I would like to know if this question is appropriate for Math Overflow, and if so, how I can phrase it better to get the best results.
I recently spoke with a very nice mathematician who gave me math career advice. Reflecting on their career, they also said to me something along the lines of
I used to think all of mathematics was useful in some deep, fundamental way, but I don't know anymore.
I generally try to briefly speak with professional mathematicians (appropriately and without being annoying) so I can learn small things from them and I've noticed that they each have their own concept of "usefulness" (e.g., they might say "Personally, I don't think X is useful"). So, my question:
What are the ways you have successfully convinced a skeptic (who is genuinely curious and not antagonistic) that your work in X is "useful" if there are not immediate, real-world applications you could otherwise quickly defend X with? Do you, as a professional mathematician, let this loose concept of "usefulness" play a role in deciding what problems to tackle, or are there good reasons for studying something solely because you and others find it interesting?
As someone interested in very abstract mathematics which, admittedly, has no immediate, real-world applications, I will probably find myself in the situation where I must convince someone that my work isn't completely useless (e.g., informally in talking to a non-math friend, and more seriously in filling out grant/fellowship applications*).
Why I want to ask on Overflow: When I google this question, I see that it is generally asked in less professional forums and that it is answered by people who are clearly not professional mathematicians. I want to know what professional mathematicians think, especially the ones in very abstract areas whose footsteps I am following in my math journey. And I want to ask this question without offending others, and without coming off as opinionated in one direction or another. I simply hold no opinion at the moment because I am just a math undergrad--I still have lots to learn. I'm just interested in hearing what mathematicians have to say. However, I feel that the second question might not be appropriate for various reasons, or it could be rephrased more appropriately. Or, perhaps it's fine.
*I'm already aware that a few questions address convincing organizations to fund mathematics.