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    Zeno and Naagaarjuna on motion
     
    [ 作者: Mark Siderits and J. Dervin O'   来自:期刊原文   已阅:9045   时间:2007-1-16   录入:douyuebo


    ·期刊原文


    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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