17 February 2009

On publishing your trash can

I'm rereading de Bruijn's book Asymptotic Methods in Analysis (which, sadly, appears to be out of print again!) -- one of the great mathematical expositions, of asymptotic methods in analysis as they stood at midcentury. It's one of the most readable math texts I know.

de Bruijn writes in the preface:
Many things in this book are not presented in the shortest possible form, as an attempt has been made to reveal, to a certain extent, the motives that lead to certain methods. Naturally one cannot go too far in this respect; a mathematician cannot possibly publish his waste-paper basket.
This seems worth remembering; terseness is not always a virtue.


Anonymous said...

Too bad about de Bruijn's book. That's sad when even the Dover edition goes out of print.

Anonymous said...

Yes, but I find that terseness can be a virtue: When something is done really crisply and clearly (e.g. as in Walter Rudin's analysis textbooks), it tends to actually be easier to understand than a long and less "fancy" explanation. Terseness allows the central ideas to jump out strongly. Sometimes a proof would also look significantly worse if written out in full detail, e.g. many of the theorems in Hartshorne's Algebraic Geometry.

On the subject, it seems that non-elementary proofs are much clearer than elementary proofs. I have heard this is true for the prime number theorem (I am familiar with the Zagier analytic proof and have not even attempted to learn the elementary proof). As another example, the Riemann-Roch theorem comes out very cleanly from Serre duality. William Fulton has a proof in his book on curves which is more elementary (since Fulton does not cover schemes and cohomology), but it is much harder (at least for me) to pick up.

However, terse sketchy proofs (as appear in many of Serge Lang's otherwise nice books), cause serious problems.