The Latimer & Stevens (1997) target article on wholes, parts, and their perception is analyzed and extended on the basis of topology involved in the Lie transformation group theory of neuropsychology and the symmetric difference model for cognition.
1. Latimer & Stevens (1997) offer a rationale for neuropsychological research based on Rescher & Oppenheim's (1955) formal analysis of the part-whole relationship. For this purpose, topology, invariance, and "complexes" are invoked, with the latter possessing "features" that are invariant under transformation by the Z-subgroup of transpositions. The purpose of this commentary is to explore the topology of perception and cognition in this context more fully. The commentary leans heavily on known topology and transformation group properties embodied in a long series of papers originating with Hoffman (1966). Latimer & Stevens mention this paper in passing as a failed attempt but appear unaware of subsequent publications in this series and the experimental substantiation of the theory by Dodwell and his school and others (Hoffman, 1994b: 23).
2. A topology consists of a set of neighborhoods which combine into a space in much the way they do in tissues according to the Cell Theory in biology and in microscopic and macroscopic brain structure and function according to the Neuron Doctrine in neuropsychology. In the present context, "space" will denote, for perception, the visual manifold (Hoffman, 1989), or, for cognition, the category of simplicial objects (Hoffman, 1980, 1996a, 1997). For the latter, part-whole decomposition and composition both stem from the symmetric difference and its complement. In this connection it is worthy of note that the symmetric difference is one of the two fundamental operations of set topology (Kostrikin & Shafarevich, 1900: sec. 20).
3. Interrelationships between the mathematics and structural neuropsychology will be "factored" (mathematical jargon for "classified") by perception and cognition. A thing is not just a thing. It has form and meaning. In the case of perception, Pribram's (1960) posterior intrinsic systems, which "mediate invariant properties of specific sensory modalities," will be taken as key structures; for cognition, it will be his frontal intrinsic systems (FISs). Analysis of the latter will rely heavily on information processing psychology and its isomorph (Hoffman, 1997) in Riegel's (1973) dialectical psychology. The decompositions involved will, for perception, be by way of the transformation groups of the constancies and the associated mathematical entity of fibre bundles; for cognition, as mentioned above, the symmetric difference and its complement, which together generate the essential structure of information processing psychology and dialectical psychology (Hoffman, 1995, 1996a, 1997).
4. Begin with the constancies and their associated transformation groups. As Rock (1980) has put it, "...the perceptual constancies ...are at the very core of, and...virtually synonymous with the field of perception." Latimer & Stevens cite two of these transformational invariances: transpositions (= mathematical "translations') and contractions/dilations. The latter constitute the dilation group governing size constancy, and translations are involved in pitch constancy and the rectilinear displacement component of shape constancy. The other constancy transformations (Hoffman, 1966, 1977, 1978) comprise rotations (which together with translations generate affine transformations), pseudo-Euclidean rotations for binaural localization and afferent and efferent binocularity, and the Lorentz group for motion invariant perception. The invariant perceptions generated by constancy preprocessing of stimuli thus embody, in toto, the action of the conformal group CO(1,3) (Hoffman, 1989, 1994b) in subjective spacetime.
5. Continuous transformation groups (which is to say, "Lie transformation groups") have both a local and a global structure. These correspond respectively to the neuronal and psychological levels of perception. The local structure is provided by the so-called "infinitesimal transformation" (Steeb, 1996); the global structure, by the orbifold (a manifold consisting of transformation group orbits or path-curves) of transformation group induced flows (1994b). The infinitesimal transformation itself is generated by a Lie derivative, which "drags the flow" along an orbit. The transition from local to global is given by the exponential map (Eisenhart, 1961: 35-6; Dodson, 1980: 106-8). A function, like a visual contour defined over the perceptual manifold or a Latimer & Stevens' "structural whole" consisting of "assignments of parts to positions," is invariant under the transformation group of a Lie derivative if it is annulled by the latter. This appears to be the role of inhibition in the CNS - cancellation by a specific afferent of the non-specific input from the Ascending Reticular Activating Formation. In keeping with this, an afferent volley of optical nerve impulses terminates mainly in layer 4c of V1 under inhibition rather than excitation. This termination takes place by the pericellular nests of the afferent axons winding around and around the stellate cells. The pericellular nests have a spherical nature characteristic of a cortical embodiment of the infinitesimal transformations of the group of rotations, the stellate cells, that for the group of dilations. Note that these are the local orbits of two orthogonal flows, which is a characteristic feature of the constancy orbits. An afferent stimulus continues vertically along a cortical microcolumn while passing through small pyramid cells and horizontal plexuses of nerve fibers. The visual manifold is thus dissected (decomposed) into a grid of the fundamental patterns: polar (= spiral = rotations + dilations), hyperbolic, and rectilinear found by Gallant, Braun, and Van Essen (1993).
6. Latimer & Stevens ask (1997: sec. 27) "what sort of machine does the human system have to be in order to derive holistic attributes from prior local analysis...?" From the standpoint of form perception, the operative word here is "system." We therefore adopt the full definition of this term as given in the Geometry of Systems (Brockett, 1977): A system is a quadruple (E,X,Y,h), where p:E-->M denotes a fibre bundle; M is the state space; X --> TM is a tangent vector field such that for all u in the fibre over m, X(m,u) is in TM at m; Y is a smooth manifold and h:M-->Y is a smooth map. An input and a state together determine a vector field via the function X. Each point in M determines an output y = h(m) in Y. The apparent neuropsychological correlates are the cortex-midbrain structure for E and M. For X, the cortical "orientation response." The projections p and p-inverse are the cortical projection and lifting maps, respectively. Y is the perceptual/cognitive manifold, and h is the efferent output. Latimer & Stevens cite theoretical and experimental work in support of the tangential approximation TM to an arc (visual contour in the visual manifold M), and the foregoing is in accord with Colonnier's (1964) analysis of the tangential organization of the visual cortex. Mathematically speaking, this implies the mappings TM --> M = Union of transverse visual contours (Hoffman & Dodwell, 1985).
In the Lie transformation group theory of neuropsychology (the LTG/NP) (Hoffman, 1966, 1977, 1978,1984, 1994a), integration of the local tangent structure TM into a contour in the visual manifold can - in principle - be done in one of two ways: the exponential mapping cited above for the constancy vector field and/or its prolongations (Hoffman, 1970, 1984) or by parallel transport of the orientation-response vector field. Here is an experimentum crucis for determination of the part-whole relation for perception.
7. So, for perception we have a universal decomposition in terms of constancy orbifolds and the tangent vector field determined by the cortical orientation response. Topologically,
G x M --> T(G x M) --> TG x TM --> TM --> M
where G x M represents distortion of a form percept by viewing conditions. T(G x M) is the microscopic cortical representation and is mapped functorially to the "local" neuronal structures TG and TM corresponding to the constancy group and the orientation response, respectively. The right inverse to TG preprocesses out the constancy distortions of an image-object, and the local structure is integrated in one of the two above ways to generate images on the visual manifold M. The attributes involved are the standard psychological ones: the senses; the constancies; Gestalt properties; and memory of, and recognition for, general shapes. The neuropsychological structure is solidly based on known topology, psychology, and neuroscience and has received abundant experimental confirmation (Hoffman, 1994b: 23)
8. The role of the frontal intrinsic systems is cognitive. Therein meaning is imparted to a percept, the meaning is colored by emotion, and the intentionality and trains of thought required are thereby generated. For this purpose Latimer & Stevens adduce "complexes." These are not complexes in the usual topological sense (Dugundji, 1966: 172) but 1 to 1 mappings that are isomorphic and invariant under transpositions. The perceptual phenomena of interposition, anomalous figures, and the Gestalt principles of closure and pragnanz seem to present some difficulty for the uniqueness of the mapping. The complex "features" are required to be invariant under a "subgroup of Z transpositions." If one accepts information processing psychology and its isomorph in dialectical psychology (Hoffman, 1997), what really seem to be involved here are the categories of simplicial objects (Hoffman, 1980, 1997) and fibrations/fibre bundles (Hoffman, 1980, 1985, 1986, 1989, 1994a, 1997). Opposites (Rychlak, 1997) and counterfactuals (Johnson-Laird, 1995; Knight & Grabowecky, 1995) constitute key elements in this decomposition of cognitive structure. The structure group involved is the group of the symmetric difference, $, (Hoffman, 1996a, 1997), which, together with the complement ~$, is enough to generate the key features of cognitive psychology (Hoffman, 1995, 1996a, 1997). In the FIS, cognitive attributes range well beyond form percepts and stimulus intensity, and, as Latimer & Stevens point out, can vary over wholes that range from cosmic to microscopic. In any given context, however, they must be identifiable, although this identification may often be subjective. Given a set of attributes referred to an object/concept, the decomposition of the whole by way of the symmetric difference formalism ($,~$) first of all decomposes the whole into its non-overlapping attributers vis-a-vis the object - "what it is" and "what it is not" - and then in the second phase generates the commonality, the cognitive "whole," together with its context (Hoffman, 1966a, 1997).
9. Latimer & Stevens call for a topological methodology that offers appropriate local structures for properly decomposing a whole in consistent fashion, one which also integrates the local structure to generate the whole. This is provided in the first instance, that of form perception, by the transformation groups of the constancies and the tangential and transverse structure of visual contours. The visual field is decomposed in this way into a grid of constancy orbits, which, coupled with higher and higher differential invariants, can generate visual forms of arbitrary complexity (Hoffman, 1970, 1986, 1994a).
10. According to information processing psychology and dialectical psychology, the cognitive processing of projections to the frontal intrinsic systems is first done by the operation of the symmetric difference $ decomposing a concept into opposites. The second phase of the process, involving the complement of $, yields synthesis and context. The concepts involved have the topological structure of simplicial objects (Hoffman, 1980, 1985, 1996a, 1997). With proper identification of nodes (objects/concepts and their attributes) and their interrelationships within the structure (Dowker, 1952), the symmetric difference model can admit the most general sorts of equivalences, decompositions, integrations, and cognitive phenomena.
Brockett, R. W. (1977). 1976 Ames Research Center (NASA) conference on Geometric Control Theory. Brookline, Massachusetts, USA: Math Sci Press.
Colonnier, M. (1964). The tangential organization of the visual cortex. J. Anatomy (London), 98, 327-344.
Dodson, C. T. J. (1980). Categories, bundles and spacetime topology. Orpington, Kent, UK: Shiva Publ.
Dowker, C. H. (1952). Homology groups of relations. Annals of Mathematics, 56, 84-95.
Dugundji, J. (1966). Topology. Boston: Allyn & Bacon.
Eisenhart, L. P. (1961). Continuous groups of transformations. New York: Dover.
Gallant, J. L., Braun, J., & Van Essen, D. C. (1993). Selectivity for polar, hyperbolic, and cartesian gratings for macaque visual cortex. Science, 259, 100-103.
Hoffman, W. C. (1966). The Lie algebra of visual perception. J. Math. Psychol., 3, 65-98.
Hoffman, W. C. (1970). Higher visual perception as prolongations of the basic Lie transformation group. Mathematical Biosciences, 6, 437-471.
Hoffman, W. C. (1977). An informal, historical description (with bibliography) of the "L.T.G./N.P." Cahiers de Psychologie (Provence), 20, 135-174.
Hoffman, W. C. (1978). The Lie transformation group approach to neuropsychology. In E. L. J. Leeuwenberg & H. Buffart (Eds.), Formal theories of visual perception. Chichester, UK: John Wiley.
Hoffman, W. C. (1980). Subjective geometry and geometric psychology. International J. Math. Modelling, 1, 349-367.
Hoffman, W. C. (1984). Figural synthesis by vectorfields: geometric psychology. In P. C. Dodwell & T. M. Caelli (Eds.). Hillsdale, NJ: Erlbaum Associates.
Hoffman, W. C. (1985). Some reasons why algebraic topology is important in neuropsychology: perceptual and cognitive systems as fibrations. International J. of Man-Machine Studies, 22, 613-650.
Hoffman, W. C. (1986). Invariant and programmable neuropsychological systems are fibrations. Behavioral & Brain Sciences, 9, 99-100.
Hoffman, W. C. (1989). The visual cortex is a contact bundle. Appl. Math. & Computation, 32, 137-167.
Hoffman, W. C. (1994a). Equivariant dynamical systems: a formal model for generation of arbitrary shapes. In O. Ying-Lie, A. Toet, Foster, D., H.J.A.M. Heijmans, & P. Meer (Eds.), Shape in picture: mathematical description of shape in grey-level images. Berlin: Springer-Verlag.
Hoffman, W. C. (1994b). Conformal structures in perceptual psychology. Spatial vision, 8, 19-31.
Hoffman, W. C. (1995). The dialectics of giftedness: gifted intellect and creativity. Roeper Review, 17, 201-206.
Hoffman, W. C. (1996a). The formal structure of dialectical psychology. Dynamical Psychology. http://goertzel.org/dynapsyc/1996/formstr.html.
Hoffman, W. C. (1996b). A topological interpretation of cognition. In G. W. Cottrell (Ed.), Proc. 18th Annual Conf. of the Cognitive Science Soc., 775. Mahway, NJ: Erlbaum Associates.
Hoffman, W. C. (1997). Mind and the geometry of systems. In S. O'Nualla'in, P. McKevitt, & E. Mac Aogain (Eds.). Two sciences of mind: readings in cognitive science and consciousness. Amsterdam: John Benjamins.
Hoffman, W. C. & Dodwell, P. C. (1985). Geometric psychology generates the visual gestalt. Canad. J. Psychol., 39, 491-528.
Johnson-Laird, P. N. (1995). Mental models, deductive reasoning, and the brain. In M. S. Gazzinaga (Ed.), The Cognitive Neurosciences. Cambridge, Mass: MIT Press.
Knight, R. T. & Grabowecky, M. (1995). Escape from linear time: prefrontal cortex and conscious experience. In M. S. Gazzinaga (Ed.), The Cognitive Neurosciences. Cambridge, Mass.: MIT Press.
Kostrikin, A. I. & Shafarevich, I. R. (Eds.) (1990). Algebra I. Berlin: Springer-Verlag.
Latimer, C. & Stevens, C. (1997). Some remarks on wholes, parts and their perception. PSYCOLOQUY 8(13) part-whole-perception.1.latimer.
Pribram, K. H. (1960). A review of theory in physiological psychology. Annual Rev. Psychol., 11, 1-40.
Rescher, N. and Oppenheim, P. (1955). Logical analysis of Gestalt concept. British Journal for the Philosophy of Science, 6, 89-106.
Riegel, K. (1973). Dialectic operations: the final period of cognitive development. Human Development, 16, 346-370.
Rock, I. (1980). Difficulties with a direct theory of perception. Behavioral and Brain Sciences, 3, 398-399.
Rychlak, J. F. (1997). In defense of human consciousness. Washington, DC: Am. Psychol. Assoc.
Steeb, W.-H. (1996). Continuous symmetries, Lie algebras, differential equations and computer algebra. Singapore: World Scientific.