Through a Glass Darkly

arXiv:0807.2656v1 [math.HO] 16 Jul 2008 Through a Glass Darkly Steven G. Krantz1 1 Prolegomena Education is a repetition of civilization in little. — Herbert Spencer Being a mathematician is like being a manic depressive. One experiences occasional moments of giddy elation, interwoven with protracted periods of black despair. Yet this is the life path that we choose for ourselves. And we wonder why nobody understands us. The budding mathematician spends an extraordinarily long period of study and backbreaking hard work in order to attain the Ph.D. And that is only an entry card into the profession. It hardly makes one a mathemati- cian. To be able to call oneself a mathematician, one must have proved some good theorems and written some good papers thereon. One must have given a number of talks on his work, and (ideally) one should have either an academic job or a job in the research infrastructure. Then, and only then, can one hold one’s head up in the community and call oneself a peer of the realm. Often 1It is a pleasure to thank David H. Bailey, Jonathan Borwein, Robert Burckel, David Collins, Marvin Greenberg, Reece Harris, Deborah K. Nelson, and James S. Walker for many useful remarks and suggestions about different drafts of this essay. Certainly their insights have contributed a number of significant improvements. 1 one is thirty years old before this comes about. It is a protracted period of apprenticeship, and there are many fallen and discouraged and indeed lost along the way. The professional mathematician spends his life thinking about problems that he cannot solve, and learning from his (repeated and often maddening) mistakes. That he can very occasionally pull the fat out of the fire and make something worthwhile of it is in fact a small miracle. And even when he can pull offsuch a feat, what are the chances that his peers in the community will toss their hats in the air and proclaim him a hail fellow well met? Slim to none at best. In the end we learn to do mathematics because of its intrinsic beauty, and its enduring value, and for the personal satisfaction it gives us. It is an important, worthwhile, dignified way to spend one’s time, and it beats almost any other avocation that I can think of. But it has its frustrations. There are few outside of the mathematical community who have even the vaguest notion of what we do, or how we spend our time. Surely they have no sense of what a theorem is, or how one proves a theorem, or why one would want to.2 How could one spend a year or two studying other people’s work, only so that one can spend yet several more years to develop one’s own work? Were it not for tenure, how could any mathematics ever get done? We in the mathematics community expect (as we should) the state legis- lature to provide funds for the universities (to pay our salaries, for instance). We expect the members of Congress to allocate funds for the National Sci- ence Foundation and other agencies to subvent our research. We expect the White House Science Advisor to speak well of academics, and of mathemati- cians in particular, so that we can live our lives and enjoy the fruits of our labors. But what do these people know of our values and our goals? How can we hope that, when they do the obvious and necessary ranking of priorities that must be a part of their jobs, we will somehow get sorted near the top 2From my solipsistic perspective as a mathematician, this is truly tragic. For math- ematical thinking is at the very basis of human thought. It is the key to an examined life. 2 of the list? This last paragraph explains in part why we as a profession can be ag- gravated and demoralized, and why we endure periods of frustration and hopelessness. We are not by nature articulate—especially at presenting our case to those who do not speak our language—and we pay a price for that incoherence. We tend to be solipsistic and focused on our scientific activi- ties, and trust that the value of our results will speak for themselves. When competing with the Wii and the iPod, we are bound therefore to be daunted. 2 Life in the Big City The most savage controversies are about those matters as to which there is no good evidence either way. — Bertrand Russell If you have ever been Chair of your department, put in the position of explaining to the Dean what the department’s needs are, you know how hard it is to explain our mission to the great unwashed. You waltz into the Dean’s office and start telling him how we must have someone in Ricci flows, we certainly need a worker in mirror symmetry, and what about that hot new stuffabout the distribution of primes using additive combinatorics? The Dean, probably a chemist, has no idea what you are talking about. Of course the person who had the previous appointment with the Dean was the Chair of Chemistry, and he glibly told the Dean how they are woefully shy of people in radiochemistry and organic chemistry. And an extra physical chemist or two would be nic

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