“mathematical proofs are as much observations as anything else. Just because they happen in one’s head or with a pencil on paper, they are still observations.”
I think this is better explained as:
We try to do math, but we can make mistakes.*
If two people evaluate an arithmetic expression the same way, but one makes a mistake, then they might get different answers.
*Other examples:
1. You can try to create a mathematical proof. But if you make a mistake, it might be wrong (even if the premises are right).
2. An incorrect proof, a typo, or something on your computer screen?
A proof might have a mistake in it and thus “be invalid”. But it could also have a typo, which if corrected yields a “valid proof”.
Or, the proof might not have a mistake in it—you could have misread it, and what it says is different from what you saw. (Someone can also summarize a proof badly.)
If the copy of the proof you have is different from the original errors (or changes) could have been introduced along the way.
I think this is better explained as:
We try to do math, but we can make mistakes.*
If two people evaluate an arithmetic expression the same way, but one makes a mistake, then they might get different answers.
*Other examples:
1. You can try to create a mathematical proof. But if you make a mistake, it might be wrong (even if the premises are right).
2. An incorrect proof, a typo, or something on your computer screen?
A proof might have a mistake in it and thus “be invalid”. But it could also have a typo, which if corrected yields a “valid proof”.
Or, the proof might not have a mistake in it—you could have misread it, and what it says is different from what you saw. (Someone can also summarize a proof badly.)
If the copy of the proof you have is different from the original errors (or changes) could have been introduced along the way.