Re: Simple question - what is the root of the "math" that we are doing?
It's just plain silly to make such claims when every bit of specification and all the code is open. What miners do is so basic that anyone with a bit of programming knowledge can have at least a coarse idea of the task being done by looking at the code. So I wouldn't trouble myself over it. 
From https://en.bitcoin.it/wiki/Blocks :
Each block contains all recent transactions, a nonce (random number), and the hash of the previous block. A block is "solved" (published and considered valid by peers) when the SHA-256 hash of the entire block is below the current target. This is very unlikely to occur after being hashed only once, so the nonce must be incremented and the block re-hashed millions of times until it does.
From https://en.bitcoin.it/wiki/Blocks :
Each block contains all recent transactions, a nonce (random number), and the hash of the previous block. A block is "solved" (published and considered valid by peers) when the SHA-256 hash of the entire block is below the current target. This is very unlikely to occur after being hashed only once, so the nonce must be incremented and the block re-hashed millions of times until it does.