reasoning as imagination

8/3/2019 Reasoning as Imagination

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Interchange, Vol. 16, No. 1 (Spring, 1985), 14--26

R easoning as Im agination

D a v i d N . P e r k i n sHarvard University

We think of reasoning and imagining as very different acts o f mind. The personinclined to and skilled at one may shun or handle clumsily the other. Sometimesthe two are cast not jus t as strangers to one another but as enemies. Yourimagination may foul your reasoning with fancy while your reasoning tethersyour imagining with constraints.Our everyday language testifies to the popular perception of an uneasyrelationship. It' s commonplace to say something like, " Ralph is reasonableenough, but he lacks imagination." While such statements suggest that reason-ableness and imaginativeness are independent abilities or proclivities, otherremarks treat them as contraries: "Ralph's tOO reasonable; he needs moreimagination" or "Ralph's too imaginative; he needs to be a little more rationalabout things."We can easily see why a clash of reasonableness and imagination makessurface sense; it matches our immediate associations with the two. "Imagina-tion" connotes the counterfactual and the unruly, both in a vague senseunreasonable. Reasoning connotes a realistic and rule-bound activity, whichseemingly must therefore be unimaginative. But maybe we're making a mistakehere. Perhaps imagining has a good deal to do with sound reasoning, andreasoning a good deal to do with adventurous imagining. At points in TheMind's Best Work (Perkins, 1981), I've argued the latter thesis. Here I want toexplore the former.For a first cut, one can easily argue that imagination is really a friend of thecourt of rationality. By imagining possible courses of action or possible inter-pretations of data we can help ourselves to come to well-reasoned decisions.However, to forecast some conclusions, a deeper cut reveals that the supposedrivalry of reasoning and imagining does not just dissolve into amity. Reasoningindeed depends crucially on imagining, but good reasoning requires the artfulhandling of imagination, which can as easily divert as support the course ofreasoning if left to run its course willy-nilly. Especially interesting here are thespecific hazards imagination presents and the defences reasoning erects againstthem.Before making this second cut, it 's fair to ask, Why bother? First of all,there's an attempt to get a better hold on the way things are, revising ourperception of indifference or rivalry toward a more realistic appreciation of howreasoning and imagining relate. Second, there 's a moral to be drawn about howeasily, and misleadingly, our surface perceptions of concepts such as reasoningand imagining dominate and obscure relations that are much more complex.14 Interchange 16/l9 Ontario Institute for Studies in Education 1985


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REASONING AS IMAGINATION 15

F i n a ll y , t h e r e i s e v e n p r a c t i c a l i m p o r t . E n c o u r a g i n g p e o p l e t o b e m o r e r e a s o n -a b l e i n e d u c a t i o n a l a n d o t h e r c o n t e x t s m e a n s e n c o u r a g i n g t h e m t o b e m o r ei m a g i n a t i v e i n c e r t a i n w a y s , a v i e w n o t s o o f t e n h e a r d .C o m i n g to T e r m sA g o o d b e g i n n in g r e q u i r e s a b r i e f d i v e r s i o n c o n c e r n i n g t e r m i n o l o g y . W e c a n ' tr e c k l e s s l y c o n s t r u e h o w i m a g i n a t i o n m i g h t f i g u r e i n r e a s o n i n g w i t h o u t b e i n gc l e a r a b o u t w h a t t h e t w o m e a n . S o m e t i m e s p e o p l e u s e r e a s o n i n g t o r e f e rb r o a d l y t o s y s t e m a t i c t h in k i n g. I n th e p r e s e n t c o n t e x t , h o w e v e r , re a s o n i n g w i llm e a n t h e c o n s t r u c t i o n o f a r g u m e n t s t o w a r d j u s ti f y in g o r d e r iv i n g a n d ju s t if y i n gc o n c l u s io n s . F o r i n s ta n c e , w h e n w e d e v i s e a n a rg u m e n t t o p r o v e a t h e o r e m o rd e r i v e a f o r m u l a i n m a t h e m a t i c s , w e a r e r e a s o n i n g . W h e n , i n t r y in g t o d e c i d ef o r w h o m t o v o t e o r w h e t h e r t o b u y t hi s c a r o r t h a t, w e c o n s i d e r r e a s o n sf a v o ri n g o n e c o u r s e o r a n o t h e r , w e a r e r e a s o n in g .I n o n e o f i ts s e n s e s , t o i m a g i n e i s s i m p l y to i n v e n t . W h e n w e w o r r y t h a t" R a l p h l a c k s i m a g i n a t i o n , " i t i s R a l p h ' s p o w e r s o f i n v e n t i v e n e s s t h a t fa ll s h o r t ,n o t h i s s e n s o r y i m a g e r y . B u t i m a g in i n g c a n a l s o m e a n f o r m i n g a m e n t a l i m a g ew i t h n o t r a c e o f in v e n t i v e n e s s , a s w h e n y o u i m a g in e y o u r m o t h e r ' s f a c e o r th el o o k o f y o u r l iv in g r o o m . W h e t h e r b y s u c h i m a g in in g o n e g a in s a n y t h in g b e y o n dt h e m e r e v e r b a l p o s i ti n g o f d e s c r i p ti v e i n f o r m a t i o n h a s b e e n c o n t r o v e r s i a l i np s y c h o l o g y . H o w e v e r , K o s s l y n ( 19 80 ) o f fe r s s u b st a n ti a l e v i d e n c e t h a t m e n t a li m a g e s d o o f f e r in f o rm a t i o n a n d b e h a v e i n c e r ta i n o t h e r r e s p e c t s i n th e m a n n e ro f p i c t u r e s . K o l e r s a n d S m y t h e ( 1 97 9), w h i l e n o t a c c e p t i n g K o s s l y n ' s p o s i t i o n ,a r g u e t h a t , a t t h e l e a s t , i m a g e s f u n c t i o n a s a p e r s o n a l s y m b o l s y s t e m t h a ts o m e t i m e s o p e r a t e s i n p ic t u re - li k e a n d s o m e t i m e s i n o t h e r w a y s , d e p e n d i n g o nt h e s y m b o l i c fu n c t i o n . T h e s e v i e w s t e s t i fy t h a t t h e s e c o n d s e n s e o f i m a g i n ed e s e r v e s t o b e t a k e n s e r i o u s l y a n d n o t d i sm i s s e d a s a n a m e f o r a n e p i p h e n o m -e n o n . M o r e o v e r , t h e s e n s e m i g h t e v e n b e e x t e n d e d t o in c l u d e th e f o r m a t io n o f am e n t a l m o d e l o f a n y s o r t, i n v i e w o f th e i m p o r t an t r o le t h a t m e n t a l m o d e l sa p p e a r t o p l a y i n h u m a n c o g n i t i o n ( G e n t n e r & S t e v e n s , 1 9 8 3 ; J o h n s o n - L a i r d ,1983).

O f c o u r s e , t h e t w o s e n s e s o f im a g i ne c a n c o - o c c u r ; w h e n i m ag in in g c a s t l e sin th e a i r o r a n e w t h e o r e m i n g e o m e t r y , y o u m a y f o r m a s e n s o r y im a g e a n di n v e n t a t t h e s a m e t i m e . F o r t h e p r e s e n t d i s c u s s i o n , I w i l l t a k e t h e i n v e n t i v es e n s e o f im a g i n e a s p r i m a r y , w h i l e m e n t i o n i n g i m a g e r y o r o t h e r s o r t s o f m o d e lm a k i n g f r o m t i m e t o ti m e . T h i s d e f i ni ti o n a l h o u s e k e e p i n g d o n e , l e t ' s m o v e t oc a s e s .

I m a g in a t io n in M a t h e m a t i c a l R e a s o n in gA t l e a s t in t h e d o m a i n o f m a t h e m a t i ca l r e a so n i n g , m o s t w o u l d i m m e d i a te l ya c k n o w l e d g e t h e i m p o r t a n c e o f i m a g in a ti on . T h e e v i d e n t c h a ll e ng e p o s e d b ym a n y m a t h e m a t ic a l p r o b l e m s p l a in l y ca ll s u p o n t h e p r o b l e m s o l v e r ' s p o w e r s o fi n v e n t io n . T o b e s u r e , i f a m a t h e m a t i c a l p r o b l e m a l l o w s a s o l u t io n b y s h e e rg u e s s w o r k o r s y s t e m a t i c c o m p u t a t i o n , w i th n o n e e d t o d i s c o v e r a p a t h f ro mg i v e n s t o a n s w e r , t h e n i m a g i n a ti o n n e e d p l a y n o r o le . B u t v i r tu a l ly a ll s e ri o u s


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16 DA VI D N. PERKINS

mathematical problems do not surrender so easily, else they would not count asserious.Readily granting the importance of imagination to mathematics, we mustalso feel that mathematicians' imaginings pose no insurmountable threat totheir logic. Yet, since imagining is an unruly force, how do they escape thisthreat and what defences do they deploy? A look at some elementary examplesof mathematical imagination at work will suggest an answer.The standard derivation of the formula for the area of a triangle, one half hebase times the height, involves two inventive transformat ions of the problem.Beginning with an arbitrary triangle, you attach a duplicate of it to itself alongone edge, making a parallelogram as in Figure 1. The parallelogram, of course,has twice the area of the original triangle. Then you work with the parallel-ogram, slicing a triangle off one end and moving it to the other to form arectangle with the same base, height, and, of course, a reaas the parallelogram.You know the formula for the area of a rectangle--base times the height--soyou have reduced the original triangle problem to a known one.

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F i g u r e 1 . T h e a r e a o f a tr i a n g l e t r a n s f o r m e d i n t o 1/2 t h e a r e a o f a r e c t a n g l e .

For another example, a classic elementary problem of mathematics asksyou to find the sum of the numbers from 1 to 50, that is, 1 + 2 + 3 + 4 . . . + 50 =?. Adding up all the numbers serves the purpose, but does a better way exist?Not only does it, but presumably the young Gauss amazed his teacher bydivining such a method. Folding the series in half at its midpoint between 25 and26 yields the following:

1 + 2 + 3 + .. . + 2 3 + 24 + 25 (fold here)50+ 49+ 48+ . . . +28 + 27+ 26? = 5 1 + 5 1 + 5 1 + . . . + 5 1 + 5 1 + 5 1As the illustration shows, when you fold the series, the numbers that falltogether always sum to 51, making the solution transparent --51 25 = 1275.


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T h e s a m e f o l d in g t a c ti c , g e n e r a l i z e d , c a n j u s t i f y t h e f o r m u l a f o r th e s u m o f 1 t on , w h i ch i s s i mpl y n(n + 1)/2.B o t h c a s e s m a k e p l a i n h o w i m a g i n a t i o n , u n r u l y t h o u g h i t i s , c a n l e a v em a t h e m a t i c a l r e a s o n i n g u n m a r r e d . T o b e s u r e , i m a g in i n g o f t e n b r e a k s p a t-t e r n s , b u t t h e p a t t e r n s b r o k e n n e e d n o t b e r u l e s o f d e d u c t i o n . I n t h e tr ia n g lea r g u m e n t , f o r e x a m p l e , t h e p a t t e r n o f o n e f i g u re , t h e g i v e n tr ia n g l e, i s b r o k e nb y i n t r o d u c in g a m i r ro r - im a g e r e p li c a . I n th e s u m p r o b l e m , t h e p a t t e r n o f o n ec o n t i n u o u s s e q u e n c e o f t e r m s f r o m l e ft t o r ig h t is b r o k e n b y f o l d in g t h es e q u e n c e . B u t n e i th e r m o v e i n v o l v e s a n i m p r o p e r d e d u c t i o n . O n t h e c o n t ra r y ,b y t ra n s f o r m i n g t h e representation o f t h e p r o b l e m , b o t h m o v e s d i s c lo s eo p p o r t u n i t i e s f o r d e d u c t i o n n o t f o r m e r l y a p p a r e n t .B o t h m o v e s , i n v o l v in g r e a r r a n g e m e n t s o f s p a ti a l l a y o u t , a l s o d e m o n s t r a t eh o w i m a g in i n g i n th e s e n s e o f m a n i p u l a t in g i m a g e s c a n f ig u r e i n m a t h e m a t i c a lr e a s o n i n g . I t ' s n o t s o , o f c o u r s e , t h a t i m a g e r y in it s m o s t s t r a ig h t f o r w a r d s e n s ea l w a y s a p p l ie s . I n a l g e b r a ic b r a n c h e s o f m a t h e m a t i c s , m a n y , p e r h a p s m o s t ,t h e o r e m s a r e p r o v e d p u r e l y b y t h e m a n i p u l a ti o n o f n o ta t io n a l e x p r e s s i o n s ,w h i c h i n v o l v e s i m a g i n in g i n t h e s e n s e o f in v e n t i n g b u t n o t o b v i o u s l y i n t h es e n s e o f f o r m i n g d i a g r a m s . A r n h e i m (1 96 9) h a s a r g u e d t h a t e v e n i n c a s e sw h e r e i m a g e r y d o e s n o t p l a y a n o b v i o u s r o l e i n t h e o v e r t e x p r e s s i o n o ft h o u g h t , i t o p e r a t e s c o v e r t l y a s a p r i n c ip a l v e h i c l e o f i n q u i r y a n d u n d e r s t a n d -i n g . T h i s m a y b e s o e v e n i n s u c h a n a u s t e r e d o m a i n a s p u r e a l g e b r a . I n a n yc a s e , i f b y s o m e o d d m a g ic w e h a d n e v e r h a d p o w e r s o f i m a g e r y, c l e ar l y o u rm a t h e m a t i c s w o u l d b e d r a m a t ic a l ly i m p o v e r i sh e d .

Hazards o f the Mathem at ica l Imag inat ionA l l t h is m i g h t s u g g e s t t h a t i m a g i n a ti o n n e v e r d o e s a n y m i s c h i e f i n m a t h e m a -t i cs . B u t it c a n . L e t u s l o o k a t a n e x a m p l e t h a t c o n v e n i e n t l y i l l u st ra t e s s e v e r a lh a z a r d s a t o n c e . A p r o b l e m a p p e a r e d o n t h e P r e l i m i n a r y S c h o l a s t i c A p t i t u d eT e s t ( S A T ) f o r 1 98 0 t h a t p r o v o k e d c o n s i d e r a b le m e d i a a t t en t io n b e c a u s e t h eo f fi ci a l a n s w e r t u r n e d o u t t o b e w r o n g ( D a v i s , 1 98 4, c h a p . 1 5 ) . T h e p r o b l e mr a n s o m e t h i n g a s f o l l o w s . T a k e a p y r a m i d w i t h a s q u a r e b a s e a n d t r i a n g u l a rs i d e s, a l l e d g e s o f t h e s a m e l e n g th . N o w t a k e a n o t h e r p y r a m i d w i t h a ll s id e st ri a n g u la r a n d a l l e d g e s o f t h e s a m e l e n g t h a s b e f o r e . T h e t r i a n g u la r s i d e s o fb o t h p y r a m i d s a r e , o f c o u r s e , e x a c t l y th e s a m e s iz e , b e in g b o u n d e d b y t h r e ee d g e s o f t h e s a m e l e ng th . M a k e a c o m p o s i t e fo r m b y g lu in g th e t w o p y r a m i d st o g e t h e r a l o n g a m a t c h i n g f a c e , a s i l lu s t ra t e d i n F i g u r e 2. T h e q u e s t i o n ? H o wm a n y s i d e s d o e s t h e r e s u l t i n g f o r m h a v e ?S A T ' s a n s w e r w a s s e v e n : th e s q u a r e - b a s e d p y r a m i d h a s f iv e s id e s a n d th ew h o l l y t r ia n g u l a r p y r a m i d f o u r , s o g lu i ng t h e m t o g e t h e r p u t s o n e s i d e f r o me a c h o n t h e i n si d e , l e a v in g f o u r a n d t h r e e o n t h e o u t s i d e f o r a to t a l o f s e v e n .U n f o r t u n a t e l y , t h is r e a s o n i n g o v e r l o o k s a p o s s i b i li t y . W h a t i f , w h e n y o u g l u et h e p y r a m i d s t o g e t h e r , t w o f a c e s t h a t m e e t a t a n e d g e a l s o h a p p e n t o b ep a r al le l 7 T h e n t o g e t h e r t h e y w o u l d f o r m j u s t o n e f a c e r a t h e r t h a n t w o . I n f a c t,e x a c t l y t h i s h a p p e n s .The hazard of a persuasive image. W h y w e r e m o s t t e s t -t a k e r s a s w e l l a s t h eS A T e x p e r t s d e c e i v e d ? B e c a u s e o f t h e p e r s u a s i v e n e s s o f a n i m ag e . S u p p o s e


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1 8 D A V I D N . P E R K I N S

w e i m a g i n e o r s k e t c h t h e p r o p o s e d c o n s t r u c t i o n , a s in F i g u r e 2 . T h e n , a t l e a s tt o m o s t p e o p l e , t h e s i d e s o f t h e s e c o n d p y r a m i d d o n o t look p a r a ll e l to a n y o fthe f i r s t . W e a r e mi s l ed by an i l l us ion .A s a r g u e d e a r l i e r , s t r i c t d e d u c t i o n i s o u r p r o t e c t i o n a g a i n s t u n r u l yi m a g i n a ti o n i n m a t h e m a t i c s . B u t th e p r o t e c t i o n o n l y w o r k s s o lo n g a s w e d o n ' tc o n f u s e h y p o t h e s i s w i t h p r o o f . S u p p o s e y o u c o n s i d e r th e p o s s i b i li t y t h a t t w oa d j a c e n t f a c e s m i g h t f al l i n t h e s a m e p l a n e a n d s o c o u n t a s o n e . Y o u p r o b a b l yr e j e c t t h is p o s s i b i l it y b e c a u s e t h e y d o n ' t look l ike t he y f a l l i n t he sam e p l ane .W el l , t h is i s a go od h yp o the s i s , b u t i t i s no t a p roof . I f you sh a rp ly d i s -c r im i n a te l og ic fr o m t h e a p p e a r a n c e o f a c o n s tr u c t io n , y o u a s k , " H o w c a n Iprove t h a t t h e f a c e s a r e n ' t p a r a ll e l, g r a n t e d t h a t t h e y d o n ' t l o o k p a r a l le l ? "Y o u d o n o t f a ll i n to t h e t r a p o f t h e p e r s u a s i v e i m a g e .

F i g u re 2 . A p y r a m i d w i t h a t r ia n g u l a r b a s e g l u e d t o a f a ce o f a p y r a m i d w i t h asq u are b ase .

The hazard that patterns once made are hard to break. Imagina t i on i s no ta n a r c h y . W h e n y o u im a g i n e , y o u t y p i c a ll y m a k e a n e w p a t t e rn . T h i s n e wp a t t e r n m a y b r e a k a p r e v i o u s p a t t e r n o r m a y b e f o r m e d ex nihilo. B u t t h ec a t c h i s t hi s: i f t h e n e w p a t t e r n t u r n s o u t n o t t o b e h e l p fu l , i t t o o m u s t b e c a s ta w a y . Y e t y o u ' v e i n v e s te d y o u r s e l f i n c o n s t ru c t i n g i t, y o u h a v e h ig h h o p e s f o rt h e i n s ig h t i t w i ll y ie l d , a n d c o n s e q u e n t l y y o u m a y f i n d y o u r s e l f t r a p p e d b y i t.Y o u m a y n e v e r e v e n t h i n k t o t ry t o b r e a k it. T h e f a c t th a t t h in k in g d e p e n d s o nt h e r e c al li n g o r c o m p o s i n g o f p a t te r n s w h i c h m a y , i n tu r n , n e e d t o b e b r o k e n i so n e o f t h e d e e p p a r a d o x e s o f m e n t a l f u n c ti o n i n g ( C f. P e r k i n s , 1 98 1, c h a p . 6 ).A g a in , t h e p y r a m i d p r o b l e m p r o v i d e s a n a p t e x a m p l e . S u p p o s e y o u h a v ea v o i d e d h a z a r d s o n e a n d t w o , s k e t c h i n g F i g u r e 2 b u t r e c o g n i z i n g t h a t t h ef i g u r e m i g h t m i s l e a d a n d s o s e e k i n g a p r o o f . H o w d o y o u p r o c e e d t h e n ? I t ' sm o s t n a t u r a l t o w o r k f r o m t h e c o n s t r u c t i o n y o u h a v e a n d t r y , t h r o u g hg e o m e t r y o r a l g e b ra , t o r e s o l v e th e q u e s t i o n o f t h e a d j a c e n t s i d e s. U n -fo r tuna t e ly , t h i s i s l i ke ly t o l ead i n to a t ang l e o f equ a t i ons .


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B u t i f y o u t h i n k t o a b a n d o n t h e c o n s t r u c t i o n y o u h a v e a n d s e e k a d i ff e re n tp r o b l e m r e p r e s e n t a t i o n , y o u m i g h t f in d a m o r e d i r e c t d e m o n s t r a t i o n . I n fa c t ,t he r e i s a co ns t ru c t i on t h a t sh ow s c l ea r l y t he pa r a l l e l s ides (Y oun g , 1982).I m a g in e a r o w o f p y r a m i d s w i t h s q u a r e b a s e s , a s i n F i g u re 3 . N o w t h i n k o f th es p a c e between t w o o f th e p y r a m i d s , h i g h l ig h t ed b y d o t t e d l in e s i n t h e f ig u re .N o t e t h a t t h i s i n - b e t w e e n f o r m h a s t h e r e q u i s i t e f o u r s i d e s w i t h s i x e q u a le d g e s . I t i s th e s e c o n d p y r a m i d , w i t h o n e f a c e f lu s h a g a in s t t h e f ir s t p y r a m i dj u s t a s i n F ig u r e 2 ! Y e t , w h e n y o u c o n s i d e r th e w h o l e r o w o f s q u a r e - b a s e dp y r a m i d s a n d t h e i r n e s t e d t r i a n g u l a r p y r a m i d s , c l e a r l y t h e t w o m e s h t o g e t h e rt o m a k e o n e c o n t i n u o u s ' b a r ' . S o w e s e e th a t t w o f a c e s o f t h e t ri an g u la rp y r a m i d i n f a c t f a ll p a r a ll e l t o t w o o f th e s q u a r e p y r a m i d , f o r m i n g s i n g le f a c e s .T h a t y i e l d s f iv e f a c e s i n a ll r a t h e r t h e n s e v e n .

F i g u r e 3 . A r o w o f p y r a m i d s w i t h s q u a r e b a s e s , t h e sp a c e s b e t w e e n t h e mf o r m i n g t r i a n g u l a r - b a s e d p y r a m i d s .

T h i s i m a g e , o f c o u r s e , n o m o r e p r o v e s t h a t t h e f a c e s l ie p ar a ll e l t h a n t h eo r ig i na l i m a g e p r o v e d t h a t t h e y d i d n o t . B u t i t i s a n e n t r y t o a p r o o f . W o r k i n gf r o m t h e n e w i m a g e, o n e c a n h a n d il y c o n s t r u c t t h e n e e d e d a r g u m e n t.The hazard of gestalt hunger. T h e c a s e o f t h e p y r a m i d s h o l d s y e t o n e m o r em o r a l . I n t h e p r e v i o u s p a r a g r a p h s , I s u g g e s t e d r e s o l v i n g t h e p r o b l e m b ym e a n s o f a n e w r e p r e s e n t a t io n t h a t m a d e t h e a n s w e r t ra n s p a r en t . W e a ll e n j o ys u c h r e s o l u t i o n s b e c a u s e t h e y s e e m p a r t i c u la r l y c l e a r a n d te ll in g . B u t s u p p o s ea p r o o f u s in g a l g e b r a w i t h t h e i ni ti al r e p r e s e n t a t i o n c a m e f a ir ly e a s il y . T h i sa l g eb r a ic p r o o f w o u l d d o t h e j o b , b u t w e ' d f in d it s o m e w h a t b l o o d l e s s a n d l e sssa t i s fy i ng . The f ac t i s t ha t t he v i sua l ges t a l t conv i nces us be t t e r t han shuf f l i nge q u a t i o n s .T h e g e s t a l t p s y c h o l o g i s t M a x W e r t h e i m e r ( 1 9 4 5 ) d e v e l o p e d s u c h a c o n -t r a s t a t l e n g th , q u e s t i o n i n g t h e m e r i t s o f p r o o f s b y c h a i n s o f o b s c u r e s y m b o l i ct r a n s f o r m a t i o n s w h i l e p r e fe r r in g t h o s e t h a t m a k e t h e e s s e n c e o f t h e p r o b l e m


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20 DAVID N. PERKINSc l ea r s o m e h o w , p e r h a p s b y m o t i v a t in g a n y n e c e s s a r y n o t a ti o na li zi n g w i t h ap o w e r f u l r e p r e s e n t a t i o n o r r e - re p r e s e n t a t i o n o f t h e p r o b l e m .

The sum of i n t ege r s p rob l em o u t l i ned ea r l i e r p ro v ide s a te l li ng i l lus t r a t iono f W e r t h e i m e r ' s co n t r a st . H o w d o y o u p r o v e th a t t h e su m 1 + 2 + 3 + . . . + ne q u a l s n ( n + 1)/2? One w ay in vo lve s t he ges t a l t so lu t i on o f fo ld ing t he se r i e s i nh a l f , w h i c h q u i c k l y l e a d s t o t h e f o r m u l a . B u t t h e s a m e c o n c l u s i o n c a n b ep r o v e d b y a t e c h n i q u e c a l le d m a t h e m a t i c a l i n d u c t io n . T h e r e ' s n o n e e d t oe x p l o r e t h e t e c h n i q u e i n d et a il h e r e , b u t b a s i c a l ly o n e h a s o n l y t o s h o w t h a t ( a)t h e f o r m u l a i n q u e s t i o n h o l d s f o r th e c a s e n = 1 , a n d ( b) i f i t h o l d s f o r a n y c a s en , t hen i t ho lds fo r t he case n + 1 . Th ese tw o s t ep s e s t ab l i sh t he r e su l t f o r a ll n .M a t h e m a t i c a l i n d u c t i o n i s a p o w e r f u l ta c t i c t h a t y i e l d s p r o o f s f o r a w i d e r a n g eo f c la i m s a b o u t t h e p o s i t iv e i n t e g e r s.

S o , w h a t i s t h e d i l e m m a ? I n b r i e f , w e h u n g e r f o r t h e n e a t c o m p e l l i n gg e s t a l t s o l u t i o n - - a n d r i g h t l y s o , b e c a u s e i t d o e s r e v e a l m o r e . W e f e e l , p e r -h a p s , s o m e w h a t d i s t ru s t f u l o f t h e e x t e n d e d a l g e b r ai c m a n i p u l a t io n t h a t y i e l d st h e s a m e c o n c l u s i o n . B u t o f t e n s u c h s o l u t i o n s a r e o u r o n l y r e c o u r s e . W ec a n n o t c a s u a l l y n e g l e c t c l e v e r n o t a t io n a l i n f e r e n c e , s i n c e a g o o d g e s t a lt s o l u -t io n m a y p r o v e h a r d t o c o m e b y . I n th i s s e n s e , w e c a n n o t a l w a y s a f f o r d t h el u x u r y o f f e e d i n g o u r g e s t a l t h u n g e r .I n s u m m a r y , a c l o s e l o o k d i s c l o s e s th a t m a t h e m a t i c a l r e a s o n i n g n e a r l ya l w a y s c a ll s f o r im a g i ni ng i n th e s e n s e o f in v e n t i v e n e s s a n d o f t e n c a ll s f o rimagin ing i n t he se nse o f i nven t ing an imagi s t ic r eprese n t a t i o n t ha t t r ans form so n e ' s u n d e r s t a n d i n g o f t h e p r o b l e m . B y a n d la r g e, s u c h i m a g in i ng d o e s n o tt h r e a t e n t h e s o u n d n e s s o f m a t h e m a t i c a l r e a s o n in g b e c a u s e it b r e a k s p a t t e r n so t h e r t h a n t h o s e c o n c e r n e d w i t h s t r i c t d e d u c t i v e i n f e r e n c e . B u t i m a g i n a t i o nd o e s n o t p l a y a w h o l l y b e n i g n r o le s i n c e i t p o s e s h a z a r d s o f th e p e r s u a s i v ei m a g e , h a r d t o b r e a k p a t t e r n s , a n d g e s t a l t h u n g e r . T h e s e h a z a r d s m a k e t h ec o n t r i b u t i o n o f i m a g i n a ti o n t o m a t h e m a t i c a l r e a s o n i n g a m a t t e r o f h a n d li n gimagina t ion c r i ti ca l ly , no t j us t w ie ld ing i t casua l ly .

Informa l Contrasted with Form al ReasoningT h e s a m e a r r a y o f q u e s t io n s d e s e r v e s e x p l o r a ti o n i n a n o t h e r d o m a i n o fr e a s o n in g , o n e I 'l l c al l i n f o rm a l o r e v e r y d a y r e a s o n in g . I n f o r m a l r e a s o n i n gr e f e r s t o j u s t if i c a ti o n o f c la i m s i n a n y c o n t e x t o t h e r t h a n m a t h e m a t i c a l, f o r m a ll o g ic a l, o r s y ll o g is t ic . M o s t o f t h e r e a s o n i n g w e d o i s i n fo r m a l . O c c a s i o n s o fi n fo r m a l r e a s o n i n g i n c l u d e , f o r i n s t a n c e , d e c i d in g f o r w h o m t o v o t e , w r i ti n g a ne s s a y i n w h i c h y o u a d v a n c e a n d d e f e n d a th e s i s , a r g u in g o v e r t h e m e r i t s o f ar e f e r e n d u m , f i g u r i n g o u t w h e t h e r t o b u y t h i s c a r o r t h a t , a n d d e t e r m i n i n gw h e t h e r t o t a k e a j o b t h a t h a s b e e n o f f er e d .O n e m i g h t a s s u m e a t f ir s t t h a t i m a g i n a ti o n p l a y s e x a c t l y t h e s a m e r o l e i nin formal r eason ing as i n fo rmal , supp os ing t ha t t he tw o d i f f e r on ly super f ic i a l-l y . S o m e h o l d t h a t i n fo r m a l r e a s o n i n g i s r e a l ly j u s t f o r m a l r e a s o n i n g i n c a s u a ld r e s s , w h e r e p r o b a b i li s ti c s t e p s a n d v a r i o u s f e a t u r e s o f n a t u r a l l a n g u a g ec o n c e a l t h e f o r m a l i n f e r e n t i a l s t r u c t u r e . B u t t h e r e a r e g o o d g r o u n d s f o r t h ec o n t r a r y v i e w t h a t i n f o r m a l r e a s o n i n g d i f f e r s f u n d a m e n t a l l y f r o m f o r m a lr eason ing .T h e c o n t r a s t b e t w e e n t h e t ro u b l e s p e o p l e e n c o u n t e r i n m a t h em a t i c a l a n d


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REASONING AS IMAGINATION 21i n i n f o r m a l r e a s o n i n g r e f l e c t s o m e f u n d a m e n t a l d i f f e r e n c e s b e t w e e n t h e l o g i c ald e m a n d s o f t h e tw o . M a t h e m a t i c a l a r g u m e n t , a n d f o r m a l a r g u m e n t i n g e n e ra l ,p r o c e e d s b y s t r i c t d e d u c t i o n f r o m g i v e n s . I n c o n t r a s t , i n f o r m a l a r g u m e n tp r o c e e d s b y p r o b a b l e i n f e r e n c e f r o m g i v e n s p l u s w h a t e v e r w o r l d k n o w l e d g et h e r e a s o n e r c a n b r i n g t o b e a r . T h i s f u n d a m e n t a l c o n t r a s t i n m a n n e r o fi n f e r e n c e h a s r e p e r c u s s i o n s s u c h a s t h e fo l lo w i n g :

I n f o r ma l a r g u me n t s c a n i n c o r p o r a t e d a t a f r o m a ll s o r t s o f u n e x p e c t e d s o u r c e sf o r b i d d e n to f o r ma l a r g u me n t , w h i c h mu s t w o r k o n l y w i t h t h e g i v e ns .Go o d i n f o r ma l a r g u me n t s a l mo s t a l wa y s h a v e t o a d d r e s s b o t h s id e s o f a c a s e ; ag o o d f o rma l a r g u me n t n e v e r d o e s , a s s u mi n g c o n s i s t e n t a x io ms , b e c a u s e f o r ma la r g u me n t s o n b o t h s id e s wo u l d a m o u n t t o a n o u t r i g h t i n c o n s is t e n c y .Go o d i n f o r ma l a rg u me n t s u s u a l ly n e e d t o b e h e d g e d ; t h e s t r i ct d e d u c t i o n o ff o r ma l a r g u me n t ma k e s t h i s u n n e c e s s a r y .L i n e s o f i n fo r ma l a r g u me n t h a v e t o b e s h o r t b e c a u s e , s i n ce e a c h s t e p i s o n l yp r o b a b l e , a l o n g c h a i n o f i n f e r e n c e s b e c o m e s v e r y u n r e l ia b l e ; p r o c e e d i n g b ys t r i c t d e d u c t i o n , a f o r ma l l i n e o f a r g u me n t c a n h a v e ma n y s t e p s w i t h o u tsacr i f ic ing re l iabi l i ty .An in fo rmal a rgument typ ica l ly needs many l ines o f r eason ing s ince no s ing lel ine has pe r fe c t r e li ab i li ty ; a fo rma l a rgum ent nee ds on ly one , w h ich se t t l e s thema t t e r .

A s t h e s e p o i n t s m a k e p l a i n , a g o o d i n f o r m a l a r g u m e n t h a s a s t r u c t u r e r a d i c a l lyd i f fe r e n t f r o m t h a t o f a g o o d f o r m a l a r g u m e n t . T h e q u e s t i o n r e m a i n s w h e t h e ra n d h o w , i n c o n s e q u e n c e , i m a g i n a t i o n f ig u r e s d i f f e r e n t ly i n t h e t w o .

Imagination in Informal ReasoningF o r t h e p a s t f e w y e a r s , m y c o l l e a g u e s a n d I h a v e b e e n i n v e s t i g a t i n g i n f o r m a lr e a s o n i n g , i ts p it f a ll s , a n d t h e i m p a c t o f h i g h s c h o o l , c o l l e g e , a n d g r a d u a t es c h o o l e d u c a t i o n o n i n f o r m a l r e a s o n i n g s k il ls ( P e r k i n s , 1 98 2; P e r k i n s , A l l e n ,& H a f n e r , 1983). S o m e o f t h e c o n c l u s i o n s w e h a v e r e a c h e d h e l p to d ef in e t h er o l e o f i m a g i n a t io n i n i n f o r m a l r e a s o n i n g . T o a n t i c i p a te a p r i n c i p a l c o n c l u s i o n :f a u l t y in f o r m a l r e a s o n i n g r e f le c t s a la c k o f i m a g i n a t i o n m o r e t h a n i t d o e s t h ec o m m i s s i o n o f e i t h e r f o r m a l fa l la c i e s s u c h a s af f i rming the consequen t o ri n f o r m a l f a l l a c i e s s u c h a s a d h o m i n e m a r g u m e n t .T h e e v i d e n c e f o r t hi s c o m e s f r o m a l ar g e c o ll e c t io n o f a rg u m e n t s g a t h e r e df r o m i n d i v id u a l s o f v a r i o u s a g e s a n d l e v e l s o f e d u c a t i o n . W e p o s e d t o t h ep a r t i c ip a n t s i s s u es t h a t h a d s o m e t i m e l i n e s s d u r i n g th e p e r i o d o f d a t a c o l l e c -t i o n a n d t h e n e l i c i t e d t h e i r p o s i t i o n s a n d s u p p o r t i n g a r g u m e n t s . O u rm e t h o d o l o g y e n c o u r a g e d t h e p a r ti c i p an t s t o e x p l o r e b o t h s id e s o f t h e c a s e a n dt o g i v e a s t h o r o u g h a n a c c o u n t i n g a s t h e y c o u l d o f t h e r e a s o n i n g b e h i n d t h e i rp o s i t i o n s .

B y w a y o f i ll u st ra ti o n, o n e q u e s t i o n w e u s e d w e n t a s f ol lo w s : " W o u l dr e s t o r i n g t h e m i l i t a r y d r a f t s i g n i f i c a n t l y i n c r e a s e A m e r i c a ' s a b i l i t y t o i n f l u e n c ew o r l d e v e n t s ? " H e r e a r e s o m e a r g um e n t s t y p i ca l o f t h o se w e e n c o u n t e r e d( t h e s e a r e p r o t o t y p i c a l r e n d e r i n g s , n o t a c t u a l q u o t e s ) :

Ye s , b e c a u s e a d r a f t wo u l d g i v e t h e U . S . mo r e ma n p o we r i n t h e a r my . T h eU.S . wou ld have a b igge r s t i ck to wave and fo re ign na t ions wou ld be im-p r e s s e d .


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22 DAVID N. PERKINSY e s , b e c a u s e m o r e m a n p o w e r w o u l d p u t t h e U . S . i n a b e t t e r p o s it i o n t o f ig h tl i m i te d t a c ti c a l w a r s . S i n c e e v e r y o n e i s s c a r e d o f w o r l d - w i d e n u c l e a r w a r ,s m a l l - s c a l e w a r s a r e m o r e l i k e l y a n d a m i l i t a r y w e l l - m a n n e d a n d r e a d y t oi n t e r v e n e s h o u l d p r o v i d e m o r e i n f l u e n c e .N o , b e c a u s e a d r a f t w o u l d t r i g g e r w i d e s p r e a d p r o t e s t s , a s i t d i d d u r i n g t h eV i e t n a m w a r . T h i s i n t e r n al d i s s e n s i o n w o u l d b e s e e n a s a l a c k o f u n i t y a n d as ig n o f w e a k n e s s b y f o r e ig n o b s e r v e r s .N o , b e c a u s e n o w a d a y s i t ' s c o m p u t e r s , m i s si l es , a n d th e p e o p l e w h o d e s ig n a n dm a i n t a i n t h e m t h a t r e a l l y c o u n t . I f t h e U . S . n e e d s a n y t h i n g i n t h e a r m y , i t ' sm o r e s m a r t t e ch n i c al p e o p l e . B u t a r a n d o m d r a f t w o n ' t n e t v e r y m a n y s u c hp e o p l e ,

A s m u s t b e p l a in , t h e p r in c i p a l t r o u b l e w i t h s u c h a r g u m e n t s i s i n c o m p l e t e n e s s .E a c h p u t s f o r w a r d a p o i n t w o r t h p o n d e r i n g . A r e a l l y g o o d r e a s o n e r w o u l dm a k e a l l t h e s e p o i n t s a n d m a n y m o r e b e s i d e s , p u l li n g o u t o f t h e w h o l e s o m eb a l a n c e d a p p r a i s a l o f t h e p r o s a n d c o n s . I n a d v a n c i n g t h e v a r i o u s l in e s o fa r g u m e n t , t h e g o o d r e a s o n e r a l s o w o u l d n o t e c o u n t e r a r g u m e n t s t h a t qu a l if yt h e m a n d w o u l d m a k e a p p r o p r i a t e h e d g e s . I n s t e a d , t h e ty p i c a l a r g u m e n t i no u r s a m p l e c o n c e n t r a t e s o n o n e s id e o f t h e c a s e , d o e s n o t d e v e l o p t h a t s i d ev e r y f ul ly , a n d n e g l e c ts r e l e v a n t c o u n t e r a r g u m e n t s a n d a p p r o p r i a t e h e d g e s .

I n i n f o r m a l r e a s o n i n g , p e o p l e r a r e l y h a v e d i f fi c u lt y f in d i ng a t l e a s t o n e l i n eo f a r g u m e n t p r o o r c o n . M o s t p e o p l e m e n t i o n m o r e , a n d o u r r e s e a r c h s h o w st h a t , i f p r e s s e d , p e o p l e u s u a l l y c a n t h i n k u p s e v e r a l m o r e b e s i d e s t h a t . S i n s o fo m i s s i o n c o n s ti t u te t h e g r e a t e s t w e a k n e s s o f i n f o r m a l r e a s o n in g . A n a r g u m e n tt y p i c a l l y f a i l s not b e c a u s e a p e r s o n c a n n o t t h in k o f an a r g u m e n t o r b e c a u s et h e a r g u m e n t a p e r s o n g e n e r a t e s h a s n o l o gi c al b e a r i n g b u t r a t h e r b e c a u s e an u m b e r o f o t h e r l i ne s o f a r g u m e n t , s o m e o f t h e m i n d e p e n d e n t o f t h e g i v e na r g u m e n t a n d s o m e q u a l i f y i n g i t , n e e d c o n s i d e r a t i o n t o o .

T h e s e d i f f e r e n c e s i n t h e s h a p e t a k e n b y s o u n d i n f o r m a l v e r s u s f o r m a la r g u m e n t m e a n t h a t th e t w o p o s e q u i t e d i f fe r e n t p r o b l e m s o f i m a g i n a t i o n fo rt h e r e a s o n e r . W h e r e a s i n m a t h e m a t i c s y o u n e e d to in v e n t one often evasivel in e o f a r g u m e n t , i n in f o r m a l r e a s o n i n g y o u n e e d t o i n v e n t many individuallyoften rather accessible l in e s o f a r g u m e n t . T h e r i ch t a p e s t r y o f c a u s a l i n -f l u e n c e s t h a t c h a r a c t e r i z e s r e a l - l i f e s i t u a t i o n s m e a n s t h a t t h i s i m a g i n a t i v ep r o c e s s c a n g o q u it e f a r. B y w a y o f e x a m p l e , l e t ' s t a k e o n e l i n e o f a r g u m e n tf r o m t h e m i l i t a r y d ra f t i ss u e a n d e x p l o r e i t m o r e f u ll y .

M a y b e s h e e r m a n p o w e r i s n ' t s o i m p o r ta n t n o w a d a y s . I t ' s t e c h n ic a l p e o p l e th a tg e t t h e w o r k d o n e a n d a d r a f t w o u l d n ' t s n a g s u c h p e o p l e .B u t c a n I i m a g i n e a d r a f t t h a t w o u l d s n a g t e c h n i c a l p e o p l e ? We l l , a s e l e c t i v ed ra f t , one ta rge t ted on ce r ta in k inds o f ind iv idua ls , cou ld .C a n I i m a g i n e a n y r e s e r v a t i o n s a b o u t s u c h a d r a f t ? T r a d i t i o n a l l y , U . S . d r a f t sh a v e b e e n b y l o t t e r y . Th e r e ' s s o m e t h i n g u n d e m o c r a t i c a b o u t a s e l e c t i v e d r a f t ,w h i c h w o u l d l e a d t o r e s i s t a n c e . O n t h e o t h e r h a n d , I t h i n k I r e m e m b e r th a td u r i n g t h e V i e t n a m w a r p h y s i c i a n s h a d s o m e s o r t o f s p e c i a l s t a t u s a n d w e r ea l m o s t s u r e t o b e d r a f t e d . S o m a y b e a s e l e c t i v e d r a f t w o u l d b e p o s s i b l e .S u p p o s e s o .W h a t e l s e c a n I im a g i n e m i g h t g o w r o n g w i t h s t r e n g th e n i n g t h e a r m e d s e r v i c e sw i t h t e c h n i c a l p e o p l e t h r o u g h a s e l e c t i v e d r a f t ? We l l , p e r h a p s t h e a r m e ds e r v i c e s d o n ' t r e a l l y n e e d t e c h n i c a l p e o p l e . A f t e r a l l , t h e s e r v i c e s r o u t i n e l yh i re c i v il ia n p e r s o n n e l t o d o m a n y o f t h o s e t e c h n i c a l j o b s , w i t h o u t a n y d r a f trequ i red .


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REASONING AS IMAGINATION 23Also, what will actually strengthen our armed services is one question andwhat wil l impress foreign observers another. Perhaps a larger army overallwould impress them more than a larger technical staff.

And so on. As this example makes plain, many everyday issues pursuedvigorously lead to seemingly endless ramifications. At each branch point, ourimaginations have the job o f exploring whether the argument can be cardedfurther. It 's worth adding that imagination figures in this exploratory processin its second sense as well, the sense of forming images. Our everydayreasoning often involves executing a mental play that enac ts in sketchy formthe causal relationships out of which we build much of our everydayreasoning.In examining mathematical reasoning, we asked what protected suchreasoning from the pattern-breaking character o f imagination. The same ques-tion needs to be asked here as well. In mathematics, our guardian against theravages of imagination was strict deduction, the rules of which were notgenerally imagined otherwise, whatever else our imagination was up to. But,of course, informal argument, laden as it is with probabilistic inference, doesnot depend on strict deduction. Here our guardians are dialectical reasoningand good judgment.Dialectical reasoning simply means that we check our lines of argumentby continuing to argue. If we imagine one line of argument, we then imaginehow that line of argument might go wrong, as illustrated above by the explora-tion of the draft issue. I f we develop one side of the case, then we imaginepoints on the other side of the case. Through such a dialectical process, ourimagination polices i tself against its own failings.Good judgme nt figures when we weigh the contributions of various finesof argument to the overall conclusion, or perhaps to an intermediate conclu-sion. For instance, we can easily imagine that a draft will trigger protest. Buthow much protest? And, given that much, how disunified would the U.S.really look to foreign observers? And, given that the armed services would beexpanded, the protests notwithstanding, to what extent would the "biggerstick" more than make up for, or less than make up for, the protest effect? Tobe sure, each of these ques tions might in itself lead to an articulated argument.But even if so, the regress cannot continue indefinitely. Eventually, whetherfor lack of further arguments or lack of will, one comes to the point where onesimply makes judgments, appraising the likely strengths of causal influencesand other factors. Good judgment here is really a matter of connoisseurshipabout the wa y the world works, a matter, tha t is, of having a feel for whichcurrents flow more strongly. Whereas the imagination generates possiblescenarios, it is this connoisseurship that appraises their relevance.Evidently, neither dialectical reasoning nor good judgment provides thesturdy defence against imagination gone wild that strict deduction does.Whereas wha t constitutes sound deduction is a fairly sharply defined matter o fconformity with a very res tricted set of rules, successful dialectical reasoningand sound judgment are matters of discovery and sensitive appraisal depend-ing on a host of contextual factors. One would like something better, ofcourse, but the messiness of informal reasoning is a fact of life. Indeed, themessiness of life is a fact of life, and that in turn makes informal reasoningmessy. Although dialectical reasoning and good judgment may not offer the


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24 DAVI D N. PERKINS

bulwark against wild imagination that strict deduction does, those are thedefences we have.

Hazards o f t h e I m a g i n a t i o n i n Informal ReasoningSeveral hazards of the imagination were reviewed in the case of mathemat icalreasoning. Not so well defended, informal reasoning is subject to an evengreater extent to such hazards. Moreover, the categories of hazards examinedearlier prove relevant in informal reasoning as well.T h e h a z a r d o f a p e r s u a s i v e i m a a g e . Just as in mathematics, a given repre-sentation of a Situation can carry a persuasive power that may mislead. Thepractice of everyday argument often brings into play highly charged construalsof a situation that bear such power. Consider, for example, the issue ofabortion. Someone may tell us that a b o r t i o n i s m u r d e r . Or someone may saythat a b o r t io n i s a m a t t e r o f a w o m a n ' s r ig h ts o v e r h e r o w n b o d y . Whicheverargument we hear, it evokes a general schema of murder or personal libertythat exerts a strong claim on our allegiance. So dramatic are those schemasthat they tend to pre-empt further inquiry.When you think about it, naive reasoners almost never reach conclusionscontrary to their own surface intuitions. In ano ther paper (1982), I'vecharacterized unskilled reasoners as having a "mak es sense epistemology,"meaning that what seems to make sense has almost complete priority in theirbeliefs. But the whole point of good reasoning is that you should be ablesometimes to reach a conclusion contrary to your own surface intuitions.What is our defence against misleading surfaces? In mathematics, thecommitment to strict deduction resolves the matter: the look of an imagesimply does not count as evidence although it suggests an hypothesis. Youmust have a formal proof. In informal reasoning, matters are not so simple.The good fit of an image with a situation does provide some legitimateevidence. But certainly it does not wholly just ify a conclusion; the dialecticmust continue awhile. In the case of abortion, for instance, I am not arguingthat either the m u r d e r or the p e r s o n a l f r e e d o m schema is wrong. Perhaps thecore of the matter is that abortion is murder or a matter of individual rights orboth. But, clearly, whether either of these construals applies to abortionshould itself be a mat ter of inquiry---and generally is in any serious explorationof the issue.T h e h a z a r d t h a t p a t t e r n s o n c e m a d e a r e h a r d t o b r ea k . This hazard, evidentenough in formal reasoning, pervades informal reasoning. It's already beenemphasized that informal reasoning calls for multiple lines of argument onboth sides of the case. Every time the reasoner seeks another line of argu-ment, especially one countering the argument up to that point, the reasonermust break the pattern so far. If, in exploring the abort ion issue, we have onlydiscovered the murder schema, we may ponder critically whether or not itapplies while neglecting to consider what other schemas might fit--the per-sonal liberty schema, for instance. Notice that here, breaking a schema doesnot necessarily mean discarding it, but at Mast escaping its thrall to considerothers as well.


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Consider again the expanded argument on part of the draft issue. It startedwith the no tion that the armed forces need technical people whom a draftwould n't catch. This sounded sensible enough, but three elements of it werechallenged in turn: whether a draft might indeed select technical people,whether the services actually needed technical people, and whether a genuine-ly stronger armed services was what would look stronger to outsiders. Togeneralize, the dialectical character of good informal reasoning requires pat-tern breaking over and over again.G e s t a l t h u n g e r . In mathematics, we prefer the elegant transparent proof---thesort Wertheimer celebrated--b ut at least we nearly always get a clear resolu-tion one way or the other. In informal reasoning, we often encounter a tangleof arguments on both sides of the case. We would like the murder schema orthe personal liberty schema to be right, but not both half-right in some uneasymeld and compromise.Sometimes we do find a clean resolution. Suppose, for example, yo u havea list of seemingly equally balanced pros and cons. Occasionally, you may beable to filter your list in a principled way. Perhaps you can say, "This isbasically a moral issue" or "This is basically a financial issue," cast out anumber of the reasons, and find a clean resolution. Sometimes you may beable to decide the matter on the basis of the exact nature o f the claim. I f theclaim at issue is a scientific theory and you have some evidence for and someagainst, the "c on " argument should win unless it's very tenuous because thetheory makes a universal claim that always should hold up. In these and otherways you may be able to distill a definite decision from seemingly very mixedevidence.

But not always. In many circumstances, you find nothing but a mix and amuddle, ff you have to make a decision, you can only weigh the pros and consas best you are able and follow their vote. There is something terribly unsatis-fying about this. We have that gestalt hunger clamoring to be fed.The hazard, of course, is not that we feel bad about some of our conclu-sions. We can bear that. The hazard, rather, is that our gestalt hunger pushesus into bad reasoning. This can happen in at least two ways . On the one hand,a mixed case with a leaning one way or another may not convince us as muchas it should, because we want a good gestalt; we want the reasons on theweaker side of the case to g o a w a y somehow. On the other, our gestalt hungeracts as a subtle motive against dialectical reasoning, because dialecticalreasoning often leads to realistically messy decision making.

On Being U nimaginatively ReasonableThis inquiry began with the conventional rivalry between reasoning and theimagination, a rivalry reflected in such phrases as "Ralph is reasonableenough, but he lacks imagination." Now we see that such an appraisal ofRalph, sensible sounding though it seems, is misconceived. Ralph c a n n o t be"reasonable enough" if he lacks imagination. Good reasoning dependscrucially on the vigorous exercise of imagination.

It wasn't terribly surprising to find imagination so important in mathemat-


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26 DAVID N. PERKINSi ca l r eason i ng . A shade more su rp r i s i ng i s t he no t i on t ha t i mag i na t i on rou t i ne -l y , n o t j u s t o c c a s i o n a l l y , f i g u re s c e n t r a l ly i n s o u n d r e a s o n i n g i n d a il y c o n t e x t s .M o r e i n te r e s ti n g y e t a re th e c h a r t s w e ' v e b e e n a b l e t o d r a w o f t h e h a z a r d s o ft h e i m a g i n a t i o n a n d h o w t o n a v i g a t e a m o n g t h e m .A s t hose cha r t s show , t he t r ad i t i ona l r i va l ry has somet h i ng t o i t . Imag i na -t io n d o e s i n d e e d t h r e a t e n t h e p r o b i t y o f r e a s o n i n g i n v a r i o u s w a y s , t h r o u g ht h e h a z a r d s o f t h e p e r s u a s i v e i m a g e , f in d in g p a t t e r n s o n c e m a d e h a r d t o b r e a k ,g e s t a l t h u n g e r , a n d p e r h a p s o t h e r s a s w e l l . F o r t u n a t e l y , d e f e n c e s a g a i n s tt h o s e h a z a r d s t u r n o u t to b e p a r t a n d p a r c e l o f t h e p r a c t i c e o f re a s o n i n g . I nm a t h e m a t i c a l r e a s o n i n g , s t r i c t d e d u c t i o n p r o v i d e s a b u l w a r k , w h i l e i n i n -f o r m a l re a s o n i n g , d i a l e c ti c a l a r g u m e n t a n d g o o d j u d g m e n t m o u n t a d e f e n c e ,a l t h o u g h p e r h a p s n o t a s w e l l .

S o R a l p h c a n n o t b e u n i m a g i n a t i v e l y r e a s o n a b l e . B u t w h y d o e s i t s e e mt h a t h e c a n ? W h y d o s u c h s t a t e m e n t s s o u n d s e n s i b l e , e v e n t h o u g h t h e y a r en o t ? P e r h a p s t h e e x p l a n a t i o n c o m e s d o w n t o t h i s . W e d o n ' t s e e i n f o r m a lr e a s o n i n g a s b e i n g p a r t i c u l a r l y h a r d i n c o m p a r i s o n w i t h , s a y , m a t h e m a t i c a lr e a s o n i n g , w h e r e w e r o u t i n e l y a c k n o w l e d g e t h e n e e d f o r i n g e n u i t y a n d t h es e r v i c e o f i m a g in a ti o n . O f c o u r s e , t h e s p e c ia l s t a t u s o f m a t h e m a t i c s i s ani l lusion. G o o d i n fo rma l r eason i ng i s qu i t e a s ha rd , ca l li ng fo r v i gorous d i a l ec -t ic a n d r e f i n e d j u d g m e n t . I t d o e s n ' t a p p e a r s o h a rd b e c a u s e w e s e t t l e f o r ap e r s u a s i v e i m a g e o f th e s i tu a t io n w h i c h , o n c e m a d e b y o u r h a b i t s o f t h in k i n g ,p r o v e s h a r d t o b r e a k . W e n e e d t o s t r e t c h o u r i m a g i n a ti o n s e v e n t o d i s c e r n t h er a d ic a l e x e r c i s e o f i m a g i n a ti o n t h a t e v e r y d a y r e a s o n i n g d e m a n d s .

Re f e r enc e sAnaheim, R. (1969). Visual th inking. Berke ley: U niversi ty of Cal ifornia Press.Davis, R. B. (1984). Learn ing mathemat ic s : The cogn i t i ve sc ience approach toma t h e m a t i c s e d u c a t i o n. Norwood, NJ: Ablex.Gentner, D., & Stevens, A. (Eds.). (1983). Me n t a l m o d e l s. Hillsdale, NJ: Erlbaum.Johnson-Laird, P. N. (1983). Me n t a l m o d e l s. Cambridge, MA: Harvard Universi tyPress.Ko lers , P. , & Sm ythe, W . (1979). Images, sy mb ols, and skil ls . Ca n a d i a n J o u r n a l o fPsycho logy , 33, 158-184.Kosslyn, S. (1980). I ma g e a n d mi n d . Cambridge, M A: Ha rvard U niversi ty Press.Perkins, D. (1981). The mind 's bes t work , Cambridge, M A: Ha rvard Un iversi ty Press.Perkins, D . (1982).Dif f icul ties in everyday reasoning an d their chan ge w i th educ at ion:Fina l repor t to the Spencer Founda t ion . Unpublished manusc'r ipt , Project Zero,Graduate Sch ool of Educat ion, Harvard U nivers i ty , Cambr idge , M assachuset t s .Perkins, D., A llen, R ., & Hafn er, J. (1983). Difficulties in ever yd ay reasoning. In W .Maxwell (Ed.), Think ing: The expan d ing f ron t i e r (pp . 177-189) . Philadelphia: TheFranklin Institute Press.Wertheimer, M. (1945). Produ c t i ve th ink ing. New York: Harper .Young , S . (1982). Th e mental repre sen tation o f geometrical know ledge. J o u r n a l o fMa t h e ma t i c a l Be h a v i o r, 3(2) , 123-144.

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