On math papers and general questions they need to address

Question

If you proved something new in math and want to publish it (or even before you've started proving that and before you have thoughts on publishing anything), do you need to answer questions like "why do people care about what I proved?", "why is this topic interesting?", "how does this fit into a broader picture?", "what theoretical implications does this have?", etc.? I have some background in linguistics, and in linguistics, from what I understand, it is essential to answer these questions (even before you've decided to work on some research question/problem). But I've taken a look at several math papers from arxiv, and I didn't seem to see those questions addressed. So does nobody care about them in math, and as long as you proved something new (or gave a better proof of an old result), this itself is considered good enough? If so, then I guess this question can be extended to "how do you choose a good research question in math?", but maybe it should be a separate question.

Answer 1

Yes you do have to answer these questions, and math papers do. But you may not have noticed it as a non-mathematician, because math research is extremely specialized, and the broader picture will still be incomprehensible to you (or to most mathematicians).

Suppose I wanted to read Wiles's paper providing the final piece of the proof of Fermat's Last Theorem. I would have to spend about 2 years of intense study to get to the point of being able to even understand what the basic theoretical constructs of the paper are, and I'm closer to this area of mathematics than 70% of mathematicians.

I might add, as is customary for more important papers, Wiles's paper pretty extensively addresses the bigger picture. Less important papers will have a smaller, more specialized audience, and there is no need to explain the picture as broadly.

One does not choose research questions in math; one chooses research questions on Glauznov operators on Tchaikovsky cohomology of Rubenstein spaces (example completely made up using names of Russian composers). Naturally, one explains the motivation for one's research for (the 10) other people working on Glauznov operators on Tchaikovsky cohomology of Rubenstein spaces, and the papers that introduced Glauznov operators, Tchaikovsky cohomology, and Rubenstein spaces all explained the motivations for these ideas. If you think your paper is important, your introduction will also explain the implications of your work for the overall study of Tchaikovsky cohomology of Rubenstein spaces.

I've done a cursory look into the historical chain of ideas that leads to my research. It's about 15 layers of theoretical constructs from something a non-mathematician might think about. About 200 years ago, some mathematicians took a naive question and created a theoretical construct to help understand it. They found some facts about this theoretical construct, and some other mathematicians created a further theoretical construct to understand these facts about the first theoretical construct. Repeat about 15 times to get to my research. (Take a different chain of about 25 layers of constructs to get to Wiles's proof of Fermat's Last Theorem.)