The truth is.

The proof is a mechanism to reach that consensus, by convincing other mathematicians of a specific truth. That is all it is.

There is a naive idea that a proof is a purely mechanical series of steps that provides access to truth. Last I checked, this isn't so for the vast majority of proofs in math. Such a proof would be way too tedious to construct or check by mathematicians. And if it isn't checkable, how do we know it is actually true?

Automated proofs are a subfield, and (again, last I checked) controversial because they can often not be checked by humans.

So for example, if the proof doesn't convince other mathematicians, then it's not a a proof.

Or it might convince other mathematicians and later turn out to be wrong after all.

For more on the practical aspects of math, I highly recommend The Mathematical Experience.

https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav...

I read it in German:

https://www.amazon.com/Erfahrung-Mathematik-German-P-J-Davis...