Coldplay - The Scientist Lyrics Meaning

The title of the song tells us that there is a scientist. A scientist is considered smart, someone who knows how to put things together and come up with a solution. Maybe it is him that is the scientist as he is the one who is singing, and refers to himself in the song.
(Verse 1)
"Come up to meet you,
Tell you I'm sorry,
You don't know how lovely you are."
From him saying he wants to say sorry, that gives us the idea that he has done the wrong and ended the relationship. The next line is a realization of how much he appreciates her.

"I had to find you,
Tell you I need you,
Tell you I set you apart."
He felt like he had to find her, and tell her how much he needs her in his life. He wants to tell her that it was him that set the situation and tore them apart.

"Tell me your secrets,
Ask me your questions,
Oh, let's go back to the start."
He wants her to do all the things they did before. To trust him with her secrets that no one else knows. To go back and have that reminiscent relationship and to go back to the start, to when it was good and strong.

"Runnin' in circles,
Comin' up tails,
It's only science apart."
These lyrics tell us that he's been losing his mind and pacing back and forth from the break up and that they just need to realize that it's science. They have enough chemistry to come together.
(Chorus)
"Nobody said it was easy,
It's such a shame for us to part.
Nobody said it was easy,
No one ever said it would be this hard.
Oh, take me back to the start."
He's realizing that nobody ever said it was hard or easy with break ups. Maybe he has nobody to give him advice. He thinks it's a shame, a tragedy for them to move on. The last line he is asking for time to take him back to the past, and to relive the times. Or maybe start all over again, a clean slate.
(Verse 2)
"I was just guessing,
At numbers and figures,
Pulling the puzzles apart."
This shows that he was being immature, and just messing around with the relationship, not thinking before he made decisions. Forgetting the consequences of his actions.

"Questions of science,
Science and progress,
could not speak as loud as my heart."
The obvious, all the answers and question couldn't describe what he's feeling. He's in pain.

"Tell me you love me,
Come back and haunt me,
Oh, and I rush to the start."
He's imagining her being there with him, wherever he is, and her memory is so alive to him that he is almost hallucinating. Like she's his drug. He may not even want her there but her presence is haunting him. Her image just makes him want to go back to when it first began.

"Runnin' in circles,
Chasin' tails,
Comin' back as we are."
From all his thinking, he's imagining them uniting again.
(Chorus)
"Nobody said it was easy,
Oh, it's such a shame for us to part.
Nobody said it was easy,
No one ever said it would be so hard.
I'm goin' back to the start."
The chorus is the same as the last, but notice that it changes in the last line. He's going back to the start now, whether that's imagining it unhealthily, or going to ask her to get back with him. But from the first lines of lyrics to the rest of the song, we get the idea that she didn't take him back.

The way he sings towards the end of the song shows his pain and that he's grieving.

Overall, I think this song is really touching and gives a real feel of regret and desperation to be with someone. Someone who doesn't feel the same as you, and you feel so alone in the situation that you end up living your life "going back to the start." It sends a message, that you should think before you make a decision and not to be a fool and get yourself into a mess that you can't get your self out of. Not to hurt yourself.