SEEING, TOUCHING AND DOING MATHEMATICS

Hearing about visual processes, from neuroscientists and artists alike, consistently brings mathematical thoughts to mind for me – like Samir Zeki’s descriptions of how visual images are constructed, or the Impressionist painters’ attention to the sensations in the eye rather than the subject of the painting, and, of course, Poincaré’s suggestion that visual space has more than three dimensions.

The relationship between vision and mathematics (or sensation and symbolic thought) came up again today, when I followed the links in a December 15 Vision Help Blog. The blog begins with a reference to an Oliver Sacks story about a blind individual given sight, and concludes with a paper pointing to the psychological significance of the reluctance of organ donors to donate corneas.

The Sacks vignette tells us a lot about vision, and explains why freeing the cornea of obstruction doesn’t just give us sight. We have to learn to see.  We have to learn how the many appearances of an object belong to the one object.

We achieve perceptual constancy — the correlation of all the different appearances, the transforms of objects — very early, in the first months of life. It constitutes a huge learning task, but is achieved so smoothly, so unconsciously, that its enormous complexity is scarcely realized (though it is an achievement that even the largest supercomputers cannot begin to match).

It is for this reason that the subject of Sacks’ story who, at age 50, was given back the sight he had lost as a very young child, could not integrate the various pieces of visual stimuli into a coherent image.

…with half a century of forgetting whatever visual engrams he had constructed, the learning, or relearning, of these transforms required hours of conscious and systematic exploration each day. This first month, then, saw a systematic exploration, by sight and touch, of all the smaller things in the house: fruit, vegetables, bottles, cans, cutlery, flowers, the knickknacks on the mantelpiece — turning them round and round, holding them close to him, then at arm’s length, trying to synthesize their varying appearances into a sense of unitary objecthood. […]”  (italics added)

While I cannot make any precise correspondence, I do see the visual processing mechanisms that this individual had not developed as ‘the mathematics’ of sight, as the way to capture the many facets of any sensation into a recognizable whole, one that the mind can manage.  Even some of the words used in the story call to mind things like algebra and geometry:

An infant merely learns. This is a huge, never-ending task, but it is not one charged with irresoluble conflict. A newly sighted adult, by contrast, has to make a radical switch from a sequential to a visual-spatial mode, and such a switch flies in the face of the experience of an entire lifetime.  (italics added)

But there is also a detail in the story that points to what I see as the body’s sheer tenacity, the tenacity to be, however that is accomplished.  Everyone was surprised when one of the first things Virgil could recognize was a symbolic thing – it was the alphabet:

Virgil’s first formal recognitions when the bandages were taken off had been of letters on the ophthalmologist’s eye chart…How was it that he had so much difficulty recognizing faces, or the cat, and so much difficulty with shapes generally, and with size and distance, and yet so little difficulty, relatively, recognizing letters? When I asked Virgil about this, he told me that he had learned the alphabet by touch at school, where they had used letter blocks, or cutout letters, for teaching the blind. I was struck by this and reminded of [R.L.] Gregory’s patient S.B.: “much to our surprise, he could even tell the time by means of a large clock on the wall. We were so surprised at this that we did not at first believe that he could have been in any sense blind before the operation.” But in his blind days S.B. had used a large hunter watch with no glass, telling the time by touching the hands, and he had apparently made an instant “cross-modal” transfer, to use Gregory’s term, from touch to vision. Virgil too, it seemed, must have been making just such a transfer. […]

I remembered reading about a blind mathematician, and today found more about him in a 2002 AMS article called The World of Blind Mathematicians.  In this story, representations created by touch are also noted:

A visitor to the Paris apartment of the blind geometer Bernard Morin finds much to see. On the wall in the hallway is a poster showing a computergenerated picture, created by Morin’s student François Apéry….The surface plays a role in Morin’s most famous work, his visualization of how to turn a sphere inside out. Although he cannot see the poster, Morin is happy to point out details in the picture that the visitor must not miss.

Later Morin brings out clay models that he made in the 1960s and 1970s to represent shapes that occur in intermediate stages of his sphere eversion.

The models were used to help a sighted colleague draw pictures on the blackboard…It is startling to consider that such a precise, symmetrical model was made by touch alone. The purpose is to communicate to the sighted what Bernard Morin sees so clearly in his mind’s eye.

And this:

Morin recalled that, when a sighted colleague proofread Morin’s thesis, the colleague had to do a long calculation involving determinants to check on a sign. The colleague asked Morin how he had computed the sign. Morin said he replied: “I don’t know—by feeling the weight of the thing, by pondering it.”

Morin believes there are two kinds of mathematical imagination. One kind, which he calls “time-like”, deals with information by proceeding through a series of steps. This is the kind of imagination that allows one to carry out long computations. “I was never good at computing,” Morin remarked, and his blindness deepened this deficit. What he excels at is the other kind of imagination, which he calls “space-like” and which allows one to comprehend information all at once.

Space-like imagination is tactile as well as visual:

One thing that is difficult about visualizing geometric objects is that one tends to see only the outside of the objects, not the inside, which might be very complicated. By thinking carefully about two things at once, Morin has developed the ability to pass from outside to inside, or from one “room” to another. This kind of spatial imagination seems to be less dependent on visual experiences than on tactile ones. “Our spatial imagination is framed by manipulating objects,” Morin said. “You act on objects with your hands, not with your eyes. So being outside or inside is something that is really connected with your actions on objects.” Because he is so accustomed to tactile information, Morin can, after manipulating a hand-held model for a couple of hours, retain the memory of its shape for years afterward.

Another anecdote in the article refers to mathematician Norberto Salinas

In a contribution to a Historia-Mathematica online discussion group about blind mathematicians, Eduardo Ortiz of Imperial College, London, recalled examining Salinas in an analysis course at University of Buenos Aires. Salinas communicated graphical information by drawing pictures on the palm of Ortiz’s hand, a technique that Ortiz himself later used when teaching blind students…

I particularly like this suggestion:

…blind people often have an affinity for the imaginative, Platonic realm of mathematics. For example, Morin remarked that sighted students are usually taught in such a way that, when they think about two intersecting planes, they see the planes as two-dimensional pictures drawn on a sheet of paper. “For them, the geometry is these pictures,” he said. “They have no idea of the planes existing in their natural space.

Happy New Year!