visual riemannian space versus cognitive euclidean space

8/7/2019 Visual Riemannian space versus cognitive Euclidean space

1/7

ANTONIO M. BATTRO

V I S U A L R I E M A N N I A N S P A C E V E R S U SC O G N I T I V E E U C L I D E A N S P A C E

A Rev is ion o f Gri~nbaum's Empir ic ism an d Lunebu rg ' sGeometry of V isual Space

In chapte r 5, ent i t led 'Empir icism and the g eom etry of visual space ' of hisb o o k Philosophical Problems of Space and Time, 1 Professor Gr i inbaumdeals with K. Luneb urg ' s theory of binocular v isual perception. At thet ime of his wri t ing, exp erimental psyc holog y was providing som e goodsupport for Luneburg's theory that visual space has a constant negativecurvature. Accordingly most psychologists accepted the assumption thatvisua l geom etry be longs to the L obachevsk ian hyperbol ic type .

Gr i inbaum's fundamenta l epis temologica l ques t ion about Luneburg ' stheory was s ta ted thus : " H ow do human be ings manage to ge t about soeas ily in a Eucl idean physica l envi ronmen t even though the geom etry ofvisual space is presumably hyperbolic?" (op. c i t . , p. 155) or , in otherwords: "H ow is man able to a rr ive a t an approximate ly cor rec t apprehe n-sion of the Eu clidean metr ic relations of his enviro nme nt by the use of apsychological instrument whose del iverances are claimed to be non-Eucl idean?" (op. c i t . , p . 155-156) . As Gr i inbaum says " the need toanswer these qu es t ions becom es even grea ter , i f we assume tha t o ur ideasconcerning the geometry of our immedia te physica l envi ronment a reforme d, in the f irst instance, no t by the physical geom etry of yardst icks orby the formal s tudy of Eucl idean ge ome try but ra ther b y the psychom etryof our visual sense data" (op. cit . , p. 156).

Gr i inbaum submit ted these and oth er re la ted ques t ions to A. A. Blank,an author i ty in Lunebu rg ' s theory, and rece ived back, among others , thefollowing two suggest ions that we con sider central to o ur discussion:( i) People " learn the signif icance of ever changing patterns of visualsensat ions for the metr ic of physical space by discounting much of thepsy cho me try of visual sensat ions" (op. ci t. , p. 156);(i i) "T her e are cer tain small two-dimensio nal elem ents of visual spaceSynthese 35 (1977) 423-429. All Rights ReservedCopyrightO 1977by D. Reidel Publishing Company, Dordrecht-Holland


2/7

424 A N T O N I O M . B A T T R Owhich are essentially isometric with the corresponding elements of theEuclidean space of physical stimuli . . . . We can understand how theconcept of similar figures, which is uniquely characteristic of Euclideangeometry among spaces of constant curvature, can be conveyed in thecontext of a non-Euclidean visual geometry: all Riemannian geometriesare locally Euclidean, thus possessing a group of similarity transforma-tions in the small" (op. cit., p. 157). "For example, in a plan parallel tothe line joining the rotation centers of the eyes, physical metric relationsare seen undistorted in the vicinity of a point at the base of the perpen-dicular to the plane from a point located half way between the eyes" (op.cit., p. 157).

We are then confronted with two p s y c h o l o g i c a l statements:I The subjective metric of visual space is presumably Lobachevskian.II The idea of Euclidean geometry can be formed by viewing smallfigures in front of the eyes because visual space is locally Euclidean.

Both statements have an experimental character and they are intendedto provide new and powerful scientific arguments towards the "disinte-gration of the Kantian metrical a p r i o r i of visual space" (Grtinbaum, op.cit., p. 154). At least, this is, I suppose, the philosophical reason tha t ledGriinbaum to dedicate a whole chapter to a psychological theory of visionin a book about physics.

In this article I shall try to demonstrate that (a) these two arguments, inthe light of new visual experiments and of some old psychogeneticobservations, are basically wrong. However I believe (b) that the substitu-tion of statements I and II by more reliable empirical facts may improvethe scientific arguments against Kantian metrical a pr ior i .

1 . V I S U A L S P A C E I S N O T L O B A C H E V S K I A N B U T R I E M A N N I A N

Recent experiments carried out by my colleagues and myself in big openfields show clearly that instead of a visual space of constant negativecurvature, we must face a space of variable curvature. 2 In fact, in ourexperiments, Luneburg's model, applied to visual alleys of different sizes,yields different positive or negative values of K, depending on thedistance of the observed objects from the subject, contradicting clearlythe hypothesis that for each subject the scalar curvature K is always anegative constant.


3/7

V I S U A L R I E M A N N I A N S P A C E 425We must conclude, th erefore, that the geom etry of visual space is notLobachevsk ian but Riemannian and that the assumption that visual space

is homogeneous is in fact wrong. This amounts, of course, only to atechnical detail but I believe that the idea of a general Riemanniangeom etry in a visual space of variable curvature is even mo re attractive toa philosopher of science and could eventually lead to fruitful speculationsabout a 'relativistic' approach to perceptual research if the curvaturehappens to be a function of 'visual matter'.

2. T H E E U C L I D E A N G R O U P O F S I M I L A R I T I E S D O E S N OT H A V EI T S O R I G I N I N V I S U A L P E R C E P T I O N B U T I N S E N S O R Y - M O T O R

A C T I V I T Y

The second a rgument stands on the general assumption that concepts ofgeometry can be abstracted from ('fo rmed by' in Griinbaum's words) thepsychometry of visual sense. The sensory origin of ideas abou t space isprobab ly a belief as old as philosophy, but although he didn't discuss thispoint, Grtinbaum is certainly not ignorant of the crucial argument ofPoincar6 for a sensory-motor group of human movements at the origin ofgeometrical concepts, against the pure visual perceptivism of 'classical'empiricism. 3 Even more, Poincar6 states that the general concep t ofgroup is not a fo rm of our sensibility but a form of our understanding. 4 Inthis sense (visual) perception cannot be, by any means, the source of ourgeometrical ideas. Piaget, on his side, has given enough experimentalevidence that children don't build geometrical concepts from visualexperience but from sensory-motor groups, although he differs fromPoincar6 on the innate origin of these groups. For Piaget these groups areacquired and need a long period (years) to become operative at therepresentational levels of concrete or formal geometrical thinking. 5

The problem, I believe, is only secondarily about sensory data butprimarily about sensory space. This needs further clarification: visualspace, as exper imental psychology shows, has a subjective metr ic or, aswe can say, a 'psycho-metric'. Luneburg was the first to devise amathematical model which allowed the computation of the scalar curva-ture of such a space. This characterization of the intrinsic geometry ofvisual space was indeed a remarkable contribution to Helmholtz' classicalstudies on visual geodesics and to his famous axiom of the 'free mobili ty'


4/7

4 2 6 A N T O N I O M . BA T T ROo f f i g u r e s i n s u r f ace s o f co n s t an t cu r v a t u r e . V i s u a l s u r f ace s o f n eg a t i v ec o n s t a n t c u r v a t u r e a n d L o b a c h e v s k i a n m e t r ic w e r e i n f e rr e d b y L u n e b u r gan d h i s f o l l o w er s fr o m t h e b eh av i o u r o f ex p e r i m e n t a l v i s u a l g eo d es i c s .O u r o w n i n v es t i g a t i o n s h av e s h o w n t h a t t h i s v i s u a l cu r v a t u r e i s b e t t e rr ep r e s en t ed a s h av i n g a v a r i ab l e cu r v a t u r e a n d a R i e m an n i a n m e t r i c . I t isi m p o r t a n t t o r e m e m b e r a l so t h a t h u m a n v i si o n l a c ks X - r a y p r o p e r t i e s a n dt h e r e f o r e o n l y t h e ' f ro n t a l ' p a r t o f a n o b j e c t c a n b e p e r c e i v e d a t a g iv e ni n s t an t . T h e s u m o f a l l t h e s e f r o n t a l p a r t s c an b e m o d e l l ed i n a t w o -d i m e n s i o n a l m a n i f o l d w h i c h c a n b e o f g r e a t c o m p l e x i t y . 6

A c c o r d i n g t o P i a g e t , s e n s o r y - m o t o r g r o u p s o f m o v e m e n t s ( d i s p l a c e -m e n t s o f t h e b o d y , e y e s o r l im b s ) d e v e l o p i n t o i n t e ri o r iz e d m e n t a l g r o u p so f t r an s f o r m a t i o n s o f s p ace ( au t o m o r p h i s m s ) i n c l u d i ng , o f co u r s e , t h es p ec i f i c s u b - g r o u p o f s i m i l a r i t i e s w h i ch i s o n l y f o u n d i n E u c l i d eanG e o m e t r y . T h e p h y s i c a l ( p r a c t i c a l ) e n v i r o n m e n t o f m a n i s E u c l i d e a n( G r i i n b au m ' s g eo m e t r y o f y a r d s t i ck s ) b u t h u m an v i s i o n i s d e f i n i t i v e l yn o n - E u c l i d e a n , t h e r e f o r e a n y change o r scale o f t h e o b j e c t ( r e d u c t i o n o ram p l i f i c a t i o n ) p r o d u ces n ece s s a r i l y a ch an g e i n i t s g eo m e t r i c perceivedp r o p e r t i e s (i .e . t h e s u m o f t h e an g l e s o f a n o n - E u c l i d e an t r ian g l e cea s e s t oh av e a co n s t an t v a l u e an d v a r i e s p r o p o r t i o n a t e l y t o t h e t r i an g l e a r ea ) .G r i i n b a u m h a s c l e a r l y n o t i c e d t h a t t h e m o s t i m p o r t a n t c h a r a c t e r o fn o n - E u c l i d e a n g e o m e t r y is c e r t a in l y n o t t h e a x i o m a t ic s u b s t i t u ti o n o f t h ep a r a l l e l p o s t u l a t e b u t t h e n o n - ex i s t en ce o f t h e g r o u p o f s im i l a ri ti e s , vE u c l i d ea n p r o p e r t i e s t h e r e f o r e a r e o n l y v a l i d ' in t h e s m a l l '.

I n c i d en t a l l y , i t i s r em ar k ab l e t h a t t h e s e g eo m e t r i c p r o p e r t i e s h av e n o tb e e n c o r r e c tl y u n d e r s t o o d b y m o s t e x p e r i m e n t a l p s y c h o lo g i s ts . T h e y stillb e l i ev e t h a t i t m ak e s s en s e t o s t u d y , e . g. t h e p e r ce p t i o n o f a s m a l l c i rc l e inl a b o r a t o r y c o n d i t i o n s a n d t o e x t r a p o l a t e t h e o b s e r v e d p e r c e p t u a lm ech an i s m s ' in t h e s m a l l ' t o t h e p e r ce p t i o n o f c ir c l es o f a r b i t r a r y s iz e ' i nt h e l a r g e ' . T h i s i n s en s i b i l i t y t o w ar d s t h e p s y ch o l o g i ca l e f f ec t s o f t h ech an g e o f s cal e u p o n f ig u r e s (a s p a ti a l t r an s f o r m a t i o n b y s im i l a ri t y)r ev ea l s h o w i g n o r an t a ty p i ca l p s y ch o l o g i s t is o f th e g e o m e t r i c a l co n s eq u -e n c e s o f t h e n o n - E u c l i d e a n m e t r i c o f v is u al s p a c e h e h a s d i s c o v e r e d , a n dh o w a t t a c h e d h e r e m a i n s t o t h e o n l y a n d u n i q u e c a s e w h e r e t h e c h a n g e o fs ca l e i s i r r e l ev an t , E u c l i d ean g eo m e t r y ! T h i s I c a l l a m i s l ead i n g d ep en -d e n c e o n t h e E u c l i d e a n p a ra d i g m . A c c o r d i n g t o G r i i n b a u m t h e g r o u p o fs i m i l a r i t y t r an s f o r m a t i o n s i n t h e s m a l l h a s ap p a r en t l y t h e p r i v i l eg e o fb e i n g t h e s o u r ce o f E u c l i d ean co n cep t s . B u t h o w s m a l l a r e t h e f i g u r e s


5/7

V IS U A L R IE M A N N IA N S P A C E 427supposed to be in order to provide such Euclidean information?Mathematically we always consider the 'small' at a differential level.Empirically this is impossible. Therefore we are obliged to choose aconventional size for the figures ('small') in a particular plane (frontal) at adeterminate distance ('near to') from the observer. This seems again to bea factual psychological problem and not an epistemological one. In otherwords, Griinbaum reports, apparently without any objection, Blank'sidea that as Riemannian visual space is Euclidean 'in the small' soEuclidean concepts could be formed by the perception of small figures.This is a psychological n o n s e q u it u r , in my opinion. Of course all teacherswill agree that this is the way Euclidean geometry is taught in school butPiaget, who intensively studied the s p o n t a n e o u s acquisition of Euclideanconcepts, shows that the geometrical reasoning with small figures may becorrect but that the similarity transformation of the same figures consti-tutes a problem. Children begin to follow the Euclidean paradigm only atthe age of 9 to 10, although the basic topological propert ies of space havebeen acquired by them much earlier. The principal difficultyof Euclideangeometry is again the representation of the group of similarities. 8 Conse-quently, it seems that the second argument that the Euclidean concept ofsimilar figures can be conveyed by viewing small diagrams is also scien-tifically unsound.

C O N C L U S IO N S

Similarity is the essential concept of the Euclidean paradigm. As Gausssaid "in Euclidean geometry nothing is absolutely great but exactly thecontrary obtains in non-Euclidean geometry".9 Psychologically, toaccept the Euclidean paradigm amounts to remaining 'insensitive' tochanges in the scale of objects because the only case of a geometricalreasoning i n d e p e n d e n t of the scale is Euclidean geometry itself. Thegroup of similarities is a structure that can' ot be conveyed by visual data,this group being certainly not embodied in human vision but in humanaction.

To sum up, the Euclidean paradigm is only a very particular case ofhuman perception and thinking. It cannot be attributed a pr i o r i toA n s c h a u u n g (as Kant claimed) because we have shown that visual spaceis Riemannian, nor a pr i o r i to V e r s t a n d (as Poincar6 intended) because


6/7

4 2 8 ANTONI O M. BATTR OP i ag e t h a s c l ea r l y s h o wn t h a t t h e E u c l i d e an g ro u p o f s i m i l a ri ti e s is n o tn ece s s a ry ' d ' em b l 6 e ' b u t p ro g re s s i v e l y i n co r p o ra t ed i n la t e r s t ag es o fm en t a l d ev e l o p m en t . I n t h i s s en s e , I b e l i ev e , t h e ' d i s i n t eg ra t i o n ' o fK a n t i a n m e t r ic a l a priori is ev en m o re d ra s t i c t h an i t was s u p p o s e d t o b e i nG r f i n b a u m ' s d i s c u s s i o n o f L u n e b u r g ' s n o n - E u c l i d e a n A n s c h a u u n g .In t e r e s t i n g l y en o u g h , a f t e r g en e ra l r e l a t i v i t y fo r p h y s i ca l s p ace , m o d e rnp s y c h o l o g y a ls o p r e s u m a b l y f a c e s a g e n e r a l R i e m a n n i a n m e t r i c fo r v i su a ls p a c e a n d t h u s t h e E u c l i d ia n p a r a d i g m b e c o m e s t h e e x c e p t io n r a t h e r t h a nt h e ru l e i n h u m a n p e rc ep t i o n ( s u b j ec ti v e s p ace ) an d p h y s i cal k n o wl ed g e(o b j ec t i v e s p ace ).Centro de Investigaciones Filos6ficas, Cangallo 1479 , B u e n o s A i re s

N O T E S1 Knopf. New York, 1963.2 Battro, A. M., di Pierro Netto, S. and Rozenstra ten, R. J. A.: 'R iemannian Geometr ies ofVariable Curvature in Visual Space', P e r c e p t i o n 5 (1976), 9-23.3 "What we call geometry is only the study of the properties of a given continuous groupsuch that we can say that space is a group. The not ion of this continuous group exists in ourmind previous to any experience, but the same occurs with the notion of many othercontinuous groups, i.e. the group of the Lobachevskian geometry . . . . Among the continu-ous groups that our mind can build we may choose that particular one analogous to thephysical continuum that experience has shown us to be the group of movements" (mytranslation), Poincar6, H.: 'Des fondements de la g6om6trie', R e v u e d e M d t a p h y s iq u e e t d eM o r a l e , 1899, VII.4 Poincar6, H.: L a s c i e n c e e t l ' h y p o t h d s e , Flammarion, Paris, 1902, p. 90.5 Piaget's fundamental studies related to the epistemology of space were not ment ioned byGriinbaum. The following books may help the reader to understand the acquisition ofgeometrical concepts in children, adolescents and adults:Piaget, J.: In t ro d u c t io n ~ l ' d p i s td mo lo g i e g d n d tiq ue , Presses Universitaires de F rance (PUF),Paris, 1950 (3 Volumes).Piaget, J. and Inhelder, B., L a r e pr d se n ta ti on d e l 'e s p a c e c h e z l ' e n f a n t , PUF, Paris 1947.Piaget, J., Inhelder, B., and Szeminska, A., L a g d o m d t r i e d e l ' e s p a c e c h e z l ' e n f a n t , PUF,Paris 1948.Piaget, J. et a l . , L ' ~p is t~mologie de l ' espa ce, PUF, Paris, 1964.Piaget, J. e t a l . , C o n s e r v a t i o n s s p a t i a l e s , PUF, Paris, 1965.6 It is interesting to note that E. Mach in E r k e n n t n i s u n d I r r t u m was perhaps the first to givethe example of the human skin as a model of a Riemannian surface.7 Similarity and the parallel postulate can be proved to be logically equivalent in Euclideangeometry since Wallis's demonstr ation in the XVII Century that the parallel postulate canbe substituted by the postulate of the existence of similar figures of arbitrary size. Wallis'salternative for postulate V was that to a given triangle there is always a similar triangle


7/7

V I S U A L R I E M A N N I A N S P A C E 4 2 9

h a v i n g a r b i t r a r y s i z e. I n c i d e n t a l l y , p o s t u l a t e I I I i m p l i c i t l y s t a t e s a l s o t h a t a l l c i r c l e s o fa r b i t r a r y s i z e a r e s i m i l a r .8 C f . c h a p t e r X I I ' L e p a s s a g e ~ l ' e s p a c e e u c l i d e a n . L e s s i m i l it u d e s e t le s p r o p o r t i o n s ' i nL are p rd se nta ti o n d e l ' e sp a c e c h e z l ' e n f a n t (cf . note 5) .9 Q u o t e d b y V u i l l e m i n , J . L a p h i l o s o p h i e d e l ' a l g k b r e , PU F, Pa r i s , 1 9 6 2 , p . 4 0 5 .

visual riemannian space versus cognitive euclidean space

Documents