# A Certain Open Manifold Whose Group is Unity

By Bobby Neal Winters
One thing I’ve learned
over the course of my university career is that every discipline is
different. We’ve different interests, different methods, different
strengths, and different personalities. This became clearer to me when I
visited a meaning of the Student Advisory Board for the College of Arts
and Sciences. Each of the departments has a student representative and
each of those students reminded me of faculty I knew in the department.
I think particular personalities might be drawn to particular
disciplines, but I also believe that the activities demanded by a
discipline has an effect upon the practitioners.

Pure
mathematics is a lonely discipline. You spent a lot of time at your
desk staring at a yellow pad. You spend a lot of time staring out into
space. You spend a lot of time staring at a black board. Staring seems
to be a key ingredient.

I am under the
impression that in some disciplines you spend a lot of time in
libraries reading books, taking notes, and copying quotes. You are
interested in a particular topic and you look up things, read them, try
to understand them, and create arguments or form a synthesis or
something. Regardless, the library is an indispensable element to your
research.

I do not believe this is as true for
mathematics as it is other disciplines. I did read some books. Indeed,
there were about three books that I lived with while working on my
doctorate. They were my own personal Torah, Nevi’im, and Ketuvim.
There were also a few papers. Together these formed not so much the
material for my doctoral thesis but the foundation upon which I was to
build my thesis.

My thesis topic was
decided upon in a bar. Napkins were taken out as they traditionally are
in such places and my thesis advisor outlined two or three problems.
These were problems that my advisor didn’t know the solution to but
suspected could be solved. I then began working on them.

The way I approached this process was to take field trips in
my head to look over the mathematical territory. This is about as good
an explanation as I can make at this point. It is a very visual
process. Where the pictures come from, I don’t know but I can surmise
they are products of the mathematical experiences of listening to
lectures, reading articles, and imagining. I cannot over emphasize the
imagining part. What differentiates this from writing fiction--which I
have done as well--is that the things you imagine must be translated
into statements in written language which can be verified by others
independently.

We mathematicians are a
bit arrogant.

I thought that last
sentence was so true it deserved a whole paragraph by itself. Within
the Academy, scientists are considered arrogant, but I believe
mathematicians are arrogant even among scientists, putting aside for the
moment whether mathematicians are actually scientists or simply a breed
of our own. One way we express this is by saying that mathematicians
can’t get a publication out of a failed experiment.

When mathematicians set out of work a problem, we either
solve it or don’t solve it. If we don’t solve it, we don’t necessarily
have anything left over that we can use. We are making the assumption
that scientists can at least publish that a particular technique doesn’t
work, but I think that is more a function of our ignorance of other
subjects.

In any case, my thesis involved Whitehead
manifolds. This is a class of topological spaces defined by J.H.C.
Whitehead in 1935 in a paper entitled “A Certain Open Manifold Whose
Group Is Unity.” There may be some of you whose curiosity is piqued by
that title. It is mysterious. Every word in the title is an English
word which could be used in ordinary conversation, but certain ones have
been hijacked by mathematical highwaymen: open, manifold, group, and
unity.

The standard mathematical response to
curious enquiry here is “This is technical. You wouldn’t understand.”

I wonder how we got the reputation of arrogance.

The truth is that mathematical research is a two-fold
process. The first part is taking an excursion into the imagination I
told you about. The second part is describing that excursion. You are
entering a world that exists within your mind--I will avoid a discussion
of Platonic philosophy by not saying “only in your mind”--and you are
then having to describe it. When the explorers came to the new world
and found plants and animals that didn’t exist in the old world, they
had the advantage of using the Indian name for them.

There are no natives living in our heads--modulo a
schizophrenic here and there--so we have to load new meanings onto
existing words. As we are going to have to use these words in proofs
that will have to be verified by others, we tend toward precision. The
art of backing off that precision to explain what is not necessarily a
difficult idea to a layman is more difficult that is generally realized.
Few mathematicians take the time to learn it.

All
of this to say, I am confident enquiring minds such as yours could
understand, but I will save that for a later date as it would take us
too far afield.

Mathematics is full of
people who have spent a lot of time in their own heads and like to play
with language. We tend to like making puns though, that having been
said, we are not that much fonder of hearing them than anyone else.

The time we spend in our own heads tends to exact a cost in
people skills. Though I have known some mathematicians who are
extroverted in the classical sense, there is something to be said for
the definition of an extroverted mathematician being someone who stares
at the top of your shoes.

A positive trait we have is the desire to take the
complicated and simplify it. Good mathematics is mathematics made
simple. Well-written mathematics is mathematics wherein the
mathematician, having made his own journey in perhaps a quite arduous
fashion, retraces the territory to blaze a trail so that others can make
the trip in safety with conveniently located rest-stops along the way.

The communication aspect of mathematics is, in fact, the most
difficult, but it is the most important part. Historically, there were
those who did their work in isolation, for their own pleasure, and did
not share. When this practice was common, mathematical progress was
slow, but as the sharing of information became more common, progress
hastened.

Mathematicians can transfer their skills to
other areas of university life. They can use their explanatory and
organizational skills in lower level mathematics in their teaching.
They can bring the ethic of hard work and ability to deconstruct
complex tasks into simply ones to committee work. They can, but, as
with anything else, the challenge is in the doing.

The weakness comes with the tunnel vision that is a part of
the focus necessary to do research mathematics. There is a pronounced
tendency to denigrate anything that isn’t math.

Maturity
as a scholar within the university community comes with the recognition
that the rest of the people there have something of value they are
bringing to the table as well and that you can learn from them. This
maturity is a goal that is worthy of our aim.

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