The proof is not a social construct.

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...

The Mathematical Experience is not a history of mathematics per se, but there is a lot of history and context in it. Highly recommended.

https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav... is a link to Amazon.

I'm not who you are asking, but this book https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav... hits some of the philosophy of math. It's been a long time since I've dipped into it, but IIRC, it puts mathematical discoveries and exploration into more of a cultural and philosophical context. I could see this being sampled for a good crossover course that would cover humanities requirements for STEM, and STEM requirements for humanities.

My experience is that non-mathematicians like to have GED on their coffee tables as a conversation piece. The related book that mathematicians like is https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav.... (High school math is enough to enjoy it, but the more math you have, the more you'll get out it. All of the way up to the PhD level.)

"The Mathematical Experience" goes into this in pretty good depth.

http://www.amazon.com/Mathematical-Experience-Phillip-J-Davi...

If you are interested in this topic and you haven't already done so, check out "The Mathematical Experience"

http://www.amazon.com/Mathematical-Experience-Phillip-J-Davi...