Losing my religion (or my faith is shaken)
In which the mathematician gets emo
I think it's time I come to admit this. I'm not doing as much
mathematics as I should be doing, and this suddenly doesn't seem all
that catastrophic. My brain is burning out, the spark is dying out,
and the most horrific thing is that I can't even decide if I ever had
a spark to begin with or not. Or if it's not burnout, at least it's
sleeping...
I've always liked mathematics. I've always had a bit of a knack for
it. I could understand and work through the concepts. I was enthralled
by the subject and the deep mysteries. Once in a while, I would shine.
I was looking at the past Putnam exam questions today. In other times,
I would have attempted to answer a few of them, but today, when I was
looking at it, I couldn't think of how to approach any of them. Worse,
I didn't even feel a tug, an itch, a desire to attempt to answer any
of them.
This is bad. A mathematician that doesn't feel an urge to solve
problems thrown at him is a dead mathematician (in the Erdosian sense).
Last night I was trying to puzzle out how to incorporate some boundary
conditions into some shallow-water equations I was solving numerically
in such a way that I could still use essentially the same numerical
method and still be able to get a decent solution after leaving the
computer munching on it overnight. It must have been the first time in
months since I picked up a pen and started scribbling on
paper. I can't even remember the last time that I actually spent hours
filling pages upon pages with calculations (I call this using the swap
partition, heh). It was difficult. At one point, to check some
differentiations, I used Maxima. I'm getting lazy, I guess, a virtue
in a coder, but a vice in a mathematician. And in the end, it was all
hopelessly trivial, and the only thing I could think I could do was to
recast the equations with a stupid change of variable that simplified
one of the equations in the system but complicated the other two, and
to iterate the boundary conditions on those two equations hopefully
getting convergence. There was a minor division by zero problem there,
which would arise in getting a singular matrix, but easy to work
around with.
For those of you still with me, the language above doesn't
matter. There may be unfamiliar words there, but believe me: it was
absolutely trivial. There were no deep concepts involved, no insights,
nothing. Blank. This scares me to death.
Thing is, it shouldn't be like this. It is a stereotype that applied
mathematics is less creative than pure mathematics, that it simply
involves using recipes to get answers. There may be some truth in
this, as I have spent an inordinate amount of time in the last few
months doing nothing but coding instead of creating mathematics. Sure,
there's creativity in coding, but after a while it gets quite
mechanical. But back to applied mathematics, there's lots in there
that requires creativity on the theoretical side, proving our
algorithms converge, proving our equations have solutions, proving
regularity conditions. I've been reading books on the subject, on
functional analysis, on Schwarz distributions, on Sobolev spaces, which is all the machinery behind the proofs,
say, of existence and uniqueness of the 2d Navier-Stokes equations
(the 3d case is the current holy grail in the subject, the 2d case is
apparently much easier).
The whole flavour of this subject, however, fails to excite me. I
thought it would with time, but it hasn't. It's just that with
analysis there seems to be less tangible objects to play
with. Everything is fuzzy and approximation. By this I don't mean that
it's in any way not mathematics, not rigourous, merely that the the
feel of the subject, of finding the bounding constants, of
assimilating complicated definitions of finely varying
details1, of sets of measure zero never mattering for
anything... It's not as exciting as the worlds of number theory,
geometry, or algebra, in their various forms.
In a way, this happened gradually. I was at my best during my
undergraduate days. I remember solving my number theory undergrad
assignments with relish, with spark, with ingenuity. At some point, I
thought that I would be too lonely as a pure mathematician. I'm not
sure when this happened. There was a time when I was happiest when I
was alone, that my me-time was my best time. That changed radically in
university, somehow. Now I need time with others to recharge. I can be
alone, but not for very long before I start to feel extremely
uncomfortable. Applied mathematics is a much more social activity than
pure mathematics for two reasons: 1) you get to interact with many persons
who aren't mathematicians themselves, 2) you can actually
explain the eventual purpose of what you're doing to almost anyone,
not just to other specialists. So these are the reasons for why I made
the switch from pure to applied after I graduated from McGill
University.
In which our hero sees hope for the future
Wow. I needed to get that out. I haven't quite vocalised it until
now. I've been much too afraid to admit publicly that I'm not doing
mathematics as much as I would like.
In other times, this would have been terrifying, especially since I
was a little kid I've had grand dreams of one day discovering great
mathematical truths before anyone else did. But nowadays, this doesn't
seem so bad. For one, it seems that I traded the recluse of lonely
mathematical discovery for the ability to share more of what I do with
others. Also, the fear of not doing mathematics, should I ever
distance myself from the subject more than I am now, isn't so terrible
either. Languages have long been something that I keep telling myself
I could do instead. Coding, if I manage to do it under my terms. I'll
find other things to do. Or if not, I can always go back. It's not something that once gone, is gone forever. At McGill I met many competent PhD students who started taking up the subject again after many years of absence from anything academic. I need to relax.
And above all, I can still keep focus: I must save the world. One way
or another, big or small, I'll do it. Nothing else matters all that
much, at least, not for now.
I'll be ok. I've always been ok.
1 The reason why definitions in analysis seem to be so
complicated and nested, (e.g. a vector space is more general than a
topological vector space is slightly more general than a normed
space which is slightly more general than a Banach space, these
last two of which happen to be a metric space which also is a
topological space but not a vector space and a Fréchet space
fits in this mess somewhere, as do the various Sobolev spaces, and
what am I forgetting?) is that analysts like to be able to state
theorems with as few assumptions as possible. There seems to be so
much work in analysis about seeing if we can drop just one more
hypothesis or make one constant just a bit better, so much refinement,
and the big ideas are all hidden in convoluted statements of these
refinements...