My attention was recently brought to a discussion of grid cells and spatial imagery as they relate to cognitive strengths in dyslexic individuals. The discussion takes place in the book The Dyslexic Advantage by Drs. Brock and Fernette Eide, and it amplified many of the thoughts I have expressed about the biological aspect of mathematical ideas.
The authors argue, convincingly, that while individuals diagnosed with dyslexia may have difficulty with the symbolic representation of words, they seem to excel at 3-dimensional spatial reasoning. I thought of my daughter, whose dyslexia was not diagnosed till her freshman year in high school, but who, even as a 5-year old, seemed to have a remarkable ability to know which way to go when we were driving. She could locate herself pretty easily.
The authors describe the coordinate-like action of grid cells in order to point to one of the neurological components of how we all negotiate 3-dimensional space. I wrote about the action of grid cells in a 2014 post, to suggest that cognitive processes themselves have a mathematical nature. There I made this observation:
Spatial relations are translated into what look like purely temporal ones (the timing of neuron firing). The non-sensory system then stores a coded representation of a sensory one. Here again we see, not the mathematical modeling of brain processes but more their mathematical nature.
Drs. Brock and Fernette Eide cite studies done, from various perspectives, which suggest that the presence of strong spatial reasoning in a dyslexic individual is not developed in order to compensate for verbal difficulties but, rather, it is a strength with which dyslexic individuals are born. As a result, many such individuals have chosen careers in areas that include art, architecture, building, engineering, and computer graphics.
I found one of their observations particularly interesting because it contradicted the perfectly reasonable expectation that strong spatial reasoning skills are accompanied by vivid, mental visual images. But, as it is with mathematics, so it is with the brain. Specifically, it seems that it is possible to separate knowledge of space from spatial images in an individual’s experience. The authors describe a particular case-study where the individual involved lost his ability to create clear visual mental images, but his spatial reasoning abilities were unaffected. In other words, he could manage spatial relations without visualizing them.
MX was a retired building surveyor living in Scotland who’d always enjoyed a remarkable vivid and lifelike visual imagery system, or “mind’s eye.” Unfortunately, four days after undergoing a cardiac procedure MX awoke to discover that though his vision was normal, when he closed his eyes he could no longer voluntarily call to mind any visual image at all.
MX was tested using a whole series of spatial reasoning and visual memory tasks. As a control, a group of high-visualizing architects performed the same tasks. Surprisingly, it was found that although MX could no longer create any mental visual images while performing these tasks, he scored just as well as the architects did. As he performed the tasks, MX’s brain was also scanned with fMRI technology. In contrast to the architects, who heavily activated the visual centers of their brains while solving these tasks, MX used none of his brain’s visual processing regions.
These studies suggested that while MX had lost his ability to perceive visual images when engaging in spatial reasoning, he could still access spatial information from his spatial database and apply it to Material reasoning tasks with no detectable loss of skill.
It is common place in mathematics to separate spatial information from the visual images with which they can be associated. Analytic or coordinate geometry, for example, studies geometric figures (or figures in space) using their algebraic representations (i.e. only numbers). So there we have the numerical approach to figures and the visual one. A discipline like abstract algebra creates other kinds of spaces and objects by abstracting away not just the particular numbers (like the variables we learn about in high school algebra do) but by abstracting away the previous meaning of things like addition, for example. The plus sign comes to stand for anything that obeys the same rules that addition obeys, like a + b = b + a and a + 0 = a. But a, b and 0 are not necessarily numbers. The point is that mathematics manages to keep finding other places where information exists. Mathematics explores structure as fully and deeply as possible.
The relationship between vision and structure that MX’s experience brought me back to also reminded me of the 2002 AMS article called The World of Blind Mathematicians. The article is full of interesting and unexpected observations of the accomplished blind geometer, Bernard Morin. This is just one of them:
…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.
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