Nagel Wholes, Sums, And Organic Unities

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PHILOSOPHICAL

STUDIES

VOLUME ItI

Edited by HER BERT FEI GL and WI LF RI D SELLARS with the advice and

ass/stan ce of PAUL RIEEHL, JOHN HOSPERS, MAY BRODBECK

C o n te n ts Fe br ua ry 1.952 NUMBER 2

Wholes, Sums, and Organic Unitiesby Ernest Nagel, COLUMSIAUNIVERSITY

Wholes, Sums, and Organic Unities

by ERNEST NAGELCOLUMBIA UNIVERSITY

IN CONNECTION w ith th e subje ct o f red uc tio n an d em erge nce , it is hel pfu l

to discuss a familiar not ion th at is frequ ently associated wi th these t hem es.

According to this notion there occurs in nature an important type of indi-

vidual wholes (which may be physical, biological, psychological, or social)

that are not s imply "aggregates" of inde pen den t mem bers , but are "organic

uni t ies"; and such wholes are of ten character ized by the famil iar dic tumthat they possess an organization which makes each of them "more than

th e s u m of its parts ." Examples of wholes that are "organic," and which

allegedly also i l lustrate this dictum , can be ci ted fro m m an y fields o f in-

quiry. Since such alleged facts are som etimes taken as indications o f l imits

to t he possibil ity of reduction and to th e scope of the m ethods of the physi-

cal sciences, i t is instructive to consider them with some care. And in the

course of examining the m we shall be com pel led to recognize dist inctions

tha t wi ll be useful in t he sequel .

The f i rs t point to note , however , i s that words l ike "whole" and "sum"as com m only employed are unusual ly vague, ambiguous, and even m eta-

phorical; and until their senses are clarified, it is frequently impossible to

assess the worth of s ta tements containing them. Let us make this evident

by a n exam ple. A quadrilateral encloses an area, and ei ther on e o f i ts diam-

17



18 PHILOSOPH ICAL STUDIES

eters divides i t into two partial areas whose sum is equal to the area of the

whole f igure . In this and many analogous contexts the s ta tement "The

whole is equal to the sum of i ts parts" is usually said to be not only true,

bu t necessari ly true, so tha t i ts denial is com m on ly regarded as self-contra-dic tory. On the other hand, some wri ters have mainta ined, on comparing

the tas te of sugar of lead wi th the tas te of i ts chemical components , that

in this case the whole is not equal to the sum of i ts parts. Now this asser-

t ion is intended to supply some information about the matters discussed;

and i t ca nno t be re jected wi th out fur ther ado as s imply a logical absurdi ty.

I t is dear , therefore , that in this la t ter conte xt th e words "whole , . . . par t,"

and "sum " (and perhaps "equa l") are being employed in senses di f ferent

f rom those associa ted wi th them in the previous context . We must there-

fore assume the task of dis tinguishing between a n um ber of senses of thesewords th at appear to play a role in various inquiries.

1. Wholes and par ts . Th e words "w hole" and "par t" are normal ly used

for correlative distinctions, so that x is said to be a whole in relation to

som ething y which is a co m pon ent or par t of x in some sense or other. I t

wil l be convenient, therefore, to have before us a brief l ist of certain fa-

miliar "kinds" of wholes and corresponding parts.

a . T he word " who le" is used to refer to so mething wi th a spatia l exten-

sion, and an yth ing is the n called a "p art" of such a who le wh ich is spatially

included in i t . However, there are several special senses of "whole" and

"par t" which fa l l under this head. In the f i rs t place, they may refer to

specifically spatial properties, so th at th e w ho le is the n som e leng th, area,

or volum e wh ich con tains as parts lengths, areas, or volumes. In this sense,

nei ther wholes nor par ts need be spatia l ly continuousmthus, the Uni ted

States and its territorial possessions are no t a spatially con tinu ou s wh ole ,

which contains as one of i ts spatial parts the desert regions which are also

no t spatial ly continuous. In th e second place, "w hole" m ay refer to a non-

spatial p roperty or state of a spatial ly exten ded thing , a nd "part" designatesan identical proper ty of some spatia l par t of the thing. Thus, the e lectr ic

charge on a body is said to have for its parts the electric charges on spatial

par ts of the body. In the thi rd place, though sometimes only such spatia l

properties are cou nte d as parts of a spatial wh ole which have th e sam e spa-

tial dimensions as the latter , at other times the usage is more l iberal . Thus,

the surface of a sphere is f requently sa id to be a par t of th e sphere , even

if on oth er occasions only volumes in th e sph ere 's interior are so designated.

b. The word "whole" refers to some temporal per iod, whose par ts are

tem poral intervals in i t . As in th e case of spatial wholes and parts, tem poralones need not be continuous.

c. The word "whole" refers to any class, set, or aggregate of elements,

and "par t " m ay the n designate e i ther any proper subclass of t he init ial se t

or any element in the set . Thus, by a par t of the whole consis t ing of a l l

the books pr inted in the Uni ted States dur ing a given year may be under-



WHO LES, SUMS, AND ORGA NI C UNI TI ES 19

stood either al l the novels printed that year, or some particular copy of a

novel.

d . The word "whole" sometimes refers to a proper ty of an object or

process, and "part" to some analogous property which stands to the f irstin certain specified relations. Th us, a force in physics is co m m on ly said to

have for i ts parts or components other forces into which the f irst can be

analyzed according to a familiar rule. Similarly, the physical brightness of

a surface i l lum inated by tw o sources of l ight is som etimes said to have for

one of i ts parts the brightness associated with one of the sources. In the

prese nt sense of the words, a part is no t a spatial part of th e w hole.

e . T he word "who le" may refer to a pattern of re la t ions betwee n cer ta in

specified kinds of objects or events, the patte rn b eing capable of embo di-

m en t on various occasions and with various modifications. How ever, "pa rt"

may then designate di f ferent things in di f ferent contexts . I t may refer to

any one of the e lements which are re la ted in that pattern on some occa-

s ion of i ts em bod ime nt. Thus, i f a melo dy (say "Auld L ang Sy ne") is such

a whole , one of i ts par ts i s then the f i rs t tone that i s sounded when the

m elod y is sung on a particular date. O r i t m ay refer to a class of elem ents

wh ich occupy correspond ing positions in th e p attern in som e specified

mo de of it s emb odim ent . Thus , one of the par ts o f th e melody wi ll then

be t he c lass of fi rs t notes wh en "Au ld Lang Syn e" is sung in th e key of G

minor . Or the word "p ar t" m ay re fer to a subord ina te pa t te rn in the to tal

one. In this ease, a par t of the m elody will be th e pattern of tones tha t

occurs in its first four bars.

f . The word "whole" may refer to a process , one of i ts par ts being an-

other process that i s some discr iminated phase of the more inclusive one.

Thu s, th e process of swallowing is part o f th e process of eating.

g. T he word "who le" may refer to any concrete object , and "par t" to

any of i ts properties. In this sense, the character of being cylindrical in

shape or being m alleable is a part o f a given piece of copper w ire.h. Final ly , the w ord " who le" is of ten used to refer to an y system whose

spatial parts stand to each other in various relations of dynamical depend-

ence. Many of the so-called organic unities appear to be systems of this

type. However, in the present sense of "whole" a variety of things are

customarily designated as i ts parts. Thus, a system consisting of a mixture

of two gases inside a con tainer is frequently, tho ug h no t always in t he sam e

conte xt, said to have for i ts parts on e or mo re of th e following: i ts spatial ly

extend ed consti tuents , such as the two gases and the container ; the proper-

ties or states o f th e system or of i ts spatial parts, such as the mass of th e sys-te m or th e specific heats of on e o f the gases; th e processes wh ich t he system

undergoes in reaching or mainta ining thermodynamieal equi l ibr ium; and

th e spatial or dynam ical organization to wh ich i ts spatial parts are subject.

This l i st of senses of "whdle" and "par t ," thou gh by no means com plete ,

will suffice to indicate the ambiguity of these words. But what is more



20 PHILOSOPH ICAL STUDIES

important , i t a lso suggests that s ince the w ord " sum " is used in a nu m be r

of contexts in which these words occur, i t suffers from an analogous am-

biguity. Le t us therefo re exam ine several of i ts typical senses.

2. Senses of "sum." We sha l l no t inqui re whether the word "sum" ac-tual ly is employed in connection wi th each of the senses of "whole" and

"part" that have been distinguished, and if so just what meaning is to be

associated w ith it. In po int of fact, it is no t easy to specify a clear sense for

the w ord in m any contexts in w hich people do use it . W e shal l accordingly

confine ourselves to noting only a small number of the well-established

uses of "sum," and to suggesting interpretations for i t in a few contexts in

wh ich i ts me anin g is unclear and i ts use misleading.

a. I t is hardly surprising tha t th e m ost carefully defined uses of "su m "

and "addi t ion" occur in mathematics and formal logic . But even in these

contexts the word has a var ie ty of specia l meanings, depending on what

type of m ath~lnatical and logical "objects" are being added . Thu s, there is

a famili~I op eration o f add ition fo r the natural integers; and ther e are also

identically named but really distinct operations for ratios, real numbers,

complex numbers, matrices, classes, relations, and other "enti ties." I t is

not a l together evident why al l these operations have the common name of

"addi t ion," tho ugh there are a t leas t cer ta in formal analogies betw een ma ny

of th em --fo r example, mo st of th em are com mu tative and associa tive. How-

ever , there are some important exceptions to the general rule impl ic i t in

this example, for the addi t ion of ordered sets i s not uniformly commuta-

tive, though i t is associative. On the other hand, the sum of two enti ties

is invar iably some unique enti ty which is of the same type as the sum-

mands-- thus , the sum of two integers i s an integer , of two matr ices a

matr ix , and so on. Moreover , though the word "par t" i s not a lways de-

f ined or used in connection wi th mathematical "objects ," whenever both

i t and " sum " a re employed they a re so used tha t the s ta tem ent "T he whole

is equal to th e su m o f i ts parts" is an analytic or necessary truth .However , it i s easy to construct an apparent eounter instance to this las t

claim. Let K* be the ordered set of the integers, ordered in the following

manner : f i rs t the odd integers in order of increasing magni tude, and then

the even integers in that order . K* may then be represented by this nota-

tion : (1,3,5 . . . . 2,4,6 . . .) . Ne xt let K1 be the class of odd integers and

K2 the class of even ones, neither class being an ordered set. Now let K

be the class-sum of K1 and K2, so that K contains all the integers as mem-

bers; K also is not an ordered class. But the membership of K is the same

:as that of K*, al though quite deafly K and K* are not identical . Accord-ingly, so i t migh t be argued, in this case th e whole (nam ely K* ) is not

,equal to the sum (i.e., K) of its parts.

This example is instructive on three counts. I t shows that i t is possible

to def ine in a precise m ann er the words "whole ," "par t ," a nd "sum " so

t h a t "The whole is unequal to the sum of i ts parts" is not only not logi-



WH OLES, SUMS, AND ORG ANIC UN ITIES 21

cally absurd, b ut is in fact logically true. T he re is there fore no a priori

reason for dismissing such statements as inevitable nonsense; and the real

issue is to determ ine, wh en such an assertion is made, in w hat sense if any

the crucial words in i t are being used in t he given context. But the exam plealso shows that thou gh such a sentence m ay be true on one specif ied usage

of "part" and "sum," i t may be possible to assign other senses to these

words so that the whole is equal to the sum of i ts parts in this redefined

sense of the words. Indeed, i t is not standard usage in mathematics to call

ei ther K1 or K2 a part o f K*. O n th e contrary, i t is custom ary to cou nt as

a part of K* only an ordered segment. Thus~ let KI* be the ordered set

of odd integers arranged according to increasing magnitude, and K2* the

corresponding ordered set of even integers. K~* and K2* are then parts of

K*. (K* has other parts as well , for example, the ordered segments indi-

cate d by th e following: (1,3,5,7), (9,11 . . . 2,4), and (6,8 . . .).) N ow

form the ordered sum of KI* and K2*. But this sum yields the ordered

set K*, so that in the specified senses of "part" and "sum" the whole is

equal to th e sum of its parts. I t is thus clear tha t w hen a given system has

a special type of organization or structure, a useful definition of "add ition,"

if such can be given, must take into account that mode of organization.

There are any number of operations that could be selected for the label

"summation,"bu t not a ll of them are re levant or appropr ia te for advancing

a given domain of inquiry.

Finally, the example suggests that though a system has a distinctive

structure, it is not in principle impossible to specify tha t structure in term s

of re la t ions between i ts e lem entary consti tuents , and moreover in such a

ma nner tha t the s tructure can be correctly character ized as a "sum " whose

"parts" are themselves specified in terms of those elements and relations.

As we shall see, ma ny studen ts deny, or appear to deny, this possibil ity in

con nect ion with certain kinds of organized systems (such as l iving things).

Th e present example therefore shows that thou gh w e may no t be able as ama tter of fact to analyze cer ta in highly complex "dynam ic" (or "o rganic")

uni t ies in terms of some given theory concerning thei r ul t imate consti tu-

ents, such inabil i ty cannot be established as a matter of inherent logicalnecessity.

b. I f we now t urn to the posit ive sciences , we f ind tha t here too there

are a large num ber of wel l-defined operations cal led "addi t ion." T he major

distinction that needs to be drawn is between scalar and vector sums. Let

us consider each in turn. Examples of the former are the addi t ion of the

num eros ity of groups of things, of spatial properties ( length, area, andvolume), of temporal periods, of weights, of electrical resistance, electric

charge, and thermal capacity. They i l lustrate the f irst three senses of

"whole" and "par t" which we dis t inguished above; and in each of them

(and in m any o ther cases tha t could be m ent ion ed) "sum" is so specified

tha t the whole is the sum of appropria te ly chosen parts.



22 PHILOSOPHICAL STUDIES

O n the o ther hand , there a re many magni tudes for which no opera t ion

of addition is defined, or seems capable of being defined in any useful

manner, such as density or elastici ty; most of these cases fal l under the

las t four of the above dis t inctions concerning "whole" and "par t ." More-over, there are some properties for which addition is specified only under

highly specialized circumstances; for example, the sum of the brightness

of two sources of l ight i s def ined only when the l ight emitted is mono-

chromatic. I t makes no sense, therefore, to say that the density (or the

shape) of a body is, or is not, the sum of the densities (or shapes) of i ts

parts, simply because there are neither explici tly formulated rules nor

ascertainable habits of procedure which associate a usage with the word

"sum" in such a context .

The addition of vector properties, such as forces, velocities, and accelera-

tions, conforms to the familiar rule of parallelogram composition. Thus, if

a body is acted on by a force of 3 poundals in a direction due north, and

also by a force of 4 poundals in a direction due east, the body will behave

as if i t were acted on by a single force of 5 poundals in a northeasterly

direction. This single force is said to be the "sum" or "resultant" of the

other two forces, which are called i ts "components"; and conversely, any

force can be analyzed as the sum of an arbi trary number of components .

This sense of "sum " is com m only associa ted wi th the four th of the above

dis tinctions concerning "whole" and "par t"; and i t i s evident that here th e

sense of "sum " is qui te di f ferent f rom the sense of the w ord in such con-

texts as "the sum of two lengths."

It has been argued by Bertrand Russell that a force cannot r ightly be

said to be the su m o f its componen ts . Thus, h e declared:

Let the re be three par ticles A, B, C. W e ma y say that B and C bot hcause accelerations in A, and we compound these accelerations by theparallelogram law. But this composition is not truly addition, for the

components are not par ts of the resul tant . The resul tant i s a new term,as s imple as thei r components , and not by any means thei r sum. Thus theeffects a t tr ibuted to B and C are never produced, bu t a thi rd ter m dif ferentfrom either is produced. This, we may say, is produced by B and C to-gether, taken as a whole. Bu t th e effect wh ich the y produ ce as a wholecan only be discovered by supposing each to produce a separate effect: i fthis were not supposed, i t would be impossible to obtain the two accelera-t ions whose resul tant is the actual acceleration. Th us we seem to reach anantinomy: the whole has no effect except what resul ts f rom the effects ofthe parts, but the effects of the parts are nonexistent.1

However , a l l that this argument shows is that by the component of a

force (or of an acceleration) w e do n ot m ean an ything l ike wh at we under-

s tand by a component or par t o f a l ength- - the components of forces a re

n o t spatial parts of forces. I t does not establish the claim that the addition

of forces "is not truly addi t ion"--unless , indeed, the word "addi t ion" is



WHO LES, SUMS, AND ORGANI C UNI TI ES 23

being used so restr ictively that no operation is to be so designated which

does no t involve a juxtaposition of spatial (or possibly tempo ral) parts of

the wh ole sa id to be thei r sum. But in this la t ter event m any oth er opera-

tions that are called "addition" in physics, such as the addition of electri-cal capacities, would also have to receive different labels. Moreover, no

antino my arises f rom the supposi t ion that , on the one hand, the effect

which each component force would produce were i t to act a lone does not

exis t , whi le on the other hand the actual ef fect produced by the joint

action of the components is the resultant of their partial effects. For the

supposition simply expresses what is the case, in a language conforming to

the an teceden t definition of the addi t ion and resolution of forces .

The issue raised by Russell is thus terminological at best. His objection

is nevertheless instructive. For i t calls needed attention to the fact that

wh en t he m atte r is viewed abstractly , the "su m" of a given set of e lements

is simply an element that is uniquely dete rmined by some func t ion ( in

the mathematical sense) of the given set . This function may be ass igned

a relatively simple and familiar form in certain cases, and a m ore co mp lex

and s trange form in others ; and in any event, the question whether such

a function is to be introduced into a given domain of inquiry, and if so

what special form is to be assigned to i t , cannot be settled a priori . But

the heart of the matter is that when such a function is specified, and if a

set of elements satisfy whatever conditions are prescribed by the function,

i t becomes possible to deduce from these premises a class of statements

about some s tructural complex of those e lements .2

c . W e mu st now consider a use of "su m" that is associa ted wi th t he

f i f th sense of "whole" and "part" distinguished above--a use that is also

frequently associa ted wi th the dic tum that the w hole is more than, or a t

any rate not merely, the sum of i ts parts. Let us assume that the following

sta tem ent is typical o f such usage: "A l though a melo dy ma y be p roduced

by sounding a series of individual tones on a piano, the melody is not thesum of i ts individual notes." The obvious question that needs to be asked

is: " In wh at sense is 'sum' being em ployed here?" t t is evident that the

sta tement can be informative only i f there is such a thing as the sum of

the individual tones of melody. For the s ta tement can be es tabl ished as

true or false only if i t is possible to compare such a sum with the whole

that is the m elody.

However , most people who are incl ined to asser t such a s ta tement do

no t specify wh at t hat sum is supposed to be; and the re is therefore a basis

for the supposi t ion that they ei ther are not c lear about what they mean,or do not mean anything whatever . In the la t ter case the most char i table

view that can be taken of such pronouncements i s to regard them as

simply misleading expressions of the possibly valid claim that the notion

of summ ation is inapplicable to the consti tuent tones of melodies . On the

other hand, some wri ters apparently unders tand by "sum" in this context




th e unordered class of individual tones; and wh at the y are the refore assert-

ing is that this class is not the melody. But this is hardly news, though

conceivably there ma y have been some persons who bel ieved otherwise.

In any event, there appears to b e no mean ing other than this on e which isassociated with any regulari ty with the phrase "sum of tones" or similar

phrases. Accordingly, if the w ord " sum " is used in this sense in contexts

in w hich th e w ord " who le" refers , to a pattern or conf iguration formed by

elements standing to each other in certain relations, i t is perfectly true

though tr ivial to say that the whole is more than the sum of i ts parts.

As has a lready been noted, however , this fact does no t preclude the

possibility of analyzing such wholes into a set of elements related to one

another in definite ways; nor does i t exclude the possibil i ty of assigning

a di f ferent sense to " sum " so tha t a m elody might the n be construed as a

sum of appropriately selected parts. I t is evident that at least a partial

analysis of a melod y is effected w he n i t is represent ed in t he custom ary

musical notation; and the analysis could obviously be ma de m ore com -

plete an d explicit, and even expressed with formal precision,z

But i t is sometimes maintained in this connection that i t is a funda-

men tal mistake to regard the consti tu ent tones of a melody as indepe nde nt

par ts , out of which the melody can be reconsti tuted. On the contrary, i t

has been argued that what we "exper ience a t each place in the melody is

a par t which is i tse lf determ ined by the character of the whole . . . Th e

flesh and blood of a tone depends from the start upon i ts role in the

melody: a b as leading tone to c is something radically different from the

b as tonic.TM An d as we shall s ee , similar views have bee n advanced in

connection wi th other cases and types of Gestalts and "organic" wholes .

No w i t ma y be qui te true that the effect produced by a given tone de-

pends on i ts position in a con tex t of oth er tones, just as the effect prod uced

by a given pressure upon a body is in general contingent upon what other

pressures are operative. But this supposed fact does not imply that ame lody canno t t ightly be viewed as a re la tional complex whose com po-

nen t tones are identif iable indep ende ntly of their occurrence in that com-

plex. For if the implication did hold, i t would be impossible to describe

how a melody is consti tuted out of individual tones, and therefore impos-

sible to prescribe how it is to be played. Indeed, i t would then be self-

contradic tory to say that "a b as leading to c is som ethin g radically different

f rom the b as ton ic ." For the nam e "b" in the expression "b as leading to

c" could then not re fe r to the same tone to which the name "b" refers

in th e expression "b as tonic"; and the presumable intent of the s ta tementcould then not be expressed. In shor t , the fact that in connection wi th

wholes that are patterns or Gestalts of occurrences the word "sum " is e i ther

undef ined, or def ined in such a way that the whole is unequal to the sum

of i ts parts, consti tutes no inherently insuperable obstacle to analyzing

such wholes into elements standing to each other in specified relations.



W H O L E S , S UM S , A N D O R G A N I C U N I T I E S 25

d. W e mu st f inally examine the use of "sum " in connect ion w ith wholes

that are organized systems of dynamically interrelated parts. Let us assume

as typical of such usage the s ta tement "Although the mass of a body is

equal to the sum of the masses of i ts spatial parts, a body also has prop-erties which are not the sums of properties possessed by i ts parts." The

comments tha t have jus t been made about " sum" in connec t ion wi th

pat terns of occurrences such as melodies can be extended to the present

context of usage of the word; and we shal l not repeat them. In the present

instance, however, an additional interpretation of "sum" can be suggested

which may put in to c learer l ight the content of such s ta tements as the

above.

\Vhen the behavior of a machine l ike a c lock is sometimes sa id to be

the sum of the behaviors of i ts spatial parts, what is the presumptive con-

tent of the asser t ion? I t i s reasonable to assume that the word "sum" does

not here s ignify an nnordered c lass of e lemen ts--for nei ther the c lock nor

its behavior is such a class. It is therefore plausible to construe the assertion

as mainta ining that f rom the theory of mechanics , coupled with sui table

informat ion about the actual arrangements of the par ts of the machine, i t

is possible to d educe s ta tements abo ut the conseq uent propert ies and be-

haviors of the entire system. Accordingly, i t seems also plausible to con-

strue in a similar fashion statements such as that of J . S. Mill : "The differ-ent act ions of a chemical com pou nd wil l never be fou nd to be th e sums of

actions of i ts separate parts. '5 More explicit ly, this statement can be under-

s tood to asser t that f rom some assumed theory concerning the const i tuents

of chemical compounds, even when i t i s conjoined with appropria te data

on the organizat ion of these const i tuents within the compounds, i t i s not

in fact possible to deduce s ta tements about many of the proper t ies of

these compounds .

I f we adopt this suggestion, we obtain an interpreta t ion for "sum" that

is particularly appropriate for the use of the word in contexts in which thewholes under discussion are organized systems of interdependent parts.

Let T be a theory that is in general able to explain the occurrence and

mo des of interde pen den ce of a set of properties P1, P2 • • . Pk. M or e spe-

cifically, suppose i t is known that when one or more individuals belonging

to a set K of individuals occur in an environment E1 and stand to each

other in some relation belonging to a class of relations RI, the theory T

can explain the behavior of such a system with respect to i ts manifesting

some or a l l of the proper t ies P. Now assume that some or a l l of the indi-

viduals belonging to K form a relational complex R~ not belonging to R1in an environ men t E2 which m ay be dif ferent f rom El , and that th e system

exhibits cer ta in m odes of behavior which are formu lated in a se t of laws L.

Two cases may then be dis t inguished: f rom T, together with s ta tements

concerning the organization of the individuals in R2, i t is possible to de-

duce the laws L; or secondly, not all the laws L can be so deduced. In the




first ease, the behavior of the system R2 may be said to be the "sum" of

the behaviors of i ts component individuals; in the second case, the be-

havior of R2 is not such a sum. I t is evident that in a currently accepted

sense of "reducible," the conditions for the reducibility of L to T are satis-fied in the first case; in the second case, however, although one of these

conditio ns may b e satisfied, the othe r is not .

I f this interpreta t ion of "sum" is adopted for the indicated contexts of

i ts usage, i t follows that the distinction between wholes which are sums

of their parts and those which are not is relative to some assumed theory T

in terms of which the analysis of a system is undertaken. Thus, as we have

seen, the kinetic theory of mat ter as developed dur ing the ninete enth cen-

tury was able to explain certain thermal properties of gases, including cer-

tain relations between the specific heats of gases. However, that theory was

unable to account for these relations between specific heats when the state

of aggregation of molecules is tha t of a solid rather t han a gas. On the other

hand, modern quantum theory is capable of explaining the facts concern-

ing t h e specific heats of solids, and presum ably also all othe r the rm al prop-

erties of solids. Accordingly, although relative to classical kinetic theory

the thermal properties of solids are not sums of the properties of their

parts, relative to quantum theory those properties are such sums.

3. Organic wholes. W e mu st n ow briefly consider wh at is the distinctive

feature of those systems which are 'commonly said to be "organic unities"

and which exhibit a mode of organization that is often claimed to be in-

capable of analysis in terms of an "additive point of view." However, al-

though l iving bodies are the most frequently ci ted examples of organic

wholes, we shall no t b e now conce rned specifically with such systems. For

it is generally admitted that living bodies constitute only a special class of

systems possessing a struc ture o f internally related parts; and it will be

an advantage to ignore for the present special issues connected with the

analysis of vital phe nom ena .Organic or "functional" wholes have been defined as systems "the be-

havior of which is not determ ined by t hat of thei r individual e lements , b ut

where the part-processes are themselves determined by the intr insic nature

of the w hole. '6 W ha t is distinctive of such systems, therefore, is that their

parts do not act, and do not possess characteristics, independently of one

another. On the contrary, their parts are supposed to be so related that

any alteration in one of them causes a change in all the othe r parts. 7 In

consequence, functional wholes are also said to be systems which cannot

be bui l t up out of e lements by combining these la t ter seriatim wi thou tproducing changes in al l those elements. Moreover, such wholes cannot

have any par t removed, wi thout a l ter ing both that par t and the remaining

parts of the system,s Accordingly, it is often claimed that a functional

wh ole cann ot be properly analyzed from an "additive poin t of view"; tha t is ,

the character is t ic modes of functioning of i ts conti tuents must be s tudied



WHO LES, SUMS, AND ORGA NIC UNITIES 27

in situ, and the structure of activities of the whole cannot be inferred from

properties displayed by i ts consti tuents in isolation from the whole.

A purely physical example of such functional wholes has been made

familiar by Koehler. Consider a well-insulated electric conductor of arbi-trary shape, for example one having the form of an el l ipsoid; and assume

that electric charges are brought to it successively. The charges will im-

mediately distr ibute themselves over the surface of the conductor in such

a way that the electric potential wil l be the same throughout the surface.

However, the density of the charge (i .e. , the quanti ty of charge per unit

surface) wil l not in general be uniform at al l points of the surface. Thus,

in the case of the el l ipsoidal conductor, the density of the charge will be

greatest at t he points of greatest curvature and will be smallest at the points

of least curvature2 In brief , the distr ibution of the charges will exhibit a

characteristic pattern or organization--a pattern which depends on the

shape of the conductor, but is independent of the special materials of i ts

construction or of the total quanti ty of charge placed upon i t.

But i t is not possible to build up this pattern of distr ibution bit by bit,

for example by bringing charges first to on e part of the cond ucto r and th en

to another so as to have the pattern emerge only after al l the charges are

placed on the conductor. For when a charge is placed on one portion of

the surface, the charge will not remain there but wil l distr ibute i tself in

the ma nne r indicated; and in consequence, the charge-density a t one point

is not independent of the densities at al l other points. Similarly, i t is not

possible to remove some part of the charge from one portion of the surface

without altering the charge-densities at all other points. Accordingly, al-

though the total charge on a conductor is the sum of separable partial

charges, the configuration of charge-densities cannot be regarded as com-

posed out of independent parts. Koehler thus declares:

The natural structure assumed by the total charge is not described if

one says: at this point the charge-density is this much 'and' at that pointthe densi ty is that much, e tc . ; but one might a t tempt a descr iption bysaying: the density is so much at this point, so much at that point, al lmutu al ly interdepend ent, and such that the occurrence of a cer tain densi tyat o ne po int d etermines t he densities at al l other p oint s? °

Many other examples--physical, chemical, biological, and psychologi-

cal -could be c i ted which have the same intent as this one. There is

therefore no doubt that there are many systems whose consti tuent par ts

and processes are "internally" related, in the sense that these consti tuents

stand to each other in relations of mutual causal interdependence. Indeed,some writers have found i t diff icult to distinguish sharply between sys-

tems which are of this sort and systems which al legedly are not; and they

have argued that al l systems whatever ought to be characterized as whole

whic h are "organic" or "func tiona l" in so me degree or other. 11 In poin t

of fact, many who claim that there is a fundamental difference between



28 PHILOSOPHICA L STUDIES

functional and nonfunctional (or "summative") wholes , taci t ly admit that

the distinction is based on practical decisions concerning what causal in-

fluences may be ignored for certain purposes. Thus, Koehler ci tes as an

example of a "summative" whole a system of three s tones , one each inAfrica, Austral ia, and the United States. The system is held to be a sum-

mative grouping of i ts parts, because displacement of one stone has no

effect on the others or on thei r m utua l re la tions .TM However , i f current

theories of physics are accepted, such a displacement is not without some

effects on the othe r s tones , even if the effects are so min ute that they can-

not be detected wi th present exper imental techniques and can therefore

be practically ignored.

Again, Koehler regards the tota l charge on a co nduc tor as a su mm ative

whole of independent par ts , though i t i s not a t a l l evident that the e lec-tronic constituents of the charge undergo no a l terat ions w hen parts of t he

latter are remo ved from it. Accordingly, al thou gh th e occurren ce of systems

possessing distinctive structures of interdependent parts is undeniable, no

general cri terion has yet been proposed wh ich m akes i t possible to id entif y

in an absolute way systems which are "genuinely functional" as dis t inct

f rom systems which are "me rely summ ative. ' la

Moreover, i t is essential to distinguish in this connection between the

question whe ther a given system can be over t ly constructed in a piecem eal

fashion by a ser ia t im juxtaposit ion of parts, a nd the question w heth er th e

system can be analyzed in terms of a theory concerning i ts assumed con-

st i tuents and thei r interre la t ions . Th ere undo ubted ly are wholes for wh ich

th e an swer to th e first ques tion is afffirmative--for exam ple, a clock, a s alt

crystal , or a molecule of water; and there are wholes for which the answer

is negativ e--for example, the solar system, a carbo n atom , o r a l iving bod y.

However , this di f ference betw een systems does no t correspond to th e in-

tended dis t inction between functional and summative wholes; and our

inabil ity to co nstruc t effectively a system ou t of i ts parts, w hich in som e

cases may only be a consequence of temporary technological l imitations,

cannot be taken as evidence for deciding the second of the above two

questions.

But let us turn to this second question, for i t raises what appears to be

the fundamental i ssue in the present context . That i ssue is whether the

analysis of "organic un ities" necessarily involves th e adop tion of irreducible

laws for such systems, and whether their mode of organization precludes

th e possibili ty of analyzing th em from th e so-called additive p oint of view.

T he main di ff iculty in this conn ection is that of ascer ta ining in wh at wayan "additive" analysis differs from one which is not. The contrast seems

to hinge on the c la im that the par ts of a functional whole do not act

independently of one another , so that any laws which may hold for such

parts wh en they are not mem bers of a functional whole cann ot be assumed

to hold for them when they actual ly are members . An "addi t ive" analysis



WHO LES, SUMS, AND ORGAN I C UNI TI ES 29

there fore appears to be on e whic h accounts for the properties of a system

in terms of assumptions about i ts consti tuents , where these assumptions

are not form ulated w ith specific referenc e to th e characteristics of t he

consti tuents as elements in the system. And a "nonadditive" analysis seemsto be one which formulates the characteristics of a system in terms of ie-

la tions between cer ta in of its par ts as functioning elem ents in th e system.

However, i f this is indeed the distinction between these al legedly differ-

en t mo des of analysis, th e difference is no t o ne of fund am ent al principle.

W e have already no ted tha t i t does not seem possible to distinguish sharply

betw een systems that are sa id to be "organic uni ties" an d those w hich are

not. A ccordingly, since even th e parts of sum ma tive wholes stand in rela-

tions of causal interdependence, an additive analysis of such wholes must

include special assum ption s abo ut the a ctual organization of parts in those

wholes when i t a t tempts to apply some fundamenta l theory to them. T here

certainly are ma ny physical systems, such as th e solar system, a carb on

atom, or a calcium fluoride crystal , which despite their complex form of

organization lend themselves to an "additive" analysis; but it is equally

certain tha t c urre nt explanations of such systems in terms of theories about

thei r consti tuent par ts cannot avoid supplementing these theor ies wi th

sta tements about the specia l c i rcumstances under which the consti tuents

occur as e lements in th e systems. In any event, the m ere fact that the parts

of a system stand in relations of causal interdependence does not exclude

th e possibili ty of an additive analysis of th e system.

T he dis t inction betwe en addi t ive and nonaddi t ive analysis is sometimes

supported by the contras t com mo nly drawn betw een th e par tic le physics

of classical mechanics and the f ield approach of electrodynamics. I t wil l

therefo re be instructive to dw ell for a m om en t ,on this contrast. Acco rding

to Newtonian mechanics , the accelerat ion induced in a par t ic le by the

action of other bodies is the vector-sum of the accelerations which would

be produced by each of these bodies were they acting singly; and the as-sumption under lying this pr inciple is that the force exer ted by one such

body is independent of the force exer ted by any other . In consequence, a

mechanical system such as the solar system can be analyzed additively. In

order to account for the characteristic behavior of the solar system as a

whole, we need to know only the for6~ (as a function of the distance)

which each body in the system exerts separately on the other bodies.

Bu t in electrodynam ics th e si tuation is different. For th e action of an

electrically charged bod y on an oth er dep ends no t o nly on their distances,

but also on their relative motions. Moreover, the effect of a change inmotion is not propagated instantaneously, but with a f ini te velocity. Ac-

cordingly, the force on a charged bod y due to the presence of other such

bodies is not dete rmin ed by the posit ions and velocit ies of the la tter, bu t

by the condi t ions of the e lectromagnetic "f ie ld" in the vic ini ty of the

former. In consequence, since such a f ield cannot be regarded as a "sum"



30 PHILO SOPH ICAL STUDIES

of "partial" fields, each due to a distinct charged particle, an electromag-

netic system is com mo nly said to be incapable of an additive analysis. "T he

field can be treated adequately o nly as a un it," so i t is claimed, " no t as the

sum total of the contributions of individual point charges.TMTwo br ief comments must be made on this contras t . In the f i rs t place,

the notion of "field" (as used in electromagnetic theory) undoubtedly rep-

resents a mathematical technique for analyzing phenomena which is dif-

ferent in many important respects f rom the mathematics employed in

particle mechanics. The latter operates with discrete sets of state variables,

so that the state of a system is specified by a finite number of coordinates;

the former requires that the values of each of its state-variables are speci-

fied for each point of a mathematically continuous space. And there are

further corresponding differences in the kinds of differential equations,

the variables that enter into them, and the l imits between which mathe-

matical integrations are performed.

But in the second place, though i t is true that the electromagnetic f ield

associated with a set of charged particles is not a "sum" of partial fields

associated with each particle separately, it is also true that the field is

uniquely determined (i.e., the values of each state-variable for each point

of space are unequivocally fixed) by the set of charges, their velocities,

and the ini t ia l and boundary condi t ions under which they occur . Indeed,

there is a technique employed within f ield theory, in the l ight of which

the electromagnetic f ield is simply an intermediary device for formulating

th e effects of electrically charged particles u po n ot he r such particles. 1"~

Accordingly, though i t may be convenient to treat an electromagnetic f ield

as a "unit," this does not signify that the properties of the f ield cannot be

analyzed in terms of assumptions concerning i ts consti tuents. And though

the field may not be a "sum" of partial f ields ' in any customary sense, an

electromagnetic system is a "sum" in the special sense of the word pro-

posed previously--namely, there is a theory about the consti tuents of thesesystems such that the relevant laws of the system can be deduced from

the theory. In point of fact, i f we take a f inal glance at the functional

whole i l lustrated by the charges on the insulated conductor, the law which

formulates the distr ibution of charge-densities can be deduced from as-

sum ption s concerning the behavior of charged particles. 16

The upshot of this discussion of organic unities is that the question

whether they can be analyzed f rom the addi t ive Point of view does not

possess a general answer. Some functional wholes certainly can be analyzed

in tha t man ner, while in t he case of others (for example, living organisms)no fully satisfactory analysis of that type has yet been achieved. Accord-

ingly, the mere fact that a system is a structure of dynamically interrelated

parts does not suffice, by itself, to prove that the laws of such a system

cannot be reduced to some theory developed initial ly for certain assumed

consti tuents of the system. This conclusion may be meager; but i t does



W H O L E S , S U M S, A N D O R G A N I C U N I T I E S 31

s h o w t h a t t h e i s s ue u n d e r d i s c u s si o n c a n n o t b e s e t t l e d , as s o m u c h o f e x-

t a n t l i t e r a t u r e o n i t a s s u m e s , i n a w h o l e s a l e a n d a p r i o r i f a s h i o n .

N O T E SBertrand Russell, The Principles of Mathematics (Cambridge: Cambridge University

Press, 1903), p. 477.2 An issue similar to th e one raised by Russell has been raised in conn ectio n with the

addition of velocities in relativity theory. Let A, B, C be three bodies, so tha t the velocityof A wit h r espect to B is VAB, tha t of B w ith respec t to C is vBo (wh ere the direc tionof vBois parallel to the direction of VAB), and of A with respect to C is vAe. Th en, ac-cordin g to classical mechanics, vAo ~ vA~ q- vBc. Bu t ac cordin g to th e special relativitytheory,

VAB ~- VB(~

VAO --VAB • VBC

l + - -

c~

where c is the velocity of light. It has been argued that in the latter we are not "reallyadding" velocities. However, this objection can be disposed of in essentially the samemanner as Russell 's argument.

Fo r an interesting sketch of a generalized formal analysis of G estalts such as melodies,cf. Kurt G relling and Paul Oppenheim , " De r Gestaltbegriff in Lichte d er neuen Logik,"Erkenntnis, 7 :211-25 (1938) .

' Max Wertheimer, "Gestalt Theory," in Willis D. Ellis , A Source Book of GestaltPsychology (New York: Harcourt, Brace, 1950), p. 5.

~J. S. Mill, A System of Logic (Lo ndon , 18 79), Bk. III, Chap . V I, Sec. 2, Vol. I,p. 432.

Wertheimer, "Gestalt Theory'," p. 2. Cf. also Koffka's statement: "Analysis if itwants to reveal the universe in its completeness has to stop at the wholes, whatever theirsize, which possess functional reality . . . Instead of starting with t he elements andderiving the prop erties of the wholes fro m them , a reverse process is necessary, i.e. totry to understand the properties of parts from the properties of wholes. The chief con-tent of Gestalt as a category is this view of the relation of parts and wholes involvingthe recognition of intrinsic real dynamic whole-properties." K. Koffka, "Gestalt," inEncycIopedia of the SoclaI Sciences (Ne w York, 193t ), V ol. 6, p. 645.

Cf. Kurt Lewin, Principles of Topological Psychology (New York: McGraw-Hill ,1936), p. 218.

s W . Koehler, Die physischen Gestalten im Ruhe und /In stationaeren Zustand(Braunschweig, 1924), p. 42; also Ellis, Source Book of Gestalt Psychology, p. 25.

o Mo re generally, the charge density on the ellipsoid is propor tional to t he fourth rootof the curvature at a point.

~°W. Koehler, Die physischen Cesta/ten, p. 58, and cf. also p. 166. Many otherphysical examples of such functiona l' wholes could be cited. The surfaces assumedby soap films provide an intuitively evident illustration. The general principle under-lying the analysis of such surfaces is tha t, su bjec t to the boun dary c onditio ns impose don the surface, its area is a minimum. Thus, neglecting gravity, a soap film boundedby a plane loop of wire will assume a plane surface; a soap bubb le will assume the shapeof a sphere, a figure which has the minimum surface for a given volume. Now considera part of the surface of a soap bubble bounded by a circle. If this part were removablefrom t he spherical surface, it would n o longer retain its convex shape, but would becomea plane. Thus, the shape assumed by a part of the film depends on the whole of whichit is a part. Cf. the accounts of soap film experiments in Richard Courant and HerbertRobbins , W ha t Is Mathematics? (New York: Oxford, 1941), pp. 386ff.

n This is the contention of A. N. Wh itehe ad's philosophy of organism. Cf. hisProcess and Reality (Ne w York: H uman ities Press, 1929 ), especially Part II, Chaps. II Iand IV.

Koehler, Die physischen Gestalten, p. 47.a8 Thi s suggestion that the distinction between functional an d nonfunc tional w holes



32 P H I L O S O P H I C A L S T U D I E S

is not a sharp one is born out by an a t tempt to s ta te more formal ly the charac ter of an"organic" whole. Let S be some system and K a class of properties P1 . . . P= which Smay exhibit . Assume, for the sake of s implicity of exposit ion, that these properties aremeasurable in some sense, so that specific forms of these properties can be associated

with the values of numer ical variables; and assume, also for th e sake of s implicity, thatstatements about these properties have the form "At t ime t the property PI of S hasthe va lue x" or more compactly , "Pt (S, t ) ~- x ." W e now define a property in K, sayP~, to be " depe nde nt" on the remaining propert ies in K w hen P1 has the same valueat different t imes if the rem aining properties have equal values at those t i m es -- th at is,wh en for every property Pt in K if Pt(S,t~) • PI(S,t~ ), then P~(S,t~) = PI(S,t~).Moreover we shall say that the class K of properties is "interdependent" if each prop-er ty in the c lass is depend ent on the remaining propert ies in K -- th a t i s, when for everyPI and Pj in K, if Pt(S,t~) ~- P~(S,t~) the n Pj(S ,t l) = Pj(S,t~ ).

On the other han d, we can def ine th e c lass K to b e an " ind epen den t" c lass i f noproperty in K is dep end ent on the re main ing properties of K. To fix our ideas, let S bea gas, V its volume, p i ts pressure, and T its absolute temperature. Then according tothe Boyle-Charles law, V is dep end ent o n p and T ; a nd also this class of properties isan interdependent class of properties . Again, if S is an insulated conductor possessing adefinite shape, R the curvature at any point, s the charge-density at any region, p thepressure at any region, then p is not dependent on R and s , and the set p,R and s dono t form an in terde pende nt class , though they do not form an indep enden t class ei ther .Fo r this analysis , and further details involved in i ts elaboration, see the papers by Ku rtGrel l ing, "A Logica l Theory of Depe ndence ," and by K urt Grel l ing and Paul Oppen-helm, "Logica l Analys is of 'Ges ta l t ' and 'Fun ct ion al W ho le ' , " repr inted for m embersof the Fif th Interna t ional Congress for the Un i ty of Sc ience, he ld in Cambridge , Mass. ,in 1939, f rom the Journal of Unified Science, vol . 9 . This volume of the ]ournal wasa casual ty of World War II and never appeared.

However, if now we define a system S to b e a func tiona l whole w ith respect to aclass K of properties if K is an interdependent class, and also define S to be a summa-rive whole if K is an ind epe nd ent class, two points should be n oted. In the first place,whether a property will be said to depend on certain others will depend in part on thedegree of experimental precision with which values of the properties in question can beestablished. This is the point already made in the text. In the second place, though Smay not be a funct ional whole in th e sense def ined, i t need no t therefore be a summa-t ive whole ; for some propert ies in K may be dependent on the remaining ones , thoughno t all are. Accordingly, there m ay be various "degrees" of interd epen denc e b f parts of

a system.is Peter G. B ergmann, In t rod uct io n to the Theory of Relativity (New York: Prent ice-

Hal l , 1942) , p . 223. I t would be point less to ask in the present context whether any

"physical reali ty" is to be assigned to electromagnetic fields or whether as some writersmain ta in the la t te r is only a "m athemat ica l f ic t ion." I t i s su~ci ent to note tha t what-ever i ts "ult imate s tatus," the field concept in physics represents a mode of analysiswhich can be distinguished from the particle approach.

The technique to which reference is made is the device of retarded potentials . Cf.the remarks in Max Mason and Warren Weaver , The Electromagnetic Field (Chicago:University of Chicago Press, 1 929 ), Intr odu ction .

~ Cf. , for example, O. D. Kellogg, Foundations of Potential Theory (Berl in , 1929) ,

Chap . VII .

Nagel Wholes, Sums, And Organic Unities

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