Nearly for any subject I try to always think with two opposing voices: one that says there's a solution and one that says it's impossible, and I always explore both sides of the argument.
If at some point some position sounds more plausible than the other, I try to put into words why is that. Either I convince myself of some position, for example that it is impossible, or the hardness of getting convinced at one position encourages me that maybe it's the other position that's true.
Either way I don't have anxiety because either I'm confident in some direction and making progress, even if it is at proving something is impossible, or the doubt from failing to prove it is impossible makes me more confident it is possible.
Even if you're making progress at the direction you didn't prefer, you should still see it as progress, and if you're confident in one direction you see it as an indication that you can make progress.
Anyways, it's not enough to just hold a general feeling that something is impossible, you need to put it into words and see if it's truly convincing. When it's just general intuitive unspoken feeling you can't change it, but when you make it concrete, you can understand what's possible.
The two opposing voices is reminiscent of game theoretic proof methods. Terry Tao has a great answer on mathoverflow that puts it in personal terms about how he solves problems [0]:
> One specific mental image that I can communicate easily with collaborators, but not always to more general audiences, is to think of quantifiers in game theoretic terms. Do we need to show that for every epsilon there exists a delta? Then imagine that you have a bag of deltas in your hand, but you can wait until your opponent (or some malicious force of nature) produces an epsilon to bother you, at which point you can reach into your bag and find the right delta to deal with the problem. Somehow, anthropomorphising the "enemy" (as well as one's "allies") can focus one's thoughts quite well. This intuition also combines well with probabilistic methods, in which case in addition to you and the adversary, there is also a Random player who spits out mathematical quantities in a way that is neither maximally helpful nor maximally adverse to your cause, but just some randomly chosen quantity in between. The trick is then to harness this randomness to let you evade and confuse your adversary.
If at some point some position sounds more plausible than the other, I try to put into words why is that. Either I convince myself of some position, for example that it is impossible, or the hardness of getting convinced at one position encourages me that maybe it's the other position that's true.
Either way I don't have anxiety because either I'm confident in some direction and making progress, even if it is at proving something is impossible, or the doubt from failing to prove it is impossible makes me more confident it is possible.
Even if you're making progress at the direction you didn't prefer, you should still see it as progress, and if you're confident in one direction you see it as an indication that you can make progress.
Anyways, it's not enough to just hold a general feeling that something is impossible, you need to put it into words and see if it's truly convincing. When it's just general intuitive unspoken feeling you can't change it, but when you make it concrete, you can understand what's possible.