Sentence 1 says "the mountains" because the person who says this sentence is having in his/her mind some specific mountains (maybe which he/she has climbed already or else) when he/she says this sentence. Yes, I read it the same way. Sentence 2 says "mountains" because he/she does not / would not climb any mountains.
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Anonymous1. I don't climb the mountains because I like themTo me, the interesting thing about sentence #1 is its ambiguity -- does the speaker climb the mountains or not?
Anonymous"He has three daughters whose hair are brown."This is not right; this one is: