What is it like to be a blind mathematician?

(Cross-posted at NewAPPS, and in the spirit of the "Count me in" campaign.)

I am now working on a paper on the role of manipulating notations in mathematical reasoning, which reminded me of an issue that interested me a few years ago: how do blind mathematicians do their work, given that the usual manipulation of notations (which obviously appeals crucially to vision) is not an option to them?

I then sent a query on this topic to the FOM list, and got some very interesting replies. In particular, many people mentioned this fascinating notice published by the American Mathematical Association on blind mathematicians, which I highly recommend to anyone interested in how mathematics 'comes about'. An excerpt:

A sighted mathematician generally works by sitting around scribbling on paper: According to one legend, the maid of a famous mathematician, when asked what her employer did all day, reported that he wrote on pieces of paper, crumpled them up, and threw them into the wastebasket. So how do blind mathematicians work?

The notice focuses in particular on Bernard Morin, the mathematician who formulated the first visualization of the eversion of the sphere (also known as Smale's paradox) by construing clay models of the process. (Here is an educational video on the topic.) Quoting from this site:

To evert a sphere is to turn it inside-out by means of a continuous deformation, which allows the surface to pass through itself, but forbids puncturing, ripping, creasing, or pinching the surface. An abstract theorem proved by Smale in the late 1950s implied that sphere eversions were possible, but it remained a challenge for many years to exhibit an explicit eversion.

The bit that interests me most in Morin's achievement is that he himself claims that being blind was an asset he had over other mathematicians on this specific problem. I quote from the AMS notice:

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.

To me, Morin's eversion of the sphere represents a case study of the impact of perceptual interaction with the environment (including bits of notations) in mathematical reasoning. On account of being blind, he seemed to have a different perception of objects, including their 'insides'. Thus, we seem to have a case of a difference in perceptual conditions actively making a significant difference for the kind of mathematical knowledge produced.

The usual story goes that mathematical reasoning is 'purely abstract' and completely divorced from sensorimotor processing, but in recent years people like Rafael Nunez and David Landy, among others working within an embodied, extended perspective, have emphasized the role of perceptual grounding for mathematical reasoning. There are many ways in which this can be cashed out, and my own focus has been on the kind of engagement we seem to have with notations when 'doing math'; are we merely reporting independent, prior cognitive processes by means of notations, or is the manipulation of the notation itself part of these cognitive processes? Unsurprisingly, I side with the second alternative, and in that case the perceptual properties of the notation will have a significant impact. Now, this obviously does not mean that people who do not manipulate notations in this manner, e.g. a blind mathematician such as Morin, cannot do mathematics; rather, the point is that they apparently do mathematics in a different way, at times 'seeing' things that the rest of us cannot see.

(When I have the time, I will elaborate further on these ideas with a post on Jason Padgett, a man with acquired savant syndrome who 'sees' everything geometrically as fractals. So stay tuned!)