Buddhist Meditations
Zeno and Naagaarjuna on motion
by Mark Siderits and J. Dervin O'Brien
19/07/2010 19:34 (GMT+7)
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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

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