"Ten Lessons I wish I Had Been Taught", Gian-Carlo Rota 1997:
4. You are more likely to be remembered by your expository work
Let us look at two examples, beginning with Hilbert. When we think of Hilbert, we think of a few of his great theorems, like his basis theorem. But Hilbert's name is more often remembered for his work in number theory, his Zahlbericht, his book Foundations of Geometry and for his text on integral equations. The term "Hilbert space" was introduced by Stone and von Neumann in recognition of Hilbert's textbook on integral equations, in which the word "spectrum" was first defined at least twenty years before the discovery of quantum mechanics. Hilbert's textbook on integral equations is in large part expository, leaning on the work of Hellinger and several other mathematicians whose names are now forgotten.
Similarly, Hilbert's Foundations of Geometry, the book that made Hilbert's name a household word among mathematicians, contains little original work, and reaps the harvest of the work of several geometers, such as Kohn, Schur (not the Schur you have heard of), Wiener (another Wiener), Pasch, Pieri and several other Italians. Again, Hilbert's Zahlbericht, a fundamental contribution that revolutionized the field of number theory, was originally a survey that Hilbert was commissioned to write for publication in the Bulletin of the German Mathematical Society.
William Feller is another example. Feller is remembered as the author of the most successful treatise on probability ever written. Few probabilists of our day are able to cite more than a couple of Feller's research papers; most mathematicians are not even aware that Feller had a previous life in convex geometry.
Allow me to digress with a personal reminiscence. I sometimes publish in a branch of philosophy called phenomenology. After publishing my first paper in this subject, I felt deeply hurt when, at a meeting of the Society for Phenomenology and Existential Philosophy, I was rudely told in no uncertain terms that everything I wrote in my paper was well known. This scenario occurred more than once, and I was eventually forced to reconsider my publishing standards in phenomenology.
It so happens that the fundamental treatises of phenomenology are written in thick, heavy philosophical German. Tradition demands that no examples ever be given of what one is talking about. One day I decided, not without serious misgivings, to publish a paper that was essentially an updating of some paragraphs from a book by Edmund Husserl, with a few examples added. While I was waiting for the worst at the next meeting of the Society for Phenomenology and Existential Philosophy, a prominent phenomenologist rushed towards me with a smile on his face. He was full of praise for my paper, and he strongly encouraged me to further develop the novel and original ideas presented in it.
"Ten Lessons for the Survival of a Mathematics Department"%20ten%20lessons%20for%20the%20survival%20of%20a%20mathematics%20department.pdf):
6. Write expository papers
When I was in graduate school, one of my teachers told me, "When you write a research paper, you are afraid that your result might already be known; but when you write an expository paper, you discover that nothing is known."
Not only is it good for you to write an expository paper once in a while, but such writing is essential for the survival of mathematics. Look at the most influential writings in mathematics of the last hundred years. At least half of them would have to be classified as expository. Let me put it to you in the P.R. language that you detest. It is not enough for you (or anyone) to have a good product to sell; you must package it right and advertise it properly. Otherwise you will go out of business.
Now don't tell me that you are a pure mathematician and therefore that you are above and beyond such lowly details. It is the results of pure mathematics and not of applied mathematics that are most sought-after by physicists and engineers (and soon, we hope, by biologists as well). Let us do our best to make our results available to them in a language they can understand. If we don't, they will some day no longer believe we have any new results, and they will cut off our research funds. Remember, they are the ones who control the purse strings since we mathematicians have always proven ourselves inept in all political and financial matters.
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Followup to: Illusion of Transparency: Why No One Understands You, Expecting Short Inferential Distances
A few years ago, an eminent scientist once told me how he'd written an explanation of his field aimed at a much lower technical level than usual. He had thought it would be useful to academics outside the field, or even reporters. This ended up being one of his most popular papers within his field, cited more often than anything else he'd written.
The lesson was not that his fellow scientists were stupid, but that we tend to enormously underestimate the effort required to properly explain things.
He told me this, because I'd just told him about my experience publishing "An Intuitive Explanation of Bayesian Reasoning". This is still one of my most popular, most blogged, and most appreciated works today. I regularly get fan mail from formerly confused undergraduates taking statistics classes, and journalists, and professors from outside fields. In short, I successfully hit the audience the eminent scientist had thought he was aiming for.
I'd thought I was aiming for elementary school.
Today, when I look back at the Intuitive Explanation, it seems pretty silly as an attempt on grade school:
(Then again, I get a roughly equal number of complaints that the Intuitive Explanation is too long and drawn-out, as that it is too short. The current version does seem to be "just right" for a fair number of people.)
Explainers shoot way, way higher than they think they're aiming, thanks to the illusion of transparency and self-anchoring. We miss the mark by several major grades of expertise. Aiming for outside academics gets you an article that will be popular among specialists in your field. Aiming at grade school (admittedly, naively so) will hit undergraduates. This is not because your audience is more stupid than you think, but because your words are far less helpful than you think. You're way way overshooting the target. Aim several major gradations lower, and you may hit your mark.
PS: I know and do confess that I need to work on taking my own advice.
Addendum: With his gracious permission: The eminent scientist was Ralph Merkle.