Think ahead

Worse than being blind, is to see and have no vision. — Helen Keller

It is really easy to get bogged down in the details of some work and not recall the purpose of what one is actually doing; thus it is good to pause every now and then and recall why one is pursuing a particular goal.

For instance, if one is trying to prove a lemma for one reason or another, take a few moments to ask yourself questions such as:

• If the lemma were proven, how would it be used?
• What features of the lemma are most important for you?
• Would a weaker lemma suffice?
• Is there a simpler formulation of the lemma?
• Is it worth trying to omit a hypothesis of the lemma, if that hypothesis seems hard to obtain in practice?

Often, the exact statement of the lemma is not yet clear before one actually proves it, but you should still be able to get some partial answers to these questions just from knowing the form of the lemma even if the details are not yet complete.

These questions can help you reformulate your lemma to its optimal form before sinking too much time into trying to prove it, thus enabling you to use your research time more efficiently.

The same type of principle applies at scales smaller than lemmas (e.g. when trying to prove a small claim, or to perform a lengthy computation) and at scales larger than lemmas (e.g. when trying to prove a theorem, solve a research problem, or pursue a research goal).

See also “ask yourself dumb questions”.