Daniel Lemire's blog

, 1 min read

Solving new and difficult theory problems… without looping into oblivion

Sometimes you decide that you need a specific theoretical result. For example, you may need a closed form formula for a given quantity. Yet, you barely know how to begin.

Maybe, you end up looping: you keep on revisiting the same ideas, again and again. You don’t seem to make any progress at all! What do you do? Don’t be hasty to conclude that you can’t do theory. Mathematicians are just ordinary people.

Some advice that has worked for me:

  • Be patient. Work incrementally.- At first, don’t try to prove conjectures. Begin by trying to disprove them! It is often much easier to disprove something than to prove it. And most conjectures end up being false.
  • Find the simplest non-trivial variation on your problem, and work from there instead. What you will learn from solving a closely related problem might teach you how to break the real problem. A related strategy is to work on specific examples.
  • Be paranoid. Check every single fact or claim twice. And then check it again.
  • Run computer simulations to check your mathematics, or to suggest new conjectures.
  • Draw pictures. Use your visual cortex.
  • Try to abstract out the problem: find a more general problem. Sometimes, specific problems appear more difficult than general ones because you are thrown off by irrelevant details.
  • Be thorough. When trying to solve a difficult theoretical problem, be neat. For example, work as if you are trying to explain your problem to someone else.