Skip to main content
Source Link
Andy Putman
  • 44.8k
  • 14
  • 186
  • 272

This is too long for a comment, but doesn't exactly answer the question. However, I've had enough eggnog this Christmas that I'm going to post it anyway (despite knowing almost nothing about category theory).

Reading the question and skimming over the comments, I see a lot of romantic descriptions of the practice of mathematics that bear little relationship to how it is actually practiced. It is truly wonderful when a single elegant idea can completely illuminate and render transparent some part of the subject. However, these ideas are usually the end product of a long development that starts with a hacked together, complicated mess of arguments. And they are discovered by people who are deeply immeshed in the subject.

To put it another way, while it is great to have a strong philosophical take on what mathematics is and how it should be practiced, if that philosophy is not informed by the actual practice of mathematics, then it is unlikely to lead anywhere. Philosophical clarity comes at the end and not at the beginning.

To be successful at research, you have to be willing to get your hands dirty. If you don't enjoy the ordinary craft of doing mathematics, then it is unlikely that you will be happy as a research mathematician. But it is a craft. I strongly disagree with various comments that make it sound like you have to be some kind of crazy romantic hero taking superhuman risks or something. I certainly am not like that, but I have been able to make a career out this.

Now, it is impossible for us to give you personal advice on what you should do with your life or what direction your research should take. We don't know you. But I can say that everyone goes through periods of doubt and frustration. What I always do in those situations is to take a brief break from the front lines of research and go back to the sources that drew me to mathematics in the first place. Read some great mathematics, be refreshed, and then get back at it.

Post Made Community Wiki by Andy Putman