Commentary: Thinking Of Mathematics

This commentary contains some of my responses to the questions that were raised during the Math Intelligencer review of my article on doing mathematics in an eyes-free environment.

Thinking Of Mathematics (HTML) Thinking Of Mathematics (PDF)

The blogosphere is probably the best place to carry on the rest of this conversation.

Date: Wed, 16 May 2007 09:45:59 -0700

Hi Chandler,

First off, thank you for 

A) Encouraging me to write this article, and
B) for all your feedback.

See specific responses  in-lined below.

 | 1) it's unfortunate that he's chosen the "steps in creativity" format for 
 | both the whole and the parts. It's not a bad way to frame the challenges 
 | he has faced, but I think he's forcing it too much.  His first examples -- 
 | days of the week, Rubik's cube -- don't fit into it very well. Of course, 
 | he thinks they do, but a) they aren't really mathematics and b) it forces
 | him to leave out things that would be of more interest to MI
 | readers.

I find the assertion "It's  not Mathematics" with regard to the
Rubik's cube amusing to say the
least. Let's just agree to disagree.

 | For example, some issues that are of great importance, and which he should 
 | discuss in much more detail:
 | a)  the role of memory; didn't he need a very good memory to calculate 
 | those days of the week? And to remember them? Not all mathematicians have
 | good memories. Did being blind sharpen his memory (and his auditory 
 | capabilities), or does one have to have a good memory to be a blind
 | mathematician?

One of my goals in writing this article was to encourage students
who are  blind, and even more importantly, teachers of students
who are blind to encourage those students to pursue
mathematics. To that extent, I dont believe I am special in any
way, and making assertions of the form "I have good memory
therefore I can ..." is not a productive thing to say. There is
probably some truth to the assertion that I have developed my
memory to an added extent given that I dont have the luxury of
scratch memory; but I  would stop far short of ascribing 
every individual  trait of mine to the fact that I cannot
see. Alas, that is how the world tends to think; however I see no
reason to encourage the world in that mistaken belief.

 | b) the substitution of touch for sight. The most interesting thing about 
 | his discussion of the Rubik cube is that touch was a better guide than 
 | sight. This comes up again later with LaTeX and Zome Tools; I would urge
 | him to explore this.

Explore it is what the article does to the best of my present ability.

 | c)  Zome Tools aside, how does one do geometry when blind? Does being 
 | blind preclude some fields of research?

Again, my belief is that being blind does not preclude any field
of research; nor does it necessarily make you especially stronger
for something else.  The whole point of the article was to point
out that once you compensate for the fact that you cannot see,
doing Math is no more special when blind than say "brushing your

 | d) Could, in his opinion, someone blind from birth become a
 | mathematician?

As a scientist, I cannot objectively answer that question without
first reliving my life as someone who is born blind. That said,
if you believe my assertion that once one has compensated for not
being able to see, there is "nothing special", the answer should
be obvious.
 | e) Does he prefer reading mathematics in Braille (can LaTeX be translated 
 | into Braille?), or does he prefer aural systems like the one
 | he devised? 

The advantage with Braille is that it provides random access. But
ability to perform incremental searches when using electronic
documents, and the fact that Braille can be both slow -- and for
computer Braille, expensive has led me to use speech over Braille
in the last 10+ years. But this is an individual preference and
generalizing this to everyone would be a mistake.

 | What are the relative advantages and disadvantages of each?

See above.
In summary:

A) Braille on paper: static, has random access.
B) Electronic documents: dynamic, multiple views.
C) Computerized Braille: expensive

 | I'm sure there are more things one might suggest, but these will do for a
 | start.

These are indeed good suggestions and I'll take the fact that the
article raised these questions as a sign of it's quality and
appropriateness, rather than as a negative.

 | PS Also I would remind him that MI readers know that mathematics is more 
 | than scratchwork, that false illumination is, alas, a common experience,
 | and that LaTeX is often useful!

Yes, but what percentage of MI readers realize that what applies
to the average mathematician is equally true for someone who is
blind? The final P.S. above most clearly reflects the dichotomy
we face; on the one hand, MI appears to be looking to find out
how someone who is blind is different from everyone else; to that
end, you want to eliminate things that are shared, and focus
exclusively on the differences. I on the other hand come from a
background where I've tried my utmost to integrate into the
mainstream. I dont believe that this exercise can be used to
bridge that dichotomy.
 | 	Now I have many questions about how blind mathematicians manage.
 | Both the referee & I have plenty of questions we would like to
 | put to you.

That sounds like a good fireside chat for another time and
another place. Perhaps it's something that will happen on the

 | On the one hand, this means we felt your piece didn't cover the subject; on
 | the other hand, the questions do include some we might not have thought of
 | before reading your piece-- and if a piece stimulates questions in the
 | reader that is a plus!

That's exactly how I view the piece.

 | 	One possible tack we might take is this.  Your existing draft is
 | well written and involves the reader in your meeting your challenges; so I
 | might imagine you publishing it somewhere else, more or less
 | as is.  

Yes. I would like to thank you for encouraging me to write this

  | If you do, I would still hope to interest you in contributing
 | something else on the subject to The Intelligencer. Perhaps (in
 | line with the thoughts I expressed above) it could be in the form
 | of an interview-- either with a naive sighted mathematician or
 | with an older blind mathematician. Which? Or both?

That is a separate conversation -- and probably not something
I'll be able to get to in the near future --- remember, this is
not a fulltime activity, --- or even a small portion of my
fulltime activity.

 | 	Another tack is the one the referee wants: You might take all the
 | above as a bunch of suggestions for revision, and simply set to work writing
 | a rather different draft.  I would be optimistic about that.

Sorry, you'll have to find someone else for that I'm
afraid. Let's hope the conversation on the Web nets you the right

 | 	In any case, I'm glad my initiative has interested you in trying
 | to give the sighted public some notion of these matters, and the end product
 | will be an important piece of communication between
 | mathematicians.  Thanks

I agree. This is why I'm publishing this on the Web, let us hope
that it serves its intended purpose.