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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.
If at any particular epoch of mathematical history no low-hanging fruit remains on some particular mathematical tree, then mathematicians may choose to plant, cultivate, and harvest the fruit of entirely new trees. Indeed, when frustrated, they have often done exactly that.
So what is going on? Why is mathematical practice today not dramatically more successful than a century ago? Why is there no spectacular contemporary flourishing of the art, with entirely new fields opened up by ten times as many Poincarés, Hilberts, Cartans, and Noethers?
A mathematical proof of the truth or falsity of this argument might be very helpful to Mr Gunnarson’s point!
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