Welcome back to "Doggerel," where I ramble on about words and phrases that are misused, abused, or just plain meaningless.
The completely honest answer is simple: We don't know that science can find an answer to a problem. We won't know until we looked for answers and found one. We can't know whether or not something is impossible, so I prefer the optimistic route with science: Keep dreaming up good ideas, and never be content with bad ones.
Of course, just because scientists are having a hard time finding an answer doesn't mean that unscientific modes of thought stand a chance. Usually when people endorse "other ways of knowing," they're asking us to be governed by biases. Science is about the removal of bias so that we can have genuine confidence in the conclusions we arrive at. Gut instinct, "common sense," and so forth can be useful for snap judgments, but when you're exploring the unknown, the weird, and the unanswered, you can't let your thinking by handicapped by your usual biases.