A review of the work of David Finkelstein and others on quantum topology is given, the intention being to present physical ideas and a progress report, which will help readers with the more detailed papers. Some new approaches involving walks on graphs are presented.
Although the idea that cognitive structure changes as we learn is welcome, a variety of mathematical structures are needed to model the neural and cognitive processes involved. A specific example of bodily-kinaesthetic intelligence is given, building on a formalism given elsewhere. As the structure of cognition changes, previous learning can become tacit, adding to the complexity of cognition and its modeling.
Nervous systems are intricately organized on many levels of analysis.The intricate organization invites the development of mathematicalsystems that reflect its logical structure. Particular logical structures and choices of invariants within those structures narrowthe ranges of perceptions that are possible and sensorimotorcoordination that may be selected. As in quantum logic, choicesaffect outcomes.Some of the mathematical tools in use in quantum logic havealready also been used in neurobiology, including the mathematicsof ordered structures and a product like a tensor product. Astheoretical neurobiology is (...) developed on its own foundation, wemay expect a rich dialogue between theoretical neurobiology andquantum logic. (shrink)
Although Phillips & Singer's proposal of commonalities seems sound, information theory and artificial neural network modeling omit important detail. An example is given of a distributed neural transformation that has been characterized mathematically and found to have both overall commonalities and differences of detail in different regions. P&S's contextual field is compared to inclusive regions in a formalism relevant for modeling bodily-kinaesthetic intelligence.
Just as physics determines physically viable movements, the spatial distribution of input excitations allows the cerebellum to choose physiologically viable beams. Cerebellar–motor coherence implies that the ordering and modes of combination of cerebellar beams reflect (1) the way movement invariants are ordered and combined in movement and (2) the way physical principles are integrated in learning to move.
The mathematical approach to such essentially biological phenomena as perseverative reaching is most welcome. To extend these results and make them more accurate, levels of analysis and neural centers should he distinguished. The navigational nature of sensorimotor control should be characterized more clearly, including the continuous dynamics of neural processes hut not limited to it. In particular, discrete conditions should be formalized mathematically as part of the biological process.
The authors' review of alternative models for reading is of great value in identifying issues and progress in the field. More emphasis should be given to distinguishing between models that offer an explanation for behavior and those that merely simulate experimental data. An analysis of a model's discrete structure can allow for comparisons of models based upon their inherent dimensionality and explanatory power.
Many mathematicians have a rich internal world of mental imagery. Using elementary mathematical skills, this study probes the mathematical imagination's sensorimotor foundations. Mental imagery is perturbed using body position: having the head and vestibular system in different positions with respect to gravity. No two mathematicians described the same imagery. Eight out of 11 habitually visualize, one uses sensorimotor imagery, and two do not habitually used mental imagery. Imagery was both intentional and partly autonomous. For example, coordinate planes rotated, drifted, wobbled, (...) or slid down from vertical to horizontal. Parabolae slid into place or, on one side, a parabola arm reached upward in gravity. The sensorimotor foundation of imagery was evidenced in several ways. The imagery was placed with respect to the body. Further, the imagery had a variety of relationships to the body, such as the body being the coordinate system or the coordinate system being placed in front of the eyes for easy viewing by the mind's eye. The mind's eye, mind's arm, and awareness almost always obeyed the geometry of the real eye and arm. The imagery and body behaved as a dyad, so that the imagery moved or placed itself for the convenience of the mind's eye or arm, which in turn moved to follow the imagery. With eyes closed, participants created a peripersonal imagery space, along with the peripersonal space of the unseen environment. Although mathematics is fundamentally abstract, imagery was sometimes concrete or used a concrete substrate or was placed to avoid being inside concrete objects, such as furniture. Mathematicians varied in the numbers of components of mental imagery and the ways they interacted. The autonomy of the imagery was sometimes of mathematical interest, suggesting that the interaction of imagery habits and autonomy can be a source of mathematical creativity. (shrink)