Doing Mathematics

I have come to question a significant portion of the culture in academia, even while I have developed a deeper connection with other parts of that same culture or at least the culture that we could have. While I am deeply committed to mathematics as a creative occupation, and to teaching and mentoring in mathematics, my experience in academia after re-entering it seven years ago has strengthened my rejection of the many parts of that culture because they hinder the best research and teaching.

There are many aspects I could discuss, but here I am singling out four: the question of what makes a mathematical result or paper worthy of recognition together with the place of exposition in mathematics,  the value of awards and recognitions in mathematics, and the effects of federal funding on mathematics and academia.

As opposed to trying to do some sort of statistical study — a study which would only be meaningful if there were sufficient numbers of people following the ideas I propose, and there is not! — I will invoke common sense and intuitions that are commonly agreed on, but usually discarded as a guide for actions because of the economic realities of higher education; the institutions that pay us expect and reward the defective model and very few actively step outside those bounds.


We start with a relatively innocuous idea that papers that answer questions completely, are best.

What comes from the idea that results are best if they are definitive? Frankly speaking, I believe this idea is part of a cluster of ideas that impoverishes mathematics and mathematical culture.

I first thought about this when reading Bill Thurston’s 1994 article On Proof and Progress in Mathematics. In this article he contrasted how he approached his first work on foliations (resolve all questions, definitively!) versus his later work in geometry and the huge difference a more generous approach made in creating a rich, open, inspiring environment that many others got involved in, rather than the pinnacle of achievement that was admired from a distance.

Instead of maintaining a museum of monuments, we should propagate a countryside filled with rich, diverse gardens of ideas and a zoo of people tending and changing and expanding and creating new gardens.  While the first model leaves a trail of impressive facts, fit for admiration and worship, the second model is defined by engagement and inspiration for widespread creativity.

When Henry Helson visited Poland after the war, he was struck by the purity and simplicity of the mathematical culture that was also very generous. As he relates in his 1997 Notices article, Mathematics in Poland after the War, he was struck by the combination of generosity and fun that pervaded a culture that was serious about mathematics, but happy to publish things that did not aim to grab and own whole swaths of mathematical territory. Rather they published relatively short papers, each of which presented one new idea very clearly.

That exposition has been neglected, in spite of all the lip service to the contrary, can be seen in the response to the astrobites.org site, which has gained a lot of attention in the astrophysics community because of the large contrast between the high quality exposition that astrobites.org offers and the usual difficulty that non-experts have in reading scholarly papers.

I am now convinced that the high art of exposition should be valued as highly as the construction of brand new theorems, that publishing in such a way as to leave much to others is better than cleaning up an area and creating a monument: that what gets considered valuable mathematics ought to be greatly broadened. If anyone finds value — maybe because of explanations that require original thought, maybe because it brings the ideas to new audiences, maybe because it helps students see something clearly, maybe because it brings the understanding to the general public, and yes, possibly because it is completely original and surprising in construction — then it is valuable mathematics, worthy of the deepest respect. In this new model, the quality of the writing becomes very important. (I suspect that some will take issue with that statement saying that this is not a new model, but I will disagree and point to the enormous quantity of poorly written articles and books, some of which are also very valuable, even though they are not written very well. Of course, there are papers and books that are very, very well written. But it seems that this is considered a cherry on top, rather than something that should always, before anything else, be there.)

I am not urging that there be an effort to police exposition, but rather that this be given a great deal more attention at every level of education and practice. If we must have awards, let them go to those that have explained things well, have written things well. Better yet, train students to pursue the intrinsic rewards of doing anything well, from explaining derivatives to a confused calculus student to proving some new, highly technical theorem.

To encourage such changes, we would need to revisit how we reward and support the mathematical enterprise. This brings us to the consideration of the last two cultural components I said I was going to discuss: awards and federal funding.

Why do mathematics? For me, it is another form of art and at the same time, an exploration of the universe we live in. Knowing and understanding and explaining and inspiring others to do the same, exercises deep creativity and generosity; this is an occupation worthy of human beings that value themselves and others. Of course, there are an enormous number of occupations that can beneficially occupy the human mind and spirit. And each one can be as satisfying and beautiful and useful in its pursuit. By useful, I mean useful as an occupation, not useful as a tool to bend the world to my will. It is the occupation itself that is valuable. What happens to us and those we teach and share with, when we occupy ourselves (in a healthy environment!) is the greatest justification for any occupation.

From this position it becomes clear that awards and honors that many aspire to are actually a distraction. The reward is in the occupation itself. There are of course honors that have more to do with real appreciation rather than ranking and fame, and for such honors there is a place in a healthy culture. But the greed that masquerades in all of us as something more beautiful, seeks fame and fortune as a substitute for love and respect, whose lack actually gives room to that greed in the first place.

When the American Mathematical Society proposed the status of Fellow of the society, the negative side effects of such a program were pointed out rather eloquently by multiple individuals. In particular, I remember that Frank Morgan’s argument against the establishment of the program, and Neal Koblitz’ refusal of the offer of the status of Fellow. Of course, there is also the curious case of Perelman who refused the Fields Medal, the mathematical equivalent of the Nobel prize, whose recipients are given a demi-god status. For an interesting telling of the story and more, see Sylvia Nasar and David Gruber’s article Manifold Destiny in the August 28, 2006 issue of the New Yorker. (In the story, they quote Gromov, another prominent mathematician. Even though I very much doubt Gromov’s explanation of Perelmans refusal as a result of some great purity on Perelmans part, it is a story worth reading and thinking about.)

The influence of federal funding in mathematics, while it has enabled a great expansion of the enterprise, has led to a degradation of the culture, and not only in mathematics. It is well known that federal funding has turned academia into a serious addict, willing to do anything for the next fix of federal funds. That, combined with, spurred on by, the neglect of higher education in the public sector, has led to the very bad state of affairs in which grant money reigns supreme, fame (which can be turned into money!) comes second and teaching, for all the lip service it is given, occupies the lowest realms of academia. Proof of this diagnosis is not needed by anyone in academia (other than administrators who profit from illusions proposing some other reality), but if proof is needed, one need not look any further than the way adjuncts and instructors, who do a great deal of the teaching, are treated. Both in terms of the dismal pay and the insecurity of their jobs, we are saying that teaching is not what a university is really about — it is just what we have to do to keep up the charade.

But this is also where the tragedy lies; it lies in the immense impoverishment that results when teaching is not given top priority. It is a law of nature that real greatness, true stature, is proportional to the service to others that an entity or person actually provides. You may prefer to see this as my definition of greatness and stature. Either way, assuming this to be true, we have traded real nobility for a meager, greedy existence when we accept the perverted system of values that we currently have at research universities — and even, in some ways at teaching universities.

While small liberal arts college do in fact value teaching, they still take advantage of the situation generated by research universities and often pay their adjuncts obscenely low wages. It is tragic and funny at the same time that such colleges are usually full of people who think that businesses ought to raise the minimum wage, provide health care and longer paid vacations, and all sorts of other good ideas, but when it comes to the situation they have power over, they turn a curiously blind eye. But there is also this idolization of research universities, of elite institutions and this admiration pulls in some of the poison that they could otherwise easily avoid.

But, as I wrote in the previous post in this blog,  Learning to Think and to Act, research is a critical piece in education. It inspires and illuminates and brings a freshness and vitality that should be insisted on. On the other hand, research without teaching becomes selfish and elitist and aimed at goals that can at times be silly and irrelevant in their isolation.


 

What then, can we do? If the system is so far astray, what can be done?

In my opinion, the most powerful thing you can do is inspire change in your own sphere of influence by a focus on the place of freedom you actually have. Having your principles and philosophy aligned with life and love, and consistently acting in accordance with them, has always been the most powerful thing anyone could do.

Creative exploration and teaching, with a deep sensitivity for those that struggle; the pursuit of both pure and applied research, with generosity, and an acute sense for which applications are morally admirable; a discipline of simplicity, eliminating the pursuit of rank or awards or status or recognition — these are still the fundamental components of a culture worth immersing myself in, worth spreading to others. Taken together, they create a deeply rewarding occupation, an occupation that quietly, powerfully, moves us forward, and higher.

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