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.