After reading the O'Reilly book and perusing online resources, I'm still confused about the value that the actual mathematical mining provides to Network (I understand it does provide value, but not clear how).
It seems like once a miner has verified transactions, the really valuable work for the network has been done. And when it starts hashing to win a competition for 25 btc, it's just playing a game that has the sole purpose of adding currency to the network (which I understand is important, no central govt. etc). I don't see how this hashing to find a value less than the difficulty actually provides a real service to the network such as verifying transactions.
Before I jumped into learning about bitcoin, I just assumed the math problems the miners were solving related to detangling or decrypting transactions, but I see now that's not the case. If the competition was a race to verify transactions, that too would make more sense.
Can anyone explain the value of this mathematical race?