NATURE AND SYSTEM6 (1984),195-215.
Wholes,Parts,and Laws of Motion
R. f'. Hassing
Pnrracr
There are deservedlywell-known studieson the laws of motion in Newton
The works of Boas,Jammer,Hall, Cohen,Koyre, Gabbey,
and Descartes.
and Westfall are among the most noteworthy.r All haveprovided carefully
riuft"a and highly instiuctive accountsof the structureand developmentof
thi physicalttiought of Newton and Descartes.Eachhasproceededwithin,
and'contriUutedtlo, the history of scienceor the philoslPhV of science'Yet
to.thelawsof
nonehasaddressedthe questionof the relation of living being-s
philosophy.of
called
be
could
of
what
"huta"teristic
motion, a concern-or.
'living being'mustinvolveat leastthis: a special
nature.2Now the meaningof
."tuiiot of parts and while, in wtrictr the former are what they are only in
termsof, arrdatr thus not neutral to, the latter.l This of courseappearsto be
fundamentally incompatible with the tradition of physics deriving from
Newton. Is itieally incompatible?The presentinvestigationseeksto clarify
this issueby dispelling ""ituin mistaken or incompleteconceptionsof the
philosophicalimplicationsof physics.
L Two Lews or lNrnrt.l
| .l Inrroduclion. The aim of Part I of this paperis to showthat, contraryto
mostcurrentinterpretations,thereis no onelaw of inertia.+Rather,Newton's
first law is distinci from Descartes'in its implicationsfor our understanding
of nature. The crux of the argumentis this: The law of inertia forbids selfiniti"t"O motion; no body uit"tt can put itself in motion.5This standsin
obuioo, contradiction wiitr the phenomenaof living beingsinitiating their
own local motion. The law of inertia thus cannotstandon its own. To saveit
from the phenomena,it must be supplementedby other principles.[t, thereiore, becomeso$€ elementin a systemof concepts{orm]ng.a1explanatory
whoielargerthanihe law of inertia by itself,and in light of whichthe law must
than
be interpieted.This explanatorywholeis different in Descartes'physics
unitwo
other
the
to
is
linked
law
first
Newion's
i" N"*t".t. Specificaily,
and
of
action
law
the
iaw,
third
the
that
see
Wi
shall
motion.
of
versallaws
reaction,removesthe contradiction betweenliving beingsand the first law.
Althoug'h they are compatiblewith a corpuscularaccountof matter, these
threelaws of motion do not logically entail suchan account.By themselves,
;h;t "; neutral with respecito ihe distinction, made within ordinary
theyare thus comeiperi"nce, betweenliving ind nonJiving. By themselves,
holism,.in living
with
generally,
puiiUt" wiih substantial-form or, more
law of
an
accompanying
lacks
inertia
of
fieirrgs.uIn contrast,Descartes'law
The
joined
corpuscularism.
to
is
necessarily
and
reaction,
action and
ue5l
Copyrightolg84by Natureand SystemInc'
All rights of reproductionin any lorm reserveo'
196
NATURE
ANDsysrEM
contradiction with certain sensiblewholes,i.e., animals,is thus removedbv
recourseto subsensibl.g
na.tr-.The resultingtheory is incompatiblewith ani
holistic principle. tt follows that, concerningthe relation of wholesand partq
the confrontation betweenthe pre-modernunderstandingof natureand our
own post-Newtonian,and not cartesian,tradition of physicsis morecomplex
than isjften thought to be the case.This is developeditt rart II of this pup"r.
| .2 The.Animalsversu!the l-aws of Inertia. Deicartes'first law of nature,
as presentedin Iz Monde, is:
. .-..eachparticularpart of mattercontinues
alwaysto be in the samestateunless
collisionwith othersconstrains
it to changethat state.Thisis to say,if thepart has
somesize,it will neverbecome
smallerunleisothersdivideit; if it isrounaoisquare,it
will neverchange
thatfigurewithoutothersconstraining
it to do so;ii it i;topped in
someplace,it will neverdepartfromthat.place
unless
othlrschase
it away;andlfit has
oncebegunto move,it will alwayscontinuewith an equalforceuntit ottrerssropor
retardit.7
one "part of matter" is whatevercontainsno relativemotion of sub-parts,or
whatevermay be so consideredrelative to certain problem contexts.erhe
'true form
and . . . essence"of matter is, of course,extension.erhe law is
assertednot simply of motion, but of nature, which is here identical to
matter.l0The "state" in which a part of matter continuesthus includessize,
figurg-and arrangementof parts,aswell aslocal motion.rr It is more general
than Newton's first law: "Every body continuesin its state of rest, or of
uniform motion in a right line, unlessitis compelledto changethat stateby
forcesimpresseduponit."tz rhis law is only aboutthelocalm6tion of a body
as a whole.As a consequence,
we may abstractfrom its internalparts and
theirrelativemotions.tr rhus thelocalmotionof somethingextendidmaybe
represented
by the local motion of an unextendedpoint (tn a right line;;.u
A comprehensiveanalysisof the first laws of Niwton and Deicartesand
their respectiveconceptsof force and inertia is not our intention.15Essential
for our-purposeis the following obviouspoint, commonto both laws of
ingrti-a:No body can beginto moveby itself. yet animals,beingsthat we call
'alive',
appearto do so all the time. The animalsappearto viola'tethe laws of
inertia. How is this issueresolved?consider a cat atreston the mat. when it
getsup and goes,what agencyexternalto the cat effectsthe changeofstate?
Cartesianphysicsand Newtonianphysicsgive different answers."
1.3 The corpuscularign
!.esolyilon of Descartes.The cartesian response
to the phenomenaof animalson behalfof the first law is givenin chapter 7 of
Iz Monde:
Now, eventhoughin mostof the motionswe seein the true world we cannot
perceive
that thebodiesthat beginor cease
to movearepushedor stoppedby some
others,wedo not therebyhavereasonto judgethatthesetwo rutes hrst lawand
[tti
thelawof conservation
of motion]arenoibeingobserved
exactly.Folrit iscertainthat
thosebodiescanoftenreceivethiir agitationTromthetwo elements
of air andfire,
whicharealwaysfoundamongthemwithoutbeingsensed.
. .rr
Sensiblewholes must be understoodin terms of the subsensibleparts of
matter. Thus, in terms of Descartes'law of inertia, the cat, as it accilerates
from rest, may not be consideredas one part of matter. The motion of this
HASSING:WHOLES,PARTS,AND LAWS OF MOTION
I97
phenomenalwholeiscausedbythemotionsofunseenpartsofmatterpresent
i "; exampleof a mixedbodv' a1d suchbodies
it. il;;;t
;ilhffi;touno
,,in themselv"rrl*r qualitiest'hat are contrary" and "that tend to
contain
mixed
change."rzinO".O"'"tt tttebodiesthat appear.aboutusare
-"ii"lin"-l
not
would
seems,
it
law,
first
The
to
corruption."rs
u"oi"u.;""i
;;;;-il;;ii'"
for
the
reason
part'
the
;.h a uody as a whole.thit it, at leastin
;ilit;
"insofar
statelater
the
in
undivided"
and
asit is simple
iric'tusionof the phraie
andan adequate
mentof theI awin Principlesll,31.reThiiissueis importa-nt,
term'simple'in
the
of
and.use
meaning
the
"iuiif'
nuu.'to
accountof it would
therefrom in
deriving
Cartesianscience,u"iin. ,tp" ,if conceptfoimation
the
however'
moment'
the
For
subsequent,"i"n"e-noi-a timpt" task'zo
the
for
law,
first
in
Newton's
appears
p-hrase
t is that no such
.rr."ilJp"it
Newton's
of
"o.tp:::d
system.
explanatbry
the
within
issuedoes not arise
section'
laws of motion. ftow ttris is io will be sliown in the following
three
-il;;ilil;;
ili;;ili;;;;
of matterin termsof whichcorporeal
Unlike
are to be explainid came.to be called corpuscles'zt
ph;;;;;;
AristoUnlikg
determinacy'z
ultimate
no
tit"V
Democriteanatoms,
iott",t
is incompatible with
telian matter, they ut"-iu'ff' ""tual. Corpuscularism
'living'must now refer to
t"t(indeed
ihr
substantialform in fininl$ings
with holism' the notion-thatthereexist
or, ;;;;#;ily,
mere appeurance)
to elimentaryparts,to partswhichareneutral
wholesin naturenon-reducilble
of the contradiction
to the wholeswhich thJy.o-po*. Descartes''resolution
implications
fundamental
contains
betweenanimals anOhis firsi law thus
concerningmatterandcausality.Inparticular,themotionofanywholemust
governed fut9 laws of
be a sum of the motions of simple parts, parts.
!J
*nature."23How Aoesii stanCinitre physici derivingfrom Newton'l
mustbeginwith an
| .4 Causal Neutr altty ii Newt on's'Iiw s of M otioi. We
physics'and'the
phrur",
as'pbst-Newtonian
such
id;;;;;uiiii"uti6".
place
of 'Newton's
are useddetberately in
;'hil,*
'pityti.t'.;;rl"ing f.o. it"*to"'
Newton'that is' to underFor wJshallnot t.t. utttmpt t-ol.nterpre-t
be attemptedbrieflyonly
(Thiiwi[
himself.
standNewton as he r;J;;;t;td
recentwork hasbeen
Much
rule.)
parallelogram
in section2.4 below, o" it "
of.physicsis
directedto this goal,oiifro"i *ftich n6 adequatecomprehension
which'
physics
U.r", t o*.n"t, *" ut. "on"""'"d '." itl",tpttl-that
possible.2a
Euler'
Leibniz'
Newton'
by
(whether
.
tV- io ,u.".rrfut a.u"ioprntnt
argument
tl,e
in
evidence
provide
to
claimi
#Alembert, or L,agrange),
beginsfrom, and incorporates,Newton's
This;hf#clearly
;g;i"rfi;ifu
laws of motion.
changeof
Newton's first law requiresthat, for any body undergoing.a
impressed
tobe
In
order
uoay.
ttre
upon
velocity,therebe u rortr-i-pr"rr.d
bv another bodv
be'exerted
must
roit'
ir'i'
;;;;'-i;;;;*1.'"ti;;;;ev,
from rest,what
externalto the former]* Wfr"n the cat on the mat accelerates
agentcan
suchforceand
no
If
onit?
ttrecat,imfiissesaforce
;;;,;;;t""tto
forceis
this
Now
law.
first
Newton's
of
be identified,trr" arrriliis ii violation
mat
the
by the third law of motion:
ri""in.a, ""4 the diifi;ltfery.9ved,
"[t]o
an
opposed
always
is
everyactionthere
ilpt.tt"i " f"r"" on lh" "ut. For
are
other
each
upon
bodies
t*o
equal reaction: or' ihe mutuat "iiiont oi
uitnuvtequal,and directedto contrary parts'"26
198
NATUREAND sysrEM
It surelyseemsto us that the action of the cat on the mat is not of the same
sort asthe reactionof the mat on the cat. It is not clearin what sensethe latter
is action at all. This, however,is a problemfor the ptritosoptryoirrature, not
for the mathematicalprinciplesthereof.2?In termi of Newton'sthree laws,
actionsand reactions-are
of equalstatus.Both aresimplyforces,eactrquantitatively.related by the second law to a simultaneouscrrangein uito"ity
(assumingtheyarenetforces).And a forcein classicalmechanic-s
idust a push
or a pull, that is, whatevereffects,or tendsto effect,a changein the velociiy of
a body.zs11tereis no further specificationof causes.Thus liewton's threelaws
of motion are neutral with respectto the question of how actions and
reactionsare initiated,whetherin the movedbr in another.In this regard,
notethat Newton'sfirst law doesnot requirethat the cat'spushing onthemat
be exlernally caused.It requiresonly that the cat's chinge of velocity be
related(in accordancewith-thesecondlaw) to the simultanedusexternalpush
or pull. unlike Descartes'firstlaw, Newton'sbearsno obligation to account
for a changeof statein general,suchas the cat'sbeginningio push,but only
for a changeof velocity.
If.causal.priority is given to the
cat's action over the (non-living)
mat's reaction, it is not becauseof-(living)
Newion's three laws of motion, but on
grounds of ordinary experience.on the basisof the three laws of
motion
alone, 'the cat's alive, the mat's dead', 'the mat's alive, the cat,s dead',
'everything's
'everything's
alive',
dead'are all acceptablesiatements.This is
the meaningof the causalneutrality of Newton'suniversallaws of motion.
In sum,insteadofjoining hisfirstiaw to corpuscularismand qualifyingit in
.
terms oJ the 'simpleand undivided',Newton positsor discoversa mathe-fulfills
matically representablereaction force which
the requirement for
force in the caseof animals,but whosecausalityis left unspeci3n
fqnSessed
lied.2erhis guaranteesthe universalityof his first law. unlike Descartei',it
appliesto everybody (corpusomne)tlken asa whole,as it presenisitself to
our senses,
living and non-living.rowithin the systemconsisiingof the three
laws of motion, Newton'sfirst law, unlike Desiartes'corpuscrilarism,does
not.ruleout the possibilityof self-initiatedmotion. It is compatiblewith either
holismor reductionism.Newton'sthreelaws of motion aie, by themselves,
merelyguidelineswhich the measurableaspectsof all forcesand accelerations
must satisfy_.
They haveno fundamentalimplicationsconcerningmatter and
causality. within the post-Newtoniandevelopmentof physici, therefore,
sucr implications can arise only on grounds oi a larger argument,one that
includesbut goesbeyondthe threeliws of motion.
This completestheaccountof two lawsof inertia and their implicationsfor
our understanding
ofnature. obviously,however,we cannotsiop here.For
our-analys-is
has led to_a.largerquestion,one which bearscruciilly on the
problemof the statusof living beiirgsin relation to physics.what precisely
ls
the.argumentfrom post-Newtonianphysicsfor rlductionism and against
holism?
II. Tsr RnoucuoNrst AncuunNr FR.Mposr-NrwroNre* pHvsrcs
2.1 Introduction. The goal of part II of this paper is to set forth and
evaluatetheargumentfor reductionismbasedon post-Newtonianphysics,or
HASSING:WHOLES,PARTS'AND LAWS OF MOTION
I99
classicalmechanics.Now classicalmechanicsis (especially,although not
exctusivety)the mathematicaldescription of the local motion of bodies
betweenwhole
effectedbv forces.The argumentthui turns on the relation
of motiSn and force. Can the force and motion of a whole
i"i;;i
;il;;rt,
;il"y. b" e.rived from theforcesand motionsof its simplep-arts,and thoseof
inviioning bodies?If so, reductionismis true and holism refuted.Needlessto
say,the phrase'simplepart is crucial.
A final decisionon this questionwill prove to lie beyondthe reachof the
pt"r*t .nO"avor. For whiie it is not difficult to set forth the reductionist
its truth, will leadinto a nexus
argument,the attempt to evaluateit, to assess
I, concerningthe meaningof
Part
to
notes
in
the
of issues,at first submerged
We shallcircumscribethese
law.
and
matter
of
part',
the
coricepts
and
simple
issuir and conclude(Sec.2.s; wittr a summaryof what hasbeenclarified,and
an unresolvedquestion.Let us indicatehow this occurs'
The questionof ttre relation betweenwholesand parts in termsof motion
and force leadseasilyto threeitemsfamiliar in Newtonianphysics.-Thefirst
namely,Newton'sthreeuniversallawsof motion.
*.-ftuu" "fr"ady discussed,
The secondisihe idea oiforce-law, exemplifiedaboveall by the law of
uniu.,.,"1gravitation, and without which no traiectoriescan-be determined.
The third is the rule by which theforceexertedby a compound.isrelatedto the
forcesexertedby its cbnstituentparts,forcesand partswhich,in someunclear
yet crucial sense,must be simpleor elementary.lhe be^st-knownrepresentarule for compositionof forces.Theseare the
iion of ttris is the parallelogram
^of
ittreeingreaients the arlument for reductionismfrom classicalmechanics
It
sometiriescalledNeWonian-Laplaciandeterminism. The third ingredient
independence
dynamical
of
will be found to involve a certain assumption
Our
reductionism.
and
holism
betwein
issue
the
of
which goesto the core
to
attempt
an
into
turn
then
will
argument
reductionist
of
the
evaluai-ion
clarify this assumption(Secs.2'4 and 2.5).
2.i Force Lawi and Traiectories.Newton'sthree laws of motion are, by
barrenfor the descriptionof moving bodies.It is only whenthese
themselves,
laws, the iecond law in particular, are combined with force laws that
equations of motion r"rult. Equations of motion coupled with initial
yield trajectories.
conditions-law
is i specification of the force on a body, under given
A force
function of time, and of thebody'spositionand
a
mathematical
as
conditions,
velocity, Thus, most generally,F = F(r, v, /), whereboldfacelettersdenote
ti pr.."ppor"s the concept of force presext in the three laws of
;;;;.
motion, uuf is not aerivablefrom them. It also involvesmeasurableproperThe most famous
liir, oi'ai^"nsions of bodies,suchas massand charge.32
exampleis the law of gravitationalattraction.
Thi calculation of irajectories of cometsand planetsfnm the laws of
Irloti,on and gravitation was the first great achievement of classical
mechanics.It wasseenasevidenceagainstthe pre-modernunderstandingof
nature. For the motions of thesebodies could be describedto (almost)
arbitrarily high accuracywithout regardto formal causality,final causality,
oi oiuin".ig"i"y.r, 6n ,n's basistheclaim for reductionism
' wasbuilt. For our
follows'
as
expressed
be
best
can
ctaim
purposes,itrat
200
NATuRE
ANDsysrEM
Thereis no essentialdifferencebetweenthe motion of an animal, and that
of planetsin the solarsystem.All aredeterminedby fixed and universallaws.
The most basicprinciplesby which wecalculatethi trajectoriesof planetsare
also thosewhich must determinethe motions of animals;biology is logically
entailed.by physics.The differenceis one of calculationatcoriptexit!, ani
this is witho-utphilosophical.significance.
For in neithercaseis thire any need
to speak of causesof motion other than those contained in the laws of
attraction and repulsion,the paradigmof which is gravitation.
The core of this claim is the idet that any systemwhatsoever,thus any
body,is composedof smallerpartswhichinteiaci and movealongtiajectoriei
determinedby force lawq just as planetsin the solar systemby-the-forceof
gravitationalattraction. Thus, acCordingto Nefion, .;natureis exceedingly
simple and conformable to herself.whatever reasoningholds for grearcr
motions should hold for lesseronesas well."3aAnd so Liplace declaredthat
'the regularity
which astronomy showsus in the movemints of the comets
doubtlessexistsalso in all phenomena.,'35
This is an account familiar to students of physics and the history of
s9ienc9.Implicit thereinis-thethird ingredientof the reductionistargument,
the-rule for compositionof forces.without it, the argumentcannotstand.In
order to make.it explicit, it is usefulto presentthe riductionist argumentin
the mathematicalterms of classicalmechanics.
.2.3composition of Forces.consider any body,b, of constantmassrnb.we
wish to investigatethe local motion of b (in any inertialframe).itre uasisis
the secondlaw of motion:
m$r(t) = Fu(l)
tll
whereFu(r)is the net force impressedon b at time / by externalbodiesacting
or reactingon b, and ar(l) is theaccelerationof the centerof massof ,. If F(i
can be shownto be in principledeterminedfor all time r, thenar(l) is determinedasa function of time which can be integratedto yield the position and
velgcrty,thus the trajectory,of b, for given inltial "onditions.
To illustratemostdirectly what is involvedin thecompositionof forces,we
considerfirst the paradigmaticexample,the solarsystem.Thereafter,weturn
to thecaseof an arbitrarysystem,or compound,andforcesof attraciionand
repulsionin general.
r-et D be a planet moving under the gravitationalattraction of the sun and
otherplanets.ThenFu(l)represents
thelotal forceon Dat time I exertedbv the
suna-ndotherplanets.To preparethe mostgeneraldiscussion,weintrodrlcea
pecufia.rlocution: Fr(t) representsthe force or action on D of a compound
consistingof parts, namely,the sun and other planets,which parts aiso act
upon eachother. what,
1ow,is theform, fixed in time and exprissedin terms
of algebraicsymbols,of Fu(r)?we know from the law of univirsal gravitation
that any^onepart of this compound,sayplanetj, of massm3,takinby itself,
i.e., as if it wereisolatedfrom all otheri,ixerts i force on b-of magnitude
fla: G(mimol&o')
t2l
where Rp is the (timedependent)distancebetweenj and b, and G is the
universalgravitational constant.But when many intiracting parts, namely,
ANDLAwsoF MorIoN
HAssINc:wHoLEs,PARTS,
201
the sun and other planets,togetheract on b, what is the one resultantforce
which affectsthe motion of b?In the caseof gravitation,the answeris simple:
just assumethat eachpart actsindependentlyof all the others.Then we can
add up forces using the parallelogramrule introduced for this purposeby
Newton immediatelyfollowing the threelaws of motion.36For example,the
resultantforceon Dby the sun,of massm,, and planetTis shownin Fig. I , and
written out in Eq. 3:
fju
b
Fb
o
Frc. I
sun
R(t) = G(mjmblRjb\rpI G(m,mulRr2)nu= moa(t)
t3]
The
distances
where 1u and r,u are unit vectorsappropriatelydirected.
Rp and Rsuare timedependentsince all three bodies are in motion
relative to each other. The solution for the motion of b, therefore,
requires that the trajectories of i and s be calculated as well.
Thus Eq. 3 must be supplementedby similar equationsfor the motions of
the sun and planet j. Although the resulting system of coupled
differential equations is, in general, too complicated for solution in
closed, analytic form, the motion of b is, in principle, determined.3T
'[n principle' means according to the physics used to constitute
theseequationsof motion.
The point, now, is to make clear the crucial step, representedby Eq. 3,
in whiih the force exerted by the gravitational compound is composed
of the elementaryforces of its parts. In this step we discoverthe assumption that each part acts independently of the others, independently of
the relations in which it standsin composingthe whole. Let us call this
of the parts of a gravitational
the assumptionof the dynamicalindependence
system. Given this assumption, the parallelogram rule mPY be applied
to the elementaryforces t3tGmimrfR;u' and r*bcm"mblR,r',in order to
determine the conjoint action of the sun and planet 7 on planet b.
Considerthe alternative.If it were the casethat the force of attraction
exerted by the sun on b ceased to be given by Gm"malR,u2and
becamesomethingelse,that is, if the form of the force law changeddue
to the presenceof planet j, then we could not use this law in
determining the resultant gra-vitationalforce exerted by a compound
of many gravitating bodies.'o Some force would be exerted on a
given test body, producingacceleration,yet the form of that force would
not be determinedby composition of elementaryforces of the form
R,*t.In addition,difficultiescould arisein the definitionof mass.3e
Gmimv.f
But this alternative does not occur. For gravitation, the assumption of
and the consequentapplicability of the parallelodynamicalindependence,
giam rule to elementarygravitational forces,is warrantedby the successof
the resultingcalculations,and entailedby the conceptof gravitationalmass'
about which we shall havemore to say.Our fundamentalconcernis with the
universalityof thesenotions.Do theyapply to allthe actionsof a//the wholes
in nature?Let us completethe exposition of the role playedby composition
of forcesin the reductionistargument.
202
NATuRE
ANDsysrEM
Let b in Eq. I be an arbitrary compound body, subjectto the action or
reactionof an arbitrary forceFu(l).we canthenimaginethat b is what wecall
an'animal',whichappearsto moveitselfby pushingor pullingon environing
bodies.For our purposehere, the latter bodieJmay be issumed to be
c-omposed
of dynamicallyindependentparts.our purposeis to presentthe
demonstrationthat F(l) is completelydeterminedby classicalmeihanics,so
that therecan be no essentialdifferencebetweenliving and non-living. Both
have no other causesand principles of motion than those contai-nedin
Newton'slaws of motion and force.
To probe the force F6(l) impressedon D, it sufficesto introduce a test
particle, a, of mass ftta. orr which b acts by exerting a force Fu"(t), as
represented
in Fig. 2. If Fu(l) can be determined,then the motion of D is
Fb"
Frcunn 2
determined.For, by Newton'sthird law, the force Fuaexertedbv b on a is
equaland oppositeto the force F"uextertedby a on b: Fu(t) = -F"r(t). The
motion of b in the presence
of a is then,by the secondlaw:
-Fu"(t).
=
mbs6(t)
If Fu"(l)is in p_rincip-le
determined,then the net force impressedon D by (the
actions or reactionsof) any set ofenvironing bodiescan be obtainedis the
vector sum, i.e., parallelogramcomposition, of forces exerted by an
appropriateset of test bodies.ro
- Now the programfor specifyingn"(r) is straightforward.Fu"(l)is givenby
the vectorsum,i.e.,parallelogramcomposition,of forcesfi" exertedby thi
i-th particle of 6 on a. Theseparticles are understoodto be elementaryor
simp.lei7 that they interact by an elementaryforcelaw, f,^,analogousto the
gravitationalinteraction,Eq. 2, in possessing
a fixed form, eipressedin
terms of position, velocity, and measurableproperties. This conceptual
reduction9{ b !o parts which are 'simple' in relation to a force law is representedby Fig. 3:
Frcunr 3
and by the equation
Fu"(l)= 2f,
t4l
The values of the forces fi; dependon time through the positions and (in
general)velocitiesof particlesi andj:
HASSING:WHOLES,PARTS,AND LAWS OF MOTION
203
fx = fi;(n,4, vi, vl)
t5l
The indexTruns over a and all the particlesof 6 other than the l-th. Finally,
the motions of the particlesof D are determinedby
r',/lit,: 2fit * f.'
t6l
jtl
togetherwith initial conditions1(lo),vi(to)for all i. Thesystemof equations4,
theargumentfor universaldeterminismby reductionof any
5, and 6 expresses
body to its sufficientlysimpleparts.
The parallelogramrule (more precisely,polygon rule, sincehere we are
adding many forces,not just two) for compositionof forces,and the associated conceptsof simple part and elementaryforce law, are explicit in the
precedingEq.4, in Eq.4 itself, and again in Eq. 6' The
three sentences
parallelogramrule forms the crucial linkfrom elementaryor simpleforces
and particleslo the force exertedby the compound.Without this link, Eqs.4
and 6 must be withdrawn, and therewith the reductionistargument from
classicalmechanics.
As in the caseof gravitation,theconjoint employmentof the parallelogram
compositionrule and the conceptof elementaryforce law necessarilyentails
the assumptionthat a part of the compound in no way modifies its active
characterin function of its relation to other parts.The partsaredynamically
independent,and neutral to the wholeswhich they compose.They are what
theyare,and act astheydo, independentlyofthe whole.Thus no wholeactsas
such,but only as a sum of neutral parts. This is the refutation of holism'
The parallelogramrule for compositionof forcesforms the doorway to a
seriesof questionscrucial for the interpretationof physics.As Newton said,
upon "this Corollary . . . dependsthe whole doctrine of mechanics. . . ."41
What, then, is the status of this rule? In its role as link from sufficiently
simple parts to the force or action of the whole, is it applicableto all the
wholesin nature?Are all wholesin nature composedof dynamically independent parts? Are all forces or actions in nature either elementary,or
composedof elementaryforces,forces governedby fixed force laws?Any
attemptto answerthesequestionsmust,at somepoint, undertakea thorough
review of the opinions of others, beginningwith Newton and his contemporaries.Here we can make only the barestbeginning.
2.4 Preliminary Historical Inquiry. How doesthe parallelogramrule for
compositionof forcesstandin Newton'sPrincipia?It is introducedby way of
two "corollaries" following the three laws of motion.a2Is it, therefore,
logically entailedby thoselaws?If the rule is denied,that is, assertedto be
inapplicableto certaincompoundsin nature,is therea contradictionwith the
lawsof motion?Is therea contradictionwith the law of universalgravitation?
We shall argueherethat the existencein natureof forcesand compoundsto
which the parallelogramrule is not applicableleadsto no contradictionwith
the lawsof motion. On the other hand,weshallseein thefollowing section(as
implied in the previousone) that denial of the rulepr gravitationalforces
would indeed contradict the law of universal gravitation' Let us then
examineNewton'saccount.In sodoing, wearefortunate to haveat hand the
work of Max Jammer, a scholar noteworthy for his attention to an issue
generallyneglectedin the secondaryliterature.ar
204
NATURE
ANDsysrnM
Corollary I is the parallelogramrule for composition of velocities,not
forces,resultingfrom two simultaneouslyactingimpulsiveforces.44
We quote
the statementof the corollary and the first sentenceof the explanation:
A body,actedon by two forcessimultaneously,
will describethe diagonalof a
parallelogram
in the sametime as it would describethe sidesby thoseforces
separately.
If a bodyin a giventime,bytheforceMimpressed
apartin theplace24,shouldwith
anuniformmotionbecarriedfrom.4to 8, andby theforceNimpressed
apartin the
sameplace,shouldbecarriedfromA to C,lettheparallelogram,4dCD
becompleted,
andby bothforcesactingtogether,it will in thesametimebecarriedin thediagonal
fromA to D.as
The remainingfive explanatorysentences
involve the directionalpart of the
secondlaw, aswell asthe first law (for the constancyof the resultantvelocity
alongAD). The essentialpoint is this: Newtonassumesthat, althoughacting
simultaneously,eachforce, Mand N, actsindependentlyof the other.a6The
independenceof effects(traversalof distancesAB and AC in a given time)
followsfrom the independence
ofthe agenciesproducingthoseeffects.In this,
thecompoundagent,arting by the two componentforcesM and-l/,is like the
gravitationalcompound.Its partsaredynamicallyindependent.Corollary II
then draws the conclusionfor composition(and resolution)of dynamically
independentforces:
And henceis explained
thecomposition
of anyonedirectforceAD, out of anytwo
obliqueforcesAC and,CD;and,onthecontrary,theresolution
of anyonedirectforce
AD into two obliqueforcesz4Cand CD: whichcompositionand resolutionare
abundantly
confirmedfrom mechanics.aT
This is not further proven,but ratherillustratedby meansof its applicationto
a typical mechanicsproblem. The dynamical independenceof component
forcesis assumed.It is not entailedby the lawsof motion, or by the conceptof
Validation of this
force aswhatevereffects,or tendsto effect,acceleration.at
assumption,and thus confirmation of Corollaries I and II, consist in the
success
of the resultingdescriptionin eachparticularcase:". . . theuscof this
Corollary spreadsfar and wide,and by that diffusiveextentthe truth thereof
is further confirmed."ae
Is there here any basis for assertingthat all compoundsin nature are
composedof dynamically independentparts?We believethe answeris no.
Considerthe opinion of Jammer:"Newton'sderivation of the parallelogram
theoremof forcestacitly assumesthat the action of one force on a body is
independentofthe actionofanotherforce,an assumptionthat is far from selfevident."soIndeed,the assumptioncan be withdrawn without contradicting
the laws of motion. In the languageof Corollary I, we would then havethe
following situation:The agentsactingby forcesM andy'y'when
apart modify
eachother whenactingtogetherso that the resultantforce, P, is not givenby
HASSING:WHOLES,PARTS,AND LAWS OF MOTION
205
producedbyM,
the parallelogramrule appliedto M andN. Theaccelerations
i/, and P all satisfythe secondlaw. Furthermore,eachforce can be resolved
into calculationallyconvenientcomponents.Yet P is not the parallelogram
sum of M and N, since thesethreeforces do not act simultaneouslv.For
concreteness,
wecanthink of Pas the forceexertedby a live animal;M andN'
the forcesexertedby the two parts obtainedby cutting the animal in half.
Here (l) the whole hasa modeof action,a specificactivity, (l), which is not
fully reducibleto the actions,M(t), N(t), of its parts;stand (2) this is not
inconsistentwith the three laws of motion.
On this interpretation-and Newton's own languagesupports it-the
parallelogramrule for compositionof forcesstandsin Newton'sPrincipia as
an hypothesisto betestedin eachcase.This claim cannot,however,beunderstood apart from the notion of dynamical independence.Specifically,
whenevera wholeconsistsof dynamicallyindependentparts,then the forces
exertedby the partsare relatedto the forceexertedby the wholeaccordingto
the parallelogram(polygon)rule. A reductionistdescription,deterministicin
the senseof Laplace,is possiblewhen,in turn, the dynamicallyindependent
parts are sufficiently simple to interact by elementaryforce laws analogous
truth that
to the law ofgravitation. It is not, however,an apriori or necessary
all wholesin nature consistof dynamically independentsimpleparts in the
sensedescribed.
The possibility of forcesand compoundsto which the parallelogramrule
would not apply wasknown to lgth Centuryphysics.The Frenchelastician
Barre Saint-Venantdescribedthe possibilitythat, at the molecularlevel,
resultantcomon a particleis notexactlythegeometric
. . . thetotalforceimpressed
posedby thestaticruleof theparallelogram
or polygon,whichweknow,of all the
by the otherparticlesif eachof them
forcesimpressed
[on the particle]separately
until ourday;thisrulewill notbemoretruethan
existedalonewithit, aswasbelieved
distances,
whosestrengthis that of universalgravitation,
for actionsat perceptible
to actionsat
compared
andalwaysnegligible
inversely
asthesquareof thedistance,
pressure
and
capillarity,collisions,
whichproduceelasticity,
imperceptible
distances
actionsarenot subjectto thestaticruleunder
vibration;andtheselastandenergetic
discussion.52
"On the Conservationof
At the time of writing his well-known memoir,
Force" in 1847,Hermann Helmholtz adheredto what he calledthe principle
of the "completecomprehensibilityof nature":
are to be reducedto the motionsof matterspossessing
. . . naturalphenomena
forcesof motion,whichforcesdependonlyon spatialrelations.. . . The
unchanging
intothe
exertoneachothermustberesolved
force,however,
whichtwowholemasses
goesbackto theforcesof
forcesof all theirpartson oneanother;therebymechanics
materialpoints,that is, to thepointsof spacefilledwith matter.. . .
thus:to reduce
is specified
Finally,then,thetaskof thephysicalnaturalsciences
to unchanging
attractiveand repulsiveforces,whosestrength
naturalphenomena
of thistaskis,at thesametime,thecondition
Therealizability
onthedistance.
depends
of nature.s3
of thecomplete
comprehensibility
In his appendicesof 1881,however, Helmholtz retreatedfrom his earlier
position. His revisedopinion includedthe following statement:
206
NATUREAND
SYSTEM
of all the
. . . Thattheforcesof motionare, astheyaredefinedin Newton,resultants
individualforcesconstructed
according
to theparallelogram
law,thattheyemanate
present-thisI canstill onlyacknowledge
from all theindividualmasselements
asa
naturallaw foundby experience.
. . . I canno longeracknowledge
theprincipleof
comprehensibility
as sufficientfor theconsequence
that theeffectarisingfrom the
conjointactionof two or morecauses
of motionmustnecessarily
befoundthrough
the(geometrical)
summation
of thoseeffectsof theindividualcauses.sa
Perhapstheseopinions and our assessment
of the parallelogramrule in
Newton's Principia suffice to resolve the issue between holism and
reductionism. Holistic systemsin nature are compatible with, and not
reducible to, physics. This is possiblebecause,from the beginning, the
reductionistargumentagainstthe possibility of holism wasfalse.It erred by
its overhastyuniversalizationof the parallelogramrule and the associated
conceptof dynamicalindependence.
Where, then, would things stand?We know by the successes
of physics
that compoundsexist (clocks,cars,the solarsystem)which admit reductionist and thus deterministicdescriptions.If thesesystemsareterrestrial,sothat
we can get at them, they can be disassembledand reassembled,precisely
becausetheir parts are neutral to the whole. The crucial question,however,
concernsthe comprehensiveness
that can in principle be claimed for the
reductionist theory. For there remain systemsin nature-animals, for
example-for which any attemptto lay baretheir partsto the extentrequired
to specifyforce lawsirreversiblydestroysthe characterof the whole.According to the account of the composition of forces given so far, the postNewtoniandevelopmentof physicsvalidly provides(l) threeuniversallawsof
motion which are, however,neutral to the issuebetweenholism and reductionism, and (2) a deterministicdescriptionbut only at a particular level of
nature.As to the further specificationof holistic principles,it is a goal lying
beyond the scopeof the presentendeavor.Our intention here is simply to
considerhow such principles might be possible.As to the preciserelation
betweenholistic principlesand the content of physics,it must sufficefor the
presentto remark that Niels Bohr suggested
a principle of complementarity
for organisms analogous to that for wave-particleduality in quantum
mechanics.In the words of Werner Heisenberg:
. . . Bohrhassuggested,
thatourknowledge
ofa cellbeingalivemaybecomplementary
to thecomplete
knowledge
of
of its molecularstructure.
Sincea complete
knowledge
thisstructurecouldpossiblybeachieved
onlyby operations
that destroythe life of
of its
thecell,it is logicallypossible
that life precludes
thecompletedetermination
physico+hemical
underlying
structure.55
We have here suggestedthe logical possibility of a principle of complementarity in the realm of classicalphysics.56
Specifically,in the caseof an
organism, the parallelogram rule for composition of elementary forces
cannotbeapplied.For the forceweperceiveexertedby thewholeassuch,and
the elementaryforceswe revealby our experimentalinterventiondo not act
simultaneouslyand cannotbecompared.Knowledgeof thecompoundasone
whole and knowledge of parts acting according to simpleforce laws are
'complementary'.Knowledgeof the one precludesknowledgeof the other.
The precedingparagraphcould serveas the conclusionof this paperwere
HAssINc:wHoLEs,pARTs,ANDLAwsoF MorIoN
207
it not for certainfurther considerations,previouslyadvertised,involving the
conceptsof simplepart, matter, and law.
2.5 Conclusior. Descartes'first law of motion governsthings insofar as
they are simpleand undivided,thus governsthe individual parts of matter of
whichall thingscorporealareconstituted.Theelaborationof theterms'part'
and'matter'requires
an immediateturn to corpuscularism,
in whichpartsare
definedneither in terms of ultimate parts, nor in termsof the wholeswhich
they compose.They are defined rather in terms of the Cartesiansimples,
figure, extension,and motion. The resulting account rules out a type of
whole-part relation which we have called holism, and which includessubstantial form in living beings.In this sense,the Cartesianaccount is not
causallyneutral.
Newton's first law governsall bodiesas we know them in the ordinary
courseof nature,yet doesso asoneelementin the systemof the threelawsof
motion. Thesethree laws require no supportingaccount of whole, part, or
matter. Entailing no commitmentconcerningwhole-partrelations,they can
accommodate
casesin whichtheactionof thewholeis irreducibleto theactions
of theparts.Theyarecompatiblewith both holismand reductionism,
and,in
this sense,they are causallyneutral.
Yet, Newton'sthreelawsof motion are,for the most part, only the stageon
which the real playersmust appear.Principal among them are the laws of
Newtonianactionat a distance(and,later, of propagatingforcefields).Let us
again considerthe premier force law, that of universalgravitation, for we
mustmakeit clearthat it is not causallyneutralin thesensehereemployed.sz
We havemadetwo distinct claimsabout it. First, in Sec.2.3 it wasasserted
that the dynamicalindependenceof the parts of a gravitational system(and
resultingvalidityof the parallelogramrule) is entailedby the conceptof mass.
Second,in Sec.2.4it wasassertedthat the denialof the parallelogramrule for
gravitational forceswould contradictthe law of universalgravitation.Thus,
in the case of gravitation, a reductionist relation of whole and part is
intimatelyinvolvedwith ( I ) the conceptof matteras(gravitational)mass,and
(2) the conceptof force law associatedwith mass.Let us try to clarify this.
The mundanefact that the weightof a body is the sumof the weightsof its
partsprovidesthe simplestintroduction to the issuein question.stThe weight
of a body is the gravitational force of the earth on it. By the third law of
motion, it in turn attracts the earth toward it with an equalforce; whatever
elseit does,thisis thegravitationalportion of thebody'sactivity.ssIt follows
(from the additivity of weight and the third law of motion) that the gravitational portion of any body'saction is the sum of the gravitationalactionsof
its parts.This sumcanonly bethe compositionof forces(here,theattractions
of the pafts of the body on the earth) by the parallelogram rule. In the
languageof presentdayphysics,the gravitationalattraction of a distribution
ofmassis correctlygivenby the vectorsumofthe gravitationalattractionsof
its masselements.This is the calculationwhich Newton had to carry out in
order to establishthat, for bodieson the surfaceof the earth, gravitational
force is proportional to the inversesquareof the distanceto the centerof the
earth:
208
ANDsYsrEM
NATURE
AfterI foundthattheforceof gravitytowardsa wholeplanetdidarisefromandwas
compounded
of theforcesof gravitytowardsall its parts,andtowardseveryonepart
fromthepart,I wasyetin
of thedistance
wasin theinverseproportionof thesquares
did accurately
asthesquareof thedistance
doubtwhetherthat proportioninversely
of somanypartialones;for it
hold,or but nearlyso,in thetotalforcecompounded
enoughtookplacein greaterdistances
mightbethattheproportionwhichaccurately
of the
shouldbewideof thetruth nearthesurfaceof theplanet,wherethedistances
particlesareunequal,andtheirsituationdissimilar.s
Indeed, for bodiesof arbitrary shapein proximity to each other, it is not
clearwhat distance,R, shouldbeusedin the Newtonianformula Gmrmzl R';
R is the distancebetweenwhich two pointsof the bodies?Thepossibilityof
finding an answerto this questionconsists,as Newton indicates,in the
rltt, t/t2,which act "accurately
reducibilityof any body to parts,of ma^sses
enough"accordingto the law Gmrmzf R'.61Sucha masselementis the "one
part" to which Newton refersin the passageabove.Herewefind the presence
of Descartes.'Part' means part, not of a whole, but of maller, matter
conceived as an algebraically representablemeasurable property, or
dimension, of all the wholes in nature. Yet, in contrast to Descartes,the
dimensionis quantity of matter, or mass,not extension,and a part thereofis
onenot in termsof motion, but in relationto a law of actionspecificto mass.62
Whentwo piecesof mass,ntr, //tz,zre-situatedsuchthat their sizesare small
their
comparedto the distance,R, betweenthem, thenGmrmz/R2describes
"one singlecorpuscle"
attraction.o3Eachis thenonepart or, asNewton says,
of mass.orNewton'sgravitational physicsis a corpuscularismof massand
law. Any body, insofar as gravitationally acting, is merelyan aggregateof
corpusclesof mass,eachof which actsby the law of attraction. Accordingly,
the simpleparts of massadd up arithmeticallyto the massof the whole,and
the correlativeelementaryforces add up vectorially, by the parallelogram
rule, to the force of the whole.
To the extent that this way of thinking is adequateto nature,reductionism
is true and holism refuted.To what extent,then, canwe generalizethe gravitational accountof body and action to the following universalprinciple: All
wholesin nature, insofar as active and thus knowable,are sumsof simple
The
partsof algebraicdimensions,which are,and act, by simpleforcelaws?65
large
resolution
of
a
question
to
the
in
turn,
contribute
would,
answerto this
issue:Can the tradition of physicssinceNewton claim to be the comprehensiveaccountof nature?
In arriving at our presentvantagepoint we haveestablishedthe following
three points, to our knowledgenot previouslymade explicit: (l) Newton's
laws of motion differ fundamentally from Descartes'in their implications
concerningmatter and causality.The former arecompatiblewith holism and
thus with the possibility of an essentialdistinction between living and
non-living; the latter are not. (2) The argumentfor reductionismfrom postNewtonian physicsdependscrucially on the parallelogramrule for composition of elementaryforces.(3) The universalemploymentof the parallelogram rule in reducing the action of a whole to the elementaryforces of
a way of conceivingmatter, simplepart, and
deterministicparts presupposes
law, first exemplifiedin Newton'stheory of gravitation.
Centerfor Naval Analyses
HAssrNc:wHoLEs,pARTs,ANDLAwsoF MorIoN
209
AcrNowtnncEMENr: No acknowledgementcould be as generousas the
instruction of Richard Kennington which informs this work, and without
which it could not havebeenwritten. For the guidanceand encouragement
of William A. WallaceI am most grateful.
NOTES
l. M. Boas,"The Establishmentof the MechanicalPhilosophy,"Osirrsl0 (1952):412-541;
Max Jammer, Conceptsof Force: A Study.in the Foundationsof Dynamics (Cambridge:
Harvard UniversityPress,1957);A. R. Hall, From Galileolo lfewron (New York: Dover, 1963
University
and l98l); AlexandreKoyre,"Newtonand Descarles,"NewtonianStudies(Chicago:
of Chicago Press,1965),pp. 53-l 14; R. S. Westfall, Force in Newton'sPftysrcs(New York:
American Elsevicr, l97l); Alan Gabbey, "Force and Inertia in the SeventeenthCentury:
Descartesand Newton," DescartesPhilosophy,Mathematicsand Physics,S. Gaukroger,ed.
(Totowa, N.J.: Barnesand Noble, 1980),pp.230-320,arevised
and expandedversionof"Force
and Inertia in Seventeenth
Century Dynamics,"Studiesin History and Philosophyof Scicnce2
( l97l): l{7; I. B. Cohen,TheNewtonianRevolution(Cambridge:CambridgeUniversityPress,
r980).
2. The terms'living being','organism',and'animal'appearrarely,if at all, in thesestudies.An
"Problemsof Living Things." Although Hall
exceptionis Hall, op. cil., in which Ch. 7 is titled
seemsto favor mechanismin the disputewith vitalism(p. 196),thereis no detailedexamination
of the arguments.
3. This can, of course,be denied,so that thereis no essentialdistinction betweenliving and
orepiphenomenonofthe latter.Sinceit is the
nonJiving.The former becomesa mereappearance
purposeofthis paperto examinethe reductionistposition,that positionwill not bepresupposed.
of a unitydifferent in kind
We thusbeginin the commonexperienceof livingbeingsaspossessed
from that of the non-living.
4. Sincethe conceptsof massand inertia aswe know andusethemarenot found in Descartes,
oneshouldsuspedat lirst glancethat differencesmustexistbetweenthelawsof inertiain Newton
"It is in fact anachronisticto ascribea'principle of inertia' toDescartes,or indeed
and Descartes.
(Gabbey,p. 288.)The indepth studyof
to anyoneprior to . . . Newtoniandynamicalrcsearches."
the meaning of this surfacedistinction forms a large part of Gabbey'sexpert account; see
especiallyPart 4, whereit is shownthat the function and context of the law of inertia differ in
Newton and Descartes.
N. R. Hansonhas describedsomeof the logical problemsinvolved in Newton'sfirst law in
*Newton's First Law: A Philosopher'sDoor into Natural Philosophy," Beyondthe Edge of
Certainty, R. Colodny, ed. (New Jersey:PrenticeHall, 1965),pp. 6-28.This is an expanded
versionof the earlier"The law of Inertia: A Philosopher'sTouchstone,"Philosophyof Scicnce
30 (1963):t07-21.Hansonshows,in particular,that differentchoicesofthe logically primitive
'uniform', and'rectilinear'leadto "differentsemantic
andderivativeamongthe terms'forcefree',
platforms"within the samemechanicaltheory(p. l2). Hanson'sanalysisthus revealsa logicalor
semanticsensein which thereis no onelaw of inertia.Our accountis distinctfrom, and generally
compatiblewith, both Gabbey'sand Hanson's.See,however,note 48 below.
5. This is in no way at variancewith the canonicalinterpretationof the law of inertia,namely,
that uniform rectilinearmotion requiresno causeor mover,and is a slareratherthana change,.or
process:"It is preciselythe institution of the conceptof slatusof motion for actualmotion that
enablesDescartes-and will enableNewton-to assertthe validity of his lirst law. . ." (Koyre'
op. cit., p.69. We shall arguethat more is involvedin validatingthe law.
6. This is not to saythat post-Newtonianphysicssupportsno argumentagainstholismandfor
universalreductionism.Thedeterminismproclaimedby Laplaceis well known.Thepoint is that
the threelawsof motion in Newtonarenot thesufficientbasisfor that argument,whichargument
will be discussedin Part II of this paper.
By'holism'we shall mean the idea, partially stated in the Preface,that whole and part
determineeachother,sothat the wholeis and actsnot asa sumofneutral parts;and the partsare
what they are,and act asthey do, only in termsofthe relationthrough which they constitutethe
maybeunderstoodintermsof parts,asufficient
whole.Thus,althoughtheirnecessaryconditions
descriptionofthesebeingswould requireprinciplesspecificto the whole,principlesofthe whole
as such.
2r0
NATURE AND SYSTEM
.The best known proponentsof holism are Aristotle and the medievalcommentators,for
whom substantial form is the holistic principle. For example, Aristotle states that ..the
continuousand limited is a wholewhenit is a unity composedof manyparts,especiallywhenrhe
parts arepotentiallypresentin it. . . . And ofthesethingsthemselves,
thosewhicharesonaturally
are moretruly wholesthan thosewhich are so artificially. . . ." (Emphasisadd,ed.)Metaphysici,
Loeb classicalLibrary, H. Tredennick,irans. (cambiidge: Harvard
lk: v' ch' 26,1023b33-36;
university Press,1975),p. 281.And rhomas Aquinas:"all parts are relaredto the ;hole asthe
imperfectto the perfect,which is, indeed,the reiation ofmitter to form. . . . flour is calledthe
matterof breadbut not insofarasit standsundertheform of flou r." Commentaryon Aristotle's
Physics,.Bk.II, Lectio 5, Sec. 184;Blackwell,Spath, and Thirlkel, trans. (New Haven: yale
Univ.ersity
Press,19631p. 90. And in Bk. IV, Leitio 4, sec.436,Biackwell'etal., p. 200:..the
whole hasthe nature ofa form in respectto the parts. . . ."
Clearly,in Aristotle and thecommentatorialdevelopment,substantialform is a principleof a//
n-atural(asopposedto artificial) beings,both livingand nonJiving,i.e.,theelemenis.wiiliam A.
Wallacehasshownthat the issueof substantialform as a principle of motion in the non-living
carriedover into the formative period of early modernsciince.See,for example,..causesand
Fo^rces
at the collegio Romano," william A. wallace, prelude to Galileo(Dordrecht: Reidel,
l98l), pp. I l0-26, especiallypp. I l3 and 122.The presentinvestigation,however,will not enter
into the.question of substantialform as a principle of motion in the non-living. Rather,
substantialform will be taken as one exampleofa holistic principle,and it will sufficefor our
the sourceof strongestevideniefor holism.
o._:.ryT.t_,:"*l_1.]i"1.s
1-elas
The argumentbetweenholism and reductionismremainsalive within contemporaryscience:
"Most biologists
are in gcneralagreementthat vital processes,
like non-living ones,oicur only
under determinatephysico-chemicalconditionsand form no exceptionsto physico-chemical
laws.Someof them nevertheless
maintain that the modeof analysisrequiredfbr understanding
living phenomenais fundamentallydifferent from that which obtainsin the physicalsciencesl
opposition to the systematicabsorption of biology into physicsand chemistryis sometimes
basedon_thepractical ground that it doesnot conform to ihe correct strategyofbiological
research.However,suchoppositionis oftenalsosupportedby theoreticalargumlntswhichlim
to show that the reduction of biology to physico-chemistryis inherentlyimpossible."Ernest
Nagel, 7he Structureof Sciznce(New York: Harcourt, Brace,196l), pp. feA-ie.
evidencein supportof holism,it is of two kindi: (l) ordinary expenence
-As.for,contemporary
of living beingsand our inability to takethemapart and put themtogetheiaswecunrn"chin"rit remainsthe casethat living organismshaveniver beenartificially producedout of non-living
material;(2) scientificevidence,aspresentedfor exampleby Jamei A. Shapiro,..variation asi
Engineering Process," Evolution Jrom Molecules to Man, b. s. Bendall, ed.
_G_enetic
(cambridge: cambridge University Press,1983),pp. 253-70.Shapiro finds that .,a detailed
investigationof the variational process[in cells] revealsthe coordinatedaction of biochemical
systemswhosespecificitiesand regulationarebeyondsimplechemicalexplanations"(p. 254).He
then detailsthreeexamples"relevantto the formulation of evolutionarytheories"<i. zsqi.
Among th€ most incisive-post-Newtonianformulations of holism are Hegel's:;the notion
of thewholcis to containparts:but if thewholeis takenandmadewhat its notion implies,i.e.,if it
is divided, it at onceceasesto be a whole.Thingsthereare, no doubt, which correspondto this
relation:butforthatveryreasontheyarelow...existences....Therelationofwholeandparts
' . . comesvery easyto reflectiveunderstanding;and for that reasonit often satisfieswhen the
questionreally turns on profounderties.The limbs and organs,for instance,of an organicbody
are not merely parts of it: it is only in their unity that they are what they are, ani they ari
unquestionablyaffectedby that unity, as they also in turn affect it. Thesl lmbs and oigans
becomeparts, only when they passunder the handsof the anatomist,whoseoccupation,6e it
remembered,
is not with the living body but with thecorpse.Not that suchanalysisisillegitimate:
we only meanthat theexternaland mechanicalrelationofwhole and partsis not sufficientfor us,
i_rwe
11111o study organic life in its truth;' EncyclopediaLogtc,sec. 135,Note, in Hegel's
zogra william wallace, trans.(oxford: oxford Universitypress,l9z5), p. 19l. seealsoSec.3g,
Note(p.63) and Sec.126,Note,(p. 183).
7. Descartes,I-e Monde, qh. 7. M. S. Mahoney, trans. (New york: Abaris, 1979),p. 60;
hereaftercited as Le Monde.I havechangedMahoney'stranslationslightly for greaterliteralness. Descartes' statement of the first law in principles of philosopny, pai ll (hereafter
Principlesll),37,is: "the first [law of nature]is:everything, insofarasit is iimple and undivided,
HASSING: WHOLES. PARTS. AND LAWS OF MOTION
2tl
remains,sofar asin it lies,alwaysin thesamestate,and neverchangesexceptby externalcauses."
vol.8 (i), p.62. Hereafterthis
Oeuvresde Descartes,AdamandTannery,eds.(Paris,1897-1913),
edition will be referred to as AT. The latest English translation is Descartes,Principlesof
Philosophy,V. R. Miller and R. P. Miller, trans.(Dordrecht: Reidel, 1983);the first law is on
p.59.Hereafterthiseditionwillbereferredtoas MM. Ourtranslationwillnotalwayscoincidewith
Monde.Most
theirs.Thelaterformulationofthefirstlawdiffersinseveralwaysfromthatof-Le
important for our purposesis the addition of the two phrases"Quantum in se est," and
'quatenusestsimplexet indivisa."The lirst joins Descartesto Newtonand is discussedby I. B.
Cohen, ' 'Quantum in se est': Newton's Concept of Inertia in Relation to Descartesand
Lucretius,"Notesand Recordsof the Royal Societyof Inndon, l9 (1964):l3 l-55. The second
distinguisheshim from Newton and will be discussedfurther in sections1.3and 1.4. For the
moment,let us notethat the phrase"insofar asit is simpleand undivided"makesexplicit a signi
ficant point presentinLe Monde,aswe shallsee,but not therecontainedwithin the statementof
the law.
8. Le Monde, pp. 20-22.Seealso PrinciplesII, 25; AT, Vol. 8 (i), p. 54; MM, p. 5l. The
definition containswell-knowncircularity.
9. Iz Monde, p. 56. Also PrinciplesII, 4; AT, Vol. 8 (i), p. 42; MM, p. 40. As usedby
Descartes,the phrase'part of matter' is possessed
of fundamentalsignificance.To beginto see
why, considerwhat is said in PrinciplesII, 8: "Thus . . . we may considerthe whole nature of
corporealsubstancewhich is in a spaceof ten feet,althoughwe do not attendto this measureof
tenfeet;becauseit isclearthat rhethingunderstoodis thesame.inanypart ofthis spaceasin the
whole" (emphasisadded).AT, Vol. 8 (i), p. ,$; MM, p. 43. To graspthe natureof matter-here
identicalto extension-in any part is to graspits naturein any whole,in fact, to graspits nature
simply.Within this understanding,partsarenotparts of wholes,justpartsof marer. Suchparts
are obviouslyneutral to, and not determinedby, the wholeswhich they constitute.This way of
conceivingbody-as-matterdiffers fundamentallyfrom that of the precedingtradition. It is
involved with the notion of simplicity in Descartes(seenote 20, below) and will figure in our
discussionof the argumentfor reductionismin sect.2.5.
10. Iz Monfu, p.58.
ll. Iz Monde,p.40.
| 2. Newton, MathematicalPrinciplesof Natural Philosophy,A. Motte and F. Cajori, trans.
(New York: GreewoodPress,1962),p.13.All references
will betakenfrom this edition,hereafter
citedas MathematicalPrinciples.The l,atin formulation (p. 6{{) is preferablesinceit contains
the important word perseverarewhich is not adequatelytranslatedby'continues':Corpusomne
perseverarein statu suo quiescendivel movendi undormiter.in directum, nisi quatenusa viribus
impressiscogitur statum illum mutare." I. B. Cohen,op. cit., (1980),p. 187,remarksthe more
restricted character of Newton's Mathematical Principles comparedto Descartes'Principles.
| 3. This is perfectlycompatiblewith Descartes,PrinciplesII, 25, regardingabstractionfrom
internal relativemotions.
14. This is compatiblewith, but doesnot entail,the viewthat the ultimatesourcesof activity are
actually unextendedbeingsin empty space,a notion apparentlyfirst proposedby Boscovich,
Theoryof Natural Philosophy,J. M. Child, trans.(Cambridge:MIT Press,1966),p. 45.
15. TheexcellentstudiesofGabbey,Westfall,andCohencitedinnotel,above,areofespecial
relevance,
16. Iz Monde, pp. 66{8. The parallelaccountin PrinciplesII will be discussedmomentarily;
seenote 19.below.
17. Le Monde, pp. ztOand 42, emphasisadded.
18. Iz Monde,p.42.
19. For the statementof the law, seenote 7. Following the statementis an explanationof how
phenomenaof self-terminatedmotionsdo not in reality violatethe law: "However, becausewe
inhabit the earth,which is so constitutedthat all motionswhich occurnearto it ceasein a short
we oftenjudged,from the
while,andfrequentlyfrom causeswhichareconcealedfrom our senses,
beginningofour life, that thosemotionswhichthusceasedby causesunknownto us,did sospontaneously.And we tend to hold in all caseswhat we think we have observedin many cases,
namely,that motion ceasesby its own nature,or tendstoward rest.Now this is most opposedto
of self-initiated
thelawsof nature."AT, Vol.8 (i), pp.62{3; MM, p.59.Thecaseforphenomena
motionsmustbethe same.In both cases,whatapp€arsto bethecase"is mostopposedto the laws
of nature,"and the resolutionis in termsof the strictly lawful behaviorof unobservableparts.
20. TheconceptofsimplicityarisesinDescartes,RegulaeadDirectionemlngenn',Crapulli,ed.
212
NATURE AND SYSTEM
(The Hague: Martinus Nijhoff, 1966).Rule VI introducesthe simple naturesas intuitively
immediatebeginningpoints of knowledge(pp. l8-19).Indeed,in Rule VIII it is assertedthat wL
'can havecertain
experienceonly of the entirely simpleand absolute"(p. 27, lines 22-23).ln
Rule XII we learn that the simpleis known completelyor not at all, that is, the leastknowledge
thereofsufficesto know the whole(p. 47, lines20-23).Extension,asdescribedin Principlesll,8
(seenote 9, above) is thus, for Descartes,simple. Rule XII statesthat "in the order of our
knowledge,eachsimple thing should be vieweddifferently than when consideredas it really
exists"(p.45, linesl7-19).Simplicityis thusa propertymore ofconceptsthan ofreal thingi.
Indeed,in Pr.rnciplesll,4l,motion is assertedto besimple;AT, Vol. 8 (i), p. 65;MM, p.62.yet a
mereconceptcannotbeactedupon,collide,accelerate
and decelerate.
Simplicitymustthen refer
to bodiesas well as to concepts.(The possibilityof suchreferenceis the possibilityof Cartesian
mathematicalphysics.)The passage
cited in Le Monde, notes17and 18,above,indicatethat in
applying.the notion of simplicitywithin physics,certainbodiescannotbeconsideredas"simple
and undivided" in respectto certain of their changes.Thesebodiesand changesmust then be
reducedto motions of parts of matter which can be so considered.
21. The term'corpuscle' is Boyle's.See,for example, Origin of Forms and eualities (The
TheoreticalPart), (Manhattan Beach:The SheffieldPress,1976),p. vi.
22. The parts of matter aie not of infinite but of indeterminate divisibility; k Monde, p. 16;
Principbs,l,26;AT,Vol.8(i),pp.14-15;MM,p.13.ForDescartes,anyoneparrof
anyof the
threeelementscan be transformedinto a part of anotherelementthrough modification of size
and speed;Iz Monde,pp.34-48. Thereis no essentialheterogeneityofelementsastherewasin
ancientatomism.Thereare alwaysthreeelements,due, not to the beingofthe el€m€nts,but to
laws and processes
which facilitatea "dynamicalequilibrium." Thus, for Descartes,it is more
accurateto say that thereare threeconditionsof matter, rather than threeelements.The same
absenceof ultimate determinacyin the corpuscularismof Boyle hasbeenclearlydiscernedby
ThgmasKuhn, "Robert Boyle and Structural Chemistryin the SeventeenthCentury," Isls 4i
(1952),17,19,22,32,and33.It is dilficult to see,however,howthiscouldbeAristoteiian,since
both Descartesand Boyle rejectthe Aristotelian principleof potency.
23. As we shall see,this assertioncan apply aswell to post-Newtonianphysics.It is, in fact, a
concisestatementof reductionistdeterminism.To repeat (note 6, above), the point is that
Newton'sthreeuniversallawsof motion are not a sufficientbasisfor it. This will be discussed
in
Part II..
24. Seethe studieslistedin note l, above.J. E. McGuire hasarguedpersuasivelythat thereis
more to Newton's thought about nature than the rational mechanicsof the Mathematical
Principles;seeMcGuire,'Force, Active Principles,and Newton'sInvisibleRealm,"AmbixXy
(1968),154-208.Gabbeya9rees:op cit., p.240.
25. Newton'ssecondlaw is then the basisfor the quantitativerelationbetweenimpressedforce
and acceleration.Assumingconstantmass,and in its presentformulation, F(t\= ma(t).lndeed,
"force asweunderstand
it is the logicalcorrelateofthe principleofinertia, an externalactionthat
altersthe stateof a body unableto initiate sucha changeof itself." R. S. westfall, op. cit., p.301.
We would modify Westfall'sstatementin two ways:(l) the word'state'shouldbechangedto the
morenarrow'velocity';(2) whethera bodyis able or unableofitsefto initiatea changein velocity
is somethingabout which Newton'sthreelawsof motion are,by themselves,
silent.ihe secondii
the point of the presentsection.
26. MathematicalPrinciples,pp. 13and &4.The mat is, of course,stuck to the floor of the
house,and the houseto the earth.
27. concerning forces, "[i]n Principia lNefion] is not concernedwith investigationsof
anything other than their mathematicaldescription.. ." Gabbey,op. cit., p.239. Seealso the
author's *The use and Non-Use of Physicsin Spinoza'sEthics," southwesternJournal of
Philosophy,ll (1980),p.50.
28. Gabbey:"Newton'sreacrioisatthesametimeanactio,the're'expressingboththefactof
dynamical opposition and the fact that one of the bodiescan alwaysbe taken as that which
resists,"op. cit., p.271.
Newton'sgcneralizationof the force conceptto include both impulsiveand continuously
actingforcesis describedin I. B. cohen, op. cr'1.(1980),sect.4.4.That Newton'sconceptofforce,
asit first app€arsintheMathematicalPrinciples,is reallyimpulse,which we representas Iiy't, is
alsoshownby Brian Ellis, "The Origin and Natureof Newton's[,awsof Motion," Colodny,ed.,
op. cit., pp.2948.
29. The reactionforceis not simplya fiction. It is supportedby certainexperiences.
If you walk
HASSING: WHOLES,PARTS,AND LAWS OF MOTION
2I3
inadvertantlyinto a wall, asI havedone,it feelsexactlyasifsomeoneshovedyou. (Thiscanonly
be an experience,not an experiment.)
30. This is the sensein which Newtonian physicscan rightly be said to be common sense
sharpenedup, a remark often madebyteachersofthe subject.The tendencyinrecentscholarshipto
analyze each of Newton's laws singly, in terms of
its historical development, leads to an emphasis on Newton's relation to D€scartes
and obscurcsa view of the three laws of motion as a completedinterlocking whole. See,
for example, Koyre, op. cit., p. 65, and I. B. Cohen, op. cil.' (1980)' p' 185; also
J. Herivei, The fuckground to Newtan's Principia: A Study of Newton's Dynamical
Researches
in the Years1664-1684(Oxford: Oxford University Press, 1965).Thesestudies
accordingly suggestthat Newton owed more to Descartesthan he acknowledged.Yet, to
theextentthat Newtonunderstoodhisfinal productin themannerheredescribed,heowedrather
lessto Descartessincehe arrived at a basically different physics.
31. It is Newtonian in that it incorporatesthe three items listed, all of which appearin the
MathematicalPrinciples.It is Laplacianin that Laplaceseemsto havebeenthe first to proclaim
it publicly in A Philosophical Essayon Probabilities,F. W. Truscott and F. L. Emory, trans.,
(Niw Yoik Dover, l95l), p. 4. Newtondid not proclaimit, and we mustpassoverthe intriguing
questionwhetherhis caution wasprudentialor theoretical.
32. Descartes,Regulae,Rule XIV; Crapulli, p. 67.
33. Approximations are necessarydue to the many-body problem, and generalrelativistic
correctionsto Newtoniantheoryarewell :stablished.Thequestionof final causalitycannotbeso
easilydispensedwith. While trajectoriescan be obtainedwithout referenceto final causes,they
cannot be obtainedwithout a choiceof initial conditions.What then are the particular initial
conditionsthat led to a solar systemstableover a human time scale?The laws of motion and
but not sufficientto accountfor the way
gravitation(Newtonianand Einsteinian)are necessary
the world is.
34. Newton, "Unpublished Conclusionof the Principia," lJnpublishedScientific Papersof
IssacNewton,A. R. and M. B. Hall, eds.(Cambridge:CambridgeUniversityPress,1962),p. 333.
Seealso Newton's prefaceto the first edition of Mathematical Principles, p. xviii.
35. Laplace,op. cit., p.6.
36. MarhematicalPrinciples,CorollariesI and II, pp. 14-15.
37. It is well known that suchsolutionscanbefound only for the 2-bodyand restricted3-body
central force problems. As indicated, the complexity of the many-body problem is of no
-philosophicalsignificance.
'many-bodyforce' rather than a
38. Inthe languageof physics,gravitation would then be a
'two-body force'. Many-body force effectsin the known forcesare virtually alwaysassumed
of the resulting
away.ThL assumptionis obviouslyjustifiedby, and to the extentof, the success
calculations.It is equallyobviousthat this doesnot meanthe assumptionis universallytrue.
39. See,for example, IsaacTodhunter,,{ History of the Theoryof Elasticity,K' Pearson,ed.
(Cambridge:CambridgeUniversityPress,1893),Vol. 2, pt. l, p. 183'note 2'
40. This makes use of the assumptionthat the bodies on which b acts are composedof
dynamically independentparts, and thus subject to the parallelogram rule. The crucial
applicationwill bethe reductionof the forceexertedDyb asa wholeto the forcesexertedby the
parts of D.
4l- MathematicalPrinciples,p. 17.
42. Ibid., CorollariesI and II, pp. 14-17.
43. Jammer,op. cit., Ch. ?. Gabbey'sexcellentstudy will also be cited, op. cit., pp.28l'86'
Jammer,howevlr, coversthe history of the compositionof forcesmore extensively,and, we
believe,more corr€ctly.
44. That the forcesareimpulsiveand not continuouslyactingis impliedby theconstancyof the
resultingcomponentvelocities.Seealso note 28,above.Coroltary I is reportedin Jammer,op.
cir., pp.1:O-li, and Gabbey,op. cit.,pp.28l-82. Westfall,op. cr?.,Ch.8, citesthe earlyversions.
of the parallelogram rule appearing in Newton's researchesleading up to the Mathematical
hinciples.
45. MathematicalPrinciples,Corollary I, p. 14.
116.Newton,z{ddMS.3965.6,f.86,quotedbyWestlall,op.cit',pp.479andlZ3,makesitclear
'if
of forcesM and .ff.' the forcesM and
the dynamicalindependence
that Newton presupposed
N be impressed. . . as though they wereseparatelygeneratingmotions . . ."
47. MathematicalPrinciples,Corollary II, p. 15.
214
NATURE AND SYSTEM
48. Jammerasserts,concerningthe parallelogramrule,that *Newton .
. . wasconvincedthat its
validity canbederivedfrom^the_verycbncept
oiforce, asis shownbv trt a".onrt."iion following
Corollary I." op. cit., p.128. we d-onot seethe reasonfo. ttretasi ptrrase
oi-tt ir."nt"n.., "na
suggcstthat the conceptof force neednot entail the dynamicalirir.[ra"""r
"ialt agentsin
nature.
"the
Gabbey
statesthat
respectiveactionsof Mand iy'. . . areindependen
,
t by definiron.M and
N.1r9 vires impressae,and so.in rhe corolary are geometricailt c;;;;iurJio'Jion"nt
ro.".,
whichcannotinfluenceeachother whenactingsimullneously. Ifthey did,
this woild contradict
that part of the Secondlawwhich refersto direction]' op..ri., p. zeo.
w.,rgg"stGat Mand il
are independentb-y.hypothesis
and wilr explain momentarily why we doilt asree that the
orrecuonarpart or the secondlaw is incompatiblewith a notion of modified
actioi.
westfall, op. cit., ch.8, refergfrequentlyto the parailelogramrure,
.
uut oynamicatindepenwithoui comment.westfalt, iia."o" i.."r"t;'N;;;",.
strussle
*lf"^,-:::t1tlL:ughout
problem, namely, how to sort out inherent force (vis insita) from
Ii^l^lT:l1m€diate
tmpressed
lorce(vis impressa),bothcalled'force'byNewton,yetpossessingdifferent
relationsto
motion.
49. MathematicalPrinciples,-p. 17.
50. Jammer,op.cit.,p.132.
5l' Theactionofthewholeispartiallyreducibletotheactionsofparts.Forexample,thegravitational portion of theanimal'i actionis so reducible.Supposethai the live
animal;sweightis l5
lbs.,,sothat its gravitational activity is to attract the earth toward it
with a force of 15 lbs.
o^bviously,the sumofthe weightsofihe parts ofthe deadanimal is l slus.
ciearly, tt "n, trr" ,,rof the gravitationalactionsoflhe partsequalsthe gravitationalaction of
the whole.The further
iw-estigationof this caseforms thl contentof the-concludingsection.f
thi, ;;;;;.
"De l^a
constitution desAtomes,"(lg7g), quoted in Todhunter,op.
12. Btry saint-venant,
crt- p.,185;andJammer,o4_r_il.,p.133;
translationUy enca Hassingi
"Ueber
die Erhaltungder kraft," wissenichaftlicheAbhandlungen
.-)J' HermannHelmholtz,
(I*ipzig, 1882),Vol. I, pp. l5-16,my translation-.
54. Ibid.,pp. 68{9, my translation.
wcrner Heisenberyreporting_lohr'sopinion in Heisenberg phvsics
and phitosophy(New
,
.55'
York rlarper & Row, 1958),pp. lft-105. For Bohr'sdescriptionof'the notion of comptemen_
tarity. in quantum physics,.see,for_example,"Discussion,"lttt Binrt"irr-on
epi.lt"-orog,"ut
Problems^in Atomic Physics,' Albert iinstein philosopher-scbntist, p.
A.'Schrlpp, ed.,
(IaSalle: Open Court, 1969),pp. 210 and 236.
What about quantummeihanics-has it not madethewholeissueof determinism
-56. .
obsolete?
Not in a way that would makeclearthe statusof holistic principlesin nature.
it e iear question
then is, what is the relation betweenlrolismand quantumtheory?.fo "nr*".
thirlarge question
requiresclarity about quantum mechanics.Yet t-heinterpretatibnthereofremains
oien cespite
copenhagen(Bohr-Heisenberg)
school.s.r, f";;;;;;il;i.
E].Balentine,
l[111;naancv,ofthe
-r ne statrstrcallnterpretationof euantum Mechanics,-Reviewsof Modein physics
42 (1970):
358{1. seealsothe works of David Bohm. Furthermoie,it is weuinown irraiqri"ni,rrn
pr,y.r",
remainsdependenton crassicalphysicsin a way that seemsnot to be removabre
by further
"A
changesof interpretation.
more generalthe;ry can usually be formulated in a logically
complete.manner,independentlyof a iessgeneraltheory which iorms a timiting
caseof it. Thus
relativisticmechanicscanbeconstructedon the basisoflts own funaamentatpiin"ipies,
without
any referenceto Newtonianmechanics.It is in principleimpossible,rr"*.".1,
i" il.1nulate the
basic conceptsoj q'r"ntum mechanicswithoui using classicalmechanics.
The fact that an
*tefinite.path meansrhat it hasatso,ii itsetf, ,,o othe.
luantitaitvJynarnicat
:11.:::lT
cnaracrcnstrcs.
Hencelt is clearthat,for.asystemcomposedonly ofquantum objects,it would be
entircly impossibleto constructany logicailyindepenientmechanici.The possibiliiy
ofa quantitative descriptionof the motion of an electroniequiresthe presence"rrf or
frtyJi""l objects
which obeyclassicalmechanicsto a sufficientdegreeofaccuraiy. . . . Thusquantum
mechanics
occupiesa very unusrnl placeamong physicallheories:it contains.f".rili
r..-ftunicsas a
umttlng cirse' yet at the same time it rcquires this limiting casefor its own
formulation."
L. D. Landau and E. M- Lifslrltz, euanium Mechanics,Ji n. sykes and
J. S. B"tt, t."rrr.,
(London:
Press,
1958),
pp.
2-3.The
attempt to understandclassicarphysicsis a
lergamon
necessary
first step.
57' Both
leyron and post-Newtonianphysicsrefrain from specifyinga causeof gravitation,
and stay with the law of its effect.In this sense,Newtoniangravitation-altheory
is ,iot a cuusal
I{ASSING: WHOLES, PARTS, AND LAWS OF MOTION
2t5
theory. The senseof causalneutrality usedin this paper,however,givesspecialemphasisto the
whole-partrelation.
58. Seenote51,above.
59. It attractsall the other bodiesin the universe,but we can neglectthis here.
60. MathematicalPrinciples,Bk. III, Prop. VIII, p. 415. Seealso Hall, op. cit., p.295.
61. Newton usedProps.75 and76 of MathematicalPrinciples,Bk. I, and their corollariesto
carry out the neededcalculation.Usingmodernvectorintegration,the procedureis standardin
"Thelawofgravitationasformulatedlcmfi2l R'lis^pplicableonly
mechanicsandfieldtheory:
to particlesor to bodieswhosedimensionsare negligiblecomparedwith the distancebetween
them;otherwisethedistance[R] is not preciselydelined,nor is it immediatelyclearat what points
and in what directionsthe forcesact. For extendedbodies,we must imagineeachbody divided
into piecesor elements,smallcomparedwith the distancesbetweenthe bodies,and computethe
forces on eachof the elementsof one body due to eachof the elementsof the other bodies."
K. R. Symon,Mechanics(Reading:Addison-Weslcy,1960),p. 257.
62. ls this not the presenceof Bacon?". . . in nature nothing really existsbesidesindividual
bodies,performingpure individual actsaccordingto a fixed law. . ." New Organon,Spedding,
Ellis, and Heath,trans., F. H. Anderson,ed., (Indianapolis:Bobbs-Mcrrill, 1960),p. 122.
63. How small should the ratio of their sizesto their separationbe?As small as requiredto
yield the accuracyneededfor the calculationin question.The law seemsto be intimatelyrelated
to practice.
64. MathematicalPrinciples,Bk. I, Prop. 75, p. 197.
65. A completelypassivebeingcould not act on our senseorgans(or on any sensinginstrument)
and would thus not be perceptible.
The mental productionof algebraicmagnitudes,and the implicationsof their employmentin
the scienceof naturearetreatedby JacobKlein, GreekMathematicalThoughtand the Origin of
Algebra,E. Brann,trans.,(Cambridge:MIT Press,1968),a work asimportant asit is difficult to
understand.The meaningof 'law'-the fundamentalintelligiblein the modernunderstandingof
nature-remains to be fully explicated.