Mary B. Hesse Models and Analogies in Science

 

Library o f Congress Catalog C a r d N um u m be b e r 66-14364 Copyright

© 1966

Notre Dame

University o f Notre Dame Press

Indiana

T h e introduction a n d first

three chapters w e r e first published i n England i n

1963 b y S h e e d a n d   \Tard Ltd same title © :Mary B. Hesse.

in a volume of the

Manufactured i n t h e United States o f America

 

 ntrodu tion

a scientific the the o r y  s to give a n  explanation of experimental data,  s i t necessary f o r t h e t h e o r y to b e understood i n t e r m s o f s ome model o r som e anal analog ogy y

 

with events o r o b j e c t s al alrr ea ead d y f a m i l i a r ? Does

ex expl plaa-

nation

imply a n account of t h e n e w a n d unfamiliar

i n t e r m s of t h e f a m i l i a r a n d intelligible, o r does i t

i n v o l v e o n l y a correlation of data a c c o r d i n g t o some other criteria, such  s mathematical economy o r elegance? Q uest uestio ion n s o f t h i s s o r t h a v e fo r ced th ems elv es upon scientists a n d philosophers a t va vari riou ouss stages o f t h e development of scientific t h e o r y , a n d particularly since t h e latter half of t h e nineteenth century when

physicists found themselves obliged to abandon t h e s e a r c h f o r m e c h a n i c a l m o d e l s o f t h e ether  s explan a t i o n s o f t h e phenomena o f light a n d e,lectromag:netism. I n 1914, i n hi hiss book   a  heorie physiqu

t he French physicist a n d philosopher Pierre Duhem contrasted two k ind ind s o f sci ent ific mind i n which h e also saw a contrast between t h e Continental a n d English temperaments: o n t h e o n e hand t h e abstra st ract ct,, logi logica cal, l, sy syst stem emat atiz izin ing, g, geometric mind typical of Continental physicists, o n t he other the visualizi n g, i m a g i n a t i v e , incoherent mind t y p i c a l o f t h e English-in Pa Pasc scal al s wor ord ds, t h e  strong a n d narrow  brr o a d a n d against t h e  b

 

weak.

Correspondingly

2

MODELS A N D A TALOGIES I N SCIENCE

D u h e m distinguished t w o kinds o f theory i n physics; the

a ~ s t r a c

a n d

_systematic o n t h e o n e hand a n d o n

t h e other theories using familiar mechanical models

H e explains t he h e d iiss ttii n nc c ttii o on n i n t e r m s o f electrostatics: This w h o l e t he heor ory y o f e l e ct r os t a t i cs c o ns nsti titt ut utee s

a group of a b s t r a c t ideas a n d general propositions f o r m u l a t e d i n t h e c l e a r a n d pr eci se lang u a g e o f g e o m e t r y a n d algebra a n d connected w i t h o n e another by t h e r u l e s o f s t r i c t logic

This w h o l e fully satisfies t h e r e a s o n f o r a French physicist a n d his taste f or cl a r i t y s i m p l i c i t y a n d order Here is a book [b [by y Oliver Lodge] intended to expound t h e modern t he heor orii e s of e l e c t r i c i t y a n d

to expound a n e w t h e o r y

I n i t a re nothing b u t

strings which move around pulleys wh whic ich h ro roll ll around d r u m s w wh h i c h go through pearl beads toothed whe e l s w h i c h a r e geared t o o n e another a n d en enga gage ge hooks W e thought we w er e entering t h e tranquil a n d neatly ordered abode of reas on b u t we find o u r s el ve s i n a factory 1

 

D u h e m admits that s u c h m o d e l s drawn from familiar mechanical gadgets m a y b e useful psychologi c a l a i d s i n sug sugge gest stin ing g the th e or o r ie i e s although h e thinks this happens less often than is generally supposed

B u t this admission implies n ot o t h iin ng ab bo ou utt t h e t r ut h

o r significance o f t h e models f o r many t h i n g s m a y  

 

TllI o orrie jJhy jJhysiqu sique, e, ch 4 5

INl RODUCTION

3

b e psycho log ical aids to discovery including astro logi lo gica call beli belief efss dreams dreams o r e v e n tea leaves without implying that t h e y a r e o f a n y permanent significance ob i n relation t o scientific t h e o r y . Duhem s main ob jection to m e c h a n i c a l m o d e l s  s that t h e y a r e

~ n c ? -

herent a n d superficial a n d tend to di dist stra ract ct t h e mind from t h e s e a r c h f o r lo gi ca l order. H e  s n o t much

concerned s other w r i t e r s h a v e b e e n with t h e pos sibility that m o d e l s may m i s l e a d b y being t a ~ e n t ~ o the e ph phen enom omen ena a a n d so h e literally  s explanations of th

does n o t object t o fundamental mec mechan hanica icall th theo eori ries es where t h e attempt such  s that of Desc escarteS teS  s made to reduce al alll phenomena to a few mechanical princi ples i n a sy sysste tem matic way. B u t f o r Duhem t h e essence o f s u c h a t h e o r y lies n o t so m u c h i n its an anal alog ogie iess with familiar mec mechan hanica icall obje object ctss a n d processes b u t rather i n i ts e con con o m ic a n d sy syst stem emat atic ic charac character ter.. T h e ideal p h ys ic al t h e o r y w o u l d b e a mathematical system with deductive structure similar t o E u c l i d s unencumbered by extraneous analogies o r imaginative representations. These a n d similar views w e r e d ire ire c tly tly challenged by t h e E n g l i s h p h ys i ci st N . R . C a m p b e l l i n hi hiss book

  hysics t he he

l em e m en e n ts ts

published i n 1920. A f ootn ootno o te

i n which Campbell r e f e r s t o national te tend nden enci cies es.. to theo eory ry suggests prefer mechanical o r mathematical th that h e has Duhem among others i n mind i n mounti n g his at atta tack ck although h e does n o t mention Duhem

b y n a m e . Campbell s main target models a r e

r

w r ~

 s

t h e view that

aid s to the the o ryry- c o n str str u ct ctii o n which

 

 

MODELS A N D ANALOGIES I N SCIENCE

c a n b e thrown away when t h e theory h a s been devel oped a n d hi hiss attack  s based o n t wo main arguments.

  irst h e that require considers we to b e i n t e l l e c  tually satisfied b y a theory if i t  s t o b e a n explanation o f phenomena a n d th this is sati satisf sfac acti tion on implies that t h e theory ha hass a n intelligible interpretation i n terms o f a v iin ng m e err e m a att h e em ma att i c a l intela model s w e l l as h av ligibility a n d perhaps t h e formal ch char arac actt eri eriss ti ti cs cs o f simplicity a n d economy. T h e second a n d more tell i n g argument presupposes t h e d n _ a m i ~ character o f

t h e o r i e s . A theory i n its s c ie ien n tif tif i c context is n o t a static museum piece b u t is always being extended

a n d modified to account f o r n e w phenomena. Campbell shows i n terms o f t h e development o f t h e kinetic theory o f gases h o w t h e billiard ball model o f this

theory played a n essential p a r t i n its it s extension a n d

h e argues perceptively that without t h e analogy with a model a n y such extensions will b e merely arbitrary. Moreover without a model i t will b e i m

p o s s i b l e to use a t h e o r y f o r o n e o f t h e essential purposes we demand o f it namely t o make predictions i n n e w d o m a i n s o f phenomena. So h e concludes:

 

analog anal ogie iess ar aree n o t   ai aids ds to th thee es esta tabl blis ishm hmen entt of theories; they are a n u t t e r l y essential p a r t of

theories wi with thou outt which theories w o u l d be com pletely plet ely va valu luel eles esss a n d u n w o r t h y of the name.   s o ft elati ntio sounggoefsttehde th tha nat ltohgayt loenacdes tthoetht heefoorry mu mula thaetorth ye bau  s f o r m u l a t e d t h e a nalogy has se rved its p u r p o s e a n d may be r e m o v e d o r forgotte n. Such a sug-

 

I N TRO D U CTI O N

gestion

 s

 

 

absol ab solute utely ly false a n d perniciously mis-

leading. Enough ha hass been s a i d t o indicate tIle general tenor

of t h e debate. TIle actual standpoints o f D uh uh e m a n d Campbell a r e a f f e ct e d b y t h e s t a t e o f their o w n contemporary physics, a n d there  s n o need t o i n s i s t o n t h e d e t a i l s o f their a r g u m e l l t s . S o m e o f t h e s e certainly cannot survive actual e v i d e n c e o f t h e ,vork-

a bility o f n e , v kind kindss o f th theo eorr y i n modern physics, a n d i n particular th the e r est estrr ic i c tio tion n o f t h e d i s c u s s i o n t o me me-chan ical m o d e l s of which Duhem  s more guilty than Campbell requires to b e modified. B u t many physicists , v ou ou ld ld n ow ow hold i n essentials ,vith Duhem

a n d would claim tllat Campbell s po poss i tion tion ha s been

decisively refuted b y t h e a bs e ll c e o f intelligible models i n quantulll physics; indeed, many vvould claim that something like Duhem s position lllust necessarily b e t h e accepted philosophy underlying modern phys hys i ca call theo theory ry.. the e c onv onvic icti tion on tllat e en ,vri ,v ritt ten t en i n th   Th This is essay llas b een hass n o t been s de deci cisi sive vely ly clos oseed, all alld d that t he he d eb eb at at e ha a n element o f truth remains i n Campbell s insistence

t h a t w i t h o u t m o d e l s t h e o r i e s cannot f ulf i l a l l t h e functions traditionally required o f t h e m , a n d i n particular that they cannot b e g e n u i n e l y predictive.

T h e chapter which follows ha hass been cast i n t h e form

o f a debate between modern disciples o f Duhem protagon agonist istss fin final ally ly a n d Campbell, respectively. T h e prot  

Physics t h e   lements p. 129.

 

 

MODELS A N D ANALOGIES I N SCIENCE

a g r e e f a i r l y a l n i c a b l y t o di f f er b u t during t h e think succeeded course o f t h e argument they have i n clarifying a n d settling s o m e o f t h e issues  vhich

often befog t h i s t o p i c TIle Campbell i a n h a s also made SOllie sort o f a case f o r greater attention to b e paid i n t h e philosophy o f sc scie ienc ncee to logi logica call questioI1S

about tIle nature alld validity of anal analog ogic ical al argument fronl lliodels T h e subsecluent chapters a t t e l l 1 p t t o pursue some o f t h e s e l o g i c a l questions

albeit i n a

preliminary a n d elelnentary fashion should like t o e x p re ress s m y grateful thanks t o ProBraithwaite a n d   r G Buclldahl f o r fessor R discussioI1s whicll have inspired some o f t h e points Inade here although probably neither o f them  viII he a rrg gu um me en n ttss p u t illto t h e mouths o f In recognize t he Iny y disputants as being po posi siti tion onss whic whicll ll they would ever

have defended

T o avoid

great bulk o f

a footnotes have ha ve co coll llec ecte ted d i n t h e suggestions f o r further read-

i n g most o f t h e re refe fere ren n c es t o published  vork that   have found valuable t h e logic o f analogy

 

t h iin nk kii n g a b bo ou utt models a n d

 

T h e Function o f Models:

A D ia lo gu e

  ampbellian I imagine that along with m o s t c on-

temporary ph phil ilos osop ophe hers rs o f science y o u would wish

to say that the use o f m o d e l s o r analogues is n o t e s s e n t i a l to scien tific theorizing a n d that theoret-

ical explanation c a n b e d e s c r i b e d i n t e r m s o f a purely formal deductive system so so m e o f w h o s e c o n sequences c a n b e interpreted into observables a n d hence em empi piri rica call lly y t es ted b u t that t h e t h e o r y as a w h o l e does n o t r e q u i r e to b e interpreted by m e a n s of a n y m o d e l.

/ Dllhemist es I d o n o t deny of c o u r s e that models m a y b e u sefu sefull g uide uidess i n su sugg ggest esting ing th theo eori ries es b u t I d o n o t think they a r e essential even as psychological ic al aids ids a n d they a r e certainly n o t logi lly essential f o r a t h e o r y to b e accepted as scientific. When we have f o u n d a n a cc ccep epta tab b le t h e o r y a n y model that m a y h a v e l e d us to i t c a n b e thrown away. Ke Keku kule le is

s a i d to h a v e arrived a t t h e structure of t h e b e n z e n e ring after dreaming o f a s n a k e with its t a i l i n its it s mouth b u t n o account of t h e snake appears i n t h e textbooks o f o r g a n i c chemistry.   ampbellian I o n t h e other hand want to argue that models i n s o m e sense  re e s s e n t i a l to t h e logic o f scientific t h e o r i e s . B u t first l e t us a g r e e o n t h e sense i n which we a r e using t h e w o r d m o od de ell when

7  

8

MODELS A ND N D A NA NA LO LO G GII ES ES I N SCIENCE

we assert o r deny that m o d e l s a r e essential. I should l i k e to explain lny sense o f t h e v o r d b y taking Camp b e l l s w ell- wo rn e x a n l p l e of t h e d y n a m i c a l t h e o r y o f gases. When   ve t a ke a c o l l e c t i o n o f billiard balls i n random motion as a model f o r a gas we a r e n o t asserting t h a t b i l l i a r d bal ls a r e i n al l re spec ts l i k e gas pa part rtic icle less for for billiard b al l s a r e r e d o r  vllite a n d hard a n d shiny a n d we a r e n o t intellding to suggest that gas n10lecules h a v e these prop proper erti ties es.. W e a r e i n fact act sayirlg that gas m o l e c u l e s a r e analog llS to bil

liard balls a n d t h e relation o f a n a l o g y m e a n s t l l a t there a r e S lne properties of billiard balls which a r e

n o t f o u n d i n molecules. L e t us cal l those properties

we k n o w b e l o n g to billiard balls a n d n o t t o mole neg g tive tive n logy of t h e mo cules t h e ne model del.. l\tlo tlotion a n d impact o n t h e other hand a r e just th the e pr prop oper erti ties es o f billiard balls that we d o want t o a s c r i b e to m o l e c u l e s posit sitiv ivee n li n o u r model a n d t hes e we c a n call t h e po

ogy N o w t h e i m p o r t a n t t h i n g a b o u t this kind o f model thinking i n science is that there   Till generally b e some properties of t h e m o d e l about which we d o n o t y e t know whether t h e y a r e p o s i t i v e o r negative

a nalog ies; t hese a r e t h e interesting properties be cause as I sha l l a r g u e t h e y a l l o w us t o make n e w predictions. L e t us ca ll t h i s third set o f properties etltrr l n logy I f gases a r e r e a l l y l i k e collec t h e n etlt t i o n s o f billiard balls except i n regard t o t h e known negati nega tive ve analog logy then from o u r k n o w l e d g e o f t h e m echa echani nics cs o f billiard b al l s we m a y b e a b l e t o make n e w predictions about t h e e x p e c t e d b e h a v i o r o f

 

T H E FUNCTION O F l\10DELS

9

gases. O f course t h e predictions m a y b e , v r o n g , b u t then ,ve s h a l l b e l e d to conclude that  ve have t h e wrong n10del. Dllhemist Your te tern rnli lill llol olog ogy y of po posi siti tiv ve, ne nega gati tive ve,, a n d neutral analogies  s useful b u t  s there n o t still about t h e sense o f  model ? aY opuos shiabvl ee ambiguity mentioned gas n l 0 l e c u l e s a n d billiard balls. When y o u s p e a k o f t h e model fo forr gases, d o

y o u mean t h e billiard balls, p o s i t i v e a n d negative analogy a n d all, o r d o y o u m e a n w h a t we i l n a g i n e when we t r y t o picture gas In Inol olec ecul ules es  s gl10stly little

objects having some b u t n o t al alll t h e properties of bi lliard balls?   should say that both senses a r e ,videly used  among many others), b u t i t distinguish them.   ampbellian

 s

important t o

a g r e e t h e y sl1 0 u ld b e d i s t i n -

conv nven enii e ent ntll y by guished, a n d   think we c a n d o so co Ine ans o f m y terminology. L e t us a g r e e tha,t ,vhen we s p e a k o f a model i n it itss prilnary sense i n t h i s discussion l e t us c a l l i t model we a r e n o t speaking o f objj e ect ct w h hii c ch h can,  s i t were, b e built o r im im-another ob  

agined alongside t h e phenomena we a r e investigatpe e rrff e c t c o p y t h e billiard ing. T h e model s t h e i m p

balls) mintlS t h e known negative analogy

so that

y co on n ssii d e err iin n g t h e known po we a r e o n lly posi sitiv tivee anal analog ogy, y, prop oper ertt iies es ab abol olll t a n d t h e p r o b a b l y o p e n ) class o f pr which i t is n o t y e t known whether t h e y a r e p o s i t i v e o r ne nega gati tive ve ana analogi logies es.. W he n we consider a theory based o n a model  s a n explanation f o r a s e t o f

phenomena, we a r e c o n s i d e r i n g t h e positive a n d  

10

lVIODELS A N D i\NALOGIES I N SCIENCE

neutral analogies  v

n o t t h e negative analog analogy y

\vhich

already kno \v vve call disc discar ard. d.

A r e y o u n o t confusing   model with tIle tl tlle leor ory y itself? The re is n o difference between t h e D uhemist

theory a l l d tIle l l l o d e l   as y o u n o w explain it so  \vhy

use tIle \ v o r d   model at all? Calnpbellian Partly because there is a t e n d e n c y particularly arllollg people o f your school o f thought to use tIle word   tI tIle leor ory y to cove coverr only  \vllat I \vould c al l tIle k n o v n p o s i t i v e a nal og y neglecting t h e features o f t h e model  \v  \vhich a r e its growing points namely its neutral analogy. My whole argument is

going to d e p e n d o n t h e se f e a t u r e s

a n d so I want

to make i t clear that I a m n o t dealing with static a n d fo form rmal aliz ized ed t h eori eoriee s corresponding only t o t h e known po posi siti tive ve a na l og y b u t w i t l l t h e o r i e s i n the process o f gro growt wtll ll.. Also sinc sincee y o u disagree with m e that models a r e e s s e n t i a l t o t h e o rie rie s y o u wi will ll necesi n a wider sense than s a r i l y use t h e word  theory m y  model t cover formal deductive systems which have only a partial interpretation into ob ob--

servables. My models   o n t h e other hand a r e t o t a l interpretations o f a deductive system depending o n t h e positive a n d neutral ana analog logies ies vvith t h e  copy.

S i n c e I s h a l l also want t o t a l k about t h e second object o r copy that includes t h e ne nega gati tive ve a na l o g y l e t us a g r e e as a shorthand e x p r e s s i o n t o c a l l t h i s  modeI 2 I f i t is indifferent which sellse is meant  

I s h a l l s i m p l y use  model. L e t us ll W t r y to produce a reconstruction o f t h e

 

T H E FU N CTI O N O F lVIODELS

 

usee o f lllodels a n d analogies i n a familiar example us

vave m o d e l s f o r sound a n d f o r ligllt. A t a n elementary l e v e l we c a n se t u p t h e f o l l o v vii n g cor  th

respondences: OUND

LIGHT

Produced   mo tion o f   vater par-

Produced   motion o f gongs

Produced   mov i n g flame etc.

ticles

strings etc.

Properties o f re-

Ec h oe s etc.

R e f l e c t i o n i n mirrors etc.

Properties o f dif-

Hearing round

Diffraction

fraction

corners

through small

WATER

WAVES

flection

slits etc.

Amplitude

Loudness

Brightness

Frequency Medium: Water

Pitch Medium: A i r

Color Medium:   Ether

indicate

T h e f i rst t l l r e e r o w s some re spe cts i n which these three processes appear to b e a l i k e t o

fairly fai rly supe superf rfic icia iall obse observ rvat ation ion.. T hey are f o r example t h e kind o f properties that   vould go i ll B a c o n s Tables o f Presence o r M i l l s A g r e e m e n t s . I n a l l

three cases there a r e p r e s e n t m o t i o n

sOlnething

indi dire rec c ttly ly fr from om o n e p l a c e t o another b y transmitted in hitting a n obstacle a n d a bending round obstacles.

T h is suggests that t h e three processes a r e perhaps alike i n more fundamental respects a n d i n o r d e r t o

investigate th this is poss possib ibil ilit ity y

we look IIlore cl clo osel ely y at

t h e o n e o f t h e three about which   ve know most namely water waves. W e postulate with Huygens

that a disturbance o f o n e particle communicates

 

 2

MODELS A N D ANALOGIES I N SCIENCE

i t s e l f to neigllboring particles i n such a vay that r i p p l e s s p r e a d f r o m t h e center o f disturballce i n concelltric ci circ rcle less a n d b y means of t h e elementary   ath-

able

ar m mo on nii c m o ott i o on n we a r e erepresent ll1atics oft hsi sinl nlpl plee h ar to e amplitude a n d frequellcy o f t h e  \Taves a n d to derive t h e la vs o f r e f l e c t i o n a n d diffractiol1.

W e   ave t i l e n a theory o f water ripples consisting o f equations o f t h e type

y where

 

s i n   Tfx

y is t h e height o f t h e v a t e r a t t h e point

measured horizontally

 

is t h e 11laxilllUm height o r

amplitude o f t h e ripples a n d   is their frequellcy. From this mathematical theory s o m e laws o f t h e

process such as t h e equality o f t h e angles o f incialld ld refl reflec ecti tion on c a n b e d e d u c e d . dence al

So f a r we h a v e

t\

sources o f information to a i d

o u r construction o f th theo eori ries es fo forr sound a n d fo forr l i g l l t namely their observed properties a n d their observed

analogies with water waves a n d i t is inlportant to notice that both o f them a p p e a l o n l y t o descriptions o f  observable events. W e m a y define o serv tion st tements as

those descriptive statements

whose t r u t h o r falsity i n t h e face o f given empirical circumstances would b e agreed upon b y a l l u s e r s o f English with o r without scientific training. L e t us also introduce t h e term expli ndllm f o r t h e se sett o f obser obs ervat vation ion statem statement entss connected with t h e phellom-

e n a we a r e attempting t o explain b y meallS o f a t h e -

o r y - t h a t is i n th this is case t h e observed properties o f

 

T H E F UNC T I ON O F MODELS

 3

sound o r o f l i g h t . A l l u s e r s o f English might not t h e ana analog logie iess bet\Veell t h e t h r e e processes until they a r e poillted out a n d u p t o t h i s

o f COllrse

noti

point they

l 1 1 ay

 

have n o ll lllo lore re si sign gnif ific ican ance ce than t h e

fact that t h e class of fingers o n a halld a n d petals o n a buttercup a r e similar i n that botll h a v e five Inem-

bers. B u t when t h e analogies have beell pointed speci cific fical ally ly sci scient entifi ificc out n o esoteric insight a n d n o spe k no no w wll e ed dg ge e a r e required to reco recogn gniz izee that tIley ex exis ist. t. I t is n o t quite t h e same  v i t h t h e m a t h e l n a t i c a l knowll ed edg ge of theory o f water waves, f o r here s o m e know trigonometry is required b u t there is n o difficulty i n understanding tIle tern l height o f  vater frequency o f waves, etc. into which t h e mathematical

symbols a r e interpreted. I n t h i s sense tIle mathematical s y s t e m is  about nt e err p prr et et a att iio on in has its i nt terms of) obse observ rvab able le e v e n t s .

  o vconsider w ha ha t h a p p e n s w h he e n we make u se o f

t h e known theory o f water vvaves a n d t h e a11alogies

between them a n d sound i n o r d e r t o construct a theory o f sound. T h e an anal alog ogii es sugg sugges estt that sound is produced by t h e motion o f a i r particles propagated i n concentric spherical waves from a center o f disturbance. S i n c e w e know that t h e greater t h e disturbance o f water t h e greater t h e amplitude o f t h e waves, a n d t h e greater t h e disturbance o f gongs,

strings, hammers etc., t h e g r e a t e r t h e n o i s e p r o duced i t is easy t o i d e n t i f y l o u d n e s s o f sound with amplitude o f sound waves, and similarly, experiences with s tr trii ngs ngs o f va vary ryin ing g le leng ngth thss pe pers rsua uade de us that

 

14

l\10DELS A N D ANALOGIES I N SCIENCE

pitch o r sound is t o b e identified \vith frequency o f sound  Naves. I n S O l n e such \vay \ve c011struct aIl Ilee-t -to o o n e correspondences betvveell t h e observable prop-

e r tie tie s o f sound   the th e ex expl plic ica1 a11c 1clu luln ln)) a n d tllose o f \vater  the e 111odel: ), al v  vaves  th alld ld \ve a r e tllen i n a position t o test t h e ma math thel ell1 l1at atic ical al ,vave theory as a theory o f sound. Further tests o f t11is kind, o f c o u r s e , n l a y

o r 111ay n o t sho\v t h e theory t o b e satisfactory. l a I n n o t clainlillg that t h e us usee o f a l l a l o g y l e a d s u s t o a n infallible theory, only that i t is used i l l t h i s \vay t o suggest a t h e o r y .   d o 110t suppose y o u \vill ,vant

to dispute t h i s so far. Dllhemist N o ,   have n o o bje jecctio tiol1 to your recon struction o f t h e wa y t h i s particular 1110del nlight b e

used. B u t   a m u n h a p p y a b o u t tI1e sellse i n \vhich yOll say t lla lla t t h e initial analogies a n d t h e illterpretat i o n s o f t h e Inathelnatical wave theory i n terlllS o f water c a n b e s a i d t o be ob obse serv rva ab ble le,, as contrasted,   suppose, with t h e a i r particles ,vhicl1 a r e n o t observ

able.

  cannot

see that there is a n i l n p o r t a n t differ

ence here. StIrely, to  observe a silnilarity between t h e behavior o f ripples a t t h e e d g e o f t h e S\Vimlning b a t h a n d t h e behavior o f sound i n a mountain valley is a f a r from superficial observation. I t requires a very sophisticated framework o f p h y s i c a l i d e a s i n which, f o r e x a m p l e , t h e phenomena o f echoes a r e described i n terms o f a train o f p h y s ic icaa l cau ses initiated b y a shout, rather than i n terms o f a n imitative spirit o f t h e mountains.   ampbellian Yes,   agree \vith this, a n d your ex-

 

T H E F UNC T I ON O F lVIODELS

15

alnple illdicates that contrary t o \ v l l a t SOl11e elnpir- ~

icist philosophers

to have held observation descriptions a r e n o t \vritten 01 1 t h e face o f e v e n t s t o s ~

l n

b e transferred directly into language b u t a r e already

 interpretations

o f events

a n d tIle kind o f illter-

pretation depends o n t h e frallle\Vork o f aSSUll1ptions

o f a language COlll111Ullity I t c a n plallsibly b e ar ar-gued that there is n o descriptive stateillellt n o t even

tIle   blue-here-no\v

beloved o f se sens nsee da data ta tl tlle leor oris ists ts

1vhich do does es n o t go beyond \Vllat is  given il illl t h e a c t o f o bs bs erv ervii ng ng . B u t I d o n o t vvisIl t o pursue tl1is argu ment here. Would y o u b e prepared t o a g r e e that scie sc ient ntif ific ic tl tlle leor orie iess bring something n e w into o u r descriptiollS o f evellts a n d that i t is t11erefore po poss ssib ible le t o l l l a k e a d is i s ttii n nc c ttii o on n be ett w we ee en n t h e observation statelllents o f a given language conll11Ullity sl slla lari rill llg g a

fralnework of assump ass umptio tions ns alld alld t h e statemellts going

beyond this shared frame\vork \vhich a r e illtroduced i n s c ie l l ti f i c th theo eori ries es?? I t is i n c o n t r a s t v i t h these

novelties 1vhich 111ay b e c a l llee d containing

theor eti

l t

r m s ~

theoreti

l s t

t

m

n t s ~

that certain a t present

descri cripti ption on m a y b e called observable. agreed kinds of des

T h i s is t o m a k e t h e distinctioll a pragmatic one relua g e COlna t i v e t o t h e assumptions o f a g i v e n l a n g ua munity

b u t i t does 11 t m e a n t h a t t h e traditional

empiricist problem o f t h e relation betweell theory a n d observation disappears. T o realize that every

h e n wa wass once a chicken is n o t t o absolve oneself from t h e task o f f i n d i n g o u t h o w a h e n gives birth t o a chicken.

 

16

MODELS A ND N D A NA NA L LO O GI GI E ES S I N SCIENCE

Dllhem ist O u r dispute does n o t turl1 o n t h e prehe o bs bs er er va va t iio o n la11guage a n d cise nature o f t he

 

viII

accept your pragmatic description o f i t . B u t   have anotl ano tller ler obje ob ject ctio ion n to your account o f t h e g enes i s o f a

theory o f SOU d Y o u s e e m t o ilnply tllat t

ere a r e t\ sorts of theory construction going o n here. First there is t h e theory of  vater  vave vavess vIlic}l is arrived a t b y making a h y po poth thes esis is about t h e propagation of disturbances expressing this i n Il1 l1aath them emaatic ical al lanlan-

nd d e ed du uc ci n ng g from i t th guage a nd the e obse obserr ve ve d properties

o f water  vaves. There is n o l11ention o f a n y analogies o r m o d e l s h e r e . B u t i n t h e case o f sound i t is said that one to one correspondences betwec11 properties of  vater a n d properties o f sound a r e se sett u p first a n d then t h e mathematical wave theory is transferred t o sound. This m a y vvell b e t h e  vay   w l l i c h th theo eori ries es a r e o f t e n arrived a t i n practice b u t y o u h a v e s a i d nothing to s h o w that reference t o tIle  vater model

is essential o r that there is a n y differellce i n principle bet veen t h e r e l a t i o n s o f theory a n d observation i n

t h e t vo cases. Both t h e o r i e s c o n s i s t o f a deductive

system together with a n i n ntt e err p prr e ett a att iio o n o f t h e terms occurring i n i t i n t o observables a n d from both systems c a n b e deduced relations  vhich vhen so interpreted correspond to observed relations such as t h e l a w of r ef efle lect ctio ion. n. This is a l l that is required o f a n explanatory t h e o r y . Y o u h a v e implicitly acknowl

 

edged i t t o b e sufficient i n t h e case o f water waves a n d i t is also also su suff ffic icien ientt i n t h e case o f sound waves. I f

 

T H E FUNC T ION O F MODELS

17

,ve 11ad never heard o f  vater  \Taves ,ve shoulcl s t i l l b e a b l e t o use t h e san1e information about sound t o obtain t h e sarrle resu result. lt. T h e information consists o f t h e observed production o f sound b y certain motions he r e ell a att iio on nss b e ett v e ee e n t h e lnagnitudes soli lid d bo bodi dies es,, t he o f so o f these motions an.d t h e louclness o f tIle sound a n d bet\veen lengths o f stri strin n gs a n d pitch o f note, a n d t h e

p11enon1ena o f echoes a n d bending. A l l o f these c a n b e deduced from a nlathematical it h a p pp p rro op prr iia a tte e i n tte e rrp p re re tta a ttii o n with wave theory w it

o u t me11tioning t h e water-wave model and

v11at is

more important vithout sllpposing t11at there is anything connected  v  vii t h t h e t ran ranss mi m i ss ss io io n o f sound

o r light which is analogous to   vater that is without supposing there a r e SaIne

11 11id idde den n motions o f particles having t h e same relation t o t h e s e observed prope r t i e s o f sound o r light that t h e m o t i o n s o f water

particles have t o t h e properties o f water waves. I n fact, i t w o u l d b e very misleading t o suppose a n y s u c h t h i n g b e c a u s e s o m e o f t h e further conse quences derived from a theory o f water waves turn

o u t n o t to b e true i f t rra an nss ffe e rr rr e ed d

b y t h e one-to-one

correspondence t o sound a n d light transmission.   ampbellian N o b u t t h e r e a s o n f o r this i n t h e

case o f sound a t l e a s t is n o t t h a t tl1ere is n o ,vave model b u t that r i p p l e s a r e t h e wrong ,vave model. T h e oscillation o f particles cons constit tituti uting ng sou sound nd ,vaves lo n ng g t h e direction o f tr trans ansmis missi sion on of t h e tak es pl plac acee a lo sound like t h e motion o f a piston a n d n o t a t r i g h t

 

 

ND A NA NA L LO O GI GI E S I N SCIENCE MODELS A ND

a n g l e s t o that direction,

as with

ripples. B u t what I

have just described is i t s e l f a m o d e l o f t h e m o t i o n s of a i r particles d e r i v e d b y analogy with observable

events such as t h e action o f b u f f e r s o n t h e t r u c k s o f a train o r  t  to o t a k e Huygens example th thee t r a n s m i s s i o n o f p r e s s u r e along a line of billiard balls when t h e b a l l a t o n e e n d is struck i n t h e direction of t h e line a n d t h e ball a t t h e other e n d mo v e s off b y i t s e l f i n t h e same direction.   uhemist

I d i d n o t intend to say that i n many cases an alte al tern rnat ativ ive e model cannot b e f ou o u n d w he he n t h e first b r ea eak k s do down wn b u t only that mention o f a model logical structure o f a n explanatory theory a n d that i t is n o t even always a u s e f u l device f o r f i n d i n g such a t h e o r y f o r i t ma may y p os osit itii ve vell y is n o t part o f t h e

suggest t h e wrong theory. I t is a question o f l o g i c I should like your react i o n s to. I a m impressed y o u see by t h e situation throughout a l a r g e part o f modern physics where i t is i m p o s s i b l e to find a n y model like t h e model o f a i r motions f o r sound, a n d where nevertheless t h e criteria f o r a deductive theory which I h a v e outlined a r e st stil illl sa sattisfied a n d theory construction a n d testas

i n g go o n much before. I t m ay b e less s a t i s f a c t o r y to t h e imagination t o h a v e n o p ic ic tu t u ra ra bl b l e model a n d more d if f i cu lt t o construct theories without it

but

t h e continuance of phys ics i n t h e sa sam m e l o g i ca call shape as before shows that t h e model is n o t logically

n ecessary t o t h e process.

  ampbellian I a m n o t convinced that there is

 

T H E F U N CTIO N O F wIODELS

 

such a n a b s e n c e o f m o d e l s i n moderll physics as y o u

suggest a d I m a y come back t o that later. Also i t is a lit little tle misle mis leadi ading ng to s p e a k o f  pictures as if they  vere synonymous  vith 11lodels f o r I  vould say f o r exan1ple that a tl tl1r 1ree ee dirr dirrle lell llsi sion onal al sp spac acee curved i n a

fourth dimension is a p e r f e c t l y g o o d m o d el i n rela-

t i v i t y t h e o r y b u t i t is certainly n o t pictur ble A model for Ine is allY system vllether buildable picturable imaginable o r Ilone o f these vhich has ha s t h e il

characteristic o f making a theory predictive a s ense I s h a l l d e s c r i b e later w I l e n I t r y t o substantiate

gic a l l y e s s e nt ntii a l f o r m y claim that models a r e l o gic theories.

B u t l e t u s s t i c k f o r t h e moment t o

ample because i t   s

s ilup iluple le exe a s i e r to bring o u t t h e difference O lr

between us there. I f I llnderstand you y o u a r e sayi n g that i n t h e case o f sound  vaves there is n o point even i n speaking about motions o f a i r particles be-

c a u s e t h e s e a r e n o t part o f t h e observed data y o u list a n d  ve c a n explain these data equally well b y means o f a Iuathematical theory some o f v ho s e c o n sequences c a n b e interpreted t o g i v e relations bet w e e n t h e observables. Y o u  viII a t least a d m i t t h a t here there is a di diff ffer eren ence ce b e t w e e n t h e t w o t h e o r i e s I described those f o r ripples a n d f o r sound i n that t h e m o t i o n s o f water particles

 r

observable i n t h e

pragmatic sense we h a v e a g r e e d o n a n d so a l l t h e symbols i n t h he e e qu q u at a t io i o ns n s o f t h e ripple t h eo eo r y a r e i n terpretable as observables. I n t h e case o f sound

ho vever we cannot  observe

i n t h i s s ense t h e am

 

20

MODELS A N D ANALOGIES I N SCIENCE

plitude a n d frequency o f t h e  vaves

indeed  ve c a n -

110t o b s er erv v e v a v e s a t all ve c a n DIlly infer theIll froll1 data such as ill1pact of ha l l l 1 ne r   g011g alld vibration of stri strin n gs gs.. D o y o u  v  vii sh to say t11at a t l l e o r y o f s o u n d n e e d n o t Inention   vaves a t all since

tllese a r e n o t observable?

  llhemist

am

that convenient

a n d u l l i v e r s al l l l o d e s o f speecl1 such as tllis s h o u l cl

necessarily b e dropped b u t l e t us see what exactly  ve lllearl b y talking about sound waves. W e d o n o t lllean just t h e same as we d o when talking about  vater  vaves vaves bec because ause as w e h a v e see n s o u r l d vaves a r e longitudinal a n d n o t transverse. TIle  vord persists beca becaus usee both theories us usee t h e same Inathemat

i c a l fO fln alism

which  v  vee c a l l

t h e  vave equation

differently applied i n t h e tvvo cases. vVllat ripples

a n d sound  vaves have i n COlllIllon is completely contained i n t h e mathematical formalism a n d i t is t h i s  \ve  \ve POillt t o b y continuing to u s e t h e w o r d  wave.

O f course

a m n o t denying that i t is legitinlate t o

thillk o f t h e propagation o f sound i n terms of p u l sa satti n g spheres o f a i r particles so long as v l l a t we mean

b y t h i s is controlled b y what we k n o w f r o m observation about sound other

a n d n o t b y reference t o s o m e

process.   suppose this c a n b e expressed i n

your terminology b y s a y i n g that i f t h e po posi siti tive ve anal anal-ogy og y between sound a n d a model o f pulsating spheres

is believed t o b e c o m p pll e t e then this model is iden tical with o u r theory o f sound a n d there is n o harm i n using t h e l a n g u a g e o f t h e Inodel   as a n interpre-

 

T H E FUNCTION O F MODELS

 

tation o f tIle n 1 a t h e m a t i c s o f tIle tI1eory. B u t I a m

d e n y i n g t h a t we c a n always g e t t h i s s o r t o f m o d e l

a n d t h a t w h e n we can t we s o m e h o \ v h a v e less o f alll explanation. al   ampbellian I a m surprised y o u a r e prepared to

allo\v as ll1uch as t h i s f o r cornmon m o d e s o f s p e e c h a n d I a m n o t sure y o u ar are e cons consis iste tent nt i n doing so so.. I f

y o u Ilad regarded a l l ta talk lk o f

oscillatio11s o f a i r par ticles as ll1isleading a n d dispensable I should have respected your C011sistency b u t I should tilen I ave a t t a c k e d y o u o n t h e grounds that y o u d o n o t give a plausible account of tI e meaning of theoretical terms. O n w at I t a k e to b e t h e co con nsist iste1 e11 1t for formal malist ist the theory

only

i n thi view thiss case corlsists nla1 opula f alate fted od rn1aacl a11i 1ipu deductive systeln-rtlarks o n paper nl cording t o certain rules-together with t h e i n te te rp rp rre e

tations i n terms of o b s e r v a b l es so that t h e only meaning that call b e given to f o r instance t h e pa pa rameter   i n t h e wave equation is ill il l teril S of inten s it y o f sound a t t h e point where that is re recc o r d e d . There is nothing to say about   during t h e t i m e

\vll \v llic ich h el elap apse sess between t h e banging o f t h e g o n g a n d t h e reception of t h e sound a t some distant point. I c a n say o n t h e other hand, that   ha hass a n interpreta

tion a t a l l t i m e s durillg t h e passage o f t h e

SOU11d

namely i t is t h e alnplitude o f o s c i l l a t i o n s o f a i r par ticles even though these a r e  llnobservable. T h u s I h a v e a solution to t h e so-called problem o f t h e  meaarli1  me i11 1g of theoretical terms. cou u r s e all kinds of defi defini niti tion onss Dtlhemist Well, o f co  

22

lVIODELS A N D ANALOGIES I N SCIENCE

o f t l l e o r e t i c a l t e rI n s h a v e b e e l l s l l g g es t e d to cover cases like this, a n d i n t h e case o f SOUlld \vaves a c o n ditional definition i n ternlS o f observables rnight b e given i n t h e forrl1:

F o r al alll  x y t

\vave a t  x y

the th e a IInp npli litl tlld lde e o f a sound

i f a rnicrophone placed a t  x y a t time t records sound o f intensity proportional to a2 • is a

Bllt i t is n o t al,vays p o s s i b l e t o g i v e d e f illi illitt io ion n s even

o f t h i s conditional kind arld when i t is not

am

content to say that t h e lneaning o f   anlplitude o f sound wave

is given indirectly b y t h e position o f

i n tIle deductive system, a n d tIle f a c t that some a consequences of t h e system,  vhen irlterpreted have ordinary er ern n pirical meaning.

Campbellian: So when y o u spoke o f  pulsating s p h er eres es o f a i r particles, y o u were   t silluggling i n a reference to a n y model: , b u t only intended these w o r d s t o b e a way o f speaking about tIle mathemat ical ic al symbols? According to y o u i t would b e  vrong t o look u p tlleir meaning i n a dictio11ary o n this occasion-what is required is t o look u p t h e position o f t h e correspondillg symbols i n t h e d e d u c t i v e

systelll. T hi s is surely a v e r y s t r a n g e account of  meaning ? I t irllplies t h a t i n d dii r e c t meaning c a n b e given to a n y word   like to coin b y i l l s e r t i n g i t i n a deductive system,

  ~

r

exarllple i n t h e syllogisrll:

A l l toves a r e   hite

 

T H E F U N C T I O N O F I\10DELS  

c a r is a t o v e therefore M y c a r is 1vhite,

23

tIle conclusioll o f 1vllich is observable.  1- oves no no,v ,v llas indirect meaning il illl your sense. Du h e mist Tllis account o f illdirect lllealling must

b e regarded as necessary bllt n o t sllfficiellt. 1. 0 nlake i t sufficient   should l1ave t o adcl that f o r a theoreti-

c al ternl t o have scientific meallillg i ll this   ay, i t l11USt occur i n a deductive syste tell ll1 1 1vhi 1vhich ch is seriously considered i n science, that is is,, o n e 1vllich Ilas many obser obs erva vabl ble e co cons nseq eque uenc nces es i n different cirCulllstances, a l l o f vvhich a r e c o n f i r m e d b y o b s e r v a t i o n a l l d none

refuted. T hi s is entirely a question f o r sc scie ient ntif ific ic re

s e a r c h , all empirical n o t a l o g i c a l question a n d so t he he c o on n d iitt i o on n s f o r a theoretical terlll t o l l a v e s c i e n  tific meaning carlnot b e lo logi gica call lly y fo forma rmaliz lized ed.. B u t i t is clear t h a t y o u r syllogislll about toves would n o t qualify. Campbellian This s t i l l seeIns t o I ne very strange t h e more so b e c a u s e y o u h a v e agreed t o accept a n aCCOullt o

observational a l l d t l l e o r e t i c a l terms i n

ti o n bet1veen theIn is n o t logical whicll t h e d i s t i n c ti

b u t pragmatic. I f y o u a c c e p t t h i s y o u IllUSt allow

f o r t h e frontier between them t o s l l i f t as s c i e n c e pro-

gresses. This is done i n m y

aCCOll11t

by saying that

we discover that sound waves a r e pulsating spheres t h e ordinary sense o f these words o f a i r particles a n d i f t h i s is accepted   y everybody i n t he he l a an ng gu ua g ge e

cOlnmunity (as I suppose i t is i n ours

i t does n o t

 

 

MODELS A N D ANALOGIES I N SCIENCE

much lnatter where t h e line o f  observability

is

d r a , v n . A d n 1 i t t e d l y i t would b e o d d i n o r d i n a r y

speec

t o talk about  observing

a i r pulses, b u t a

statelnent about thell1 Inig t well function as a n ob ob servation staten1ent i n a particular scientific experi

ment, that is t o say, e v e r y o n e would a c c e p t it s truth

o r falsity as t e final   Ollrt o f appeal without deduc i n g further   observab observable le cons conseque equences nces from it. O n your account I d o n o t see h o w  p  pu ulsati11g sph spheres eres o f a i r particles

ever gets i n t o o r d i n a r y la11guage,

because y o u have spec specif ific ical ally ly denied that t ese ,vords a r e used i n t h e i r o r d i n a r y sense.   llhemist l\tIy account is n o t i n t h e l e a s t i n c o n 

sistent with what we h a v e p r e v i o u s l y agreed-ill fact,

I h a v e g i v e n a n account o f h o w t h e frontier o f ob serv se rvab abil ilit ity y s h if t s w h i l e y o u h a v e n o t . TI1e essence o f t h i s s h i f t is surely t h a t o r d i n a r y la11guage itsel itselff changes-when we talk about a i r p u l s e s we a r e f l t u s i n g t h e words i n exactly t h e se se they previously had, a n d what I h a v e done is p r e c i s e l y t o explain h o w ordinary language is extended t o t a k e i n n e w

senses o f t h ese w o r d s, depending o n t h e structure o f t h e scientific theory i n which they o c c u r . Y o u , o n t h e contrary, have n o t e x p pll a i n e d h o ow w t h e ordinary

senses o f w o r d s c h a n g e . M o r e o v e r , I think y o u have smuggled i n a quite different issue here, namely, t h e

question o f t h e  reality

of t h e a i r pul se s. Y o u s e e m

to imply that I a m comlnitted to a 110nrealistic view, t o s a y i n g that they a r e fictional entities o r heuristic devices o r what-110t, b u t this is n o t t h e case. F o r m e ,

 

T H E FU N CT I O N O F MODELS

 

t o say t l l a t   a i r p u l se sess ex exis istt means just ,vhat I h a v e explained-they a r e entities referred t o b y  th  thee v a l u e s o f v a r i a b l e s i n ) a deductive syst em h a v i I l g all all  

charac char acte teri rist stic icss of a n acc accep epted ted sci cien entt i fi c tlle tlleor ory y that I h a v e d e s c r i b e d . I d o hold that models a r e heuristic o l d in in g t ha ha t the devices, b u t I a m n o t committed to h ol oretical entities understood wholly as interpretations o f a n accepted Il Il1a 1ath them emat atic ical al theory ar aree also. I f y o u like, lik e, In Iny y t11eoretical e nt n t iiti tie e s a re re related to your mod els i n having t h e known positive   n logy only Campbellian Y o u h a v e certainly made your posi tion clearer, a n d I a g r e e that   v e need l l o t d i f f e r o n eo r et et iic c a l entities. W e t h e subject o f existence o f t h eo differ o n what i t is that is a s s e r t e d t o exi st . I say that

t o a ss e rt a theory is to a s se r t a modell positive a n d neutral analogy; y o u say i t is t o a s se r t t h e positive

analog anal ogy y o nl nly, y, a n d according t o y o u t h e neutral anal ogy is merely a heuristic device. Dllhemist O f course, t h e theory illay n o t b e de scribable i n terillS o f models a t all, i n cases ,v-here I deny tl14t there a r e models. T h e n i n order t o a s s e r t t h e e x i s t e n c e o f a th theo eore reti tica call enti en tity ty,, ,ve Inust either coin n e w words o r give o l d   vords a n e w sigllificance b y t h e method o f indirect m ea e a n i n g i n deductive syst e l n s I h a v e d e s c r i b e d . T o go back to your origillal

examples, t h e word  ether,

,vhich you have p u t i n

quotes i n t h e third column o f your t ab abll e , was surely a word .adopted a n d gi give vell ll sign signif ific ican ance ce i n just this

way,

that is t o say, there were s o m e t h e o r i e s seri

ously considered a t o n e stage i n physics i n w h i c h t h e  

 

M O D E L S A ND ND A N A AL LO G GII ES ES I N SCIENCE

ether h a d a we l l - d e f i n e d pl plaa c e i n a deductive system a n d t h e o b s e r v a b l e c o n s e q u e n c e s o f its properties could b e empirically tested.   ampbellian I a m n o t satisfied that this is suffi

c i e n t . I want   sa say y that t h e we well ll-d -def efin ined ed pla place ce i t h a d was d u e to its being understood i n t e r m s o f wave

models a n d t h a t its meaning was g i v e n b y a series o f analogies of t h e f o r m: water waves water p a r t i c l e s ..

sound waves a i r particles

light waves

..

ether particles

I d o n o t really understand h o w  meani n g s a r e g i v e n b y a n a l o g i e s i n t hi s way a t all. A r e y o u s a y i n g s i m p l y that when there is a modeL f o r a theory as i n t h e case of t hi s theory o f l i g h t then Dllhemist

 air

and

ether

interpretations

the

a re s a m easnedt of symbo ymbolls i n th thee t h e o r y a i r i n t h e caso e fof sound ether i n t h e case o f l i g h t ?   so I a gr gree e that we m a y a cq cq ui u i re r e a n int intuit uitive ive understanding understanding o f   ether

in this

indirect way

by an anal alog ogy y with t h e a i r modeL. B u t s i nc e I d o n o t regard models as part of t h e logi c o f a n no n o t r eg eg ar ar d t h i s sense o f   meaning as theories I c an interesting f o r t h e logician.   ampbellian I d o mean by m y analogical relations

what you sugg ggeest b u t I also mean something more conv nvin ince ce y o u   part o f t h e logic o f which I hope to co

theories. L e t us go back to t h e example a n d t r y t o fill o u t m y a c c o u n t o f t h e way t h e t h e o r y o f sound is arrived a t . I a m prepared t o c o n c e d e your objection t h a t g i v e n a l l t h e observational information I h a v e  

T H E F U N CT I O N O F J\;IODELS

27

allo ved l11yself,   could have gone straight t o a math-

ematical ,v ,vav avee tl1eory frOlll ,vhic11 t h e observations

could b e decluced,  vithout going through t h e proc ess o f finding one to one cor corres respon ponde1 de11ce 1cess Nith  vater

,vaves. T h e r e ,viII generally b e a n indefinite llumb e r o f s u c h mathell1atical theories b l l t   agree ,vitl1 y o u that there is n o g u a r a n t e e that tI1e ,vater-,vave

model: ,viII lead t o t he he c o rr r r ec e c t theory so y o u rightly a s k  vhether   c a n h a v e a n y reasons f o r using this

analogy e x c e p t t h e c o m ffo o r t a b l e feeling that   have

seen t h e Inathematics before. Well

tllink   11ave

a reason a n d   c a n explain i t b y taking a slightly different situation. Suppose ,v ,vee a r e no tV atten1pting t o construct a

theory o f l i g h t . Your procedure ,viII b e t o find, n o

matter how a mathematical system from w h i c h t h e observed properties o f t h e explicandum say reflec tion a nd n d r ef e f ra r a ct c t io i o ll l l c an a n b e deduced a n d f o r this

y o u will only demand interpretations o f t h e formul a e which yield t h e observable relatio11s y o u w i s h t o explain. Suppose b y whatever method o r l a c k o f

method y o u use, y o u d o choose t h e n1atherrlatical

,vave theory o u t o f t h e in.definite nun1ber o f possi

bilities.   s11all arrive a t t11e san1e theory b y noticing

t h e analogies between light a n d sound

a n d setting

u p a model o f light transmission i n t e r m s o f oscil lation o f particles i n a medium. No tV w e must  

d is i s ti t i ng n g ui u i sh s h b et e t we w e en e n t h e v a r i o u s results

 v

c a n ob-

t a i n . Y o u  tV ill b e able t o deduce trle simple laws

o f reflection a n d refraction b y using space coordinate

 

 

MODELS A N D ANALOGIES I N SCIENCE

a n d intensity observables b u t t ese  viII b e t h e only terlllS i n t h e t h e o r y w h i c h y o u  viII interpret as

obss e r vab ob vab le less . So f a r y o u viII h a v e deduced geometr i c a l o p t i c s fr or n a mathematical  v  vaave th theo eory ry.. I f y o u  vant to d o lllore than t hi s y o u w i l l h a v e t o interpret s i n   7Tfx s  veIl t h e syrnbol f i n y o u r equation  s t h e sYlnbols   a n d x N o w y o u m a y h a v e observational information that w i l l a l l o v y o u to d o t his his dire direct ctly ly.. F o r example y o u m a y derive frOlll your theory equations relating t o t h e passage of l i g h t t h rro ou ug g h a p r i s m iilll w h i c h y o u notice that t h e a n g l e o f refraction depellds o n t h e value o f   I f y o u also h a v e exp experi eriln lnent ental al data da ta o n t h e

production o f a spectrurn o f c o l o r s b y t h e prism

it

will b e r e a s o n a b l e t o s e t u p a o n e t o o n e c o r r e s p o n d e n c e b et et w e e n values o f   a n d colors i n t h e spectrum. T h e theory will then have shown itself capable o f explaining t h e laws o f dispersion as w e l l as t h o s e o f geometrical optics. B u t suppose y o u d o

n o t know t h e prism experiment o r a n y other relating

t o colors. H o w is   to b e interpreted? You Yo u m a y o f course make a guess that since there a r e lights o f is

different colors a n d there a n available parameter   i n t h e theory i t would b e  vorth investigating

whether th the e iden id enti tifi fica cati tion on o f v a l u e s o f   with differ-

corr rres espo pond nden ence ce bet bet veen ve en thee n t c o l o r s w i l l y i e l d a co o r y a n d experiment. O r y o u m a y d e c i d e t l l a t   is

uninterpretable; i t is part o f t h e m a c h i n e r y o f t h e d e d u ct ct i ve ve t h e o r y b u t h as n o observable correlate. I n t hi hiss case y o u will n o t b e able t o include disper

 

T H E F U N CTIO N O F ~

sian i n your theory. Have procedures correctly?

 

D E L S

 

described your possible

  llhemist

Yes

viII accep acceptt that i n principle

 

should h a v e t h e s e three possibilities i n tIle case o f a hitherto uninterpreted term i n t h e theory. O f

course t h e example y o u a r e using hardly brillgs o u t t h e p o i n t s i n a re real alis isti ticc way be beca caus usee t h e  v  vaave equ equ a tion was n o t i n t r o d u c e d i n t o op opti tics cs Ulltil after t h e

facts o f color dispersion were already kn kno o vn alld alld so there was little difficulty about this particular iden tification. B u t   c a n see that i n other cases there I n i g h t b e n o obvious identification o f a theoretical decide de to term; a n d then o n e might s y o u sug gest deci leave i t uninterpreted s i n t h e case o f a Schrodinger in

quantum

or

lJI-function

s o m e scho o ls o f physics; o n e might make what y o u c a l l a guess b u t   should prefer t o c a l l a h y p o t h e s i s about its it s illterpretation

a n d investigate t h e experilnental consequences o f t h e hypothesis. What   cannot see  s that y o u a r e a n y bet t e r off when i t c o m e s t o interpreting a feature o f your m o d e l . Y o u w i l l o f c o u r s e know that   s what corr co rres espo pond ndss t o t h e frequency of waves i n t h e model

b u t i n t h e absence o f a n y observations connecting color with t h e laws o f geometrical optics which y o u

have already explained b y t h e theory h o w does that ident entif ify y freq fr eque uenc ncy y o f waves with color? help y o u t o id Y o u h a v e t h e s am amee ch choi oice ce that   have either t o l e a v e

  uninterpreted a n d hence  frequency o f waves u n correlated with anything i n your t h e o r y o f l i g h t o r t o resort to guess v vo ork.

 

 

MODELS A ND N D A NA NA LO L O GI GI E S I N SCIENCE

I used t h e vvord  guess ratller tllan  llypothesis t o bring   t t h e fact that o n your ac he o r i e s y o u c a ~ n o t giv ny C O U I l t o f tIle l 1 a t u r e o f t he   a ~ n p b e l l i a n

r

ne illterpretation sons f o r choosing t o e x a n l i n e o ne

ratiler t l l a n a n y other. A n d I notice tllat y o u d i d n o t

give allY actual exanlple o f a theoretical terlll being interpreted 1vithout t h e help o f a model. I t is n o accidellt that i t is d i f f i c u l t t o tllink o f al l exalnple i n practice  r because   there suggest al,vays reasons f o r exal11illing a h y p o t l 1 e t i c a l irlterpretatioll aIld these r e a S O l l S a r e dra1vn fronl lllodeis. give allY reasons before havillg carried o u t experinlental tests?   cannot give a n y reasons f o r c l l o o s i n g aIl e t l l e o r y rather than a n   llhenlist

WIlY sI sI10 10ul uld d

 

other llntil   have t e s t e d i t , a n d tIle illt illter erpr pret etat atio io11 11 o f a particular tlleoretical term is only a n elenlellt

i ll a t i l e o r y , t o b e considered as part o f tIle w h o l e.

B u t y o u llave n o t answered illy questioll about your   1Vl1 procedure. l-Io\v

does YOllr rn rnod odel el h.elp y o u t o

give gi ve re reas ason onss f o r your interpretation? C a n ~ p b e l l i a r L This is , v l le lerr e   appeal to t h e anal ogy bet \veen t h e 111 del a n d tIle phell0111ena to b e l I S f i r s t s ee h V explained.   c a n interpret t h e paranleter   o f t h e t h e o r y , 1vllich is already corre·

lated i n IllY model with t h e amplitude o f t h e vvaves.

suggest t11at tIle lllodel: imnlediately makes i t rea sonable t o suppose that  nlagnitude o f t h e ,vaves corresponds  vith   lllagnitude o f t h e light alld i n lnagIlitude means brightness. t h e case o f light  

J u ~ t

as a greater ,vave disturbance meallS a louder

 

T H E F l J N C T I O N O F lVI0DEI-JS

 

sound, so d o e s a greater vvave disturballce meall a

brigl1ter light, althoug11 this this cannot b e illvestigatecl directly since vve cannot  make a greater \vave dis turbance b y lnovillg a body as \ve c a n \vitl1 soul soulld. ld. T h e hypothesis that tllis is t h e case cOllles frOlll a n a n a l o g y o f tI le f o l l o \ v i n g k i l l d :

loudness properties   sound

.. ..

brig11tness

properties o f light

s ll gg e st tl1at th this is analogy is fOUIld i n t h e language before a n y ,vave theory is thought of. I t is indepelld e n t   t h e particular theory o f light we a r e consider  

i n g a n d so c a n b e u s e d t o develop this theory. Duhernist O n e ll llli ligl gllt lt,, surely surely,, just as plausibly sug

gest that br brig ight htll lles esss is correlated vvith shrilllless, o r loudness with purple o r scarlet  calleel, b e i t noted,  loud colors .   anlpbellian Adrnittedly there lnay b e s o m e a m 

biguities   this

b u t if w e c on. sid er t h e p o i n t s o f

s i l n i l a r i t y o f lOlldness an.d brightness-tIle scale o f

intensities from absence o f sound o r light t o i n d e f i  nitely large degrees o f i t , t h e analogies betweell their

effects o n o u r sense organs   deafening

a n d  blind

ing ), a n d so o n , t h e suggested correspondence seelns t h e mo most st pl plau ausi sibl ble. e. Dehtlmist

A l l r i g h t , b u t ,vhat about t h e cor

respondence between pitch, frequency, a n d color which y o u must claim if y o u r m e t h o d is t o ,vork f o r interpreting t h e symbol   is,, admittedly, more difficult. ampbellian This is

 

 

MODELS A ND N D A NA NA L LO O GI GI E ES S I N SCIENCE

I d o no nott ffor or exam exampl plee sseee h o w t h e correspo11dence o f f r e q u e n c y of waves with p i t c h c o u l d have been arrived a t without o b s e r v e d cor cor re rela lati tio o n s in inv v ol olvi vin ng s u c h t h i n g s as vibrating strings. I n t h e case o f sound

used

a model f o r light there is some pl plau ausi sibi bili lity ty i n claiming a pre re--scie ien nti tiffic ana analo logy gy:: as

pitch properties o f sound

 

color properties o f light

i f we think o f t h e v a r i o u s met me t ap ap ho ho rs rs fro fro m sound to light-Locke s blind man s  scarlet sound of trum pets a n d t h e use o f s u c h t e r m s as   harmony and appealing to a n a l o gi e s of p l e a s u r e a n d pain  clash i n their effects o n o u r sense org rgal alls ls.. Dllhemist I a m n o t a t a l l c o n v i n c e d that this roundabout way o f r e c o g n i z i n g a n a l o g i e s c a n b e enti tire rely ly arbi ar bitr trar ary, y, b u t even s h o w n to b e other than en

if i t can y o u seem t o m e o n l y t o h a v e g i v e n m e o n e way of making m y  guess

ntt e err p prr e ett a att iio o n of a at an in

t h e o r e t i c a l t e r m . You h a v e n o t shown that i t consti t u t e s a n y r son f or expecting a guess made by t h i s method t o b e a right o r even fruitful one.   ampbellian I hope y o u w i l l wa ive f o r t h e fil0 ment t h e question of whether a n y o b j e c t i v e a n a l  ogies o f t h e kind I d e s c r i b e a c t u a l l y exist b beecau s e I hope t o go into this i n more d e t a i l l a t e r . M e a n  while I should l i k e t o examine t h e o b j e c t i o n y o u have just made. There a r e two t h i n g s I should like to say about it. Fi r s t I claim that t o assert a n anal 

T H E FUNCTION O F MODELS

 

ogy between amplitude o f waves a n d loudness o f light ht even even be befo fore re a n y experisound o r brightness o f lig mental correlation is k n o w n t o give a r e a s o n f o r t h e interpretation o f t h e symbol a o f a kind which c a n never b e g i v e n o n y o u r account of t h e matter.   uhemist L e t m e interrupt y o u b e f o r e y o u go a n y

further. O f course

i t is p o s si b l e t o find a m o d e L

this th is case a n interpretation derived from t h e m o d e L c a n b e s a i d to h a v e t h e reason that i t is s i t is i n

from the

derived modeL a n d t h i s dist distin ingu guis ish h es i t from a n y interpretation I might d e c i d e to m a k e B u t

this

is pure

evasion I cannot a c c e p t a r e a s o n i n t e r m s

  a m o d e l f o r I claim t h a t n o model is required. I a m a s k i n g f o r a r e a s o n f o r a s s u m i n g that t h e model   required o r even that i t is l i k e l y to l e a d to a better interpretation than o n e I m a y m a k e ampbellian O f c o u r s e I cannot expect y o u to b u t what a c c e p t a r e a s o n appealing to cie a en mtis want to point o u t is that  s sci tiosdtse l use t h e wordI  reason

i n this context they will a c c e p t r e a so n s ap-

pealing to m o d e l s This c a n b e se e n i n t h e way t h e y

m a k e p r e d i c t i o n s f r o m models a n d use them  s tests o f theories A prediction w i l l b e thought t o b e rea-

s o n a b l e i f i t follows from a n  obvious interpretation g i v e n to a theoretical term b y appeal to a t h e prediction c o me s off t h e theory a n d model its m o del del 1 will b e regarded  s strengthened whereas i f i t fails to c o m e off t h i s m a y b e r e g a r d e d  s suffi-

c i e n t l y s e r i o u s t o refute t h e t h e o r y a n d t h e m o d e l 1 together F o r example the c o r p u s c u l a r m o d e l o f  

 

34

MODELS A ND ND A NA NA LO LO G GII ES ES I N SCIENCE

light ,vas regarded as refuted when t h e obvious i n t er er p prr e ett at a t iio on th ha a t t\

,corpuscles falling o n o n e spot

,vould produce tvvice t he he i n ntt e en n si s i t y o f l i g l l t produced b y o n e ,vas s h o \ v n t o b e contrary t o diffraction

experiments. T h a t t h e model l e d t o t h e , v r o n g in in

terpretation ,vas i n this instance a  reason abandoning t h e ,vhole theory. Dllhemist

for

a m n o t clear \vhy o n your account i t

should be be,, f o r y o u h a v e already allo,ved f o r t h e pos sibility tllat a model; m a y n o t correspond t o t h e

phenomena il illl  

respects. W h y call1 ll1lo lott tIle feature \vhich fails i n this instance b e rell 10 ved t o t h e llega tive analogy a n d t h e rest o f t h e corpuscular model l retained?   ampbellian T o answer this ,,,auld certaillly re re

 Iuire fllrther analysis. Roughly, i t ,voulel turn a l l l s ~ a r e luore t h e fact that some properties o f n l o  essential than others, tllat is t o say a r e causally

more closely connected o r

t o c o - o c c u r more

frequently. F o r e x a m p l e , c o l o r is n o t a n essential property o f a billiard b a l l f r o m t h e point o f view o f mechanics, b u t momentulu is.

a prediction de rived f r o m c o l o r fails, this does n o t ess essent entiall ially y af afffec ectt

a me mech chan anii cal c al mo del delll b u t i f something elerived from momentum fails, t h e m o d e l is refuted. Duhemist B u t such refutation still depends o n  

t h e a s s u m p t i o n tllat a theory must have a model, \vhich   a m denying. A n d your example plays into m y hands, f o r we know that t h e  essential property

y o u have appealed t o i n t h e case o f t he he c o orr p pu u ssc cu ull a arr

 

T H E FU N CT I O N O F MODELS

35

t h e o r y o f light  s   ~ t n o v a l l o v vee d t o r e f i t t e that theory. T h e quantulll t h e o r y o f radiation accommo-

dates both diffraction experiments a n d model talk about light particles. B u t t h e \vay part particl icles es alld other models a r e used i n quantuln theory  s quite consiste n t with m y account. T h e theory  s regarded  s satisfactory i f i t  s p o s s i b l e to deduce o bs bs e r ve ve d r e s u l ttss from t h e mathematical formalisln plus interpretam o

l s ~

tion o f some o f its a r e u s e d as its terillS alld only mnemonic a n d heuristic devices  vhen convenient. I n this theory models

 

need n o t even b e con-

sistent  vith o n e another to b e useful. want t o c o m e b ac Campbellian ac k t o t h i s question

of l l l od odeeis i n quantum theory later b u t b e f o r e t h a t l e t us look a t this question o f prediction more carefully f o r this  s m y second point i n a n s \ v e r t o your challenge to m e t o produce reasons f o r using models. have suggested that m y m o d e l enables lile t o m ak ak e p re r e di d i ct ct io io ns n s because i t l e a d s t o Ile v a n d obvi 

o u s interpretations o f some theoretical terms  vhich m a y then b e used t o derive n e v relations bet veen obse ob serv rvab able les. s. Yo You u reply that a ny assigllment o f a n e v interpretation with o r  \vitll0ut t h e us usee o f a model viII enable y o u to ma mak k e p re re dic dicti tio o ns ns a n d that there  s n o reason to have more confidence   my predictions than i n yours.   agree that   have n o t y e t g i v e n

a n y reason b u t   still want f o r a moment to pursue m y point that t h e kin

o f pre i tion req lired c a n

o n l y b e obtained b y using models.   take i t that we both agree that a criterion f o r a

 

36

MODELS A N D ANALOGIES I N SCIENCE

theory is that i t should b e falsifiable b y empirical tests. Fa Fals lsif ifia iabi bili lity ty is closely c on on n ne e ct c t e d v i t h predic

t iv ivee p o ve ve r although they cannot quite b e identified without further analysis. I  vant to point o u t tl1at us a ge o f t h e cr crit iter erio iorl rl o f f a l s i f i a b i l i t y cover s a t least three requirements o n theories only t h e strongest o f which is s u f f ic ien t t o esta establ blis ish h t h e superiority o f m y

theory-plus-model over your f o r I n a l t h e o r y . L e t us c on on ssii d de er th hrr e ee e types o f falsifiability a n d three corre eory ry sponding typ es o f t h eo and   TY

stat atem emen entt ha hardl rdly y ever I n s c i e n c e a s i n g l e observation st

descri ribe be only o n e unique event b u t t h e purports to desc of

that would b e observed under

set

events

suffi

cien ci entl tly y s iimi mila larr circumstances a t a n y time. Hence a n

may y always b e s a i d t o b e falsi observation statement ma fiable i n t h e sense that t h e circumstances i t describes o r sufficiently similar circumstances m a y al vays be e r e p e a t e d ; h e n c e i t is c o n c e i v a b l e i n principle b hass been confirtned i n t h e that a statement which ha

past m a y b e falsified i n t h e future Questions about what  vould constitute  sufficiently similar circumstances a n d w h a t we should b e disposecl t o say

about a n unexpected falsification o f this kind need

n o t detain us o f   falsifiable

because i t is clear that such a sense is f a r to too o weak t o satisfy t h o s e w h o

w i s h t o say that a condition fo forr s c i e n t i f i c t l l e o r i e s is hey ar aree falsifiable. A theory must d o more than that t hey predict that t h e same ob obss e erv rvat atio ion n s ttat atem emen ents ts that

 

T H E FUNCTION O F l\10DELS

 

have been confirmed i n t h e pas pastt ,viII, i n sufficiently similar circumstances, b e c o n f i r m e d i ll t h e future A sci en ti fic th theo eory ry is required to b e falsifiable i n t h e sense that i t l e a d s to new ob obse serv rvat atio ion n statements

w h i c h c a n b e te st ed , that is is,, that i t l e ad s t o n e w a n d perhaps unexpected a n d interesting predictions.   u t is

here there a n ambiguity T h e weaker sense o f such a requirement is that n e w co corr rrel elat atio ions ns ca can n be

found between t h e same ob obser servat vation ion pr pred edic icat ates es;; t h e stronger sense is that n e w correlations c a n also b e found which i n v o lve lve n e , v obse observ rvat atio ion n predicates. I t

will b e convenient t o introduce some notation here. I want t o argue o n t h e basis o f your o w n account, b e c a u s e I think i t does provide some necessary con ditions that theories must satisfy; what I deny is that they a r e Sllfficient. L e t us c o n s i d e r a n obser vation language containing observation predicates

 

l

~

is

O Suppose there a set o f o bs er  vation s t a t e m e n t s e a c h o f which is accepted that is 1

t o say, e a c h member o f t h e se sett expr expres esse sess a n empirical correlation between some o f t h e O s a n d p s which, a t a g i v e n stage o f use o f t h e l a n g u a g e , is a c c e p t e d

as true

I f t h e se t also e x h a u s t s a l l such accepted

statements i t will b e called t h e accepted set. I t represents then a science o f these particular observables a t t h e stage o f empirical generalizations, before explanatory th thee o r i e s h a v e been introduced I t m a y n o t , o f co u rs e , exhaust all t h e true statements con and

there m a y b e

taining O s whichPre because s,main correlations rema in un unno noti tice ced d a t th this is stagsome e.

 

38

lVIODELS A ND N D A NA N A LO LO GI G I E S I N SCIENCE

N   V consider a s e t o f tlleoretical p r e d i c a t e s ( t h e T s a n d a theory cont.ainillg theln \vhich h a s as c o n sequences a l l those observation staternellts o f t11e accepted s e t which cOlltain O s a n d only O s . Tllat xp p lla an na a ti ti o on n is t o say t h e theory is i n your sense a n e x o f t h e accepted statemeIlts containing only D s . This

tlleory m a y o r m a y n o t , i n addition, contain stateInents with observation predicates ot11er thall t h e D s namely t h e P s. Falsifiability i n senses A a n d B

can

  ov

b e explained as £ 11 1vs TY

A

S u p p o s e t h e t h e o r y does   o t contain a n y P s. T l l en i t c a n have n o consequences relating t o predicates

o t h e r t h a n t h e O s. T h u s i t c a n l l o t b e used t o ex ex-plain tIle remainillg statements o f t h e accepted s e t containing a n y o f t h e P s, n o r c a n i t b e u s e d t o predict correlations between them \vhich a r e true bllt

n o t yet a c cc cept e ed d . T h a t is t o say i t is n o t falsifiable

i l l t h e strollger se11se I t may however, b e possible t o u s e i t t o predict correlations bet1veen t h e O s which a r e true b u t n o t y et et a c cc ce ep p tte ed d.. Such a theory will b e s a i d t o b e lve veak akly ly falsifiable o r w e a k l y p r e d i c t i v e a n d 1vill b e called a f o r m a l t h e o r y . l\ Iany o f t h e socalled  mathematical models o f modern COSlllOlogical economic, a n d psychological theory a r e o f t h i s

kind; they a r e mathematical hypotheses d e s i g n e d t o

fit experimental data, i n whicll either there a r e n o t he h e or or e ett ic i c al al t e err m mss o r i f t.here a r e such terms, t h e y a r e n ot o t f u rrtt h e err i11terpreted i n a m o e l ~

 

T H E F UNC T I ON O F l\;fODELS

39

TY

Sllppos Sll ppose, e, h01vever, t h e tlleory cloes contain SaIne o f t h e P s. W e m a y dismiss t h e case i n 1vllicll i t contai S theIn o n l y

s t a t e m e n t s wllich c o n t a i n n o T s ,

f o r then th.ese statements cannot properly b e said t o b e part o f tIle t l l e o r y , altll011gh t i l e y I n a y b e part

o f a s c h e m e o f empirical ge gene nera rali liza zatt io ioll lS 1 vh i c h re re

main , vh vh ol ol ly l y , vi v i tth h il i l l t h e observatioll lallguage. 1- he tlleory may, however, contain some o f t h e P s i n sonle

statemellts vvhich a l s o c O l l t a i n SOllle T s . Such a theory lllay then yield as consequellces observation

statements containing a n y o f these P s

a n d l1en.ce

Inay explain members o f tIle accepted s e t COlltain i n g theIn a n d   ay predict l e V correlations bet1veerl strong ongly ly falsifiable them. I t w i l l t h e l l b e said t o b e str o r st stro rong ngly ly pred predic icti tive ve.. N o w consider h o v statements containing T s a n d

P s   call them P-statements

could   Oille t o b e i n -

trodllced into t h e tlleory. TIley a r e n o t illtro luced b y considering t h e o bs bs er e r va v a bl b l e r el e l at a t io i o ns n s b et e t ,,v v ee ee n t h e

wass P s because w e have supposed that t h e t h e o r y wa designed i n t h e first place as a n explanation o f t h e O s n o t the

T h e y a r e n o t introduced arbitrar-

P s.they were there woulel b e n o reason ily, because if w h y a n y particular s ta t a tte em me en ntt s h o ou u lld d b e introduced

rather than a n y other, a n d sllch a theory could n o t b e t a k e n seriously as a predictive theory. Also, i t would not, as a whole, b e falsifiable, because falsi

fication o f o n e a r b i t r a r i l y introduced P-statement

 

40

lVI0DELS A N D ANAI-lOGIES I N SCIENCE

could b e dealt with by replacing i t  vith anotller, leaving t h e r e s t o f t h e theory unaffected. T h e only

other possibility is that P-statements a r e introduced f o r reasons internal to t h e theory rrhese reasons,

Inoreover, cannot b e c o n c e r n e d m e r e l y with t h e formal properties of t h e theory for e x a m p l e , its f orInal symm symmet etry ry o r simplicity because they must b e r ea so ns f o r a s s e r t i n g particular tllings i n t h e the oreticallanguage about particular observation pred

icates ( t h e P s , a n d though t h e theoretical predicates m a y b e seen from t h e formalist point o f v i e v  s u n interpreted symbols, e v e n f r o m this point o f view th e o b bss e r v a att i o n p r e d i c a t e s m a y n o t . Hence t h e se sett o f P-statements must b e interpreted i n terms o f t h e theory I t is this interpretation w h i c h , I maintain is given  y t h e model, a n d ,vhich requires t h e whole theory t o h a v e a model interpretation. Duhemist I a m n o t sure I h a v e f o l l o w e d your sy mbol i sm. Sure Surely ly t h e P s a r e a l r e a d y interpreted since t h e y a r e observation predicates?

Campbellian Yes, b u t I a m concerlled ,vith h o w they g e t into t h e theory.   y t h e c o n d i t i o n s o f my

problem they a r e n o t introduced i n virtue o f their

correlations with other observation predicates, hence they must have a n i n ntt e err pr pr e ett a att iio o n i n a model 2 which also p r o v i d e s a n interpretation o f t h e theoretical predicates. Consider m y example o f sound a n d light orr y waves, where sound ,vaves a r e a model   f o r t h e t h e o o f l i g h t. Here t h e O s might b e position coordinates a n d in inte tens nsit itie iess of light a n d t h e P s color predicates.

 

T H E FU N CTI O N O F MODELS

41

T h e theory of re refl flec ecti tion on a n d refractio11 e x p l a i n s t h e accepted O-statements b u t says n o t h i n g a b o u t t h e

is,, 11 V d o t h e P s g e t into t h e the P s T h e question is o r y to enable i t t o 111ake predictions about color? They have t o g e t i n i n t h e f o r m o f PP-st stat atel eln n ent ent s correlating t h e P s with v a l u e s o f t h e parameter   a n d   is a theoretical predicate. N o w t h e   n o e t ~ comes i n as a n interpretation o f a l l t h e T s into predicates referring to sound waves.   is t h e frequency o f

sound waves, o r pitch. This model: , together with m y su sugg gges este ted d an anal alog ogy y  pitch cor res esp pol1ds to color give.s t h e interpretation   / i n t h e theory o f light corresponds t o color

a n d t h e theory n o w yields

predictions about color. This c a n b e represented

schematically: schematicall y: THEORY

INTERPRETATIONS

 con  c on taining   etc. as theoretical predicates)

i n sound model2

I N T E R P R E TATIONS

in li

t

observables

a

t

loudness   pitch ~

 

 

 

brightness color   ~

O statements

Observation

P statements

 geometrical

statements

 colo  co lorr dis dispers persion ion,,

opti op tics cs,, etc. etc.))

f o r sound

etc.)

42

lVIODELS A N D .A.NALOGIES I N SCIENCE

Here signs o f equality indicate interpretatio11s \vit11i n a theory o f theoretical predicates into observation predicates; double arrows indicate t h e d i r e c t i o n o f deduction; a n d double arr 1VS indicate observable

relations o f analogy. Duhe mist cal1 see that y o u a r e asking f o r a dou b l e i11terpretation o f t h e P s, once into observables

a n d once into t h e mode1 2   a n d this is because y o u  vant t o predict t h e observable P-statements b y u s i n g the

r r l o d e L ~

B u t l e t m e returl1 t o your argument i n f a v o r o f t h e step involving t h e analogies between light a n d sound.

  c a n see from your diagram that t h e analogies as ve Il as t h e interpretations a r e o f tvvo ki11ds TIle first kind a r e t h e one to one co corr rres espo po11 11de denc nces es bet\veen tl1eoretical predicates a n d predicates o f tIle I n o d e l : ~ o n t h e one hand

a n d bet\veen t he he o orr e ett iic ca all p rre ed dii c ca a tte es and

light observables o n t h e o t h e r giving one-ta-one cor-

respo11dences between predicates referring t o light a n d t o sound i n virtue o f t h e saIne formal theory. This take it is t h e conventional us usee o f a n a l o g y i n mathematical physics as when Kelvin exhibited analogies between fluid flow heat flow electric i n -

duction

electric current

a n d magnetic field

by

showing that a l l a r e describable b y t h e sarne equa tions  vith a p p ro r o p rrii a tte e in ntt e err p prr e ett a att iio o n s i n each case.

B u t y o u a r e asking f o r something i n addition t o this namely a s e n s e o f analogy i n terms o f w h i c h y o u c a n make these one-to-one correspondences before y o u

have g ot t h e theory

y some kind o f prescientific

 

T H E F U N CTI O N O F IVIODELS

43

recognition o f a nalo nalogi gies es such such as pitch: c o l o r . Is t h i s correct?

C e r t a i n l y . M y h o l e point is that i t is n e c e s s a r y to h a v e t h e s e correspondences before t h e theory other1vise t h e theory is n o t predictive o r falsi  a n ~ p b e l l i a r l

fiable i n t h e strong sense. Duhernist I think t h e weakest part o f your argum e n t is where y o u d i st st i n g u i s h your senses   a n d falsifiability. Even i f   admit f o r a m O i l l e n t that cannot accept that strong falsifiability is required n t rro od du uc ce ed in ntt o theories b y P-st Pstat ateme ement ntss can ca n only b e i nt   of

m e a n s o f y o u r d u b i o u s analogies. There a r e other  vays o f extending theories which d o n o t deserve

y o u r e p i t h e t  arbitrary. guarantee o f success

They give

o f course

no

b u t neit11er do does es your model

method.   ampbellian If y o u think there a r e other n1ethods

1vhich w i l l d o al alll that m y m o d e l s do I think i t is u p to y o u t o exhibit them. I h a v e already s a i d I d o n o t think merely formal considerations o f simplicity suff ffic icie ient nt because they d o n o t b y them a n d so o n a r e su

selves s u p p l y a n interpretation o f t h e theory as u p pl p l y p re re d ic i c t iio o ns ns i n extended a n d hence d o n o t s up a n e w field o f observables. I f   silnplicity were ex-

tended t o apply also t o interpretation

then I think

y o u would find y o u were after a l l U il g a m o d e l . Duhemist I think I c a n d o better than to appeal t o a vaguely defined sense o f  simplicity.

W e might

realize strong falsifiability i n t h e f o l l o w i n g way. Suppose w e a r e g i v e n a number o f a c c e p t e d state

 

44

NIODELS A N D   N

I ~ O G I

S

I N SCIENCE

ments correlating so me o f t h e P s ,vith sorlle o f t h e

O s. I f tIle c o n s e q u e n c e s of t h e fornlal t h e o r y a r e developed, i t m a y b e tIle case that t h e structure of

  lne

o f thenl appear fo forn rnla lall lly y sirrlilar to that of tIle

acce ac cept pted ed stat statem emen ents ts,, i n t h e sense that a o n e- t o- o n e correspondence between s y mbo ls o f t h e theory a n d t e r m s o f t h e o b s e r v a t i o n s t a t e m e n t s c a n b e found. I t ,viII then b e p os s i b l e to i d e l l t i f y some o f t h e P s with s y mb o ls o f t h e theory. T h e theory c a n then b e s a i d to explain t h e ac acce cept pted ed corr correl elat atio ions ns c O ll llta tain inin ing g thes th esee P-pred P-predica icates tes,, a n d i t m a y also b e c a p a b l e of gen erating ne,v a n d as ye yett unaccepted s e n t e n c e s con taining t h e P s a n d t h e r e f o r e o f making gerluinely n e w predictions. I t ,vas s u r e l y i n   lne s u c h ,vay equa uatio tions ns,, de deve velo lope ped d f o r explanation that Maxwell s eq

of el elec ectr trom omag agne neti ticc phenomena, w e r e s e e n t o explain also t h e t r a n s m i s s i o n of l i g h t , b e c a u s e their solutions w e r e wave equations formally similar t o equations of t h e wave t h e o r y o f l i g h t .   ampbellian I c a n see s on le o b j e c t i o n s to this . First, i t is n o t clear ,vhat is meant b y   t h e structure o f some o f t h e co cons nseq eque uenc nces es of t h e t h e o r y being for similar

that

the

mally of acce ac cept pted ed st stat atem emen ents ts.. I n a case s u c h to as that o f M a x w e l l s equations i t was clear that there was such a similarity o r isomor he i ssom omor orph phii ssm m consisted in. B u t phism, a n d what t he i t is n o t easy t o say i n general h o w o n e ,v ,vol ollid lid recog

n i z e a situation o f i s o m o r p h i s m , f o r e x a m p l e , h o w much formal manipulation o f t h e t h e o r y would b e admitted before t h e id iden entif tific icat atio ions ns were were found t o b e  

T H E FUNCTION O F MODELS

45

possible? I t might even b e p os s ib l e t o s h o w that t h e occurrence o f isomorphism is trivial i n t h e sense that   y sufficiently rich theory could b e made isomor

phic   vi v i t h a n y gi give ven n ac acce cept pted ed st stat atem emen ents ts espe especi cial ally ly

if these were simple a n d few i n number.

You Yo u might of course b e a b l e t o e v a d e t h i s o b je c  tion b y tightening u p t h e formal criteria of i s o m o r  phism i n so some me way b u t even then i t is n o t clear that

success i n finding a n isomorphism would b e suffi cient i n i t s e l f t o c o n f i r m t h e v i de de r a pp pp l i c ab ab i l it it y o f t h e theory. Mere f o r m a l a p p e a r a n c e o f t h e wave

equation i n t w o different systems would n o t suffice

t o s h o w a correlation i n o n e theory unless s with Maxwell s equations there we werr e som e interpretation which made i t plau·sible t o a s s u m e that o n e s e t o f phenomena the optical was produced b y t h e other t h e electromagnetic-the interpretation i n t h i s case

being that o f wave propagation i n t h e material ether.

Whittaker gives the the example o f Mathieu s Equation which appears i n both t h e theory o f elliptic mem branes a n d t h e theory o f equilibrium o f a n acrobat i n a balancing act. I t would n o t b e suggested that any unific unificati ation on o f theory is accomplished b y noticing

this th is fact. Again f o r your program t o work significantly there must already b e a f ai airr ly well well-d -dee ve velo lope ped d sy stem o f relations i n t h e observation language. T h e less developed this is t h e more difficult i t will b e t o en  sure that a n a p p pa ar e en n t isomorphism is n o t accidental o r arbitrary. This means that t h e program will n o t

 

 

l\10DELS A N D ANALOGIES I N SCIENCE

b e universally applicable arId n o t applicable a t a l l t o

alreacly part o f s u e l l a systern i n t h e observatioll language. I t a lIllost s e e m s as though f o r t h e forillalist prograln t o  vark a t all a

observation predicates

previous s t a g e

 

t

theories s c i e n c e lllaking us usee   dels is necessary ill order t l l a t a s u f f i c i e nt l y c o m plex observatioll language sllall h a v e b e e l l bllilt u p .

Tllat tllis is t h e case 1voulel b e a d i l l i t t e d b y tll0se w h o regard clas asssical ph phys ysic icss as a n observation lallg u a g e f o r whicll n o further theoretical rnodels a r e

possible even though c la ssica l p hy s ic s itse itself lf COIISists o f theories with models f r o m t h e POillt o f view o f t h e observation language o f commOl1 disco discollrse llrse.. T h e d e sc sc r ip ip t i o l l y ou ou n o v g i v e o f t h e forll1alist program do does es l10t i n a n y case provide ne nece cess ssar ary y cr criiteria f o r a t h e o ry fo r o n t he he f or or ma ma li li s t vie v i t c a n never b e more than a that a satisfactory isomorphisnl is found. Whenever i t is found there is a spectacular unificatio11 o f t w o o r l l l o r e p re re--

disco11 disc o11ne necte cted d fi fiel elds ds as i n optics a n d electroInagnetisln b u t s u c h t h e o r e t i c a l d e v e l o p l n e n t s a r e

ViOllSly

exceptions 1vhich cannot b e systematically sought for.   llhemisl B u t o f course   a l l know tl1at t h e

progress o f s c i e n c e is n o t a mechanically systematic affair b u t depends partly o n hunches intuitions ancl guesswork lucky accidents i f y o u like a n d I d o n o t think m y account i n v o l v e s a greater proportion o f these than anybody else s. I a m i n fact prepared t o accept that much o f t h e progress o f s c i e n c e d o e s d epend o n these things a n d t o say that t h e requirement  

T H E FUNCTION O F lVIODELS

47

o f falsifiability i n sense   is t o o strol1g i f i t is taken

t o Inean that theories o f this kind c a n b e SOllght f o r systen1atically. After all

spec sp ecta tacu cula larr predi pr edict ctio ions ns

i n ob obsserva ervati tio o11 11al al dOll1ains ou outs tsic icle le tI tIlle o ri r i gi g i na na l r an an ge ge o f a t l l e o r y a r e i n fact rare i n sc scie ierl rlce ce aIlcl callnot b e

regarded as Ilec Ileces essa sary ry logica logicall conditions f o r a t h e o r y .

suggest that whet11er a theory is recluired t o b e falsifiable i n this strong sense wi will ll depend   11 t h e i11itiall cOln tia cOlnpl plex exit ity y o f th the e corr co rrel elat atio ions ns i n t h e observatio11  

he p re r e di di ca ca te te s o f ordi language. I f this C 11tains only t he ar1d

nary language, prescientific correlations betvveen them, i t is likely th.at weak falsifiability  viII n o t b e

sufficient f o r a genuine t11eory. F o r if correlations between only a f e w O s a r e k n o w n , 1 10 theory o f type   will b e a b l e t o predict a n y more, a n d a theory ex-

plaining tile correlations b e t w e e n t h e O s remaills imprisoned within t h e same limited observational situations. I f however t h e o b bss e err va v a ti t i on on l a an n gu gu a ag g e is already co mp l ex -i f i t is f o r exan1ple t h e language o f classical p hy h y si s i cs c s -t - t h en e n i t is p o s s i b l e that t h e form a l theory m a y go O I l f o r a long time pro rovi vid di1 i11g 1g in-

t er er es es ti t i n g c or or re re la la ti ti o ns ns b e t w e e n

n e w observational

situations w hi h i c h a r e still still de desc scri ri·b ·bed ed b y tIle sa same me pred icates b e t w e e n , f o r e x a m p l e , v a r i o u s k i n d s o f par

ticles described   charge,

tum

t h e classical predicates  lnass

and  spin.

IJ:? echaniC>s

Parts o f t h e theory o f quan

m a y well b e purely formal, a n d y e t

falsifiable i n th this is sense.   ampbellian T h i s is a n interesting suggestion

a n d i t w o u l d n e e d a f a r more detailed investigation

 

48

MODELS   N D  N

IJOGIESI N SCIENCE

than we c a n undertake now. B u t I should like t o introduce s o me exam examp p le less from quantum physics to there

of

indicate that may be more model thinking i n i t than a r e r e c o g n i z e delemel1ts b y your school of thought

I t  s usually claimed that a t least o n t h e

so-called Copenhagen view qualltunl tlleory  s a n exam pIe o f a n accepted a n d useful theory i n which m o d e l s h a v e been abandoned a n d   vhich therefore proves that models a r e n o t e s s en t i al to t h e progress of theo theorr ie ies. s.   nd n d i t s certainly t r u e t h a t t h e Copen hagen view c a n b e regarded  s a f o r m a l i s t v i e w o f

quantum theory i n that i t re refr frai ains ns fro fro nl making a n y interpretations of t h e fo form rmal alis ism m of t h e t h e o r y e x c e p t  s

can be made

such directlyus i nthat t e rwhat m s o fstands classicianl physics. I t need n o t trouble p l ac e o f t h e observation language here  s n o t ordi nary descriptive language b u t t h e l a n g u a g e o f clas siccal physics which  s from a n o t h e r p o i n t o f v i e v si lr e ea ad dy y a g rre e e d that highly t h e o r e t i c a l f o r we h a v e a lr

what counts  s a n observation language  s pragmatically re rela lati tive ve.. B u t i t does n o t follo v that because t h e adheren ts o f t h e Copenhagen view refrain from

maki ma king ng in inte terp rpre reta tati tion onss when talking

  ollt

quan-

impli licit cit int interp erpret retati ations ons t u m t h e o r y they they also avo avo id imp

when actually using i t i n t h e process o f research. Many examples could b e g i v e n from t ec ech h n i cal cal pa

p e r s to s h o w that they d o n o t i n fact a v o id interpretations. L e t m e d e scr scr ib ibee a comparatively si sim m p le o n e which is t y p i c a l o f t h e kind o f argument that can n o t b e avoided when developments o f t h e t h e o r y a r e suggested.  

T H E FU N CTI O N O F MODELS

49

I n t e r m s o f classical physics acting   cre as t h e observation language i t is s o m e t i m e s po s s i bl e t o

describe certain phenomena as effects of charged particles for for exa exalnp lnple le elec electr tron ons. s. I t is never possible h o v ev e r to spea speak k i n classical terIllS of identifying a n i n d i v i d u a l electron   11 di diff ffer eren entt occasiolls o r i n particular o f di dist stin ingu guis ishi hill llg g t h e s t a t e o f a sys te m containing two elect electro rons ns i n gi give ven n positio11s from that i n w h i c h t h e el elec ectr tron onss h av avee changed places. Ac Acco cord rd-in.g to t h e Copenhagen vie v t en v Inust n o t

Inake a n y interpretation implying anythiIlg about t h e identity of individual el elec ectro trons. ns. If ho howe weve verr w wee d o n o t adhere t o t h i s view there a r e tw o p os si bl e

interpretations o f a situation i n which a n object can-

n o t b e re identified o n e ex exem empl plif ifie ied d by th e model 2

of identical billiard balls a n d t h e o t I l e r b y t h e model 2 of pounds shillings a n d pence i n a bank balance. I n t h e case of identical billiard balls iiff we a r e n o t i n a p o s i t i o n to o b s e r v e thenl continuously we cannot i n practice distinguish a situation i n which two b all s 4re i n t w o g iv iven en p o c k e t s from a situation a t a later time i n which t h e y h a v e changed places. B u t t h e t w o s i t u a t i o n s a r e i n f a c t d i f f e r e n t a n d i f we were concerned  vith t h e number of arrangements

o f t wo b al ls i n t h e two p ock ets we we should h a v e t o

differ diff eren entt arra arrang ngel elne nent nts. s. With pounds shillings a n d pence i n a bank balance how-

count them as

is

t\

the

vve

n o t nlerely that cannot i n it ever p r a c t i c e r e iid d e n t i f y a g i v e n case pound appearing i n t h e credit column b u t that there is n o sense i n speaking

o f t h e s e l f iid d e n t i t y o f t h i s pound

a n d of asking

 

 

N D A NA N A LO LO GI GI ES E S I N SCIENCE MODELS A ND

 vhere i t reappears i n another column o r  vhetller

i t is t h e pound paid over t h e counter yesterday. I n

t hi hiss case t h e n U l n b e r o f vays o f ar arra rall llgi girl rlg g t vo Llnit pounds i n different places i n a column ·is just one a n d there is n o sense i n sp spea eaki kill llg g of anotller arra11ge Inent i n  vhich tIl tIley ey h.ave changed places. Units  vhich behave i n t h i s w a y conform to t h e so-called FermiDirac statistics alld n o t t o t h e s t a t i s t i c s o f o b j e c t s

h a Villg self-iden tity. I f tIle Copenhagen v i e w w i t l l regard t o electrons were adhered to we should b e unable t o say  vhich

o f t h e s e t vo models o f indistinguishability  v  vaas atpo cau s e w e should n o t   e i n a position propriate b e cau   ~

a n y models a t all. B u t i n f a c t w e f i n d t h e follo v

i n g argument very frequelltly used. W e a r e unable t o identify individual electrons hence i t is meaning-

less t o s p e a k o f t h e s el elff-id iden enti tity ty of elec electr tron onss hence shillings a n d pence i n a balance a n d n o t like indistinguishable billiard balls a n d hence they conform t o Fermi Dirac statistics. T h e last step o f t h i s argument c a n b e m a d e t o yield observable predictions since there a r e v a r i o u s waxs electrons a r e like pounds

i n which t h e behavior o f e n t i t i e s satisfying Fermiati s t i cs is different i n classically observable Dirac s t ati

ways from t hose hose sa sati tisf sfyi ying ng t h e statistics o f ordinary objects. B u t t h e argument ill il l spite of i t s a gn gno o s t i ci ciss m about w ha ha t c a an n no n o t b e observed does i n fact involve

a n interpretation a n d a c h o i c e between t w o different

models a nd n d w iitt h o u t th this is c l l 0 i c e t h e observable pre dictions cannot b e derived. T h e crucial step from

 

T H E F U N C T I O N O F MODELS

51

interpretation i n t h e argument occurs / \Then for ma l v}1at i sm t otIle o bs bs e err v ve er ca all l n no o t do-narllely nlake cert ce rtai ain n dis distir tirlct lction ion-is -is taken to b e a property o f tIle

interpreted system namely tl1at there is 1 1 0 su suel elll distinction. Such a r g gu u l n e llll t s a rre e very COIll111 nly used i n quantuln theory t o derive observable resu results lts alld. a r e sufficient t o sho v tl1at t h e theory is n o t as a

  vh v h o l e a counter-exaluple to t h e vie\v that interpre tations are

f o r predictions.

Another essential exanlple c a n b e g i v e n t o indicate t h e

inadecluacy o f tIle Copenhagen vie\v \vhi \vhich ch was developed to deal \vitl1 t h e paradoxes of elenlelltary quantum theory a n d ha hass never been consistently ad ad--

hered t o i n t h e later developments o f quantum field

theory. I n t h e case o f Dirac s predictiorl o f t h e posi-

tron

n o t Drlly 1vas a n i n ntt e err p prr e ett a att iiv v e t11eory success-

ful b u t also tIle s a m e theory treated formally would ee n r e eff u utt e ed d a n d discarded. TIle su have b ee succ cces essf sful ul preprediction arose as f01101VS. T h e equations o f motions o f both classical a n d q u a n t u l l l charged particles admit o f s o oll u utt iio on nss representing particles with either

positive o r negative energy. I n clas classical sical pl plly lysi sics cs ho vthe e oc occu curr rren ence ce o f negative energy solutions c a n ever th b e ignored since i n class assical ph phys ysic icss energy values chang ch ange e co cont ntin inuo uous usly ly a n d if a particle is once taken t o h a v e p o s i t i v e energy i t c a n never reach a neg neg at ativ ivee energy state. I n qllantum physics 11owever energy

c ha hang nges es t a k e p l a c e discontinuously. T h u s a n electron m a y j ump from o n e energy state t o another a n d negative states a r e as ac ce s s i b l e as p o s i tiv tiv e . Now i f

 

52

MODELS A N D ANALOGIES I N SCIENCE

t h e theory o f these equations o f motion is takerl

i n a f o r m a l s e n se

t h e nonappearance o f negative

energy particles i n a:qy known experiment would

theo eory ry.. Dirac Dirac ho veve veverr count as a refutation o f t h e th nlade a n interpretation o f t h e theory which depended o n t h e idea that e a c h o f t h e pos possibl siblee n.eg .egativ ativee states is already filled b y a n electron wllich is n o t

observable as long as i t r e l l l a i n s ill t h i s s t a t e b u t  vh  v h i c h b e c o m e s o b s e r v a b l e i f i t is knocked o u t o f t h e state leaving a  hole i n t h e n e g a t i v e sta tes which is also obse observ rvab able le.. By conlbining t h e t w o neg atives provided b y n ega ega tiv tiv e energy a n d t h e 110tion o f

  hole,

the hole carl b e expected to b e l l a v e like a

parr ttii cl pa cle e of positive energy a n d i t w i l l also h a v e posit i v e c h a r g e . This predicted particle t h e positron,

 vas il

f act o b s e r v e d a n d hence tIle i 11ter pret ed theory both made a s u c ce s s f ul predictioll a n d explained t h e previous nonappearance o f negative energy particles which threatened theory regarded formally.

to r e f u t e t h e

D Llhemist: I t m a y b e true that there a r e s t i l l s o m e preformalist argulnents used i n quantu m theory b u t you cannot m a aii n ntt a aii n that i n gelleral quantum theory supports your case that m o d e l s a r e e s s e n t i a l . T h e fact that h e r e t h e m a t h e m a t i c a l f o r m a l i s m m a ay y

sometimes b e usefully interpreted i n terlllS o f   aves a n d sometimes in terms o f p a rtic rticle less a n d t I l a t t h e s e

models contradict each otller althougll t h e formali sm is self consistent sho vs that t h e models cannot b e e s s e n t i a l t o t h e logic o f t h e theory.   ~ h e theory

 

T H E F UNC T I ON O F MODELS

 

is here t h e formalism, n o t t h e partial interpretations, such as those i n you{ examples, although t h e s e m a y b e useful f o r special a n d l i m i t e d problems.

  ampbellian I h a v e t o a g r e e that t h e situation i n quantum theory is peculiar frOln nlY point o f view.

Perhaps I c a n p u t i t t h i s way i n t h e t e r m i n o l o g y I

introduced earlier. T h e particle nlodel   m o d e l ~ hass ha s o m e p o s i t i v e a n a l o g y with atomic phenomena a n d

s o m e neg neg ativ ativee a na l og y, a n d t h e s a m e a p p l i e s t o t h e wave model 2 • Much of t h e p ar ar ttii c cll e m o od de ell s positive a n a l o g y is t h e wave model s negative a11alogy a n d vice versa, a n d this is w h y t h e t wo m o d e l s appear to b e contradictory. I f t h a t w e r e a l l there were t o say, we could simply e x t r a c t t h e t w o sets o f p o s i t i v e a n a l o g i e s a n d drop a l l talk about particles a n d waves, b u t that is n o t a l l there is t o

say, b e c a u s e i n both cases there a r e s t i l l f e a t u r e s about which we d o n o t know whether t h e y a r e positive o r ne nega gati tive ve anal nal ogi giees. A n d i t is i n arguing i n t e r m s o f t he s e f e a t u r e s that t h e particle a n d 1vave models a r e s t i l l e s s ent i a l , supplemented b y t h e hunches p hy sic i sts ha have ve acquired about wheTl to argue i n t e r m s o f o n e a n d when t h e o t h e r . A n d , as yOll h a v e s u g g e s t e d e a r l i e r , developments i n quan-

t u m theory ,vhich appear t o b e novel   i n t h e sense o f falsifiability

  m a y actually b e r e s u l t s o f 11 vel

deductions within parts o f t h e t h e o r y already inter-

preted, a n d hence b e only what I h a v e c a l l e d e xte n s i ons o f t y p e   These a r e surely going t o y i e l d

diminishing returns, a n d a n y quantum theorist w h o

 

54

lVI0DELS A N D ANi\LOGIES I N SCIENCE

adopts lIlY POillt o f vie\N o n 111 dels  v  viiII pres presuI uI1l 1lab ably ly

b e di dissa ssatis tisfi fied ed vitI vitIll t h e state o f tIle tlleory until a n e w nI0del is found illcorporatil1g t h e positive allalogies

o f bo bott llll pa parr ttii cl cl e ess a n d vvaves b u t 11 t involving tlleir contradictions. B u t   d o n t slIppase either o f u s

 vishes t o rest h i s argulnents o n Cllrrent displItes i n qualltulll theory o r o n speculatiollS about it itss future.   llhemist I t sOInetimes seell1S that o u r  v  vll llol olee dis 

 

difference opinioll about   vhat pute reduces t o kind o f tIl tIleor eory y viII pred0111i11ate il illl t h e future arld this is rather unprofitable t o speculate lIpOll

think

ho\vever

that y o u have been f o r c e d t o adnlit that ilnportant e xt xt e n nss iio on nss o f tlleory n y t a k e p l a c e \vith-

o u t t h e u s e o f models

a n d so

YOll

have effectively

a dm d m iitt t e ed d th a att models a r e n o t logically esselltial Y o u

could only continue t o nlaintain tllat t h e y a r e b y sho ving that a l l m y exalnples o f forll1al Inetllods a r e eit11er ullacceptable o r n o t purely farInal arld tllis

y o u llave n o t done. F o r m y p a r t I c a n see that i t m a y b e possible alld useful t o analyze

 

rnore detail   vhat

is involved il illl using models whel1 tIley a r e useel alld t o enquire whether there is allY justification f o r expectillg more systematic theory-construction with

tlleir a i d than without. T h i s would b e a n extension o f inductive logic i n application t o t h e hypothetico

deductive structure o f theories. I must confess that

inco concl nclus usiv ive e re resu sult ltss o f inductive logic i n view o f t h e in i n t h e simpler case o f empirical generalizations I a m

n o t very optilnistic a bout t h e success o f s u c h a n investigation.  

T H E FUNCTION O F MODELS

 

Cam pbellian I think two sor ts of problems have he g e n ne e r a l probt o b e distinguished here. The re is t he l e m o f t h e j ust ust ific ificat atii on o f induction o f which t h e just stif ifyi ying ng th the e inference to h y p o t h e s e s by problem o f ju means o f m o d e l s would b e a s p e c i a l case a n d I a g r e e

that t h e history o f in ind d uc u c ttiv ive e l og i c does n o t make t h e

p r o s p e c t s f o r t h i s v e r y bright B u t there a r e subsidia r y problems to this n a m e l y to to f i n d t h e conditions f o r t h e a s s e r t i o n o f a n analogy to e l u c i d a t e t h e nature o f arguments using models a n d analogies a n d to compare these arguments with t h os e us usua uall lly y called inductive i n a m or or e g e n e r a l sense. These problems

arise o n your view o f t h e nature o f t h e o r i e s  s well  s o n mine because even i f m o d e l s a r e merely disis

pensable ai ds t o d i s c ov e r y i t s t i l l p r o f i t a b l e to a s k h o w t h e y w o r k a n d i f t h i s is t o b e c a l l e d a p s y c h o -

logical

investigation i t m a y b e none t h e worse f o r

er tta a iin n lly y th the e use o f m o d e l s is n o t psychological that. C er i n t h e sense o f being wholly a n individual a n d subjective matter since communication a n d argument often go o n between scientists i n terms of m o d e l s a n d i f t hi s shows n o more than a uniformity i n t h e is

scientific temperament i t still worth investigating. course se foll follow ow that such a n inves  does not of cour tigation will provide anything like a n infallible method f o r t h e construction o f t h e o r i e s a any ny more than i t is t h e intention o f accounts o f m e t h o d s o f induction to p r o v i d e i n f a l l i b l e induction ma ma--

c h i n e s . A l l that

is being attempted is a n

analysis o f

what assumptions a r e made when analogies a r e used

 

56

lVIODELS   N D

i n science

N

LO OG G IE IE S I N SCIENCE

a n d h o w i t  s

that certain hypotheses

rather than others su sugg gges estt th them emse selv lves es   b y analogy. Whether t h e hypotheses thus suggested turn o u t t o be

t

u

is

s

always a matter f o r empirical inves-

tigation. T h e lo logi gicc of anal analo ogy lik likee t h e logic o f il1duction may b e descriptive without bei eill1g justi justificato ficatory. ry.

 

Mate Ma teri rial al Analogy TlVO

questions raised i n o u r dialogue m or o r e d e tta ac ch h e d investigation:

I What 2

is a n

nOlV

requIre

analogy?

anal alog ogy y valid? When is a n argument from an

I t is characteristic o f modern as opposed t o clas-

sical a n d n1ed n1edie ieva vall logi logicc that t h e a swer t o t h e first question is taken to b e either obvious o r unanalyzable

while t h e s e c o n d is taken to b e a question involving induction a n d therefore highly proble-

matic I n classical a n d medieval logic o n t h e other hand there is a certain amount o f allalysis o f types of anal analog ogy y b u t practically n o attempt a t justification o f t h e validity o f a n a l o g i c a l arguments although such arguments a r e frequently used A n d s i n c e n e i typ pes o f analogy n o r t h e sketchily ther t h e classical ty

defin de fined ed anal analog ogie iess o f Il Illo lode dern rn log log ic bear much resemblance to a n a l o g y as used i n reasoning froITI scientific m o d els els l we need to examine t h e relation o f t h i s problem t o t h e traditional di disc scus ussi sio ons I shall shall then p u t forward a definition o f t h e analogy r l tion il illl this chapter a n d go o n to consider tIle just ju stif ific icat atio ion n of anal analo o gica gicall  rgtlm nt i n t h e next. I t is as w e l l t o b e g i n b y c o n s i d e r i n g v e r y b r i e f l y I. I n this chapter t h e sense o f  model first chapter unless otherwise stated.

 

will always b e mode12 o f t h e

 

 

lVI0DELS A N D ANALOGIES I N SCIENCE

examples o f various types o f an anal alog ogy y fr fron onll t h e l i t e r a -

ture i n o r d e r to bring o u t t h e rnain issues. Example  

t\veen

t V

A n analogy l lay b e s a i d t o e x i s t b e-

objects i n v i r t u e o f their COlllrnon proper-

ties. Take, f o r e x a m p l e , tIle earth a n d t h e Uloail. Botll a r e l a r g e , so l i d , op aclue, spherical b o d i e s , receiving heat arId light frorn t h e s u n , revolving o n

their axes, a l l d gravitating t01vard otiler bodies. Tllese properties m a y b e s a i d to constitute their

positi posi tive ve allalogy. O n t h e other hand, t h e moon is srnaller than t h e earth, more volcanic, a n d ha hass n o atlnosphere a n d n o 1vater. I n th thes esee re resp spec ects ts there is negative analogy between them. T h u s t h e question o f what t h e analogy is i n t h i s case is fully answered

b y pointing t o t h e positive a n d negative allalogies,

a n d t h e dis discus cussio sion n passes inlInediately to t h e second question. Under what circul1lstances c a n we argue

from, f o r e xa xa mp m p le le , t h e p r e s enc enc e o f human beings o n t h e earth to their presence o n t h e moon?  rIle valid-

i t y o f such a n a r g u m e n t will depend, first, o n t h e extent o f t h e p o s i t i v e a n a l o g y compared  \v  \vith tIle negative  fo  forr example, i t is stronger f o r Venus than f o r t h e mOOll since Venus is more similar t o tIle earth and, second, o n t h e r e l a t i o n b e t w e e n t h e n e w property a n d t h e p r o p e r t i e s already known t o b e parts o f t h e positive o r negative a n a l o g y , r e s p e c -

tively. I f we h a v e reason t o think that t h e p r o p e r ties i n t h e p osit ositiv ivee anal analog ogy y a r e causally related, i n a f a v o r a b l e sense, t o t h e p r e s e n c e o f humans o n t h e e a r t h , t h e argument ,viII b e strollg. I f , o n t h e other

 

MATERIAL ANALOGY

59

hand t he h e p rro op pe e r ttii es e s o f t h e moon \vhicl1 a r e parts o f t h e negative analogy tend c a u s a l l y t o prevent t h e

presence o f hUlnans

 

t h e moon t h e argument \vill

b e \veak o r invalid I s h a l l return t o t h i s t y p e o f argument later

but

meanwhile tw two o features o f t h e analogy should b e noted First there  s a o n e to to o n e relation o f iclentity

o r differe11ce bet\veen a property o f o n e o f t h e ana-

logues a n d a corresponding property of t h e other and second t h e relation b et e t we w e en e n p rro op pe e rrtt iie e s of the same analogue  s that being properties o f t h e same og e ett he h e r v iitt h causal relations between these object t og properties Schematically: EARTH

  spherical · . causal

relations

atmosphere

1 1

humans

MOON

. spherical . . n o atmosphere ?

 

~

relations o f identity

o r difference

s h a l l f i n d that a common feature o f a l l t h e a n a l o g i e s w e disc uss will b e t h e a p p e a r a n c e o f t \ sorts o f dyadic relation a n d I s h a l l c a l l t h e s e horizont l a n d verti l rel tions respectively. Thus hor 

izontal relations will b e c o on n ce c e rn rn e ed d with identity a n d

difference ity

 

this th is case o r i n general with simil ra n d vertical relations will i n most cases b e

c lls l

s in

 

lVIODELS A ND ND A N NA A LO LO G GII E ES S I N SCIENCE

 

Example  

Consider next t h e s c ie ien ntifi tificc anal analog ogy y

chap aptt er er b bet etw w een e en t he he already referred t o i n t h e last ch properties of light a n d of so soun und d Here a g a i n we h a v e t\\ lists of properties with s o m e I n e n l b e r s o f o n e

liss t c o r r e s p o l 1 d i n g to s om e members of tIle other: li

 

PROPERTIES

PROPERTIES

OF

OF

SOUND

echoes

loudness

LIGHT

reflection

brightness

causal relations

1

pitch

color

detected b y e a r

detected b y eye

p ro r o pa pa g ga a te te d i n a i r

propagated i n e th t h er er

similar sim ilarity ity rela relation tionss

I n thi s e xa xam m p le

unlike   y there  s n o clear divi-

si on of t h e t \ lists into identities a n d differences since t h e p a i r s o f corresponding terms a r e never identical b u t only simil r There are o f course some terms o n both sides that have no corre corresp spond ondin ing g term o n t h e other b u t I s h a l l regard these  s special  s

he re re t h e s im im i l a r i t y relation cases o f   s i m i l a r s w he defined so s to include identities a n d differences T h e vertical relations between lllembers o f t h e same

list list a re s i n example   ca caus usal al rel relati ations ons I t ha hass been suggested i n t h e previous chapter that thiss an thi anal alog ogy y lik likee   c a n b e used i n arguments from y

similarities i n s om e r e s p e c t s t o similarity i n respect o f a p ro r o pe p e rt rt y k no n o wn w n to belong t o o n e analogue b u t  

iVIATERIAL ANALOGY

61

n o t y e t kno,vn t o belong to t h e otI1er. F o r example,

known similarities i n properties of ref l ecti oI l, re refr frac ac-tiOl1 a n d intensity m a y l e a d t o a p r e c l i c t i o n r e g a r d -

i n g color from properties involving pitch o r frorn t h e properties o f a i r t o t h o s e o f  ether.

Here, ho v-

or e c o om mp pll i c a att e d than i n exever, t h e situation is m or ample   i n that i t m a y n o t b e i n i t i a l l y obvious which property o f light corresponds witll ,vhich property o f sound   why wh y d o we m a k e c o l o r correspond with pitch?), o r i t m a y b e that a particular

property o f sound h a s n o correlate among t h e properties o f light, i n which case o n e m a y b e invented   ether

is initially n o t observed as t h e obvious cor-

relate o f air, i t is rather postul ted t o fill t h e place o f a m i s s i n g correlate among t h e p rro op pe e rrtt i e ess of l ight ight)) .

T h u s i n this example, unlike t h e first, tIle question of d e eff iin n iin n g t h e analogy relation a n d hence identify-

itss terms must c o m e b e f o r e t h e question of i n g it justification o f t h e analogical argument.   x mple C Consider n e x t a n analogy

a classifi si ficat cation ion syst ystem, em, o f a kind first statecl explicitly b y Aristotle:  ener

BIRD

FISH

wing

  n

 

lungs feathers

gills scales

Here t he he h o orr i z o on n tta a l relation filay b e o n e o r more

o r several similarities o f structure o r o f function,

 

 

1\10DELS A ND N D A NA N A LO L O GI G I E S I N SCIENCE

a n d each list m a y contain sonle items   v11icll h a v e n o

o r n o obvious

correspondent i n t h e other list; f o r

example without anatomical investigatioll i t is llot clear that birds legs correspo11d to anything il illl t h e

structure o f fish T h e vertical relatiorls m a y b e con-

ceived as n o m o r e than that o f w h o l e t o its p a r t s o r t h e y m a y b e regarded as c a u s a l relations clepending o n some theory o r interrelation o f parts determined b y evolutionary origin o r adaptation t o environ

ment. I n tllis latter case

t he analogy lnay b e u s ed

predictively as i n t h e previous ex amp l e- t o argue f o r instance frolll t h e kno vn structure o f a bird s

skeleton t o missing p a r t s o f a fish s k e l e t o n B u t again

t h e nature o f t h e analogy relation itself requires elucidation before c o n s i d e r i n g t h e validity o f t h e

argument.

Finally a n example o f a kind used a n d nlisused i n political rhetoric brings o u t b y con  x mple  

trast some importallt characteristics o f t h e three previous examples:

state citizens

father children

An

analogy

of

this

kind

is

apparently a n

assertion

that t h e relation between father a n d child is t h e

same i n many re resp spec ects ts as tl tlla latt be bet\ t\ve veen en state a n d citizens f o r e x a m p l e i n that t h e father is responsible f o r t h e m a i n t e n a n c e welfare alld d e f e n s e o f t h e child; a n d i t is further implied t h a t i t follows from this that other relations should also b e t h e same f o r

 

l\ fA TERIA L ANALOGY

63

exalnple tllat t h e citizen o ves respect antl obedience to t h e state. There a r e s ev ever eral al diff differ eren ence cess between

this example a n d tIle previous ones. First o f all, its purpose is persuasive rather tI1all predictive. I t is n o t arguing from three k n o , v n t e r m s to o n e Ullkno\vn, as

is t h e case i n t h e first three exaillples; i t

is rather

I ointing o u t t h e c o n s e q u e ll llcc e s , o f a moral o r norrl1a tive character, \ v h i c h f o l l o v from t h e relations of

four ternlS a lr lr ea ea dy dy know known. n. Second, t h e vertical rela tion is n o t sp spec eciifi fica call lly y ca caus usal al.. 1- h heere a r e i n f ac t sev eral vertical re rela lati tion ons, s, pr prov ovid ider er-f -for or,, prot protec ecto torr-of of,, a n d as-

o n ; a n d t h e argument implicitly passes from so suc c h r ela elati tion onss ,vhicll a r e already recog serting some su nized to persuading t h e hearer that other relations

( o b e d i e n t - t o , etc.) f o l l o w f r o n l these. Third, there does n o t s e e m t o b e a n y h o r i z o n t a l relation o f s im i  larity between t h e terms, except i n v i r t u e of t e fact that t h e t vo pairs a r e related b y t h e same vertical relation. T h a t is t o say, t ere is n o horizontal rela tion independent of t h e vert.ical relations relations,, a n d here

this example differs from al alll t h e other three types, ,vhere t h e h o r i z o n t a l r e l a t i o n s o f similarity ,vere independent o f t h e vertical a n d could b e recognized before t h e vertical relations were known. I t seerns t o b e e x c l u s i v e l y a n a l o g i e s o f t h i s kind that Richard Robinson is thinking o f when h e asserts that allalogy

mathem hemati atical cal propo pr oporti rtiona onalli n a n y sense other than mat ity

i s m er er e l y t h e fact that some relations have more

than o n e example, 2 that is that

 

is t o   as c is t o

e v Metaphysics V, 1952, 466.

 

 

64

~ f O

L S

A ND N D A NA NA LO LO G GII ES ES I N SCIENCE

d is merely equivalent t o asserting t e e x i st st en en ce ce o f a relation R such that a R b a n d cRd B u t Robinson overlooks analogies o f t h e otl1er kinds   v e have men JJ

tioned, w h e r e t h e r e a re re s i m il i l a r it it y r e la la ti ti o ns ns aS aScc a n d bS d independent o f t h e vertical relations. A N A L O G Y AND

MATHEMATICAL

PROPORTION

A t this point w e c a n usually draw a distinctio11 b e t,veen t h e types o f analogy w e a r e concerned with a n d t h e relation o f m at at h he em ma a ttii ca c a l p rro op po o rrtt iio on na a llii tty y.

T h e t vo kinds o f relation have often been thought t o b e closely connected, as is indicated b y t h e fact that t h e Greek word f o r  proportion is analogia

A n d t h e r e l a t i o n s d o indeed have some formal r e -

semblances ,vhich have been presupposed i n t h e

110

tation f o r t h e four-term relation already adopted i n

th e previous chapter. L e t u s represent t h e relation   a is t o b as c is t o

d

 

by

c

a

b

 

where a a n d b a r e a n y t vo terms

taken from a list representing o n e analogue i n ex ex-amples A B o r C including t h e heading, a n d c a n d he c o orr rre e ssp po on nd dii n ng g terms taken f r o m t h e other d a r e t he

list. F o r example,  pitch is t o SOU d as color is t o light, wing is t o bird as fin is t o fish o r  wing is t o feather as fin is t o scales :

pitch . . color light sound

fin

,vIng

:: fish

bird

fin feathers :: scales· wIng

 

MATERIAL ANALOGY

 

This gen gener eral aliz ized ed anal analog ogy y relation ha hass t h e follo \ving

form fo rmal al char charac actt eri eri s t ics ics i n common  vith numerical proportionality:  

W e wi sh to say that t h e a n a l o g y relation is  

reflexzve

t at

 S

a

to say b

a :: b

although this is a

t r i v i a l case o f t h e g e n e r a l r e l a t i o n . 2 W e w i s h t o say that t h e a n a l o g y relation

 S

. .   e a symmetrzcal f o r If b :: d then d :: b · 3 W e w i s h t o say that t h e analogy relation c a n c b e inverted f o r if   i :: d 4

a

then b

 

d

W e c a n c o m p a r e t h e additive property o f n u -

merical p r o p o r t i o n w i t h t h e r e s u l t s o f taking t h e logical S t l m of t e r m s o f a n analogy. Just as we h a v e ro p po o rt rt i o n t h a att i n numerical p ro

then