Yet it is impossible to argue that all these advantages enjoyed today by researchers in the mathematical sciences have led to equally spectacular improvements in the overall quality of their achievements. At best, it may be possible to argue that no steep decline has occurred.

In particular, looking just at mathematics, we find the following names associated with profound innovations made in the first two decades of the last century: Frobenius, Burnside, Poincaré, Hilbert, Minkowski, Hadamard, Cartan, Takagi, Ramanujan, Weyl, Hecke, Noether, Banach.

It is impossible to argue that the first two decades of mathematical research in this century have produced any innovations as profound as group representation theory, functional analysis, dynamical systems theory, the geometry of fiber bundles, or class field theory.

Great researchers in mathematics are certainly not ten times more numerous today than they were a century ago; indeed, it takes some audacity to argue that we have as many. (It's far from clear, for example, whether anyone alive today can bear close comparison with Poincaré.)

But if conditions today are so spectacularly more favorable to successful research in the mathematical sciences than a century ago, and the number of trained researchers has grown by at least an order of magnitude, why is there no corresponding growth in achievement?

Mathematics itself may be the most illuminating case to study, because a "depletion of low-hanging fruit" explanation of modern stagnation is least tenable there. All the fundamental laws of physics may already have been discovered, but nothing like this is true in mathematics.

Indeed, mathematics is demonstrably inexhaustible, and the exceedingly long history of the art records no fallow period during which its master practitioners believed they might be unable for fundamental reasons to discover deep new results of lasting interest.

Note that this contrasts strikingly with physics: in 1894, Michelson judged it likely that "most of the grand underlying principles have been firmly established," and that "the future truths of physical science are to be looked for in the sixth place of decimals."

No similarly eminent mathematician has mooted a similarly pessimistic view of the art's prospects. On the contrary: great mathematicians have tended to predict extraordinary things to result from the art's inevitable assimilation and refinement of recent breakthroughs.

Because mathematicians have the freedom to devise and pursue entirely new fields of research -- a freedom successfully exploited, repeatedly, by its greatest past masters -- the formidable intricacy of its current best-established fields is no bar to its further flourishing.