The Nature of Truth

[Som1Lse]

• My mother is suddenly yelling at me for stealing the cookies. I see from the corner of my eye my an evil grin on my brother's face. That's "proof" he stole the cookies. That's (probably) reasonable. But not absolutely correct, as this kind would fail me if my brother was grinning for some other unrelated reason.
  
• In a court of law, in a reasonably small list of possibilities, you being the only suspect without an alibi is "proof" you committed the crime; say court learns the committer was wearing a  red shirt, and you happen to be the only one who was wearing a red shirt that day. That's reasonable. But not absolutely correct, as this kind would fail if you're a victim of a plan to incriminate you/police did a sloppy work/unlucky/..etc.
  
• In science, getting positive results for a theory in an experiment again and again is "proof" (appropriately, evidence) the theory is correct (works). That's reasonable, but note absolutely correct. There's the known quote by Einstein: "No amount of experimentation can ever prove me right; a single experiment can prove me wrong."
  
• In formal mathematics and logic, proofs rely on solid foundation that's the closest to "the metal" as you can get. Reasonable of course, and people tend to think you can't go wrong here. But that's not absolutely correct either. There's a lot to say about this.
  

While I agree with the overall statement I would like to make a few nitpicks:

• The cookie and court example are actually kind of similar: The term commonly used is "proof beyond a reasonable doubt", which notes that it is not definitively true. The term used for the grin on your brother's face and wearing a red shirt is "evidence" to support a claim. A court will typically use these terms correctly, your mother (unless she is a lawyer/judge) probably won't.
  
• In science the term "proof" is rarely ever used. Scientists generally prefer (as you noted) "evidence", or typically "sufficient evidence" to "support a given theory". You can't even really prove something wrong because there is always uncertainty in your measurements, so you can find evidence against it.
  
• For mathematics (including formal logic) a proof is a set of statements that, given a a set of assumptions, tries to convince others of a conclusion. Those assumptions don't necessarily have to be axioms (close to "the metal") but they are simply assumptions where, if you agree with them, you have to accept the conclusion. The funny thing to note here is the statements in the middle are never necessary for the proof to be sound, and they are essentially like comments in programming, in that they help other people follow along. Here the proof is absolutely correct, if the assumptions are.