Optical Realities: Mathematics and Visual Processes
I was reading up on some nineteenth century philosophy and science for a book project of mine and I found an essay by Timothy Lenoir called The Eye as Mathematician. It is a discussion of the construction of Helmholtz’s theory of vision. The title suggests that the eye is acting like a mathematician. My disposition is to think that mathematics is acting like the eye.
One of the passages I found striking was this from Helmholtz:
I maintain, therefore, that it cannot possibly make sense to speak about any truth of our perceptions other than practical truth. Our perceptions of things cannot be anything other than symbols, naturally given signs for things, which we have learned to use in order to control our motions and actions. When we have learned how to read those signs in the proper manner, we are in a condition to use them to orient our actions such that they achieve their intended effect; that is to say, that new sensations arise in an expected manner.
Read in a particular light (perceptions as symbols, as naturally given signs for things that we learn to read in the proper manner) this description reassociates fleshy things with abstract things. What is known as Helmholtz’s constructivist theory of vision was inspired by what Lenoir describes as “Herbart’s conception of the symbolic character of space and of the deep connection between motion and the construction of visual space.” Herbart was an early 19th century philosopher whose influence was acknowledged by Riemann at the beginning of his famous 1854 lecture on the foundations of geometry.
The essay goes on to explain the way Helmholtz saw visual experience as “a symbolic shorthand for aggregates of sensory data:”
Perceptions of objects in space, for instance, link together information on direction, size, and shape, with that of color, intensity, and contrast. None of these classes of information is simply given; rather, they are the result of measurements carried out by the components of the visual system. Moreover, these data aggregates are not linked with one another by an internal logic given in experience; rather,the connections are constructed by trial, error, and repetition. The more frequently the same linkages of sensory data are effected, the more rapidly the linkages are carried out by the brain; for the conscious mind in this process, they come to have the same force of necessity as logical inference.
I was struck by a few things in these words. One is the statement that the connections are constructed by trial, error, and repetition which actually lines up well with current statistical models of learning. The other is that “for the conscious mind in this process, they come to have the same force of necessity as logical inference.” Perhaps this is because the conscious mind, in developing logical inference, is actually borrowing something from the processes that occur outside of our awareness.
In the midst of my time in graduate school, I felt strongly that mathematics was, itself, a way to direct our eyes, or what we were able to see. Before I had much training, I would say to my non-mathematician friends that different mathematical ideas were like grids you could hold up in front of your eyes that would change, in actual substance, what they saw, as if the mathematics itself was a visual process. The grid suggests that I was thinking more in terms of changing coordinates than the conceptual shifts that I was beginning to see in advanced topics. But the thought was only encouraged when I learned more about mathematics, physics and, more recently, theories of vision.
While neural imaging techniques have changed our understanding of how the body accomplishes making visual images, the insights of Herbart and Helmholtz, into the constructive nature of vision, still stand. And their language certainly suggests that the body is doing something very much like mathematics – reading signs and symbols or constructing various spaces which Herbart calls ‘the symbol of the possible community of things standing in a causal relationship.’
In one of my earlier posts, I described Poincaré ‘s observation that visual space is not Euclidean. One of the reasons for his claim is that, in his analysis of it, visual space is determined by at least four independently varying parameters. While the reasoning is somewhat different, I found more talk about visual dimensions on The VisionHelp Blog (which contains a good deal of information on vision processes, ailments, and therapies). They referenced a Discovery show about wormholes which, given the unanswered question about the number of dimensions of space, featured an opening segment on vision. In another one of their posts, a Poincaré or Bloch sphere illustrates how, in addition to the three primary vectors of stereopsis (one of our distance tools), there are actually many sub-vectors operating with movements of the eyeballs as we look at different angles.
The visual theorist Semir Zeki has made the observation that “our inquiry into the visual brain takes us into the very heart of humanity’s inquiry into its own nature,”………..as does our inquiry into mathematics!