Laurens Gunnarsen

Even if Joe and Hunter Biden were criminally corrupt, I can't see how that would exculpate Trump. Surely we can agree that no President may order the investigation of his political rivals. That's a favorite technique of corrupt despots seeking to perpetuate their rule.

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?

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.

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.

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.

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."

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.

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.

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?

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.)

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.

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.

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.

The global community of researchers in the mathematical sciences has grown over the last century by more than an order of magnitude, and technologies and circumstances obviously favorable to its success have improved and spread spectacularly.

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.)

Institutional support is also vast compared to a century ago. In the US alone, major research universities offering tenure to professors in the mathematical sciences have more than doubled in just the past 25 years. Conferences, summer schools, and prestigious prizes abound.

Access to advanced education in the mathematical sciences has become much easier and more general over the last century. Today's graduates hoping to pursue a career in research have vastly more practical options than did their counterparts of a century ago.

Perhaps partly as a result, IQ scores have risen over the past 100 years. Just from 1942 to 2008, average scores on those tests that best measure abstract pattern recognition ability -- the key requisite for mastering advanced mathematics -- have risen by a standard deviation.

General conditions, too, overwhelmingly favor contemporary researchers. Rates of extreme poverty have plummeted. Life expectancies are longer by two decades. No mathematician or physicist today dies of tuberculosis, or of peritonitis following a ruptured appendix.

Contemporary researchers in the mathematical sciences enjoy countless advantages over their counterparts of a century ago. A vastly larger and better-funded community now has convenient access to the literature, and to communication channels that make global collaboration easy.