This song is so beautifully written. Being in this situation at the moment, it's very easy to interpret, from my perspective.
I'd like to start my interpretation by stating that the "Scientist" in this song is a metaphor. I'll explain why I think so later.
Come up to meet youTell you I'm sorryYou don't know how lovely you areI had to find youTell you I need youTell you I set you apart(This is pretty straightforward. The guy is going to the girl and apologizing for his actions, trying to make up for not speaking with her. He had to find her because it was killing him inside, and "haunting" him. He hopes that by saying these words, she'll finally understand how much he loves her.)
Tell me your secretsAnd ask me your questionsOh let's go back to the start(He wants her to tell him her deepest thoughts. He wants to hear that she still loves him. He wants her to ask him questions about what's been going on so he can explain to her. He realizes that if she asks questions, it means she still cares and still wants to save the relationship. He wants to go back to the way things were and hopes that by explaining everything to her, it will clear up any misunderstanding that may have occurred during their time apart.)
Running in circles; coming up tailsHeads on a silence apart(He doesn't know where to start which is why he wants her to ask the questions. He keeps "running in circles", trying to figure out what to say to her. He flips a coin which decides their fate. Heads means he will continue to wait for her to talk to him, but he hasn't heard from her which explains the "silence apart". Tails means he has to be the one to talk to her and make things right with them. Fate has decided that it is up to him to fix their relationship, which is why the coin keeps "coming up tails".)
Nobody said it was easyIt's such a shame for us to partNobody said it was easyNo one ever said it would be this hardOh take me back to the start(He clearly doesn't want the relationship to end. He regrets not telling her and is feeling the pain it brings every day. The fact that the lyrics say "nobody", means he has talked to numerous people for advice but they always told him the same thing. They kept telling him that it wouldn't be easy but it's for the best and so he listened to them. They also told him that he'd get over her in time, but clearly he has not and he again, he wants to "go back to the start" when everything was perfect with the couple.)
I was just guessing at numbers and figuresPulling your puzzles apartQuestions of science; science and progressDo not speak as loud as my heart(This is where the metaphor of the scientist comes into play. He was trying to figure out how to make the relationship work, using logic and reasoning. Again this line suggests he asked others for advice, but they told him it would not work. There are an endless list of possibilities as to why the relationship wouldn't work. He has been trying to find the one reason as to why the relationship would work out, and realizes that it's his love for her that would make the relationship work. He realizes that no one else can see why he feels this way about her. No one else can tell him why it would work because they don't understand the relationship. They just see two completely different people, trying to make a what seems impossible relationship work. They can't understand that love for someone defies all logic and reasoning.)
Tell me you love meCome back and haunt me(Again, he wants to hear that she still loves him. He wants her to come back to him and "haunt" him. This suggests that they did not tell each other how they actually felt, and it has been on his mind since he started to fall for her. He has been analyzing her every move, trying to figure out how she feels, and in the process he fell for her even more. She has been "haunting" his thoughts.)
Tell me you love meCome back and haunt meOh and I rush to the startRunning in circles, chasing our tailsComing back as we are(All of this was explained above, except for "coming back as we are." This suggests that he has finally decided to talk to her. He tried to rush things at first "and I rush to the start", but realizes this is difficult because of all the pain that he put her through. However, after he explains his actions and answers her questions, they are slowly working things out and slowly going back to the way things were.)
Nobody said it was easyOh it's such a shame for us to partNobody said it was easyNo one ever said it would be so hardI'm going back to the start(He realizes that it isn't going to be easy to mend their relationship, but his love for her is more important than what other people think. He realizes that what other people think doesn't matter. It isn't their relationship, it's his. He knows that it's up to him to fix it, as decided by the "coin", another metaphor for fate. He learns that to "go back to the start", he needs to explain everything to her. He is willing to do anything it takes to fix their relationship, and is ready for the hard times because once the hard times pass, he realizes that it will be the perfect relationship, and that the hard times are worth for what is to come.)
Another brilliantly written song by Coldplay! :)
