-gone-to?

The "mathematical" interpretation of this argument
assumes infinitely divisible time, or a temporal
continuum. No special assumptions about the nature of
space are required, so that space may be taken as
either continuous or discontinuous. The argument may
thus be taken to correspond in function to either
Zeno's Arrow Paradox or to the Paradox of Achilles
and the Tortoise. Assume an individual, Devadatta,
who during the interval t[0]-t[1] is standing at a
given location, and at some time during the interval
t[1]-t[2] leaves that location. Then assume that
there is some time t[x] contained in the interval
t[1]-t[2], subsequent to which Devadatta is going. We
may now ask when Devadatta commenced to go. The
interval t[0]-t[x] exhaustively describes the
duration of Devadatta's not-going. And the interval
t[x]-t[2] exhaustively describes the duration of
Devadatta's going for the period that concerns us.
Then since (t[0]-t[x]) + (t[x]-t[2]) covers the
entire duration of the analysis, we must conclude
that at no time does Devadatta actually commence to
go, that is, at no time does the activity of
commencing to go take place. Similarly, where i is an
infinitesimal increment in duration (that is, a
k.sana subphase), then for any n, (t[1] + n.i ) t[x] Therefore at no time does the
commencement of going take place.
The "conceptual" interpretation of this argument
goes as follows: The gone-to, the not-yet-gone-to,
and present-being-gone-to, as temporal moments, are
not naturally occurring existents, but rather
conventional entities defined in relation to going.
It is therefore impossible to designate these three
moments prior to the commencement of going. It is
impossible to speak of going actually taking place,
however, without this division of the temporal stream
into the three moments of gone-to, etc. In other
words, a necessary condition of our recognizing the
commencement of going is our being in a position to
speak of a time where going has ceased, a time where
going is presently taking place, etc. If we are to
succeed in designating the commencement of going, it
must take place in one of these three moments--and,
of course, the not-yet-gone-to may be exluded from
our considerations as a possible locus of the
commencement of going, since by definition no going
may take place in it. Thus the commencement of going
must take place either in the gone-to or in
present-being-

p.296

gone-to. This is impossible. however. since neither
of. these moments may be designated prior to the
commencement of going.
Candrakiirti's commentary on II:13 appears to
support this interpretation: "If Devadatta is
standing, having stopped here. then he does not
commence going. Of him prior to the beginning of
going there is no present-being-gone-to having its
origin in time. nor is there a gone-to where going
should be begun. Therefore from the non-existence of
gone-to and present-being-gone-to, there is no
beginning of going."(20)
A moment's reflection will show, however. that
this interpretation is not substantially different
from the "mathematical" interpretation of the
argument, particularly the second version, which made
use of infinitesimal increments of duration. Indeed
on this interpretation the argument seems specious
unless we make the additional assumption that its
target includes a ''knife-edge'' picture of time.
Thus if one assumes that time is continuous and
infinitely divisible, then at the instant (that is,
time-point) at which going actually commences, there
is in fact no real motion, since this is just the
dimensionless dividing-line between the period of
rest and the period of motion. And no matter how many
infinitesimal increments one adds to the period of
rest after it has supposedly terminated, the same
situation will prevail. Moreover, as long as one is
unable to locate real motion, one will likewise be
unable to discern a gone-to and
present-being-gone-to. This means, however, that we
will never succeed in designating a commencement of
going. Naagaarjuna summarizes the results of II:12-13
in verse 14:

Gata^m ki^m gamyamaana^m kimagata^m ki^m vikalpyate
ad.r'syamaana aarambhe gamanasyaiva sarvathaa.

The gone-to present-being-gone-to, the
not-yet-gone-to, all are mentally
The beginning of going not being seen in any way.

In the remaining verses of Chapter II (15-25)
Naagaajuna continues his task of refuting motion by
defeating various formulations designed to show how
real motion is to be analyzed. Thus, for example, in
II:15 the opponent argues for the existence of motion
from the existence of rest; that is, since the two
notions are relative, if the one has real reference,
the other must also. In particular we may speak of a
goer ceasing to go. As Naagaarjuna shows in II:15-17,
however, the designation of this abiding goer is
even more difficult than the designation of a goer
who actually goes. There are also arguments
concerning the relationship between goer and activity
of going, and the relationship between goer and that
which is to be gone-to. None of these introduces any
new style of argumentation, however; all seem to be
variations on objections already raised. In
particular, none of the arguments presented in these
verses is susceptible to a "mathematical
Interpretation. Thus we shall bring our analysis of
MMK II to a close here, merely noting in passing that
where Zeno has four Paradoxes, one designed to refute
each permutation of the ramified

p.297

Pythagorean spatiotemporal analysis, we have
succeeded in uncovering only three such arguments in
Naagaarjuna. The 'first (II: 1) covers the case of
infinitely divisible space and infinitely divisible
time; the third (II.12-13) deals with infinitely
divisible time, and thus covers the two cases of
discontinuous space and infinitely divisible time,
and continuous or infinitely divisible space and
infinitely divisible time (already covered by II:1).
The second "mathematical" argument (II: 3), depending
on how one reads it, covers either discontinuous
space and discontinuous time (Vaidya), or continuous,
infinitely divisible space and discontinuous time
(Teramoto, May). Thus depending on which text of II:3
is rejected, the corresponding permutation of the
four possible analyses will not be covered by
Naagaarjuna's arguments.
The natural philosophies against which Zeno and
Naagaarjuna argue are surprisingly similar. It seems
likely that in each case the account in question
began as an atomism, maintaining that the universe is
additive and that it is composed of some sort of
minims or atoms; we can then suppose that each of
these theories was severely shaken by the discovery
of ? and the incommensurability of the hypotenuse of
a unit right triangle with its side, which prove the
impossibility of proper minims. However the result
of this discovery was, in each case, not the
abandonment of atomism, but an ill-fated attempt to
reconcile that atomism with the new mathematical
knowledge, an attempt which resulted in great
confusion and inconsistency.
Zeno and Naagaarjuna attack these muddled systems
for similar reasons. Neither is constructing a system
or defending a thesis of his own; each is, instead,
attacking his opponents' positions to provide
indirect proof of an established doctrine. The
doctrines defended are, however, completely different
in kind. Zeno argues against pluralism to support the
monism of his teacher Parmenides, a theory of the
same type as that being rejected. Naagaarjuna, on the
other hand, attacks pluralism, among other theories,
to support the doctrine of emptiness, a doctrine of a
higher logical order than those which he refutes.
There is a further difference between the two
philosophers, in that, unlike Zeno, Naagaarjuna
designs his refutations as much to elucidate his
chosen doctrine as to defend it: In providing a
philosophical rationale for "emptiness" he is
exhibiting the true import of this term, which occurs
essentially undefined in the Praj~naapaaramitaa
literature. In showing why all dharmas are empty,
Naagaarjuna gives the first truly formal account of
the meaning of this doctrine.
There are also important similarities between the
two philosophers' styles of argument. Both, as we
have seen, are given to the use of indirect proof.
Both make use of a "mathematical" style of argument
which accepts the opponent's premises and
demonstrates that they entail either absurdities or
consequences unacceptable to the opponent. However,
Naagaarjuna also makes use of a very different sort
of argument--one which approaches the problem in
question from a meta-level, showing the problem as
one of reification, arising from the opponent's
attempt to project his analysis out onto some

p.298

extralinguistic "reality," and to make the terms of
this analysis correspond to independent entities in
that "reality." There are other differences as well.
Zeno is far more formal and systematic in his
arguments than is Naagaarjuna in his "mathematical"
arguments; Zeno constructs Paradoxes to cover all
four possible cases of spatiotemporal continuity
and/or discontinuity, whereas Naagaarjuna has only
three arguments, and these tend to overlap. On the
other hand, Naagaarjuna seems more clearly aware of
the nature of his opponents' fallacy, the confusion
of mathematical analysis with physical occurrence and
of mathematical fictions or conventions with physical
entities.
By means of their various arguments concerning
motion, both Zeno and Naagaarjuna reach the
conclusion that no intelligible account of motion is
possible. However, the two proceed from this point of
agreement in quite different directions. Zeno
concludes that since no intelligible account of
motion can be given, and since the unintelligible
cannot exist, therefore motion itself is impossible,
and Being must be unmoving, This supports Parmenides'
doctrine that Being is one and unchanging.
Naagaarjuna concludes instead that it is impossible
to give an intelligible account of motion because to
do so is to attempt to make a description or analysis
designed to cope with a certain limited practical
problem apply far beyond its sphere of competence.
This in turn supports the thesis that metaphysics is
a fundamentally misguided undertaking. One could
only tie everything up into one neat bundle if there
were some single extralinguistic reality, "the
world," out there standing as guarantor of the
veracity of one's account. The nature of "reality,"
which is just our experience of a constructed world,
is determined by the nature of the language in which
it is described--and that varies according to the
task at hand. For this reason any rational
speculative metaphysics is impossible.
As has been noted by others, the two
philosophers' treatments of motion are remarkably
similar, despite their great separation in time,
place and culture. What differences there are between
the two can largely be accounted for by the differing
purposes of these accounts.

NOTES

1. Kajiyama Yuuichi, Kuu no Ronri (Tokyo: Kadokawa
Shoten, 1970), p. 89.

2. T. R. V, Murti, The Central Philosophy of
Buddhism (London: George Alien and Unwin, 1960),
pp. 178, 183-184.

3. Robert S. Brumbaugh, The Philosophers of Greece
(New York: Thomas T. Crowell Co.. 1964), pp.
57-67.

4. G. S. Kirk and J. E. Raven, The PreSocratic
Philosophers (Cambridge: Cambridge University
Press, 1969), pp. 292-3. The English translations
follow Gaye.

5. Kirk and Raven, p, 294.

6. Kirk and Raven, p, 294.

7. Kirk and Raven, pp. 295-296.

p.299

8. Bibhitibusan Datta, The Science of the 'Sulba
(Calcutta: University of Calcutta, 1932), pp.
195-202.

9. The use of the notion of atomic size in the
Saa^mkhya theory of time involves the conception
of a spatial minim, or a finite indivisible
length. Confer below.

10. Surendranath Dasgupta, A History of Indian
Philosophy (Cambridge: Cambridge University
Press, 1922), vol. 1, pp. 314--315.

11. Both Saa^mkhya and Abhidharma hold that
time, unlike space, is not an ultimate
constituent of reality. They appear to maintain,
like Whitehead, that our notion of temporal flow
is derivative and secondary, a product of the
occurrence of atomic occasions. This is the basis
for Naagaarjuna's rejection of the Abhidharma
theory in MMK XIX:6. But the ultimate unreality
of time does not detract from the significance of
the k.sana theory for our considerations.

12. Yamaguchi Susumu, trans., Gesshozo Chuur nshaku
(Tokyo: Kobundo, 1951).

13. Maadhyamaka'saastra of` Naagaarjuna, P.L. Vaidya,
ed. (Dharbanga: Mithila Institute, India, 1960),
p. 33.

14. Vaidya, p. 34.

15. Teramoto Enga, trans. and ed., Chuuronmuiso
(Tokyo: Daito Shuppansha, 1938), p. 42.

16. Candrakirti, Prasannapadaa Madhyamakav.rtti
Jacques May, trans. (Paris: Adrien-Maison-neuve,
1959), p. 55, n. 19.

17. Vaidya, p. 34. Not only is Yamaguchi's
translation of this passage (p. 149)
incomprehensible. it also ignores the grammar of
the original.

18. Vaidya, p. 34; Yamaguchi, pp. 145-146.

19. Yamaguchi, p. 146.

20. Vaidya, p. 37.