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?

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!)

In the 1990's, I collaborated extensively with Larry Wos -- one of the first blind math PhD.s in the US. We worked on theorem proving together. He had an uncanny knack for syntax -- he worked on a Braille terminal, and was faster than I was at absorbing proofs (and generally interpreting output on the computer). Really an amazing story. See:

ReplyDeletehttp://www.mcs.anl.gov/~wos/uofc-bio.html

How cool that you worked with Larry Wos! His name came up in replies to my query at FOM, and among other things, I found out that he is also the best blind bowler in the US, how cool is that? (For a logician, I mean... He's probably the best logician-bowler, period.) At some point you should tell me more about the way he works.

ReplyDeleteI imagine this has already come up, but Euler famously spent nearly forty years producing mathematics while totally blind, including a large number of his best works. The biographies appear to put it down to his "remarkable memory" and the help of several assistants (see http://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.html ).

ReplyDeleteYes, Euler is an interesting case, because he learned mathematics in the 'usual' way and then had to adapt to a different modus operandi later in life. In fact, all different 'variations' are interesting from the point of view of the cognitive impact of manipulating notations when doing math: those who were born blind, those who were not born blind but who were blind at the point they started learning math (that's Morin's case, blind at age 6 if I'm not mistaken) and those who were not blind when they trained to be mathematicians but became blind later in life. My prediction is that different cognitive strategies would be used in each case, in particular along the dimension of being blind or not upon being trained as a mathematician. I expect there would be less of difference between those born blind and those who became blind early in life such as Morin, prior to mathematical education.

ReplyDeleteHello, I have just found your blog this past weekend and am very glad to have done so.

ReplyDeleteAmongst many other things I am fascinated with the learning and teaching of mathematics and logic so this bit of your post really caught me eye:

"[A]re 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."

Please can you tell me anything you have written on this as I would be very interested to read it.

Thankyou very much.

Michael, from the UK.

Hi Michael, glad you liked the post. The line you singled out above is pretty much a summary of what I've been working on over the last few years. A draft of my forthcoming book on the topic is available here:

ReplyDeletehttps://sites.google.com/site/catarinadutilhnovaes/home/book-on-formal-languages

I also have a draft of a paper which may be of interest to you:

http://m-phi.blogspot.com/2011/07/mathematical-reasoning-and-external.html

Hi Catarina,

ReplyDeleteThanks very much, that's very helpful.

Best,

Michael

you might check out Dr. Robert O Shelton who published papers on non-collision singularites in the 4 body problem. Working at NASA, he created software, MathTrax, I believe, that helps blind students work with graphs and physics simulations.

ReplyDeletei like to exam in multiple equation because this type of question i easily solve and this article give us good info how to solve math question in exam thanks for sharing college readmission letter .

ReplyDelete