[impendia]

My postdoctoral mentor didn't go that far. But he did say that when he read a book, and encountered the statement of some lemma or theorem, he liked to try to figure out how to prove it himself instead of reading the proof there.

There is also the infamous "exercise" in Serge Lang's algebra book (scroll to the bottom):

https://drive.google.com/file/d/0B_pEp00B111JYWU1NmY4MjktZTN...

[atrettel]

I think your mentor strikes the right balance between awareness and originality. My criticism of Landau's idea is that it's a great way to make you think you have an original idea when you actually don't. Only a detailed literature search will verify its originality. I think that any attempt to "isolate" your thinking from others' thinking will fail because you cannot live in a total vacuum. At least some external thinking will influence you no matter what and we must acknowledge that. "Multiple independent discovery" happens all the time in science, and it's because everyone tends to operate in a similar environment.

I do agree with Landau that occasionally knowing how things are previously done could limit how you approach the problem, but that can be fixed by systematically reproducing or falsifying the previous results and drawing your own conclusions (trust but verify). You gotta get your hands dirty. It's slow and difficult but the right way in the long term.