Many people, even mathematicians, feel that pure mathematics is useless. But I would argue that at the very least, pure mathematics has always been useful to those who love it-- in keeping us happy, mentally healthy people! Healthy happy people make better life partners, better parents, and citizens who are more likely to help maintain order and fairness. In this way mathematics is a lot like art or sports.
So what happens when an artist or an athlete finally starts to make money with their art or sport? Well, it messes with our heads. We start to question our motivations. On those days when we just don't feel like thinking about math non-stop or when it feels like we are banging our heads against a wall, we feel guilty because we're not doing our job or we're not doing it well. We're goofing off, we're not productive! "Wait a minute," the brain says, "I used to goof of by DOING MATH!" The brain starts to freak out that its owner will never solve that thesis problem, will not be a good enough professional mathematician, that "gasp!" the brain will lie dormant while its owner teaches BUSINESS CALCULUS FOREVER!!! That may be every math graduate students biggest fear. So how do we recover the feeling of joy and amazement that we first had??
My first love of pure mathematics grew out of my joy at solving puzzles as a child, an excitement at having to exercise my brain, the struggle followed by the success, and a feeling that I owned the solution. I never cared whether the solution to the puzzle would lead to a cure for a disease, a better battery, or a million dollars. I did math problems for the same reasons that I spent hours every week dancing or reading everything I could about horses -- it was intrinsically appealing and satisfying! I liked to tell my friends mathy riddles, draw mathy pictures, do modular origami, and talk math with my dad on our walks around the neighborhood.
Young people who have that joy and amazement oozing out of their pours inspire us. Teaching young people, hanging out with my friends' kids, I feel like I can tap into their curiosity. And kids don't feel guilty about or try to justify their likes or dislikes. Our adult culture seems to have a vast spreadsheet of pre-approved purposes for why we should do something non-work-related. When was the last time you bought a book because you wanted to "do your small part to help the economy"? By always rationalizing the activities we do when we're not working, we negate the benefits of those activities while adding to our ambivalence about the work we're supposed to be doing. Don't feel like doing math? Not getting anywhere? Writers have writer's block, and they still think of themselves as writers. So, do something else you really enjoy for a while, and you'll still be a mathematician.
Activities whose outcome is not directly tied into my economic well-being renew my sense of learning for learning's sake. I can immerse myself for a few hours in a physical and/or expressive activity like swimming, dance, art, music, jogging, biking, or yoga. Appreciating the expressions of others through their art also fills this need to acquire "useless" information. Lastly, wandering around/exploring a city or the outdoors is"pointless" but awakens my senses and get me back into the mindset that I am on the lookout for beauty and not results.
On this note, a friend told me recently: "I was driving home today, and I saw this beautiful sunset! I thought to myself 'If only I had my camera!' But I couldn't enjoy it because I became so focused on getting home and getting my camera. When I finally got the camera and walked outside, the sunset was gone. Why couldn't I have just enjoyed it while it was there?!" I think this illustrates that fact that beauty is transient, just like the inspiration you need to do mathematics, and it doesn't emerge because someone is pursuing it.
So if I feel like biking around and taking in the scenery without a destination in mind, I just do it.
I swim back and forth in a lane while thinking about my breath and counting my strokes. I move from one yoga poses to the next even if I fall over in between. I walk out on the dance floor with a stranger and follow each movement as it arrives. I turn my thoughts into lyrics as I walk home.
Those of us who have the opportunity to put food on the table and feed our heads at the same are privileged. We can retain a child-like curiosity even as we grow into new adult responsibilities. Or we bend our intellects to fit a "grown-up" world that is, in truth, incredibly immature and irrational.