Humans do not by nature have the cognitive structure necessary to do mathematics. There are of course exceptions; genetic mutation gifts humanity with a mathematical genius at the rate of approximately ten or twenty per generation, in whose shadow the spirit of mathematics trembles. The other mathematicians have to make do with hard effort and dedication, and perhaps also the hope of having a great insight, some day. The non-mathematicians avoid the subject as if it were swine flu.
Without a doubt, the birth of modern mathematics has been a great boon to humanity. Together with science, the two give rise to technology, which has raised us out of the feudal era, saved us from starvation at the hands of the Malthusian Apocalypse (for now), and dramatically improved the quality of life for at least a few billion people. For this reason the practice of mathematics still thrives despite the incredible social and political problems that plague the field.
For a time during the rise of the philosophy of mathematics known as formalism, the idea spread that mathematics ought to be done for its own sake, like the practice of music or art, for the "enrichment of the individual". Such a view greatly underestimates the necessity of mathematical progress for the continued advancement of the human condition. Such a view has poisoned an already indifferent public against the public funding of mathematics, corroborating a similar view from a different demographic that rails against the public funding of the sciences.
If at one time, perhaps forty years ago, humanity was prepared to leave the cradle and become a full-fledged civilization, since then there has been no other time than now when we are more than happy to live our lives in the same place, in the same condition, until some happy existential disaster puts an end to the glorious serendipity that is intelligent life. It is not that we cannot solve (most of) the problems that face us. It is that the people who have the capacity to solve them 1) are ineffective at communicating that the problem is in fact solved, and 2) are ineffective at training the next generation of mathematician.
Being a mathematician myself, I'm no good at the first point. I'm almost convinced that the qualities that make one a good mathematician are antithetical to the qualities that make one a good politician. So while I can recognize that the practice of mathematics as it is now is problematic from a political point of view, I have no solution to offer, and even less a solution that would be accepted by the vastly non-mathematical majority. On the other hand, I have some hope that I can contribute this series of essays to working a solution of the second problem.
In the popular culture of mathematics there is a notion that a problem is solved in at least three stages. First, someone presents a proof of the problem, which may or may not be accepted as proof. Second, someone solves the 'exposition problem', presenting a proof or perhaps a demonstration of the subject at hand that is capable of teaching others the relevance of the result and how the result might be developed further. Finally, others create variations on the theme until a hard, tried-and-true path through the subject has solidified. In fact, this process tends to take several generations of iteration before a canonical text comes to the fore.
For example, it's generally accepted that Walter Rudin has written the canonical text on the general precepts of modern analysis, in Real and Complex Analysis. Despite that there are still several contenders -- Royden has a widely successful text that is worth mentioning, Bartle wrote a text abandoning the Lebesgue integral in favor of the simpler yet more general gauge integral, and I may be biased, but I wouldn't count out non-standard analysis yet. Despite the subject being more or less settled since the early twentieth century, there is still no clearly superior treatment of the subject.
Worse still, this incremental process of evolving presentations on the free market is by no means guaranteed to produce a superior text at all! Look no further than the omnipresent yet widely disliked calculus textbook by Stewart. I believe with insufficient evidence that Stewart's Calculus became so widespread as a result of a successful advertising campaign, and now the barrier of entry to anyone writing a better calculus textbook has risen so high that we will be stuck with Stewart's for at least two more generations, or until the academic industrial-complex collapses entirely, whichever comes first.
To be of true service to humanity, the society of mathematicians needs to find a solution to the problem of education. In professional practice there is sometimes the feeling that a solution will be found eventually; that every problem is just a matter of time. In the problem of education, this approach fails horribly — to paraphrase Galois, we don't have the time. It isn't enough to hope for a better way to teach mathematics to show up when some pedagogical genius invents it, for we needed a solution more than a century ago.
- "I have not the time": cognitive limits, running up the mountain