{"id":83,"date":"2009-03-30T08:11:07","date_gmt":"2009-03-30T12:11:07","guid":{"rendered":"http:\/\/mathgradblog.williams.edu\/?p=83"},"modified":"2009-03-30T08:11:07","modified_gmt":"2009-03-30T12:11:07","slug":"the-hardest-part-of-writing-a-math-paper","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2009\/03\/30\/the-hardest-part-of-writing-a-math-paper\/","title":{"rendered":"The Hardest Part of Writing a Math Paper"},"content":{"rendered":"

by Alex Levin<\/a><\/p>\n

When I began my first math research project a few summers ago, I thought that I was fairly good at communicating mathematics. \u00a0By 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. \u00a0Thus, 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.<\/p>\n

I spent that summer at Williams College\u2019s undergraduate research program<\/a>.\u00a0My collaborators were extremely strong, and the experience proved very educational and productive. \u00a0There 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. \u00a0Eventually we decided to make these connections the basis of a new paper, which would present our main results from the new perspective \u2013 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. \u00a0There were definitions to motivate, pages of messy induction to explain, and countless other tasks to take care of. \u00a0And indeed, throughout the coming months, we continually wrote and rewrote each of these components until finally we had a coherent technical argument. \u00a0(Thanks to a co-author\u2019s cleverness, and an unpublished paper by Noam Elkies, three pages of induction turned into just under a page of generating function magic.)<\/p>\n

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. \u00a0Surprisingly, what turned out to be the hardest part was the non-technical introduction. \u00a0While 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. \u00a0For example, it took us numerous attempts to get the first paragraph right. \u00a0We started with prose that was perfectly bland, and after a few iterations, spiced it up with an interesting example. \u00a0Unfortunately, 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. \u00a0At 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.<\/p>\n

My experience in writing the paper from that summer has given me an appreciation for solid introductions in research papers. \u00a0Reading a paper, it is easy to lose sight of the big picture when struggling through the technical details. \u00a0A glance at a well-written introduction can serve as a much-needed reminder of the context and outline of the argument. \u00a0And even if many technical details in a paper prove elusive, the introduction can help frame one\u2019s discussion with someone who would be able to help understand the paper.<\/p>\n

Now that I\u2019ve had my first experience writing a mathematical work, I hope that crafting introductions will become easier in the future. \u00a0I 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.<\/p>\n

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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. \u00a0By that time, I had written up quite a few problem sets for various classes, … Continue reading →<\/span><\/a><\/p>\n

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