I am a set theorist. Since I began to study this subject, I became increasingly aware of negative attitudes about it. These were expressed both from an internal and an external perspective. By the “internal perspective,” I mean a constant expression of worry from set theorists and logicians about the relevance of their work to the broader community / “real world”, with these worries sometimes leading to career-defining decisions on the direction of research.
For me, this situation is unwanted. I studied set theory because I thought it was interesting, not because I wanted to be a soldier in some kind of movement. Furthermore, I don’t see why an area needs defending when it produces a lot of deep theorems. That part is hard enough.
To what degree does there exist, in the various areas of mathematics, a widespread feeling of pressure to defend the relevance of the whole subject? Are there some areas in which it is enough to pursue the research that is considered interesting, useful, or important by experts in the field? Of course there will always be a demand to explain “broader impacts” to funding agencies, but I am talking about situations where the pressure comes from one’s own colleagues or even one’s own internalized sense of what is proper research.
Best Answer
Overall, people in academia in general and mathematicians in particular are very lucky in being free to study (and being able to make a good living) according to the standards of their discipline, without feeling pressure to defend the relevance of their whole subject. In fact even within our disciplines we have a lot of freedom to pursue our individual visions and tastes. (To appreciate how lucky we are compare the situation with musicians, writers, artists, film directors, actors, ...)
Relations with other areas of mathematics or outside mathematics are nice but they are one (and not necessarily a major one) among variety of criteria to appreciate mathematical progress.
I think we do have some duty to try to explain what we are doing outside our community and even outside the mathematical community. (But also this task is easier in some areas and harder in others.)
Another thing that I found useful in similar contexts is the "sure thing principle". Given an unwanted situation that has no implication on your action why worry about it
at alltoo much. For example, suppose a paper you wrote and regard as a good paper is rejected. If the rejection was unjust then the conclusion is: "Improve your paper", and if the rejection was just then the conclusion is "Improve your paper".