The health of mathematics

Abstract

How is mathematics doing? This is a question worth occasional examination. Today the answer must be: It's doing well, but at the same time, it's also doing badly. There are many indications that it's doing well, for example, in the numerous important problems which just now have been settled. For finite simple groups we now have a complete list headed by the monster. The famous Weil conjectures on diophantine equations have been settled, using a considerable part of the intended heavy new machinery of algebraic geometry. Paul Smith had conjectured that a periodic homeomorphism of a 3-sphere with prime period and leaving a knot fixed must be conjugate to a rotation of the sphere; thanks to the joint efforts of five or six people, we now know that Smith was right. The Poincar6 conjecture has been settled in dimension 4, and the original proof has been substantially clarified; this leaves the original conjecture in dimension 3 as a challenge. Many other new and interesting results have been achieved, and more mathematicians than ever before are hard at work in all sorts of directions. Some of these directions seem to show relatively little recent major progress (lattice theory, homotopy groups of spheres, etc.), but there are other specialities which exhibit spectacular current progress: algebraic geometry , mathemat ical physics (Yang-Mil ls and super Lie algebra), hyperbolic geometry (3-manifolds), computer science (will P equal NP?), and many others. Now such enhanced activity in a particular direction usually reflects a general perception that there are currently real chances for major progress in that direction. Also, occasional shifts in the dominance of particular fields have long been a feature of mathematical progress. For example, the beautiful prospects for functions of one complex variable made this a dominant field early in this century; it is rumored that in France opinion dictated that the true objective of any mathematician must be that of discovering some extension of the big Picard theorem. It was only much later that there was comparable emphasis on several complex variables. After World War I, analytic number theory deve loped and f lourished, as did abstract algebra (Noether and Artin) and linear operators on Banach space (Wiener and Banach); all these fields experienced a subsequent decline. After World War II, algebraic topology for a while held a central position which it gradually lost to differential topology and analysis. Shifts such as these are inevitable, but there can be concern that too many mathematicians, trained in some once lively field, have stuck with that field as it became mired down in complications--as, for example, the complications with the nondenumerable Abelian p-groups and the unmanageable spectral sequences which might have (but did not) compute all the u n k n o w n higher homotopy groups of spheres. Perhaps one needs a wider awareness that specialties can become unprofitable (most of us do not like to give up), plus some re t reading process for those who would like to shift fields. At present it may be that too many of us talk only to our fellow specialists at seminars and meetings concentrated on that specialty. The publ i shed literature of mathematics is in trouble. Too much is published, as may be suggested by the size of the current volumes of Mathematical Reviews. The suggestion is confirmed by the considerable number of superficial papers better left as preprints and the large number of dull preprints better left unwritten. At the same time, too little is published. In fields ranging from category theory to hyperbolics