How to not feel bad for doing math? I have a MsC and want to take a PhD in algebraic topology. Probably very few people in the world will have any interest of my thesis.  They will pay me for doing my hobby. Its the only job I can think of which doesnt contribute to bettering the world somehow. I feel like I should get a real job like doctor or garbageman. I  get bad conscience. How can I feel better for pursuing my dream?

It is worth noting that in the dim and distant past, graph theory was considered recreational math with no practical application to real life; however, now graph theory is essential to computer science, modeling the brain, and all sorts of other practical fields. So right now algebraic topology may seem abstract and nonapplicable, but in 100 years, your research could be essential to a new development in a field which has appeared, which no one foresaw.
