Visual space

Visual space is the experience of space by an aware observer. It is the subjective counterpart of the space of physical objects. There is a long history in philosophy, and later psychology of writings describing visual space, and its relationship to the space of physical objects. A partial list would inclulde René Descartes, Immanuel Kant, Hermann von Helmholtz, William James, to name just a few. Visual space is the experience of space by an aware observer. It is the subjective counterpart of the space of physical objects. There is a long history in philosophy, and later psychology of writings describing visual space, and its relationship to the space of physical objects. A partial list would inclulde René Descartes, Immanuel Kant, Hermann von Helmholtz, William James, to name just a few. The location and shape of physical objects can be accurately described with the tools of geometry. For practical purposes the space we occupy is Euclidean. It is three-dimensional and measurable using tools such as rulers. It can be quantified using co-ordinate systems like the Cartesian x,y,z, or polar coordinates with angles of elevation, azimuth and distance from an arbitrary origin. Percepts, the counterparts in the aware observer's conscious experience of objects in physical space, constitute an ordered ensemble or, as Ernst Cassirer explained, Visual Space can not be measured with rulers. Historically philosophers used introspection, and reasoning to described it. With the development of Psychophysics beginning with Gustav Fechner, there has been an effort to develop suitable experimental procedures which allow objective descriptions of visual space including geometric descriptions to be developed and tested. An example illustrates the relationship between the concepts of object and visual space. Two straight lines are presented to an observer who is asked to set them so that they appear parallel. When this has been done, the lines are parallel in visual space A comparison is then possible with the actual measured layout of the lines in physical space. Good precision can be achieved using these and other psychophysical procedures in human observers or behavioral ones in trained animals. The distinction should be made between the visual field, the area or extent of physical space that is being imaged on the retina, and the perceptual space in which visual percepts are located, which we call visual space. Confusion is caused by the use of Sehraum in the German literature for both. There is no doubt that Ewald Hering and his followers meant visual space in their writings. The fundamental distinction was made by Rudolf Carnap between three kinds of space which he called formal, physical and perceptual. Mathematicians, for example, deal with ordered structures, ensembles of elements for which rules of logico-deductive relationships hold, limited solely by being not self-contradictory. These are the formal spaces. According to Carnap, studying physical space means examining the relationship between empirically determined objects. Finally, there is the realm of what students of Kant know as Anschauungen, immediate sensory experiences, often awkwardly translated as 'apperceptions,' which belong to perceptual spaces. Geometry is the discipline devoted to the study of space and the rules relating the elements to each other. For example, in Euclidean space the Pythagorean theorem provides a rule to compute distances from Cartesian coordinates. In a two-dimensional space of constant curvature, like the surface of a sphere, the rule is somewhat more complex but applies everywhere. On the two-dimensional surface of a football, the rule is more complex still and has different values depending on location. In well-behaved spaces such rules used for measurement and called Metrics, are classically handled by the mathematics invented by Riemann. Object space belongs to that class. To the extent that it is reachable by scientifically acceptable probes, visual space as defined is also a candidate for such considerations. The first and remarkably prescient analysis was published by Ernst Mach in 1901. Under the heading On Physiological as Distinguished from Geometrical Space Mach states that 'Both spaces are threefold manifoldnesses' but the former is '...neither constituted everywhere and in all directions alike, nor infinite in extent, nor unbounded.' A notable attempt at a rigorous formulation was made in 1947 by Rudolf Luneburg, who preceded his essay on mathematical analysis of vision by a profound analysis of the underlying principles. When features are sufficiently singular and distinct, there is no problem about a correspondence between an individual item A in object space and its correlate A' in visual space. Questions can be asked and answered such as 'If visual percepts A',B',C' are correlates of physical objects A,B,C, and if C lies between A and B, does C' lie between A' and B' ?' In this manner, the possibility of visual space being metrical can be approached. If the exercise is successful, a great deal can be said about the nature of the mapping of the physical space on the visual space. On the basis of fragmentary psychophysical data of previous generations, Luneburg concluded that visual space was hyperbolic with constant curvature, meaning that elements can be moved throughout the space without changing shape. One of Luneburg's major arguments is that, in accord with a common observation, the transformation involving hyperbolic space renders infinity into a dome (the sky). The Luneburg proposition gave rise to discussions and attempts at corroborating experiments, which on the whole did not favor it. Basic to the problem, and underestimated by Luneburg the mathematician, is the likely success of a mathematically viable formulation of the relationship between objects in physical space and percepts in visual space. Any scientific investigation of visual space is colored by the kind of access we have to it, and the precision, repeatability and generality of measurements. Insightful questions can be asked about the mapping of visual space to object space but answers are mostly limited in the range of their validity. If the physical setting that satisfies the criterion of, say, apparent parallelism varies from observer to observer, or from day to day, or from context to context, so does the geometrical nature of, and hence mathematical formulation for, visual space.