tremely interested in both sports and games—the difference being that
in a sport you keep score and sweat from your own exertion, and in a
game you merely keep score (sorry, Tiger, golf is a game, not a sport). One
of my interests was chess; I studied the game avidly and read stories
about it. I recall one story about a chess grand master traveling incognito
on a train, and who is drawn into a pickup game with an up-and-coming
young phenom. At one stage, the grand master observes that as one ages,
one no longer moves the pieces, one watches them move.^2
I think there is a profound amount of both truth and generality in that
remark; I believe it applies to mathematicians as well as grand masters,
and I certainly feel that it applies to me. I am no longer capable of con-
structing lengthy and complicated proofs, but I have acquired a good deal
of “feel” for what the right result should be. Some years ago, I had the
pleasure of working with Alekos Arvanitakis, a brilliant young mathema-
tician from Greece. I have never met Alekos; he contacted me because he
had obtained results relevant to a paper that I had written, and we started
to e-mail and eventually began collaborating. He brought new insight and
considerable talent to a field I felt I had been worked out (in the sense that
nothing really interesting was left to prove). I would suggest something
that felt like it ought to be true, and within a week Alekos had e-mailed
me a proof. I felt somewhat guilty about coauthoring the paper, feeling
that Alekos had done most of the heavy lifting, but I decided that at least
I had a sense of what needed to be lifted. I could no longer move the
pieces as well as I did when I was younger, but I could watch them move
as if of their own volition.Classifying the Dead Ends
Looking back through the previous chapters, it seems to me that the
problems and phenomena we have investigated fall into a number of dis-
tinct categories.
Of these, the most ancient are the problems that we are unable to solve
within a particular framework. The classic examples of these are prob-
lems such as the duplication of the cube and the roots of the quintic. In
both cases, the problem was not so much an inability to solve the problem
as it was to solve the problem using given tools. The usual way to deal
with such problems is to invent new tools. That’s exactly how the cube
was duplicated and the roots of the quintic found: by using tools other
than those available to formal Euclidean geometry, and finding ways
other than solutions by radicals to express certain numbers.
Undecidable propositions could well belong in this category. Recall that
Through a Glass Darkly 239