Chapter 5'

EMPIRICISM AND THE GEOMETRY OF VISUAL SPACE

A very brief review of the account of our knowledge of visual space given by Carnap, Helmholtz, and Reichenbach will precede the discussion of some problems posed by very recent experi­mental studies of the geometry of visual space.

Distinguishing the space of physical objects from the space of visual experience ("Anschauungsraum"),Carnap sided with em­piricism even in his earliest work to the extent of maintaining that the topology of physical space is known a posteriori and that the coincidence relations among points disclosed by experience yield a unique metrization for that space once a specific coordi­native definition of congruence has been chosen freely.l But the neo-Kantian paTti pris of that period enters in his epistemological interpretation of the axioms governing the topology of visual space: "Experience does not provide the justification for them, the axioms are . . . independent of the <quantity of experience,' i.e., knowledge of them does not, as in the case of a posteriori propositions, become ever more reliable through multiply re­peated experience. For, as Husserl has shown,· we are· dealing here not with facts in the sense of empirically ascertained realities but rather with the essence ("eidos") of certain presentations whose special nature can be grasped in a single immediate ex-

1 R. Carnap: Det Raum, op. cit., pp. 39, 45, 54, 63.

A. Grünbaum, Philosophical Problems of Space and Time© D. Reidel Publishing Company, Dordrecht, Holland 1973

153 Empiricism and the Geometry of Visual Space

perience."2 Reminding us of Kant's distinction between knowl­edge acquired "with" experience, on the one hand, and "from'" experience, on the other, the early Carnap classified these axioms as synthetic a priori propositions in that philosopher's sense.

This theory of the phenomenological a priori was a stronger version of Helmholtz's claim that "space can be transcendental [a priori] while its axioms are not."3 For Helmholtz's concession to Kantianism was merely to regard an amorphous visual extended­ness as an a priori condition of spatial experience' while pro­claiming the a posteriori character of the topological and metrical articulations of that extendedness on the basis of his pioneerin~ method of imagining ("sich ausmalen") the specific sensory con­tents we would have in worlds having alternative spatial struc­tures.5

The phenomenological a priori will not do, however, as an account of our knowledge of the properties of visual space. For it is an empirical fact that the experiences resulting from ocular activity have the indennable attribute which is characteristic of visual extendedness rather than that belonging to tactile explora­tions or to those experiences that would issue from our possession

2 Ibid., p. 22. Cf. also p. 62. For a more recent defense of the thesis that "there are synthetic a priori judgments of spatial intuition," cf. K. Heide­meister: "Zur Logik der Lehre vom Raum," Dwlectica, Vol. VI (1952), p. 342 .. For a discussion of related questions, see P. Bemays: "Die Grund­begriffe der reinen Geometrie in ihrem Verhaltnis zur Anschauung," Natur­wissenschaften, Vol. XVI (1928).

3 H. von Helmholtz: Schriften zur Erkenntnistheorie, P. Hertz and M. Schlick (eds.), (Berlin: Julius Springer; 1921), p. 140.

4 Ibid., pp. 2, 70, 121-22, 140-42, 144-45, 147-48, 152, 158, 161-62, 163, 168, 172, 174. Helmholtz attempts to characterize the distinctive attri­bute of space, not possessed by other tri-dimensional manifolds, in the following way: "in space, the distance between two points on a vertical can be compared to the horizontal distance between two points on the Hoor, because a measuring device can be applied successively to these two pairs of points. But we cannot compax:e the distance between two tones of equal pitch and differing intensity with that between two tones of equal intensity and differing pitch" (ibid., p. 12). Schlick, however, properly notes in his commentary (ibid., p. 28) that this attribute is necessary but not sufficient to render the distinctive character of space.

5 Ibid., pp. 5, 22, 164-65. Cf. also K. Gerhardt's papers: "Nichteu­kIidische Kinematographie," Naturwissenschaften, Vol. XX (1932), p. 925, and «Nichteuklidische Anschauung und optische Tauschungen," Naturwis­senschaften, Vol. XXN (1936), p. 437.

PIDLOSOPIDCAL PROBLEMS OF SPACE AND TIME 154

of a sense organ responding to magnetic disturbances. In the class of all logically possible experiences, the "Wesensschau" provided by our ocular activity must be held to give rise to em­pirical knowledge. For the only way to assure a priori that all future deliverances of our eyes will possess the characteristic attribute which Hussed would have us ascertain in a single glance is by resorting to a covert tautology via refusing to call the resulting knowledge "knowledge of visual space," unless it pos­sesses that attribute.

Reichenbach made a particularly telling contribution to the disintegration of the Kantian metrical a priori of visual space by showing that such intuitive compulsion as inheres in the Euclideanism of that space derives from facts of logic in which the Kantian interpretation cannot find a last refuge and that the counter-intuitiveness of non-Euclidean relations is merely the result of both ontogenetic and phylogenetic adaptation to' the Euclidicity of the physical space of ordinary life.s

In very recent years, experimental mathematico-optical re­searches by R. K. Luneburg7 and A. A. BlankS have even led these authors to contend that although the physical space in which sensory depth perception by binocular vision is effective is Euclidean, the binocular visual space resulting from psycho­metric coordination possesses a Lobatchevskian hyperbolic geom­etry of constant curvature. This contention suggests several questions.

S H. Reichenbach: The Philosophy of Space and Time, op. cit., pp. 32-34 and 37-43.

7 R. K. Luneburg: Mathematical Analysis of Binocular Vision (Princeton: Princeton University Press; 1947), and "Metric Methods in Binocular Visual Perception," in Studies and Essays, Courant Anniversary Volume (N ew York: Interscience Publishers, Inc.; 1948), pp. 215-39.

8 A. A. Blank: "The Luneburg Theory of Binocular Visual Space," Jour­nal of the Optical Society of America, Vol. XLIII (1953), p. 717; "The non-Euclidean Geometry of Binocular Visual Space," Bulletin of the Ameri­can Mathematical Society, Vol. LX (1954), p. 376; "The Geometry of Vision," The British Journal of Physiological Optics, Vol. XIV (1957), p. 154; "The Luneburg Theory of Binocular Perception," in S. Koch (ed.) Psychology, A Study of a Science (New York: McGraw-Hill Book Com­pany, Inc.; 1958), Study I, Vol. I, Part III, Sec. A. 2; "Axiomatics of Binocular Vision. The Foundations of Metric Geometry in Relation to Space Perception," Journal of the Optical Society of America, Vol. XLVIII (1958), p. 328, and "Analysis of Experiments in Binocular Space Perception," Journal of the Optical Society of America, Vol. XLVIII (1958), p. 911.

155 Empiricism and the Geometry of Visual Space

The first of these is how human beings manage to get about so easily in a Euclidean physical environment even though the geometry of visual space is presumably hyperbolic. Blank sug­gests the following as a possible answer to this question: First, man's motor adjustment to his physical environment does not draw on visual data alone; moreover, these do contribute physi­cally true information, since they supply a good approximation to the relative directions of objects and since the mapping of physical onto visual space preserves the topology (though not the metric) of physical space, thereby enabling man to control his motor responses by feedback, as in the parking of a car or threading the eye of a needle; and second, the thesis of the hyperbolicity of visual space rests on data obtained under experi­mental conditions which are far more restrictive than those ac­companying ordinary visual experience. Under ordinary condi­tions, we secure depth perception by relying on the coordination of our two ocular images which we have learned in the past in the usual contexts. But in order to ascertain the laws of merely one of the sources of spatial information-stereoscopic depth perception alone-the experimenters of the Luneburg-Blank theory endeavored to deny their subjects precisely that contextual reliance: there were no guideposts of perspectives and familiar objects whose positions the subject had determined by tactile means, the only visible objects being isolated point lights in an otherwise completely dark room; in fact, the subject was not even allowed to move his head to make judgments by parallax. Since these contextual guideposts are also available in monocu­lar vision, the experimenters assumed that they play no part in the innate physiological processes governing the distinctive sensations of three-dimensional space which are obtained bi­nocularly.

Several additional questions arise in regard to the Luneburg theory upon going beyond its own restricted objectives of fur­nishing an account of binocular visual perception and attempting to incorporate its thesis of the non-Euclidean structure of visual space in a comprehensive theory of spatial learning: (1) how is man able to arrive at a rather correct apprehension of the Euclid­ean metric relations of his environment by the use of a physio­logical instrument whose deliverances are claimed to be non-

PHILOSOPHICAL PROBLEMS OF SPACE AND TIME

Euclidean? (2) how can students be taught Euclidean geometry by visual methods, methods which certainly convey more than the topology of Euclidean space and whose success is therefore not explained by the fact that the purportedly hyperbolic visual space preserves the topology of Euclidean physical space? (3) if we have literally been seeing one of the non-Euclidean geometries of constant negative Gaussian curvature all along, why did it require two thousand years of research in axiomatics even to conceive these geometries, the Euclideanism of physical space being affirmed throughout this period? (4) why did such thinkers as Helmholtz and Poincare first have to retrain their Anschauung conceptually in a counterintuitive direction before achieving a ready pictorialization of the Lobatchevski-Bolyai world, a feat which very few can duplicate even now? (5) if we took two groups of school children of equal intelligence and without prior formal geometrical education and taught Euclid to one group while teaching Lobatchevski-Bolyai to the other, why is it the case (if indeed that is the casel) that, in all probability, the first group would exhibit a far better mastery of their material?

The need to answer these questions becomes even greater, if we assume that our ideas concerning the geometry of our imme­diate physical environment are formed, in the first instance, not by the physical geometry of yardsticks or by the formal study of Euclidean geometry but rather by the psychometry of our visual sense data.

A. A. Blank, to whom the writer submitted these questions, has suggested that these questions may have answers which lie in part along the follOwing lines: First, man has to learn the signifi­cance of ever-changing patterns of visual sensations for the metric of physical space by discounting much of the psychometry of visual sensation, thereby developing the habit of not being very perceptive of the metrical details of his visual experience. Thus, we learn before adulthood to associate with the non-rigid sequence of visual sensations corresponding to viewing a chair in various positions and contexts the attribute of physical rigidity, generally ignoring all but those aspects of the changing appear­ances that can serve as a basis for action. In fact, laboratory find­ings show that for any physical configuration whatever, there are

157 Empiricism and the Geometry of Visual Spac~ an infinity of others which give the same binocular clues.9 Since we retain those aspects of visual experience which enable us to place objects in the contexts useful for action, Euclidean rela­tions can be more readily pictured (though not actually seen or made visible) than those of Lobatchevski; second, those geo­metrical judgments disclosed by binocular perception which are common to both Euclidean and hyperbolic geometryl will be true physically as well.

Moreover, there are certain small two-dimensional elements of visual space which are essentially isometric with the correspond­ing elements of the Euclidean space of physical stimuli. For ex­ample, in a plane parallel to the line joining the rotation centers of the eyes, physical metric relations are seen undistorted in the vicinity of a point at the base of the perpendicular to the plane from a point located half way between the eyes. We can there­fore obtain first-order visual approximations to the physical Euclidean geometry from viewing small diagrams frontally in this way. In a like manner, we can understand how the concept of similar figures, which is uniquely characteristic of Euclidean geometry among spaces of constant curvature, can be conveyed in the context of a non-Euclidean visual geometry: all Rieman­nian geometries are locally Euclidean, thus possessing a group of similarity transformations in the small; third, the presumed greater ease with which students would master Euclid than Lobatchevski is due to the greater analytical simplicity of the numerical relations of Euclidean geometry.

1) Cf. A. A. Blank: "The Luneburg Theory of Binocular Visual Space, op. cit, pp. 721-22, and L. H. Hardy, C. Rand, M. C. ruttler, A. A. Blank and P. Boeder: The Geometry of Binocular Space Perception (New York: Columbia University College of Physicians and Surgeons; 1953), pp. 15f£. and 39H.

1 For the axioms of the so-called "absolute" geometry relevant here, see R. Baldus: Nichteuklidische Geometrie, edited by F. Lobell (3rd revised edition; Berlin: Walter de Gruyter and Company; 1953), Sammlung Giischen, Vol. CMLXX, Chap. ii.