By late-undergrad, it was intuitive to me that "proof" means "all mathematicians who reads this agrees with it". Mathematics is unique in that, mostly, the field can achieve consensus on results, which we then call "proofs".
But similarly, it makes sense that, even if a result is is "true" in a universal, objective sense, if the mathematician cannot communicate this in a convincing fashion to the rest of the mathematics world, I don't think we can call that result "proved".
But similarly, it makes sense that, even if a result is is "true" in a universal, objective sense, if the mathematician cannot communicate this in a convincing fashion to the rest of the mathematics world, I don't think we can call that result "proved".