Why Graduate School

Rereading the personal statement I wrote when applying to grad school is always good for a chuckle.  In essence, the essay says what all application essays say: I like math, I'll be good at it, here are the mitigating circumstances for any shortcomings in my application -- in my case the lack of B.A. in math, and the fact that I had been out of school for ten years.  The essay was honest and I think it was well written, but math has turned out to be much harder than I thought it would be. Being good at math requires more than a logical mind.   It requires being able to rapidly absorb great masses of new ideas and employ them to solve difficult problems.  First year was hard.  The first month of analysis covered general measure theory, and oh my gosh did I not have the slightest clue what we were doing.  Years later measure theory seems like such an obvious way to do things, so obvious, in fact, that it feels like I've known it all my life.  What could be simpler than assigning some measure of size to a set in a way that respects countable unions and intersections?  To do so based on geometric considerations is nothing more than what we learned as a kid when we encountered words such as 'length', 'area', 'volume.'

One of the appeals of mathematics is that while it makes rigorous basic ideas we've worked with all our lives, it also introduces very new notions.  For example, the Hausdorff measure can be used to define 'length' and 'area' by setting a dimension parameter to 1 and 2 respectively.  The formal construction of the Hausdorff measure does not require that this parameter be an integer.  And so, along the way to making 'length' and 'area' clear, you also pick up all these fractional dimensional things for which the theory is as clear as it is for length, but the intuition is not clear at all.  After a while you suffer what my analysis professor might call a "professional deformation" whereby these ideas that started out as strange and foreign begin to seem like common, everyday objects.  At this point leaving math means going out into a world where almost no one thinks of fractional dimensional objects, and even fewer have effective tools for working with such things.  All these familiar ideas would drop out of existence outside of math because there is no language for them.  It isn't a happy prospect for someone who has spent considerable time struggling to understand these ideas. 

The other appeal of math is that on the few occasions I've made progress on an open problem I've gotten an enormous rush.  To have an idea come at an unexpected moment or to see the details of a new argument work out on paper in front of you is a tremendous feeling.  There is this realization that you know something about this abstract world that no one else knows.  It's a rare to be sure, but there are maybe three times when I've had this happen and it makes me feel remarkably alive. 

The thing is that the process of doing math is delicate.  Here's an analogy: As an instructor, when students came to my office hours, I would make them write on my whiteboard.  There is considerable evidence that more learning occurs when the student figures something out on her or his own than when the instructor reiterates the answer.  In those moments students are caught in a realm where they aren't sure of their footing.  They move slowly, second guess themselves, and set out on epically wrong paths.  This is remarkably like what I've observed research to be.  In meetings with researchers there is an awful lot of "I don't know," and, "I couldn't get this to work."  The analogy breaks down in that researchers have no answer keys and no one to guide them to the answer when they get lost.