I know, proof is the most crucial part of mathematics, it makes all the things be rigorous and keeps mathematics from contradiction.
In real life, there's things that we know to be true, for sure. Such as in physics, objects with opposite charges attract each other, or in biology, human can't live without oxygen or water. Of course, these are based on experimental result.
What I'm thinking is that, is there possibly something outside (or above) mathematics, that can tell us that "..." is true, in a way very different from normal mathematical proof, and we can totally believe this. I know this sounds unscientific, like religion, but seems that science haven't prohibited this to happen. Can there actually be another definition of "True" in mathematics, that we can know something to be true without really proving it?
I am personally of the opinion that the claim $2 + 2 = 4$ is a scientific conclusion of repeated experiment, to do with combining collections of size 2 and observing the size of the result. In particular, if I'm building a foundations for mathematics and after much effort I devise an axiomatic system that proves $2 + 2 = 4$, it is completely absurd to say “oh, thank goodness, it is true after all”.
Such a proof, I claim, is purely and only a proof of the adequacy of your logical system for proving true facts, which after all you'd very much hope a logical system could do. Imagine what would happen if you devised a system with axioms that you believed were true, and after all it happened that in this system $2 + 2 = 5$. You would certainly not revise your opinion of arithmetic! You'd re-inspect your axioms, or your rules of inference, or the whole idea of logical foundations sooner than you'd decide $2 + 2$ was anything other than $4$.
So, yes, I claim that there are certainly truths that precede any proof. They are perhaps not systematic or easily-identifiable, but that is after all the whole purpose of formal logic and proof, to introduce some order to the mess.