It sometimes happens during the research process that I come to a point where I think I know what I should propose or argue, but I can’t see how to prove or disprove it. This happens fairly often when I’m writing papers in the humanities, but even more so when I am trying to solve a problem in mathematics. In fact, I ran into this issue while trying to prove one of the claims on my most recent math assignment. The goal was to prove the following claim:
(I include the claim for completeness, but it’s not terribly important.) My first step was to reduce the claim to a statement that seemed easier:
which I was able to do just by using the information given in the problem. After that, I was able to get nearly three-quarters of the way through the proof just using the given information. Then I ran into something I knew should work, but wasn’t able to verify:
Because I already knew what I was trying to prove, I was able to use the assumptions I described above to complete the problem (more or less). There was still a hole in my argument, but I was able to construct a solid “scholarly narrative” for the problem (that is, finish the proof) by carefully delineating what I understood and where my understanding went off a cliff. Writing out exactly what I thought was wrong also told me what I needed to learn the next time I worked on the topic — effectively, it told me what further research I needed to do. So, even though I wasn’t quite sure what was going on, I was still able to decide what I would need before I made a second pass over the problem.
There are, of course, several caveats to what I have said thus far. First, in this case, I knew that my overall claim was correct, so I was confident I should be able to jump over the hole in my reasoning and complete my argument. In general scholarly writing, we don’t know with certainty that our thesis is correct, so a question like the one above could throw the entire premise of the argument into doubt. The second caveat is that clearly, my argument couldn’t have been called “complete” while containing the excerpt above. Since there was a hole in my argument, I hadn’t really proved anything yet. Indeed, the kind of serious gap in understanding displayed above was only permissible because the homework assignment was the equivalent of an early draft.
But despite these two cautionary notes, annotating incomplete arguments as I do above has often proved helpful (and necessary) for me in both mathematics and the humanities. By whittling down the “unknown” part of the problem to a single nugget that we are not equipped to attack with our current tools, we mark the bounds of our own knowledge, and in doing so, lay the groundwork needed to push those bounds even further than before.
–Isabella Khan ’21