by Alex Levin
When I began my first math research project a few summers ago, I thought that I was fairly good at communicating mathematics. By that time, I had written up quite a few problem sets for various classes, and had even been a TA for a freshman math course. Thus, I was very surprised by how difficult the process of writing a math paper actually was, and especially astounded by the fact that the hardest part proved to be the introduction.
I spent that summer at Williams College’s undergraduate research program. My collaborators were extremely strong, and the experience proved very educational and productive. There were many great moments throughout that summer, but one of my favorite happened during an informal group discussion one evening; we realized that a little calculation I had been doing on the side actually had some very interesting connections with some of our other work. Eventually we decided to make these connections the basis of a new paper, which would present our main results from the new perspective – and in a manner more suitable for a broad spectrum of readers. When I began writing up the results at the end of the summer, I thought that the most difficult aspect would be conveying all the technical arguments necessary to make our proofs work. There were definitions to motivate, pages of messy induction to explain, and countless other tasks to take care of. And indeed, throughout the coming months, we continually wrote and rewrote each of these components until finally we had a coherent technical argument. (Thanks to a co-author’s cleverness, and an unpublished paper by Noam Elkies, three pages of induction turned into just under a page of generating function magic.)
All in all, the technical portion of our paper went through numerous iterations (much more than I expected), but at the end, was done relatively quickly. Surprisingly, what turned out to be the hardest part was the non-technical introduction. While we knew what we liked about the work, and understood its context and main contribution, we found it hard to put these thoughts on paper. For example, it took us numerous attempts to get the first paragraph right. We started with prose that was perfectly bland, and after a few iterations, spiced it up with an interesting example. Unfortunately, once we realized that the example was irrelevant to the rest of the paper, we were forced to make our first paragraph bland once more. At the end, I think we were able to achieve a good balance; having several people work on it probably made it us converge faster on something that would both interest and inform our prospective readers.
My experience in writing the paper from that summer has given me an appreciation for solid introductions in research papers. Reading a paper, it is easy to lose sight of the big picture when struggling through the technical details. A glance at a well-written introduction can serve as a much-needed reminder of the context and outline of the argument. And even if many technical details in a paper prove elusive, the introduction can help frame one’s discussion with someone who would be able to help understand the paper.
Now that I’ve had my first experience writing a mathematical work, I hope that crafting introductions will become easier in the future. I know that it will always be a special challenge, but at least now, I have a better appreciation of its difficulty, as well as of the rewards of a job well done.
This reminds me of another paper writing conundrum I have: whether to write the introduction first. If I write it first, I end up rewriting the damn thing completely a couple of times over by the time I’m done. If I write it last, my paper has an ill-defined scope and I have trouble resisting the temptation to add other little results and change the statement of the main theorem. I’ve done it both ways and I’m still undecided.