The Music of the
Primes will attract many young people to noncommutative geometry a
la Connes. It would be great if someone would spend a year trying to
write a similar pamphlet in favour of noncommutative _algebraic_
geometry, but as I mentioned before chances are not very high as most
mathematicians are unwilling to sacrifice precision and technical detail
for popular success. Still, perhaps we should reconsider this position.
A fine illustration why most mathematicians cannot write books for a
bigger audience is to be found in the preface to the book “The
problems of mathematics” (out of print or at least out of
amazon.com) by the Warwick mathematician Ian Stewart.
Below I quote a fraction from his ‘An interview with a
(I)nterviewer : … So,
Mathematician : what delights do you have in store for us?
(M)athematician : I thought I’d say a bit about how you can get a TOP
but non-DIFF 4-manifold by surgery on the Kummer surface. You see,
there’s this fascinating cohomology intersection form related to the
exceptional Lie algebra $E_8$, and…
(I) : That’s
(M) : Thank you.
(I) : Is all that
gobbledegook really significant?
(M) : Of course! It’s one of the
most important discoveries of the last decade!
(I) : Can you
explain it in words ordinary mortals can understand?
(M) : Look,
buster, if ordinary mortals could understand it, you wouldn’t need
mathematicians to do the job for you, right?
(I) : I don’t want
the technical details. Just a general feeling for what’s going on.
(M) : You can’t get a feeling for what’s going on without
understanding the technical details.
(I) : Why not?
Well, you just can’t.
(I) : Physicists seem to manage.
: But they work with things from everyday experience…
Sure. ‘How gluon antiscreening affects the colour charge of a
quark.’ ‘Conduction bands in Gallium Arsenide.’ Trip over
‘em all the time on the way to work, don’t you?
(M) : Yes,
(I) : I’m sure that the physicists find all the
technical details just as fascinating as you do. But they don’t let them
intrude so much.
(M) : But how can I explain things properly if I
don’t give the details?
(I) : How can anyone else
understand them if you do?
(M) : But if I skip the fine
points, some of the things I say won’t be completely true! How can I
talk about manifolds without mentioning that the theorems only work if
the manifolds are finite-dimensional paracompact Hausdorff with empty
(I) : Lie a bit.
(M) : Oh, but I couldn’t do
(I) : Why not? Everybody else does.
(M) : But, I
must tell the truth!
(I) : Sure. But you might be prepared to
bend it a little, if it helps people understand what you’re doing.
(M) : Well…