TITLE: Pressure to defend the relevance of one's area of mathematics
QUESTION [118 upvotes]: 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.

REPLY [3 votes]: this notion that there is X and then there is rest of mathematics is a very very strange view of mathematics. mathematics is a fluid subject, ideas float around yielding results in many different places. partitioning mathematics into groups and putting names on them certainly has a value and purpose, but neither its value nor its purpose is to classify some of them as "normal" and some as "not normal". 
for me personally, it is incredibly difficult to say what set theory is. often i have difficulty saying whether a paper is set theory or not. perhaps it is possible to make a distinction between computational math and more abstract math, like calculus vs abstract algebra, but it doesn't work. the boundaries are blurred. there is an example above where a user says that you cannot claim that the notion of being countable is set theoretic. very well! exactly the point! mathematical concepts are fluid concepts that appear in many forms and many places. in the case of the notion of "being countable", if you are  not giving this to set theorists then you are admitting that the very action of splitting math into such boxes is silly, it is much better to think that "being countable" is a mathematical notion rather than set theoretic. 
of course, there are concepts and notions that fall into groups. it will be hard to argue that triangles are not geometric. but the point is that the boundaries of these subjects are so confused and meshed that you will ultimately run into problems. geometry has been connected to algebra more times than you can count. is a group a geometric concept or an algebraic concept? we can debate this for the sake of debating mathematical points of views, but not for the sake of defining subjects, like drawing a boundary around geometry and algebra with no intersections, such an endeavor is ultimately divisive, meaningless and above all pointless. 
there is no set theory or number theory, there is mathematical thought process, and people respond to that thought process differently when they are subjected to it, some pursue very concretely defined objects such as numbers, some pursue less concretely defined geometric shapes and some pursue the mysteries of infinity. more people pursue number theory, why? i don't know, i suspect it is because it is easier to grow in the subject, but if you are like me then you would leave number theory because number theory never made me feel satisfied with myself, i was never happy with myself while growing up in number theory, it was like i was living somebody else's life, set theoretic ideas and its mysteries did make me feel fulfilled and happy. i believe what we study has a lot to do with our personalities. i do not like the concrete, it bores me, i like what i don't understand, i like that there is independence and vagueness and my job is to clear it. i have noticed this exact characteristics in myself in many aspects of life, in my choice of music, films, books and etc. 
there are different personalities. some mathematicians are problem solvers, and they want problems you can state in a few minutes and that they spend a month thinking about it. set theory of course has many of them of various difficulty. some were mentioned earlier by others. some people like theories and stories. there are people in between. 
i cannot say that i am doing math so that i can then tell it to others, i believe this is more like a personal quest, at least that is how i got into it. however, this is a profession, and we need to be professionals. while hearing things like  "X is disconnected from the rest of math" is stressful and frustrating, there are many situations where i enjoy defending myself and explaining the meaning of what i do. i enjoy giving general audience talks or write introductions to my papers. i think this is part of the profession and is important, given that the society ultimately pays our salary.
i do agree, though, that there is too much out there claiming that set theory is somehow irrelevant. but as i hinted above, most cases of this that i have encountered were expressed by people who didn't really know what they were claiming. already the meaning of the word "relevance" is problematic for me.