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The Dimensionality of Visual Space

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Abstract

The empirical study of visual space (VS) has centered on determining its geometry, whether it is a perspective projection, flat or curved, Euclidean or non-Euclidean, whereas the topology of space consists of those properties that remain invariant under stretching but not tearing. For that reason distance is a property not preserved in topological space whereas the property of spatial order is preserved. Specifically the topological properties of dimensionality, orientability, continuity, and connectivity define “real” space as studied by physics and are the spatial properties that characterize the physical universe as being an integral whole. By contrast the geometrical analysis of VS has taken little cognizance of its topology. Instead such properties have been presupposed a priori rather than being established a posteriori by empirical means, perhaps because these properties are self-evident. Applying the method of coordinative definition expounded by Hans Reichenbach for determining geometrical and topological properties of physical space (PS), it can be shown that VS fulfills the topological criteria of being a “real” space sui generis. Though theorized to be produced by the brain, the topology of VS is not topologically equivalent (homeomorphic) with the structure and activity of the brain because, as will be shown, the topology of VS cannot be formed from the topology of the brain without tearing and/or cutting and pasting.

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Notes

  1. Translated as “visual space,” the term actually introduced by Hering was the German term Sehraum, as opposed to wirklichen Raum, or “real space” in English (Hillebrand 1929).

  2. By “metageometry” was meant what is now called non-Euclidean geometry (Russell 1959).

  3. Ophthalmologist LeGrand Hardy and his associates differentiated visual space from physical space in much the same way: “‘Physical space’ is a measurable, engineering space of the outside world. It is the space in which houses, bridges, roadways and railways are mapped and built with great precision. ‘Visual space’ is the immediate instantaneous impression we have of our environment as a three dimensional manifold; the immediate impression we have of objects about us in relative terms; farther-nearer; larger-equal” (Hardy et al. 1951: 53).

  4. Morgan cites a paper by Battro et al. (1976) which advances the idea that VS is not hyperbolic as Luneburg claimed.

  5. The physical properties Morgan lists in support of his argument are from Hinckfuss (1975). Morgan did not heed the caveat, however, which Hinckfuss added regarding these properties: “It is not being alleged, of course, that each item in the lists [of properties of space] is a true statement, but only that it has been believed to be true by someone at some time.”

  6. Cf. Mach (1906a).

  7. For a review of Luneburg’s work and that inspired by it see the analysis by philosopher Patrick Suppes (1977).

  8. Philosopher Wayne Stromberg (1975) addressed certain conceptual inconsistences in Gestalt theory’s account of 3-D visual perception in terms of figure–ground relationships.

  9. Cf. Ranucci and Rollins (1977).

  10. Of course there are no ocular muscles between the eyes but that fact may not be experienced as such, given the musculature covering the nose.

  11. From a topological standpoint these reports are significant because the cyclopean viewpoint is maintained: the projection is merely inverted or turned around, but the perspective remains otherwise unchanged. It does not result in seeing VS from behind, as it were, so that it being one-sided topologically is unchanged.

  12. I am indebted to Professor of Philosophy Graham Nerlich of the University of Adelaide for lively correspondence in the past to discuss points about visual geometry in his stimulating book The Shape of Space (1976).

  13. For criticism of Luneburg’s theory, see Hardy et al. (1951), Gibson (1959), Baird (1970) and Battro et al. (1976).

  14. Dimensionality is used here in the sense given in the OED: “[T]he condition of having (a particular number of) dimensions; dimensional quality.”

  15. Evidently Schrödinger is using “sensual” as a synonym for “sensory,” as indicated by the OED, “Of or pertaining to the senses or physical sensation; sensory. Now rare.”

  16. Cf. Koffka (1935) and Angeles (1981): “[Q]ualities, primary/secondary. 1. Primary qualities are (a) those qualities such as motion, rest, size (extension) shape (figure), solidity (impenetrability), number, structure, (b) which are believed to be inherent characteristics of matter in itself and not to depend for their existence on consciousness. 1. Secondary qualities are those (a) sensed qualities such as of color, smell, taste, sound, hear, cold, (b) which are believed to be caused in us by the primary qualities inherent in matter and (c) which depend for their existence on the operations of the mind. Primary qualities exist in reality only independently of an observer. Secondary qualities exist only as content in consciousness. This distinction has in general been made since the time of Leucippus and Democritus… and was commonly accepted during the rise of modern science by thinkers such as Galileo, Boyle, Newton. The distinction in philosophy is mainly associated with Locke.”

  17. I am indebted to physicist Nick Herbert for bringing to my attention the relevance of Mach’s views on sensations qua the fundamental elements in Mach’s phenomenalist epistemology.

  18. By complexes of sensations Mach evidently meant the physicist’s observations.

  19. Though in his presentation Ward does not refer to corresponding work on intensity and extensity in theoretical physics, cf. philosopher Karl Robert Eduard von Hartmann: “The quantity, or the extensity factor, of energy is attached to one structure and cannot be transferred to another structure without carrying with it parts of the first; but the intensity factor can pass from one structure to another.” Quoted in Jung (1972: 20).

  20. “Dimensionality without Dimensionality” is one of any number of characteristic Wheelerisms that take the form ‘A without A,’ roughly, anything presupposing A can never explain A, and was his remedy for circular reasoning or tautologies, evidently using Einstein as his point of departure who sought to formulate “force without force.” Thus Wheeler speaks of “charge without charge,” “mass without mass,” etc. Students and associates of Wheeler’s compiled an anthology of essays in his honor, and called it, appropriately, Magic Without Magic (Klauder 1972). Saul-Paul Sirag (personal communication, 1984) suggested the most fitting title for a biography of Wheeler would be “Without Without Without.”

  21. Cf. Quinn (1981): “As [Bertrand] Russell suggested, these punctual elements [idealized geometrical points] are qualitatively homogeneous; there is no inherent quality in a single point, as there is in a single color, by which it can be qualitatively distinguished from another.”

  22. Remarkably the experimental quest for a Ganzfeld of completely uniform illumination and color has yielded the paradoxical result of a “black out,” in which persons lose the power of sight temporarily, and their visual field (VS) disappears completely (Gibson and Waddell 1952; Avant 1965).

  23. Katz (1935) refuted Buhler’s proposal that our perception of empty space is the result of “air-light,” light reflected off of dust particles in the air.

  24. “Since one-sidedness is equivalent to non-orientability, we have again proved the one-sidedness of the projective plane” (Lietzmann 1965: 149).

  25. This conceptualization was developed more fully by Reichenbach (1991).

  26. Quoted from an unpublished abstract (1982), “Plato’s cave revisited: The projection of space–time from C [S4].” The abbreviation “C[S4]” denotes the symmetric-four group in algebraic group theory.

  27. The late Avrum Stroll, professor of philosophy at University of California, San Diego, pointed out that my thinking in this context paralleled that of Goodman (1977) and I remain grateful to him for that and much fruitful discussion of the language of surfaces. I am also indebted to Saul-Paul Sirag for enlightening and copious tutoring concerning grand unification schemes.

  28. I am indebted to the Derek Fender, late professor of Applied Science and Biology in Bioinformation Systems at Caltech, who first got me thinking about stereopsis in terms of topology, and envisioned that my dissertation research would be to model stereopsis as a catastrophe as theorized by René Thom (1975). This was in response to a conference paper of mine entitled “Is the engram a stereogram?” (Rosar and Perrott 1980) in which I had argued that visual mental images lack stereopsis, but that stereopsis is remembered and visualized as separate figure–ground relationships.

  29. Blank also suggested that the brain might be responsible for the putative curvature of VS: “In a metric geometry the straight lines or geodesics are paths of shortest length. This fact leads one to conjecture that the visual geodesics, the loci which give the impression of straightness, must be associated with something like a least action principle in the visual neural organization. Perhaps a physiological understanding of visual space perception may eventually be reached by pursuing this possible connection between the metric of visual space and neural function.” (1957: 234–235) However, I identified and analyzed certain problems in the Gestalt use of topological and topographic concepts relative to brain mapping (Rosar 1985).

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    William H. Rosar

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Rosar, W.H. The Dimensionality of Visual Space. Topoi 35, 531–570 (2016). https://doi.org/10.1007/s11245-016-9385-0

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