INDEX
Institute for Topological Psychology Tucson, Arizona 85742-9074
William C. Hoffman, Ph. D., Director
Why Topological?
A thing is not just a thing--it has form and meaning. Form perception is inherently geometric--the things one sees consist, as they do, of geometric objects. And as for meaning, if one accepts the quaint fancy of the Connectionists that thought processes consist of point "neurons" and the paths connecting them, then simplicial topology enters as the lines, triangles, tetrahedrons, etc. that connect cognitive "chunks" by "trains of thought."
Topology is "rubber geometry," technically, the mathematics of bicontinuous one-to-one transformations of a set of points comprising a so-called space. Associated with the points is a family of "open sets" that cover the space, like the response fields that constitute neighborhoods of the actual neurons in the brain. For present purposes the key point is that there are certain invariants associated with a topology that remain unchanged under the transformations. Kurt Lewin's "topological psychology" was different. Instead of invariance under transformation, it revolved around approach-avoidance reactions. The latter are subsumed in the present approach under the dialectical-pair model for cognition.
In the case of the visual field, the transformations are the distortions imposed by viewing conditions. The objects in the visual field are recognized as what they are in their own right no matter how their appearance may be distorted by viewing conditions: near or far, right-left, up-or-down in the field of view, rotated, moving, or viewed obliquely or binocularly. And a tune is still recognizable even if it is shifted in key or changed in loudness, or heard binaurally. These invariances constitute the psychological constancies. Lacking the constancy invariances, you would always be moving through a surrealistic world of perpetually deforming, rubbery objects. The Gestalt "laws" are still another kind of perceptual invariance. And the phenomenon we call memory is a prime example, for it represents invariance of mentation under time changes.
Point-set topology was mentioned above, but there are other kinds as well, no less important in neuropsychology. Neuropsychology is a word of several different meanings, even clinical, but used here in the sense of psychology firmly grounded in neuroscience. The neurons of the visual pathway have center-surround response fields, Mexican-hat shaped, but when the optical flow gets to the visual cortex, something new is added, the so-called orientation response. This couples a directional response with the neighborhood response. The result is what mathematicians call a vector field, that is, an arrow bound to each point of a manifold like the visual field. The invariances involved here lead immediately to a second kind of topology, differential topology. All kinds of good things now enter: flows, Lie groups, Lie derivatives, fibre bundles, etc., all of which play a prominent role in the theory of perceptual psychology set forth in the linked list of theoretical and experimental publications.
The brain divides naturally, fore and aft ("sagitally") into two
main systems, the posterior perceptual systems and the fronto-limbic higher
systems, as shown in the figure below. 
As an eminent neuroscientist, Dr. Karl Pribram, put it, the posterior
perceptual systems "mediate invariant properties of specific sensory
modalities." (There's that word again!) The frontal cognitive system
acts to confer meanings on the form perceptions that the posterior systems
provide it. Involved in the flow are Working Memory, thought processes,
foresight and planning, decision making--the essential activities of
"consciousness." Coupled in with this are the outputs of the
limbic system, the "emotional brain." Attached to every
cognitive meaning is an emotional connotation.
After the midbrain gets done processing a visual stimulus, it flows to the cortex in the posterior perceptual system, where it is preprocessed to remove constancy distortions and then processed for form, color/pitch, spatial location, state of motion. kinesthetic and sensorimotor properties, etc. For the visual system, the basic thing here is the Figure-Ground Relation--shape emerges from the (back)ground, bounded by its contours. It is axiomatic that an object is determined by its bounding contours, and it is the invariance of these under viewing conditions that determines constancy and form memory. This brings us to the blessed domain of Lie transformation groups, denoted symbolically by the mapping G x M --> G, where G is a group and M is a manifold (think of a surface). No, not that kind of group. G is a mathematical group, a set of elements that can be "multiplied" (transformed associatively) and "divided" (there's an identity element and each element has an inverse). G is also continuous and is a manifold just like spacetime. Now think of a visual contour as a path-curve generated by the transformation group action, and choose some point on it. Call this the identity element of the group. Draw a tiny tangent line to the curve at that point. This is the infinitesimal transformation of the continuous or Lie group (named after Sophus Lie, a genius mathematician of the last century who invented the things. The story goes that when Lie and Felix Klein--another great mathematician--were students together at Göttingen, they decided to divide the world up between them, Klein to take the discrete groups, Lie the continuous ones. Though this was said at the time as a joke, in the intervening 150 years or so, it has certainly come true in physics and chemistry, and the burden of the work described here is that it may well hold in psychology as well.) The infinitesimal transformation is embodied in a Lie derivative, which "drags the flow along the path-curve," the so-called "orbit"--in this case the visual contour. If £ denotes the Lie derivative and f, the visual contour, then invariance of the contour under the transformation group is shown by its being annulled by the action of the Lie derivative: £ f = 0, or by its being handed on as a "contact element" for further processing: £ f = g(f) . These operations characterize psychological constancy. For the work that proves it, see the papers listed under Theoretical and Experimental.
Now we come to the topology involved in meaning and cognition. This is where algebraic topology comes in, in the form of simplicial topology and fibrations. The latter consist of "lifting maps" and "projections"--like brain circuits--to and from a total space, the brain, to a base space, consciousness. As indicated above, this is in part related to information processing psychology, with its "point neurons" and "neural nets," but the present theory is mainly motivated by Klaus Riegel's dialectical psychology. Dialectical psychology asserts that we go through life continually encountering and resolving contradictions. Contradictions, like change, are ceaseless and inevitable. Dialectic is the apposition of opposites--thesis and antithesis--leading to a synthesis, and so, in a psychological context, to reduction of cognitive dissonance. That dialectical psychology admits a full range of cognitive phenomena is argued in the papers listed under dialectical.
The symmetric difference operation $ is one of the two fundamental operations of set topology. If you have two cognitive sets C and C', C $ C' is "one or the other but not both," that is C and C' but without their commonality. The symmetric difference is dialectical in that it embodies thesis-antithesis. The complement, ~(C $ C'), provides the corresponding synthesis, the "commonality" of the two sets together with their context, "everything else" in the momentary universe of discourse--Working Memory. This is the dialectical pair ($, ~$) model for dialectical psychology <click here>. It is the familiar similarities and differences paradigm but in the opposite order for the sake of psychological realism. It provides an algorithmic expression of Riegel's dialectical psychology, and since $ is the "sum" function for the chain mappings in the category of simplicial sets, it also describes the connectionism involved in information processing psychology.
Theoretical Papers Experimental Papers
Papers on Dialectical Psychology
CONTACT INFORMATION
E-mail: topologicalpsychology@worldnet.att.net