I don't think Graffin is arguing for science over emotion here. He is conveying the existentialist belief that anything you believe is real. Your entire reality is based on your own perception that you can never escape. The no such thing as hell line just means that hell exists because people believe it does. There is no proof necessary because everything you ever were, feel, or did is only in your mind. That is you cannot escape your mind.

This song is basically about how beliefs, proofs, and decisions based on emotion are flawed.

"My heart is not a cold cauldron of proof": this particularly applies to religion where the only 'proof' of the existence of god is an emotional presence people feel when practicing religion. Emotions are unreliable and inconclusive and therefore can't be accepted as proof: "There may be a day when emotion can be quantified but as of now there is no proof necessary."

The first line "Hit the road [...]" is just kind of setting up the complete haste and pace of his life as an active musician. It kind of says 'Shoot first...Ask questions later'. The "My heart is not [...]" line says I am not out living my life to prove what I am to you. I'm just trying to be the best person I can be.
"Mistakes are another opportunity to refine" just basically expresses that, yes he made some mistakes in life but the important thing is that he learned from them.

The "There's no such thing as hell" part is just him expressing that life is all what you make of it. . "There's no proof necessary, it's only in your mind" addresses the rampant pomposity in the punk scene.