Over the decades, the number of PhDs granted in the mathematical sciences has steadily grown. In the US alone, nearly three times more PhDs in mathematics were granted between 1995-9 than between 1960-4. (In physics, the corresponding growth factor was just over two.)
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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.
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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.
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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?
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