In the post Graduate level textbooks I I mentioned an advice given to me by a colleague many years ago:
"Do not write any books until you retire". posic commented on this:
"Do not write any books until you retire"?! One is tempted to generalize to "do not do any mathematics until you retire". Or, indeed, to "do not do anything you find interesting, important or meaningful until you retire"...
Gone are the days when Gian-Carlo Rota wrote "You are most likely to be remembered for your expository work" as one of his famous "Ten lessons I wish I had been taught". Not that I so much like this motivation, that is one's desire to have oneself remembered at any expense, but compared to people doing mathematics from the main motivation of getting tenure, grants, etc., it was, at least, leaving ground for some cautious hope. Presently I do not see any.
I am sorry for the long delay with a reply. Here are some thoughts.
The advice of my colleague does not admit such generalizations. He based it on the opposite grounds: he wanted me to do something more interesting than writing books.
He made a couple of common mistakes. First, he has no way to know what is interesting to other people, including myself. A lot of people do find writing expository works (at any level, from elementary school to the current research) to be very interesting. Actually, I do. At the same time, many mathematicians complain about lack of necessary expository writings. Some direction of research died because the discoverers are not able to write in an understandable manner, and others were discouraged to write expositions. At the same time, writing down some ideas is a creative work at a level higher than most of “Annals of Mathematics” papers.
Second, he followed a prejudice common at least in the US: expository writing is a second-rate activity compared to proving theorems. This prejudice is so strong that proving “empty” theorems is valued more than excellent expository writing. Apparently, this is a result of external with respect to mathematics influences. The main among them is the government funding of pure mathematics. There is essentially only one agency in the US providing some funds for pure mathematics, namely, the NSF. The role of few private institutions is negligible. It is not surprising that NSF has its own preferences, and the pure mathematics is not its main concern. Moreover, it is very likely that NSF is even not allowed by law to fund expository writing (I did not attempted to check this).
G.-C. Rota is right. He almost always right, especially if you at least try to read between the lines. Actually, the most cited (and by a wide margin) work of the mentioned colleague is a purely expository short monograph. So, he does not put his money where his mouth is.
Actually, I am not inclined to read G.-C. Rota so literally. He is a too sophisticated thinker for this. Whatever he says, he says it with a tongue in cheek. He wanted to encourage expository writing. The motivation he offered isn’t really the fame. It is the usefulness. You will be remembered most for things most useful for other people. For many expository writing will be much more useful than publishing a dozen of “research” papers.
I think that it will come as no surprise to you that the government agencies, supposedly to work on behalf of the people, demand a lot of work hardly useful to anybody, and do not support really useful (at least to some people) activities. I also believe that only few other mathematicians will agree.
Doing mathematics for getting tenure or its equivalent is essentially doing mathematics for having an opportunity to do mathematics. There are no other ways. If you know a way to do mathematics without an equivalent of a tenured academic position in the US, please, tell me. I do have tenure, but I am quite interested.
This is not so with "grants, etc.", especially if you have tenure. Working for grants is a sort of corruption. Unfortunately, it is so widespread. Well, some people, for example G.W. Mackey, predicted this at the very beginning of the government funding. They turned out to be correct.
G.-C. Rote wrote these words quite a while ago. Things did not improve since then. The expository writing is valued even less than at the time. Nobody cares if he/she or you will be remembered 100 years from now, or if a current paper will be remembered 10 years from now. Everything is tailored for the medicine and biology. Reportedly, almost no papers there are remembered or cited after 2 years. Anyhow, the infamous impact factor of a journal takes into account only the citations during the first 2 years after the publication. The journals are judged by their impact factor, the papers are judged by the journals where they are published, and academics are judged by the quantity (in the number of papers, not pages) and the "quality" of their publications.
Apparently, mathematicians are content with the current situation and are afraid of any changes more than cosmetic ones. Is there a hope?
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