Читать книгу The Form Within - Karl H Pribram - Страница 77

Groups can be described mathematically in terms of symmetry. A Norwegian mathematician, Sophus Lie, invented the group theory that pertains to perception. The story of how this invention fared is another instance of slow and halting progress in gaining the acceptance of scientific insights. As described in the quotation at the beginning of this chapter, early in the 1880s Hermann von Helmholtz, famous for his scientific work in the physiology of vision and hearing, wrote to Henri Poincaré, one of the outstanding mathematicians of the day. The question Helmholtz posed was ”How can I represent objects mathematically?” Poincaré replied, “Objects are relations, and these relations can be mapped as groups.” Helmholtz followed Poincaré’s advice and published a paper in which he used group theory to describe his understanding of object perception. Shortly after the publication of this paper, Lie wrote a letter to Poincaré stating that the kind of discrete group that Helmholtz had used would not work in describing the perception of objects—that he, Lie, had invented continuous group theory for just this purpose. Since then, Lie groups have become a staple in the fields of engineering, physics and chemistry and are used in everything from describing energy levels to explaining spectra. But only recently have a few of us returned to applying Lie’s group theory to the purpose for which he created it: to study perception.

My grasp of the power of Lie’s mathematics for object perception was initiated at a UNESCO conference in Paris during the early 1970s. William Hoffman delivered a talk on Lie groups, and I gave another on microelectrode recordings in the visual system. We attended each other’s sessions, and we found considerable overlap in our two presentations. Hoffman and I got together for lunch and began a long and rewarding interaction.

Symmetry groups have important properties that are shown by objects. These groups are patterns that remain constant over two axes of transformation. The first axis assures that the group, the object, does not change as it moves across space-time. The second axis centers on a fixed point or stable frame, and assures that changes in size—radial expansion and contraction—do not distort the pattern.