There is a certain very special type of unproductivity that I have experienced only in math. You are working on something and you feel almost certain your strategy is not going to work. In fact, it is more likely than not that your strategy is not even a new strategy, but a variant on something you’ve already tried unsuccessfully — or maybe not even actually a variant, but just a rephrasing you’ve fooled yourself into thinking is a variant. So you are not sure whether you are actually working on something at all. In fact, you doubt it.
And yet you keep going! Because what if it works?