[ch]poincare 1913.sciencepress mathematical.creation.in.the.foundations.of.science.ed.poincare

8/9/2019 [Ch]Poincare 1913.SciencePress Mathematical.creation.in.the.foundations.of.Science.ed.Poincare

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M a t h e m a t i c a l C r e a t i o nB y H e n r i P o i n c a r E

T h e g e n e s i s o f m a t h e m a t i c a l c r e a t i o n is a p r o b l e m w h i c h s h o u l d i n t e n s e l y i n t e r e s t t h ep s y c h o l o g i s t . I t i s t h e a c t i v i ty i n w h i c h t h e h u m a n m i n d s e e m s to t a k e l e a s t f r o m t h eo u t s i d e w o r l d , i n w h i c h i t a ct s o r s e e m s t o a ct o n l y o f i t s e lf a n d o n i ts e lf , so t h a t i n s t u d y i n gt h e p r o c e d u r e o f g e o m e t r ic t h o u g h t w e m a y h o p e t o r e a c h w h a t i s m o s t e s s en t ia l i n m a n ' sm i n d .T h i s h a s l o n g b e e n a p p r e c i a t e d , a n d s o m e t i m e b a c k t h e j o u r n a l c a ll e d L'enseignementmath~matique, e d i t e d b y L a i s a n t a n d F e h r , b e g a n a n i n v e s t ig a t io n o f t h e m e n t a l h a b i t s a n dm e t h o d s o f w o r k o f d if f e re n t m a t h e m a t ic i a n s . I h a d f i n is h e d t h e m a i n o u t l i n e s o f t h isa r t ic l e w h e n t h e r e s u l ts o f t h a t i n q u i r y w e r e p u b l i s h e d , s o I h a v e h a r d l y b e e n a b l e t o u t il i z et h e m a n d s h a ll c o n f i n e m y s e l f t o s ay i n g t h a t t h e m a j o r i t y O f w i t n e s s e s c o n f i r m m yc o n c l u s i o n s ; I d o n o t s a y al l, f o r w h e n t h e a p p e a l i s t o u n i v e r s a l s u f f ra g e u n a n i m i t y i s n o tt o b e h o p e d .A f i r s t f a c t s h o u l d s u r p r i s e u s , o r r a th e r w o u l d s u r p r i s e u s i f w e w e r e n o t s o u s e d t o i t. H o wd o e s i t h a p p e n t h e r e ar e p e o p l e w h o d o n o t u n d e r s t a n d m a t h e m a t i c s? I f m a t h e m a t i c si n v o k e s o n l y t h e r u l es o f l o g ic , s u c h a s a r e a c c e p t e d b y a ll n o r m a l m i n d s ; i f i ts e v i d e n c eis b a s e d o n p r in c i p l e s c o m m o n t o a ll m e n , a n d t h a t n o n e c o u l d d e n y w i t h o u t b e i n g m a d ,h o w d o e s i t c o m e a b o u t t h a t s o m a n y p e r s o n s a r e h e r e r e f ra c t o ry ?T h a t n o t e v e r y o n e c a n i n v e n t i s n o w i s e m y s t e r io u s . T h a t n o t e v e r y o n e c a n r e ta i n ad e m o n s t r a t i o n o n c e le a r n e d m a y a ls o p as s. B u t th a t n o t e v e r y o n e c a n u n d e r s t a n dm a t h e m a t i c a l r e a s o n in g w h e n e x p l a in e d a p p e ar s v e r y s u r p r is i n g w h e n w e t h i n k o f it . A n dy e t t h o s e w h o c a n f o l l o w t h i s r e a s o n i n g o n l y w i t h d i f f i c u l t y a r e in t h e m a j o r i t y : t h a t i su n d e n i a b l e , a n d w i ll s u r e ly n o t b e g a i n s a id b y t h e e x p e r i e n c e o f s e c o n d a r y - s c h o o lt e a c h e r s .A n d f u r t h e r: h o w i s e r r or p o s s ib l e i n m a t h e m a t i c s ? A s an e m i n d s h o u l d n o t b e g u i l t y o fa lo g i c a l f a ll a cy , a n d y e t t h e r e a r e v e r y fi n e m i n d s w h o d o n o t t r i p i n b r i e f r e a s o n i n g s u c hReprinted from Science et m 6thode, 1908.

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a s o c c u r s i n t h e o r d i n a r y d o i n g s o f l if e, a n d w h o a r e i n c a p a b l e o f f o l l o w i n g o r r e p e a t i n gw i t h o u t e r r o r t h e m a t h e m a t i c a l d e m o n s t r a t i o n s w h i c h a r e l o n g e r , b u t w h i c h a f t e r a ll a r eo n l y a n a c c u m u l a t i o n o f b r i e f r e a s o n i n g s w h o l l y a n a l o g o u s t o th o s e t h e y m a k e s o e a s il y .N e e d w e a d d t h a t m a t h e m a t i c i a n s t h e m s e l v e s a re n o t i n f a l li b l e ?T h e a n s w e r s e e m s t o m e e v i d e n t . I m a g i n e a l o n g s er ie s o f s y l lo g i s m s , a n d t h a t t h ec o n c l u s i o n s o f t h e f i r st s e r v e as p r e m i s e s o f t h e f o l lo w i n g : w e s h a ll b e a b l e t o c a t c h e a c h o ft h e s e s y l lo g i s m s , a n d i t is n o t i n p a s s i n g f r o m p r e m i s e s t o c o n c l u s i o n t h a t w e ar e i n d a n g e ro f d e c e i v i n g ou r s el v e s. B u t b e t w e e n t h e m o m e n t i n w h i c h w e fi rs t m e e t a p r o p o s i t i o n a sc o n c l u s i o n o f o n e s y ll o g is m , a n d t h a t i n w h i c h w e r e e n c o u n t e r i t a s p r e m i s e o f a n o t h e rs y l l o g i s m o c c a s i o n a l l y s o m e t i m e w i l l e l ap s e , se v e r al l in k s o f th e c h a i n w i l l h a v e u n r o l l e d ;s o it m a y h a p p e n t h a t w e h a v e fo r g o t t e n i t, o r w o r s e , t h a t w e h a v e f o r g o t t e n i ts m e a n i n g .S o it m a y h a p p e n t h a t w e r e p l a c e it b y a s l i g h tl y d i f f e r e n t p r o p o s i t i o n , o r t h a t , w h i l er e t a i n i n g t h e s a m e e n u n c i a t i o n , w e a t tr i b u t e t o it a s l i g h t ly d i f f e r e n t m e a n i n g , a n d t h u s i ti s t h a t w e a r e e x p o s e d t o e r r o r .O f t e n t h e m a t h e m a t i c i a n u s e s a r u le . N a t u r a l l y h e b e g i n s b y d e m o n s t r a t i n g t h i s r u l e ; a n da t th e t i m e w h e n t h is p r o o f is f re s h i n h is m e m o r y h e u n d e r s t a n d s p e r f e c tl y i ts m e a n i n ga n d i ts b e a r i n g , a n d h e i s i n n o d a n g e r o f c h a n g i n g i t. B u t s u b s e q u e n t l y h e t r u s t s h i sm e m o r y a n d a f te r w a r d o n l y a p p li es i t i n a m e c h a n ic a l w a y ; an d t h e n i f h i s m e m o r y f ai lsh i m , h e m a y a p p l y i t a ll w r o n g . T h u s i t i s, t o ta k e a s i m p l e e x a m p l e , t h a t w e s o m e t i m e sm a k e s l i p s in c a l c u l a t i o n b e c a u s e w e h a v e f o r g o t t e n o u r m u l t i p l i c a t i o n t a b le .A c c o r d i n g t o th i s , t h e s p e c ia l a p t i t u d e f o r m a t h e m a t i c s w o u l d b e d u e o n l y t o a v e r y s u r em e m o r y o r to a p r o d i g i o u s f o r c e o f a t t e n t io n . I t w o u l d b e a p o w e r l i k e t h a t o f t h e w h i s t -p l a y e r w h o r e m e m b e r s t h e c a r d s p l a y e d ; o r , t o g o u p a s te p , li k e t h a t o f t h e c h e s s - p l a y e rw h o c a n v is u a li z e a g r e at n u m b e r o f c o m b i n a t i o n s a n d h o l d t h e m i n h i s ~ e m o r y . E v e r yg o o d m a t h e m a t i c i a n o u g h t t o b e a g o o d c h e s s p l a y e r, a n d i n v e r s e l y ; li k e w i s e h e s h o u l d b ea g o o d c o m p u t e r . O f c o u r s e t h a t s o m e t i m e s h a p p e n s ; t h u s G a u s s w a s a t t h e s a m e t im e ag e o m e t e r o f g en i u s a n d a v e r y p r e c o c i o u s a n d a c c u r a te c o m p u t e r .B u t t h e r e a r e e x c e p t i o n s ; o r ra t h e r I e r r; I c a n n o t c al l t h e m e x c e p t i o n s w i t h o u t t h ee x c e p t i o n b e i n g m o r e t h a n t h e r u le . G a u s s i t i s, o n t h e c o n t r a r y , w h o w a s a n e x c e p t i o n , A sf o r m y s e l f , I m u s t c o n f e s s , I a m a b s o l u t e l y in c a p a b l e e v e n o f a d d i n g w i t h o u t m i s t ak e s . I nt h e s a m e w a y , I s h o u l d b e b u t a p o o r c h e s s- p l a y e r ; I w o u l d p e r c e i v e th a t b y a c e r t a i n p l a yI s h o u l d e x p o s e m y s e l f t o a c e r t a in d a n g e r ; I w o u l d p a s s i n r e v i e w s e v e r al o t h e r p l a y s ,r e j e c t in g t h e m f o r o t h e r r e a s o n s, a n d t h e n f i n a ll y I s h o u l d m a k e t h e m o v e f i r st e x a m i n e d ,h a v i n g m e a n t i m e f o r g o t t e n t h e d a n g e r I h a d f o r e s e e n .

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I n a w o r d , m y m e m o r y is n o t b a d , b u t i t w o u l d b e i n s u ff i c ie n t t o m a k e m e a g o o d c h e ss -p l a y e r. W h y t h e n d o e s i t n o t f a il m e i n a d i f f ic u l t p i e c e o f m a t h e m a t i c a l r e a s o n i n g w h e r em o s t c h e s s - p l a y e rs w o u l d l o s e t h e m s e l v e s ? E v i d e n t l y b e c a u s e i t i s g u i d e d b y t h e g e n e r a lm a r c h o f t h e r e as o n i n g . A m a t h e m a t i c a l d e m o n s t r a t i o n i s n o t a s i m p l e j u x t a p o s i t i o n o fs y l l o g i s m s , i t i s s y l l o g i s m s placed in a certain order, a n d t h e o r d e r i n w h i c h t h e s e e le m e n t sa r e p l a c e d is m u c h m o r e i m p o r t a n t t h a n t h e e l e m e n t s t h e m s e l v e s . I f I h a v e t h e fe e l in g , th ei n t u i t i o n , s o t o s p e a k , o f t h i s o r d e r , s o a s t o p e r c e i v e a t a g la n c e t h e r e a s o n i n g a s a w h o l e ,I n e e d n o l o n g e r f e a r l e s t I f o r g e t o n e o f t h e e l e m e n t s , f o r e a c h o f t h e m w i l l ta k e i t s a l l o t t e dp l a ce i n t h e a rr ay , a n d t h a t w i t h o u t a n y e f fo r t o f m e m o r y o n m y p a r t.I t se e m s t o m e t h e n , i n r e p e a t i n g a r e a s o n i n g l e a r n e d , t h a t I c o u l d h a v e i n v e n t e d i t. T h i si s o f t e n o n l y a n i l lu s i o n ; b u t e v e n t h e n , e v e n i f I a m n o t s o g i f t e d as to c r e a t e i t b y m y s e l f ,I m y s e l f r e - i n v e n t i t i n s o fa r a s I r e p e a t i t .W e k n o w t h a t t h i s f e e li n g , t h i s i n t u i t io n o f m a t h e m a t i c a l o r d e r , t h a t m a k e s u s d i v i n eh i d d e n h a r m o n i e s a n d r e l a t io n s , c a n n o t b e p o s s e s s e d b y e v e r y o n e . S o m e w i l l n o t h a v ee i t h e r t h i s d e l i c a t e fe e l in g s o d i f f ic u l t to d e f i n e , o r a s t r e n g t h o f m e m o r y a n d a t t e n t i o nb e y o n d t h e o r d i n a r y , a n d t h e n t h e y w i ll b e a b s o l u te l y in c a p a b le o f u n d e r s t a n d i n g h i g h e rm a t h e m a t i c s . S u c h a r e t h e m a j o r i t y . O t h e r s w i l l h a v e th i s fe e l i n g o n l y i n a s l i g h t d e g r e e ,b u t t h e y w i ll b e g i ft e d w i t h a n u n c o m m o n m e m o r y a n d a g r e at p o w e r o f a t te n t i o n . T h e yw i l l l e a r n b y h e a r t t h e d e t a il s o n e a f te r a n o t h e r ; t h e y c a n u n d e r s t a n d m a t h e m a t i c s a n ds o m e t i m e s m a k e a p p l i c a t io n s , b u t t h e y c a n n o t c r e a te . O t h e r s , f i na l ly , w i l l p o s s e s s i n a l e sso r g r e a te r d e g r e e t h e s p e c ia l in t u i t i o n r e f e rr e d t o , a n d t h e n n o t o n l y c a n t h e y u n d e r s t a n dm a t h e m a t i c s e v e n i f t h e i r m e m o r y i s n o t h i n g e x t r ao r d i n ar y , b u t t h e y m a y b e c o m ec r e a t o r s a n d t r y t o i n v e n t w i t h m o r e o r le s s s u c c e s s a c c o r d i n g a s t h i s i n t u i t i o n i s m o r e o rl es s d e v e l o p e d in t h e m .I n f a c t, w h a t is m a t h e m a t i c a l c r e a t i o n ? I t d o e s n o t c o n s i s t i n m a k i n g n e w c o m b i n a t i o n sw i t h m a t h e m a t i c a l e n t i t ie s a lr e a d y k n o w n . A n y o n e c o u l d d o t h a t, b u t t h e c o m b i n a t i o n ss o m a d e w o u l d b e in f i n it e i n n u m b e r a n d m o s t o f t h e m a b s o l u t e ly w i t h o u t i n t e re s t. T oc r e a te c o n s i s t s p r e c i s e l y i n n o t m a k i n g u s e l e s s c o m b i n a t i o n s a n d i n m a k i n g t h o s e w h i c ha r e u s e f u l a n d w h i c h a r e o n l y a s m a l l m i n o r i t y . I n v e n t i o n i s d i s c e r n m e n t , c h o i c e .H o w t o m a k e t h is c h o i c e I h a v e b e f o r e e x p la i n ed ; t h e m a t h e m a t i c a l f a c ts w o r t h y o f b e i n gs t u d i e d a r e t h o s e w h i c h , b y t h e i r a n a l o g y w i t h o t h e r f a c ts , a r e c a p a b l e o f l e a d i n g u s t o t h ek n o w l e d g e o f a m a t h e m a t i c a l l a w ju s t a s e x p e r i m e n t a l f a c t s l e a d u s to t h e k n o w l e d g e o f ap h y s i c a l la w . T h e y a r e t h o s e w h i c h r e v e a l t o u s u n s u s p e c t e d k i n s h i p b e t w e e n o t h e r fa c ts ,l o n g k n o w n , b u t w r o n g l y b e l i e v e d t o b e st r a n g e r s t o o n e a n o th e r .

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Among chosen combin ations the most fertile will often be those formed of elements drawnfrom domains whic h are far apart. Not th at I mean as sufficing for inven tion the b ring ingtogether of objects as disparate as possible; most combinations so formed would beentirel y sterile. But certain among them, very rare, are the most fru itful of all.To invent , I have said, is to choose; but the word is perhaps not wholly exact. It makes onethink of a purchaser before whom are displayed a large number of samples, and whoexamines them, one after the other, to make a choice. Here the samples would be sonume rous that a whole lifetime would not suffice to examine them. This is not the actualstate of things. The sterile combinations do not even present themselves to the m ind ofthe inven tor. Never in the field of his consciousness do combinatio ns appear that are no treally useful, except some that he rejects but wh ich have to some exte nt the characteristicsof useful combinations. All goes on as if the inv entor were an exami ner for the seconddegree who would only have to question the candidates who had passed a previousexamination.But what I have hither to said is what may be observed or inferred in reading the writi ngsof the geometers, reading reflectively.It is time to penetrat e deeper and to see what goes on in the very soul of the m athemati cian .For th is, I believe, I can do best by recalling memories of my own. But I shall limit my sel fto telling how I wrote my first memoir on Fuchsian functions. I beg the reader's pardon;I am about to use some technical expressions, but they need not frigh ten him, for he is notobliged to un dersta nd them. I shall say, for example, that I have found the demons trationof such a theorem un der such circumstances. This theorem will have a barbarous name,unfamiliar to many,, but that is unimporta nt; what is of interest for the psychologist is notthe theorem but the circumstances.For fifteen days I strove to prove that there could not be any functions like those I havesince called Fuchs ian funct ions. I was then very ignoran t; every day I seated mys elf at mywork table, stayed, an hou r or two, tried a great numbe r of combinati ons and reached noresults. One evening, contrar y to my custom, I drank black coffee and could not sleep.Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, maki ng a stablecombination. But the next mo rning I had established the existence of a class of Fuchsia nfunctions, those which come from the hypergeometric series; I had only to write out theresults, which took but a few hours.Then I wanted to represent these functions by a quotient of two series; this idea was

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p e r f e c t l y c o n s c i o u s a n d d e l i b e r a t e , t h e a n a l o g y w i t h e l li p t ic f u n c t i o n s g u i d e d m e . I a sk e dm y s e l f w h a t p r o p e r t ie s t h e s e s er ie s m u s t h a v e i f t h e y e x i s te d , a n d I s u c c e e d e d w i t h o u td i f f i c u l t y in f o r m i n g t h e s e r ie s I h a v e c a ll e d t h e t a - F u c h s i a n .J u s t a t t h is t i m e I l e f t C a e n , w h e r e I w a s t h e n l iv i n g , t o g o o n a g e o lo g i c a l e x c u r s i o n u n d e rt h e a u s p i c es o f t h e s c h o o l o f m i n e s . T h e c h a n g e s o f t ra v e l m a d e m e f o r g e t m y m a t h e m a t i -c al w o r k . H a v i n g r e a c h e d C o u t a n c e s , w e e n t e r e d a n o m n i b u s t o g o s o m e p l a c e o r o th e r . A tt h e m o m e n t w h e n I p u t m y f o o t o n t h e s te p th e i d e a c a m e to m e , w i t h o u t a n y t h i n g i n m yf o r m e r t h o u g h t s s e e m i n g t o h a v e p a v e d t h e w a y f o r i t, t h a t t h e t r a n s f o r m a t i o n s I h a d u s e dt o d e f in e t h e F u c h s i a n f u n c t i o n s w e r e i d e nt ic a l w i t h t h o s e o f n o n - E u c l i d e a n g e o m e t r y . Id i d n o t v e r i f y t h e i d e a ; I s h o u l d n o t h a v e h a d t i m e , a s, u p o n t a k i n g m y s e at i n t h e o m n i b u s ,I w e n t o n w i t h a c o n v e r s a t i o n a l r e a d y c o m m e n c e d , b u t I f e l t a p e r f e c t c e r t a i n ty . O n m yr e t u r n t o C a e n , f o r c o n s c i e n c e ' s a k e I v e r i f i e d t h e r e s u l t a t m y l e is u r e .T h e n I t u r n e d m y a t t e n t io n t o t h e s t u d y o f s o m e a r i th m e t i c q u e s t io n s a p p a r e n t l y w i t h o u tm u c h s u c c e ss a n d w i t h o u t a s u s p ic i o n o f a n y c o n n e c t i o n w i t h m y p r e c e d i n g r e se a rc h e s.D i s g u s t e d w i t h m y f a il u r e, I w e n t t o s p e n d a f e w d a y s a t t h e s e a s id e , a n d t h o u g h t o fs o m e t h i n g e ls e. O n e m o r n i n g , w a l k i n g o n t h e b l u f f, t h e i d e a c a m e to m e , w i t h j u s t t h es a m e c h a r a c t e r i st i c s o f b r e v i t y , s u d d e n n e s s a n d i m m e d i a t e c e r t a i n t y , t h a t t h e a r i t h m e t i ct r a n s fo r m a t i o n s o f i n d e t e r m i n a t e t e r n a ry q u a d r a t i c f o r m s w e r e id e n t ic a l w i t h t h o s e o fn o n - E u c l i d e a n g e o m e t r y .R e t u r n e d t o C a e n , I m e d i t a t e d o n t hi s re s u l t a n d d e d u c e d t h e c o n s e q u e n c e s . T h e e x a m p l eo f q u a d r a t i c f o r m s s h o w e d m e t h a t t h e r e w e r e F u c h s i a n g r o u p s o t h e r t h a n t h o s e c o r re -s p o n d i n g t o t h e h y p e r g e o m e t r i c s e r ie s ; I s a w t h a t I c o u l d a p p l y t o t h e m t h e t h e o r y o f t h e t a -F u c h s i a n s er ie s a n d t h a t c o n s e q u e n t l y t h e r e e x is te d F u c h s i a n f u n c t i o n s o t h e r t h a n t h o s ef r o m t h e h y p e r g e o m e t r i c s e r ie s , t h e o n e s I t h e n k n e w . N a t u r a l l y I s e t m y s e l f t o f o r m a llt h e s e fu n c t i o n s . I m a d e a s y s t e m a t i c a tt a c k u p o n t h e m a n d c a r r i e d al l t h e o u t w o r k s , o n ea f t e r a n o t h e r . T h e r e w a s o n e h o w e v e r th a t s t il l h e l d o u t , w h o s e fa ll w o u l d i n v o l v e t h a t o ft h e w h o l e p l ac e . B u t a ll m y e f f o r ts o n l y s e rv e d a t f i rs t th e b e t t e r t o s h o w m e t h e d i f f i c u l ty ,w h i c h i n d e e d w a s s o m e t h i n g . A l l t h i s w o r k w a s p e r f e c t l y c o n s c i o u s .T h e r e u p o n I l ef t f o r M o n t - V a l 6 r i e n , w h e r e I w a s t o go t h r o u g h m y m i l i t a r y s e r v i c e ; so Iw a s v e r y d i f f e r e n t l y o c c u p i e d . O n e d a y , g o i n g a l o n g th e s t r e et , t h e s o l u t i o n o f t h ed i f f i c u l t y w h i c h h a d s t o p p e d m e s u d d e n l y a p p e a r e d to m e . I d i d n o t t r y t o go d e e p i n t o i ti m m e d i a t e l y , a n d o n l y a f te r m y s e r v ic e d id I a g a i n ta k e u p t h e q u e s t i o n . I h a d a l l t h ee l e m e n t s a n d h a d o n l y to a rr a n g e t h e m a n d p u t t h e m t o g e th e r . S o I w r o t e o u t m y f in a lm e m o i r a t a s i n g l e s t r o k e a n d w i t h o u t d i f f i c u l t y .

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I s h a l l li m i t m y s e l f t o th i s s i n g l e e x a m p l e ; i t is u s e l e s s t o m u l t i p l y t h e m . I n r e g a r d t o m yo t h e r r e s e a r c h e s I w o u l d h a v e t o sa y a n a l o g o u s t h i n g s , a n d t h e o b s e r v a t i o n s o f o t h e rm a t h e m a t i c i a n s g i v e n in L'enseignement math~matiquew o u l d o n l y c o n f i r m t h e m .M o s t s t r i k i n g a t f i rs t is t h i s a p p e a r a n c e o f s u d d e n i l l u m i n a t i o n , a m a n i f e s t s i g n o f l o n g ,u n c o n s c i o u s p r i o r w o r k . T h e r61 e o f t h is u n c o n s c i o u s w o r k i n m a t h e m a t i c a l i n v e n t i o na p p e a r s t o m e i n c o n t e s t a b l e , a n d t ra c e s o f i t w o u l d b e f o u n d i n o t h e r c a s e s w h e r e i t i s l es se v i d en t . O f t e n w h e n o n e w o r k s a t a h a r d q u e s t i o n , n o t h i n g g o o d i s a c c o m p l i s h e d a t t h ef i rs t a tt a c k . T h e n o n e t a k e s a r e s t, lo n g e r o r sh o r t e r , a n d s i t s d o w n a n e w t o t h e w o r k .D u r i n g t h e f i r s t h a l f - h o u r , a s b e f o r e , n o t h i n g is f o u n d , a n d t h e n a ll o f a s u d d e n t h e d e c i s i v ei d e a p r e s e n t s i t s e lf t o t h e m i n d . I t m i g h t b e s a id t h a t t h e c o n s c i o u s w o r k h a s b e e n m o r ef r u i t fu l b e c a u s e i t h a s b e e n i n t e r r u p t e d a n d t h e r e s t h a s g iv e n b a c k t o t h e m i n d i ts f o rc ea n d f r e s h n e s s . B u t i t i s m o r e p r o b a b l e t h a t t h i s r es t h as b e e n f il l ed o u t w i t h u n c o n s c i o u sw o r k a n d t h a t t h e r e s u l t o f t h i s w o r k h a s a f t e rw a r d s r e v e a l e d i t s e l f t o t h e g e o m e t e r j u s t a si n t h e c a s e s I h a v e c i t e d ; o n l y t h e r e v e l a t i o n , i n s t e a d o f c o m i n g d u r i n g a w a l k o r a j o u r n e y ,h a s h a p p e n e d d u r i n g a p e r i o d o f c o n s c i o u s w o r k , b u t i n d e p e n d e n t l y o f t h is w o r k w h i c hp l a y s a t m o s t a ro l e o f e x c i t a n t, a s i f i t w e r e th e g o a d s t i m u l a t i n g t h e r e s u l ts a l r e a d y r e a c h e dd u r i n g r e s t, b u t r e m a i n i n g u n c o n s c i o u s , to a s s u m e t h e c o n s c i o u s f o r m .T h e r e is a n o t h e r r e m a r k t o b e m a d e a b o u t t h e c o n d i t i o n s o f t h is u n c o n s c i o u s w o r k : i t isp o s s i b l e , a n d o f a c e r ta i n t y i t i s o n l y f r u i tf u l , i f i t is o n t h e o n e h a n d p r e c e d e d a n d o n t h eo t h e r h a n d f o l lo w e d b y a p e r i o d o f c o n s c i o u s w o r k . T h e s e s u d d e n i n s p i ra t i o n s ( a n d t h ee x a m p l e s a l r e a d y c i t e d s u f f i c i e n t ly p r o v e ~ his) n e v e r h a p p e n e x c e p t a f te r s o m e d a y s o fv o l u n t a r y e f fo r t w h i c h h a s a p p e a re d a b s o l u t e l y f r ui tl e ss a n d w h e n c e n o t h i n g g o o d s e e m st o h a v e c o m e , w h e r e t h e w a y t a k e n s e e m s t o t a l ly as t ra y . T h e s e e f f o r ts t h e n h a v e n o t b e e na s s t e ri l e a s o n e t h i n k s ; t h e y h a v e s e t a g o i n g t h e u n c o n s c i o u s m a c h i n e a n d w i t h o u t t h e mi t w o u l d n o t h a v e m o v e d a n d w o u l d h a v e p r o d u c e d n o t h i n g .T h e n e e d f o r th e s e c o n d p e r i o d o f c o n s c i o u s w o r k , a f te r th e i n s p i r a t io n , i s s t il l e a si e r tou n d e r s t a n d . Ix i s n e c e s s a r y to p u t i n s h a p e t h e r e s u l ts o f t h i s i n s p i r a t io n , t o d e d u c e f r o mt h e m t h e i m m e d i a t e c o n s e q u e n c e s , t o a r ra n g e t h e m , t o w o r d th e d e m o n s t r a t i o n s , b u ta b o v e a ll is v e r if i c a t io n n e c e s s ar y . I h a v e s p o k e n o f t h e f e e l i n g o f a b s o l u t e c e r t i t u d ea c c o m p a n y i n g t h e i n s p i r a t io n ; i n t h e c a s es c i t e d t h is f e e l i n g w a s n o d e c e i v e r , n o r i s itu s u a l ly . B u t d o n o t t h i n k t h i s a r u l e w i t h o u t e x c e p t i o n ; o f t e n t h is f e e li n g d e c e i v e s u sw i t h o u t b e i n g a n y t h e l e ss v iv i d , a n d w e o n l y f i n d i t o u t w h e n w e s e e k t o p u t o n f o o t th ed e m o n s t r a t i o n . I h a v e e s p e c i a ll y n o t i c e d t h i s f a c t i n re g a r d t o i d ea s c o m i n g t o m e i n t h em o r n i n g o r e v e n i n g i n b e d w h i l e in a s e m i - h y p n a g o g i c s t at e .

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S u c h a r e th e r e a l it i es ; n o w f o r th e t h o u g h t s t h e y f o rc e u p o n u s . T h e u n c o n s c i o u s , o r , a sw e s a y, t h e s u b l i m i n a l s e l f p l a y s a n i m p o r t a n t r6 1e i n m a t h e m a t i c a l c r e a t i o n ; t h i s f o l l o w sf r o m w h a t w e h a v e sa id . B u t u s u a l l y t h e s u b l i m i n a l s e l f i s c o n s i d e r e d a s p u r e l y a u t o m a t i c .N o w w e h a v e se e n t h a t m a t h e m a t i c a l w o r k is n o t s i m p l y m e c h a n i c a l , t h a t i t c o u l d n o t b ed o n e b y a m a c h i n e , h o w e v e r p e rf e c t. I t i s n o t m e r e l y a q u e s t i o n o f a p p l y i n g r u l e s, o fm a k i n g t h e m o s t c o m b i n a t i o n s p o s s i b l e a c c o r d in g t o ce r ta i n f ix e d la w s. T h e c o m b i n a -t io n s s o o b t a i n e d w o u l d b e e x c e e d i n g ly n u m e r o u s , u s e le s s a n d c u m b e r s o m e . T h e t r u ew o r k o f t h e i n v e n t o r c o n s is t s i n c h o o s i n g a m o n g t h e s e c o m b i n a t i o n s s o a s t o e l im i n a t e t h eu s e le s s o n e s o r ra t h e r t o a v o i d t h e t r o u b l e o f m a k i n g t h e m , a n d t h e r u le s w h i c h m u s t g u i d et h i s c h o i c e a r e e x t r e m e l y f i n e a n d d e l ic a te . I t is a l m o s t i m p o s s i b l e t o s ta t e t h e m p r e c i s e ly ;t h e y a r e f e l t r a t h e r t h a n f o r m u l a t e d . U n d e r t h e s e c o n d i t i o n s , h o w i m a g i n e a s i e v e c a p a b l eo f a p p l y i n g t h e m m e c h a n i c a l ly ?A f i rs t h y p o t h e s i s n o w p r e s e n t s i ts e lf ; t h e s u b l i m i n a l s e l f is i n n o w a y i n f e r i o r t o t h ec o n s c i o u s s e lf ; it is n o t p u r e l y a u t o m a t i c ; i t is c a p a b l e o f d i s c e r n m e n t ; i t h a s t a c t , d e l ic a c y ;i t k n o w s h o w t o c h o o s e , t o d iv i n e . W h a t d o I sa y ? I t k n o w s b e t t e r h o w t o d i v i n e t h a n t h ec o n s c i o u s s e lf , s i n c e i t s u c c e e d s w h e r e t h a t h a s fa i le d . I n a w o r d , i s n o t t h e s u b l i m i n a l s e l fs u p e r i o r t o th e c o n s c i o u s se lf? . Y o u r e c o g n i z e t h e fu l l i m p o r t a n c e o f t h i s q u e s t i o n .B o u t r o u x i n a r e c e n t l e c t u r e h a s s h o w n h o w i t c a m e u p o n a v e r y d i f f e r e n t o c c a s i o n , a n dw h a t c o n s e q u e n c e s w o u l d f o l lo w a n a f f i rm a t i v e a n s w e r. ( S e e a ls o , b y t h e s a m e a u t h o r ,S c i e n c e e t R e l i g i o n , pp. 313 f f . )I s t h is a f f i rm a t i v e a n s w e r f o r c e d u p o n u s b y t h e f a c ts I h a v e j u s t g i v e n ? I c o n f e s s t h a t , f o rm y p a r t, I s h o u l d h a t e t o a c c e p t it . R e e x a m i n e t h e f a ct s t h e n a n d s e e i f t h e y a r e n o tc o m p a t i b l e w i t h a n o t h e r e x p l a n a t i o n .I t i s c e r t a i n t h a t t h e c o m b i n a t i o n s w h i c h p r e s e n t t h e m s e l v e s t o t h e m i n d i n a s o r t o fs u d d e n i l lu m i n a t i o n , a f te r an u n c o n s c i o u s w o r k i n g s o m e w h a t p r o l o n g e d , a re g e n e r a ll yu s e f u l a n d f e r t il e c o m b i n a t i o n s , w h i c h s e e m t h e r e s u l t o f a f i rs t im p r e s s i o n . D o e s i t f o l l o wt h a t t h e s u b l i m i n a l s e lf , h a v i n g d i v i n e d b y a d e l i c a te i n t u i t i o n t h a t t h e s e c o m b i n a t i o n sw o u l d b e u s e fu l , h a s f o r m e d o n l y t h es e , o r h a s it r a th e r f o r m e d m a n y o t h e r s w h i c h w e r el a c k i n g i n i n t e r e s t a n d h a v e r e m a i n e d u n c o n s c i o u s ?I n t h is s e c o n d w a y o f l o o k i n g a t i t, all th e c o m b i n a t i o n s w o u l d b e f o r m e d i n c o n s e q u e n c eo f t h e a u t o m a t i s m o f t h e s u b l i m i n a l s e l f, b u t o n l y t h e i n t e r e s t i n g o n e s w o u l d b r e a k i n t ot h e d o m a i n o f c o n s c i o u s n e s s . A n d t h i s is s ti ll v e r y m y s t e r i o u s . W h a t i s t h e c a u s e t h a t ,a m o n g t h e t h o u s a n d p r o d u c t s o f o u r u n c o n s c i o u s a c t iv i t y , s o m e a re c al le d t o p a s s t h et h r e s h o l d , w h i l e o t h e r s r e m a i n b e l o w ? I s i t a s i m p l e c h a n c e w h i c h c o n f e r s th i s p r i v il e g e ?

RESO NAN CE J Fe b r u a r y 2 0 0 0 9 1


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E v i d e n t l y n o t ; a m o n g a l l t h e s t i m u l i o f o u r s e n s e s , f o r e x a m p l e , o n l y t h e m o s t i n t e n s e f i xo u r a t t e n t i o n , u n l e s s i t h a s b e e n d r a w n t o t h e m b y o t h e r ca u s e s. M o r e g e n e r a l l y t h ep r i v il e g e d u n c o n s c i o u s p h e n o m e n a , t h o s e s u s c e p ti b l e o f b e c o m i n g c o n s c i o u s , a r e th o s ew h i c h , d i r e c t l y o r i n d i r e c t l y a f f ec t m o s t p r o f o u n d l y o u r e m o t i o n a l s e n s i b i l i ty .I t m a y b e s u r p r i s i n g t o se e e m o t i o n a l s e n s i b i l i ty i n v o k e d ~ propos o f m a t h e m a t i c a ld e m o n s t r a t i o n s w h i c h , i t w o u l d s ee m , c a n i n t e r e s t o n l y th e i n t el le c t . T h i s w o u l d b e t of o r g et t h e f e e li n g o f m a t h e m a t i c a l b e a u t y , o f t h e h a r m o n y o f n u m b e r s a n d f o r m s , o fg e o m e t r i c el e g a n c e. T h i s i s a t r u e e s t h e t ic f e e li n g th a t a l l r e al m a t h e m a t i c i a n s k n o w , a n ds u r e l y i t b e l o n g s t o e m o t i o n a l s e n s i b il i ty .N o w , w h a t a r e th e m a t h e m a t i c e n t it i es t o w h i c h w e a t tr i b u t e t h i s c h a r a c te r o f b e a u t y a n de l e g a n c e, a n d w h i c h a r e c a p a b l e o f d e v e l o p i n g i n u s a s o r t o f e s t h e t i c e m o t i o n ? T h e y a r et h o s e w h o s e e l e m e n t s a re h a r m o n i o u s l y d i s p o s e d s o t h a t t h e m i n d w i t h o u t e f f o rt c a ne m b r a c e t h e i r t o t a l i t y w h i l e re a l iz i n g t h e d e t a il s. T h i s h a r m o n y i s a t o n c e a s a t i s fa c t i o n o fo u r e s t h e t i c n e e d s a n d a n a id t o t h e m i n d , s u s t a i n i n g a n d g u i d i n g . A n d a t t h e s a m e t im e ,i n p u t t i n g u n d e r o u r e y e s a w e l l - o r d e r e d w h o l e , it m a k e s u s f o r e s e e a m a t h e m a t i c a l l aw .N o w , w e h a v e s a id a b o v e , th e o n l y m a t h e m a t i c a l f ac ts w o r t h y o f f i x in g o u r a t t e n t i o n a n dc a p a b l e o f b e i n g u s e f u l ar e t h o s e w h i c h c a n t e a c h u s a m a t h e m a t i c a l l a w . S o t h a t w e r e a c ht h e f o l l o w i n g c o n c l u s i o n : T h e u s e f u l c o m b i n a t i o n s a r e p r e c i s e l y t h e m o s t b e a u t i f u l , Im e a n t h o s e b e s t a b l e t o c h a r m t h i s sp e c i a l s e n s i b i l i t y t h a t a ll m a t h e m a t i c i a n s k n o w , b u to f w h i c h t h e p r o f a n e a r e s o i g n o r a n t a s o f t e n t o b e t e m p t e d t o s m i l e a t i t.W h a t h a p p e n s t h e n ? A m o n g t h e g re a t n u m b e r s o f c o m b i n a t io n s b l i n d l y fo r m e d b y t h es u b l i m i n a l s e lf , a l m o s t a ll ar e w i t h o u t i n t e r e s t a n d w i t h o u t u t i l i t y ; b u t j u s t f o r t h a t r e a s o nt h e y a r e al s o w i t h o u t e f f e ct u p o n t h e e s t h e t i c s e n s i b il i ty . C o n s c i o u s n e s s w i l l n e v e r k n o wt h e m ; o n l y c e r t ai n o n e s a r e h a r m o n i o u s , a n d , c o n s e q u e n t l y , a t o n c e u s e f u l a n d b e a u t i f u l .T h e y w i l l b e c a p a b l e o f t o u c h i n g t h i s s p e ci a l s e n s i b i l i t y o f t h e g e o m e t e r o f w h i c h I h a v ej u s t s p o k e n , a n d w h i c h , o n c e a r o u s e d , w i ll c a ll o u r a t te n t i o n t o t h e m , a n d t h u s g i v e t h e mo c c a s io n t o b e c o m e c o n s c i o us .T h i s i s o n l y a h y p o t h e s i s , a n d y e t h e r e is a n o b s e r v a t i o n w h i c h m a y c o n f i r m i t: w h e n as u d d e n i l lu m i n a t i o n s ei ze s u p o n t h e m i n d o f t h e m a t h e m a t i c i a n , i t u s u a l ly h a p p e n s t h a ti t d o e s n o t d e c e i v e h im , b u t i t a l so s o m e t i m e s h a p p e n s , a s I h a v e s a id , t h a t i t d o e s n o t s t a n dt h e t e s t o f v e r i f i c a t io n ; w e l l, w e a l m o s t a l w a y s n o t i c e t h a t t h i s f a l se i de a , h a d i t b e e n t r u e ,w o u l d h a v e g r a t i f i e d o u r n a t u r a l f e e l i n g f o r m a t h e m a t i c a l e l eg a n c e .T h u s i t is t h is s p e c ia l e s t h e t i c s e n s i b i l i t y w h i c h p l a y s t h e r 61 e o f t h e d e l i c a te s i e v e o f w h i c h

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I spoke, and that sufficientl y explains why the one lacking it will never be a real creator.Yet all the difficulties have not disappeared. The conscious self is narrowl y limited, andas for the subliminal self we know not its limitations, and this is why we are not tooreluctant in supposing that it has been able in a short time to make more differentcomb inat ions than the whole life of a conscious being could encompass. Yet theselimita tions exist. Is it likely that it is able to form all the possible combinat ions, whosenumb er would frighten the imagination? Nevertheless that would seem necessary,because if it produces only a small part of these combinations, and if it makes them atrandom, there would be small chance that the good, the one we should choose, would befound among them.Perhaps we ought to seek the explanation in that preliminary period of conscious workwhich always precedes all fruitful unconscious labor. Permit me a rough comparison.Figure the future elements of our combinations as something like the hooked atoms ofEpicurus. Dur ing the complete repose of the mi nd, these atoms are motionless , they are,so to speak, hooked to the wall; so this complete rest may be indefin itely prolon gedwithou t the atoms meeting, and consequently without any combin ation between them.On the other hand, during a period of apparent rest and unconscious work, certain of themare detached from the wall and put in motion. The y flash in every direction th rou gh thespace (I was about to say the room) where t hey are enclosed, as would, for example, a swarmof gnats or, if you prefer a more learned comparison, like the molecules of gas in thekinematic theo ry of gases. The n their mutual impacts may produce new combinations.Wha t is the r61e of the pre limina ry conscious work? It is evidently to mobilize certain ofthese atoms, to unhook them from the wall and put them in swing. We thin k we have doneno good, because we have moved these elements a thousand d ifferent ways in seeking toassemble them, and have found no satisfactory aggregate. But, after this shaking upimposed up on t hem by our will, these atoms do not return to their primitive rest. Theyfreely continue their dance.Now, our will did not choose them at random; it pursued a perfectly det ermi ned aim. Themobilized atoms are therefore not any atoms whatsoever; they are those from which wemight reasonably expect the desired solution. The n the mob ilized atoms undergo impactswhich make them enter into combinations among themselves or with other atoms at restwhich th ey struck against in their course. Again I beg pardon, my comparison is veryrough, but I scarcely know how otherwise to make my thought understood.

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H o w e v e r it m a y b e , t h e o n l y c o m b i n a t i o n s t h a t h a v e a c h a n c e o f f o r m i n g a r e t h o s e w h e r ea t le a s t o n e o f t h e e l e m e n t s is o n e o f t h o s e a t o m s f r e e l y c h o s e n b y o u r w i ll . N o w , i t ise v i d e n t l y a m o n g t h e s e t h a t i s f o u n d w h a t I ca l le d t h e good combination. P e r h a p s t h i s i s aw a y o f l e s s e n in g t h e p a r a d o x i c a l i n t h e o r i g i n a l h y p o t h e s i s .A n o t h e r o b s e r v a t io n . I t n e v e r h a p p e n s t h a t t h e u n c o n s c i o u s w o r k g i v e s u s t h e r e s u l t o f as o m e w h a t l o n g c a l c u l a t i o n all made, w h e r e w e h a v e o n l y to a p p l y f ix e d ru l es . W e m i g h tt h i n k t h e w h o l l y a u t o m a t i c s u b l i m i n a l s e l f p a r t i c u l a rl y a p t f o r t h i s s o r t o f w o r k , w h i c h i si n a w a y e x c l u s iv e l y m e c h a n i c a l . I t s e em s t h a t t h i n k i n g i n t h e e v e n i n g u p o n t h e f a c to r s o fa m u l t i p l ic a t i o n w e m i g h t h o p e t o f in d t h e p r o d u c t r e a d y m a d e u p o n o u r a w a k e n i n g , o ra g a i n t h a t a n a l g e b r a i c c a l c u l a t i o n , f o r e x a m p l e a v e r i f i c a t i o n , w o u l d b e m a d e u n c o n -s c io u s ly . N o t h i n g o f t h e s o rt , a s o b s e r v a t i o n p r o v es . A l l o n e m a y h o p e f r o m t h e s ei n s p i ra t i o n s , f r u i t s o f u n c o n s c i o u s w o r k , is a p o i n t o f d e p a r t u r e f o r s u c h c a l c u l a t io n s . A sf o r t h e c a l c u l a ti o n s th e m s e l v e s , t h e y m u s t b e m a d e i n t h e s e c o n d p e r i o d o f t h e c o n s c i o u sw o r k , t h a t w h i c h f o ll o w s th e i n s p i ra t i o n , t h a t i n w h i c h o n e v e r i fi e s t h e r e s u lt s o f t h i si n s p i r a t i o n a n d d e d u c e s th e i r c o n s e q u e n c e s . T h e r u l es o f t h e s e c a lc u l a t io n s a r e s tr i c t a n dc o m p l i c a te d . T h e y r e q u i r e d i s c ip l in e , a t t e n t io n , w i ll a n d t h e r e f o r e c o n s c i o u s n e s s. I n t h es u b l i m i n a l s e l f, o n t h e c o n t r a r y , r e i g n s w h a t I s h o u l d c a l l l i b e r t y , i f w e m i g h t g i v e t h i sn a m e t o t h e s i m p l e a b s e n c e o f d i s c ip l i n e a n d t o t h e d i s o r d e r b o r n o f c h a n c e . O n l y , t h isd i s o r d e r i t s e lf p e r m i t s u n e x p e c t e d c o m b i n a t i o n s .I s h a ll m a k e a l a st r e m a r k ; w h e n a b o v e I m a d e c e r t a i n p e r s o n a l o b s e r v a ti o n s , I s p o k e o f an i g h t o f e x c i t e m e n t w h e n I w o r k e d i n s p i te o f m y s e l f. S u c h c a s es a re f re q u e n t , a n d i t is n o tn e c e s s a r y t h a t t h e a b n o r m a l c e r e b r a l a c t iv i t y b e c a u s e d b y a p h y s i c a l e x c i t a n t a s in t h a t Im e n t i o n e d . I t s e em s , i n s u c h c as es , t h a t o n e is p r e s e n t a t h i s o w n u n c o n s c i o u s w o r k , m a d ep a r t i a ll y p e r c e p t i b l e t o t h e o v e r - e x c i te d c o n s c i o u sn e s s , y e t w i t h o u t h a v i n g c h a n g e d i t sn a t u re . T h e n w e v ag u e l y c o m p r e h e n d w h a t d i s ti n g u is h e s th e t w o m e c h a n i s m s o r, i f y o uw i s h , t h e w o r k i n g m e t h o d s o f t h e t w o e go s . A n d t h e p s y c h o l o g i c o b s e r v a t i o n s I h a v e b e e na b l e th u s t o m a k e s e e m t o m e t o c o n f i r m i n t h e i r g e n e r a l o u t li n e s t h e v i e w s I h a v e g i v e n .S u r e l y t h e y h a v e n e e d o f it , f o r t h e y a r e a n d r e m a i n i n s p it e o f al l v e r y h y p o t h e t i c a l : t h ei n t e r e s t o f t h e q u e s t i o n s i s s o g r e a t t h a t I d o n o t r e p e n t o f h a v i n g s u b m i t t e d t h e m t o t h er e a d e r .

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[ch]poincare 1913.sciencepress mathematical.creation.in.the.foundations.of.science.ed.poincare

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