Thinking with your fingers can be dangerous. For example, the first code you write, the first code that solves the problem, is never the right code. Thinking with your fingers involves refactoring. Sometimes you need to throw everything out and start again but this time, you know the problem space.
You can thrash with your fingers too. You'll know when you are thrashing. You hack and get nowhere. Everyone has experienced that sinking feeling. When that happens, turn the computer off! Go talk to someone, google or read a book. You are not alone.
There is another version of analysis paralysis that affects managers. It leads to defeating meetings, phone drone and endless email. I won't discuss that in case I get fired. Besides, I don't want to limit my advancement opportunities.